cfd inflow conditions, wall functions and turbulence models for flows around obstacles

CFD inflow conditions wall functions and turbulencemodels for flows around obstacles

Alessandro Parente lowast

Universite Libre de Bruxelles Belgium

March 19 2013

Contents

1 Introduction 5

2 Theory 621 Inlet conditions and turbulence model 722 Wall treatment 1123 Generalization of the ABL model for the case of obstacles immersed in the

flow 16

3 Applications 1931 Empty fetch at wind-tunnel and full scale 1932 Flow around a ground-mounted building 2333 Flow over complex terrains wind-tunnel and full-scale hills 34

4 Influence of stability classes 40

lowastThe present lecture notes are based on the research work performed by several Authors in the fieldof ABL flows at the von Karman Institute of Fluid Dynamics and at the Universite Libre de BruxellesIn particular the contribution by Prof Carlo Benocci Prof Jeroen van Beeck Dr Catherine Gorle andDr Miklos Balogh should be acknowledged

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LIST OF FIGURES LIST OF FIGURES

List of Figures

1 Computational domain with building models for CFD simulation of ABLflows and indication of different parts in the domain for roughness modellingBlocken et al [1] 12

2 Law of the wall for smooth and sand-grain roughened surfaces as a functionof the dimensionless sand-grain roughness height k+S Blocken et al [1] 12

3 Turbulent kinetic energy profiles at the inlet and outlet sections of an emptycomputational domain (see Figure 2b) Dashes cell value for turbulentdissipation rate and turbulent kinetic energy averaged over the first cellShort dashes cell value for turbulent dissipation rate and kinetic energyDots turbulent dissipation rate and kinetic energy averaged over the firstcell 15

4 Rough law of the wall implementation 15

5 Configurations PS1 (a) and PS2 (b) for the definition of the building influ-ence area (BIA) 16

6 Variation of the turbulence model parameter Cmicro for an undisturbed ABL(top) using a prescribed region Beranek [2] (middle) and using an auto-matic switching function (bottom) 18

7 Computational domain and main boundary conditions applied for the nu-merical simulation of the unperturbed ABL at wind tunnel (a) and full (b)scale 20

8 Profiles of velocity turbulent kinetic energy and turbulent dissipation rateat inlet and outlet section of the computational domain (Figure 7) ob-tained when applying inlet conditions given by Equations 10-12 (a-c) andEquations 1015 and 12 (d-f) STD WF = Standard Wall Function MODWF = Modified Wall Function [3] 21

9 Wall shear stress as a function of the axial coordinate Solid black linetheoretical value ρu2lowast Dashed red line Richards and Hoxey [4] profileBlue dots Yang et al [5] profile [3] 23

10 Profiles of velocity (a) turbulent kinetic energy (b) turbulent dissipationrate (c) and non-dimensional velocity gradient (d) at inlet and outlet sectionof the computational domain obtained with Equations 10 (23) and 12 [6] 24

11 Profiles of velocity turbulent kinetic energy and turbulent dissipation rateat inlet and outlet section of the computational domain (Figure 7) ob-tained when applying inlet conditions given by Equations 10-12 (a-c) andEquations 10 (29) and 12 (d-f) Results obtained with the wall functionformulation by [3] [3] 25

12 Wall shear stress as a function of the axial coordinate Solid black linetheoretical value ρu2lowast Dashed red line Richards and Hoxey [4] profileBlue dots Yang et al [5] profile 26

13 Building geometry and location of measurement planes for the flow aroundthe obstacle [7] 26

14 Computational domain and grid for the flow around the obstacle [7] 26

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15 Contour plots of non-dimensional velocity on the planes y = 0 (left) andz = 0035m (right) Experimental measurements are compared to theresults obtained applying the PS1 and PS2 model [3] 29

16 Contour plots of non-dimensional turbulence kinetic energy on the planesy = 0(left) and z = 0035m (right) Experimental measurements are com-pared to the results obtained applying the PS1 and PS2 model [3] 29

17 Experimental and numerical profiles of non-dimensional velocity upstreamover and downstream of the obstacle Solid line experimental data [3]Dashes PS1 configuration Short dashes PS2 configuration 31

18 Experimental and numerical profiles of non-dimensional turbulent kineticenergy upstream over and downstream of the obstacle [3] Solid line ex-perimental data Dashes PS1 configuration Short dashes PS2 configura-tion 31

19 Experimental and numerical profiles of non-dimensional velocity over anddownstream of the obstacle Solid line experimental data [6] DashesUABL model Short dashes PS model Dots ASQ model 31

20 Experimental and numerical profiles of non-dimensional turbulent kineticenergy over and downstream of the obstacle Solid line experimental data[6] Dashes UABL model Short dashes PS model Dots ASQ model 32

21 Local hit rates for the non-dimensional turbulent kinetic energy applyingthe ASQ model (left) and the UABL model (right) [6] 33

22 Stream-wise velocity (top) and turbulent kinetic energy profiles (bottom)in the symmetry plane against measurements obtained on the 3D hill atlaboratory scale using Fluent and OpenFOAM [8] 35

23 Simulated wall shear stress along the symmetry of the domain against theo-retical values extracted from the inlet profile (Inlet τw) and against valuesextracted from measured profiles (meas)[8] 36


25 Comparison of simulated and measured horizontal and vertical stream ve-locity (Uh and W ) and turbulent kinetic energy (k) along line-A using thecomprehensive approach [6] with α = 3 [8] 39

26 Comparison of simulated and measured vertical profiles (U and k) at thehill summit [8] 39

27 Profiles of (a) velocity (b) turbulent kinetic energy (c) turbulent dissipa-tion rate (d) turbulent viscosity and e) temperature at the inlet and outletsection of a 2D computational domain (60m high and 400m long ) and (f)shear stress at the wall Inlet conditions taken from [9] 42

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LIST OF TABLES LIST OF TABLES

List of Tables

1 Inlet conditions and turbulence model formulation 102 Fitting parameters for velocity and turbulent kinetic energy inlet profiles

according to Yang et al [5] Parente et al [6] and turbulent model parameters 193 Fitting parameters for velocity and turbulent kinetic energy inlet profiles

according to Richards and Hoxey [4] Brost and Wyngaard [10] and turbu-lent model parameters 22

4 Hit rate values for non-dimensional velocity and turbulent kinetic energy forthe prescribed BIA size approach varying the turbulence model settings [3] 28

5 Test cases and corresponding model settings for the numerical simulationof the flow around a bluff-body [7] using the prescribed BIA size approachTM=Turbulence Model 28

6 Test cases and corresponding model settings for the numerical simulationof the flow around a bluff-body [7] using the automatic switch approachBIA for the BIA TM=Turbulence Model 30

7 Hit rate values for non-dimensional velocity and turbulent kinetic energyfor the automatic switch approach varying the turbulence model settings[6] 33


9 Fitting parameters for velocity and turbulent kinetic energy inlet profilesaccording to Parente et al [6] and turbulent model parameters for the 3Dhill simulation 35

10 Hit rate values for non-dimensional velocity and turbulent kinetic energyfor the 3D hill simulation varying the turbulence model settings [8] 37

11 Normalized errors in the prediction of the separation point εSP and wakelength εWL for the 3D hill Negative and positive values sign the under-and overestimation respectively[8] 37

12 Fitting parameters for velocity and turbulent kinetic energy inlet profilesaccording to Parente et al [6] and turbulent model parameters for theAskervein hill simulation 39

13 Hit rate values for non-dimensional velocity and turbulent kinetic energyfor the Askervein hill simulation varying the turbulence model settingswithin the wake region [8] 40

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1 INTRODUCTION

1 Introduction

Computational Fluid Dynamics (CFD) is widely used to study flow phenomena in thelower part of the atmospheric boundary layer (ABL) with applications to pollutant dis-persion risk analysis optimization and siting of windmills and wind farms and micro-climate studies Numerical simulations of ABL flows can be performed either by solvingthe Reynolds-averaged Navier-Stokes (RANS) equations or by conducting large-eddy sim-ulations (LES) It is generally acknowledged that LES which explicitly accounts for thelarger spatial and temporal turbulent scales can provide a more accurate solution forthe turbulent flow field provided that the range of resolved turbulence scales is suffi-ciently large and that the turbulent inflow conditions are well characterized Shah andFerziger [12] Lim et al [13] Xie and Castro [14 15] For example Xie and Castro [14]presented a comparison of LES and RANS for the flow over an array of uniform heightwall-mounted obstacles the Authors compared the results to available direct numericalsimulation (DNS) data showing that LES simulations outperform RANS results withinthe canopy Dejoan et al [16] compared LES and RANS for the simulation of pollutantdispersion in the MUST field experiment and found that LES performs better in pre-dicting vertical velocity and Reynolds shear stress while the results for the stream-wisevelocity component are comparable However LES simulations are at least one orderof magnitude computationally more expensive than RANS [17] and the sensitivity to in-put parameters such as inlet conditions imply that as for RANS multiple simulationsare needed to quantify the resulting uncertainty in the output for realistic applicationsHence practical simulations of ABL flows are still often carried out solving the RANSequations in combination with two-equation turbulence models Consequently investigat-ing possible improvements to these models is worthwhile In RANS simulations the effectof roughness on ABL flows is generally represented with the so-called sand-grain basedwall functions Cebeci and Bradshaw [18] based on the experiments conducted by Niku-radze [19] for flow in rough circular pipes covered with sand Moreover the upstreamturbulent characteristics of a homogeneous ABL flows are generally modeled using theprofiles suggested by Richards and Hoxey [4] for mean velocity turbulent kinetic energyand turbulent dissipation rate However this modelling approach can result in an unsat-isfactory reproduction of the ABL for two main causes The first cause of discrepancylies in the inconsistency between the fully developed ABL inlet profiles and the roughwall function formulation Riddle et al [20] Franke et al [21] B Blocken [22] Blockenet al [1] Hargreaves and Wright [23] Franke et al [24] Furthermore the inlet profilefor the turbulence kinetic energy k proposed by Richards and Hoxey [4] assumes a con-stant value with height in conflict with wind-tunnel measurements Leitl [7] Xie et al[25] Yang et al [5] where a variation of k with height is observed A remedial measureto solve the inconsistency between the sand-grain based rough wall function and the fullydeveloped inlet profiles was proposed by Blocken et al [1] It consists in the modifica-tion of the wall law coefficients namely the equivalent sand-grain roughness height ksand the roughness constant Cs to ensure a proper matching with the velocity boundaryconditions This approach ensures the desired homogeneity of the velocity distributionfor the fully developed ABL but it is code dependent and does not provide a generalsolution to the problem Moreover the standard law of the wall for rough surfaces poseslimitations concerning the level of grid refinement that can be achieved at the wall This

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2 THEORY

restriction becomes particularly relevant for applications requiring a high resolution nearthe wall boundaries An additional complicating factor is the necessity to apply differentwall treatments when a combination of rough terrains and smooth building walls mustbe simulated Concerning the inlet profile for turbulent kinetic energy Yang et al [5]derived a new set of inlet conditions with k decreasing with height However the ap-plication of such a profile at the inlet boundary only provides an approximate solutionfor the system of equations describing a fully developed ABL In a recent work Gorleet al [26] proposed a modification of the constant Cmicro and of the turbulent dissipationPrandtl number σε to ensure homogeneity along the longitudinal ABL direction whenthe k profile of Yang et al [5] is applied Parente and Benocci [27] Parente et al [3]proposed a modification of the k minus ε turbulence model compatible with the set of inletconditions proposed by Yang et al [5] Such a modification consisted in the generalizationof the model coefficient Cmicro which becomes a local function of the flow variables and inthe introduction of two source terms in the transport equations for k and ε respectivelyThe limitation of such an approach consisted in the inlet profile adopted for turbulent ki-netic energy which does not satisfy all the governing simulations involved in the problemParente et al [6] Parente et al [6] addresses the aforementioned aspects by proposinga comprehensive approach for the numerical simulation of the neutral ABL First a newprofile for turbulent kinetic energy was derived from the solution of the turbulent kineticenergy transport equation resulting in a new set of fully developed inlet conditions for theneutral ABL which satisfies the standard k minus ε model This was accomplished throughthe introduction of a universal source term in the transport equation for the turbulentdissipation rate ε and the re-definition of the k minus ε model coefficient Cmicro as a function ofthe flow variables Second for the purpose of solving the flow around obstacles immersedin the flow the modelling approach derived for the homogeneous ABL was generalizedwith an algorithm for the automatic identification of the building influence area (BIA)As a consequence the turbulence model formulation is gradually adapted moving fromthe undisturbed ABL to the region affected by the obstacle Parente et al [28 3] alsoproposed a novel implementation of a wall function which incorporates both smooth-and rough-wall treatments employing a screening algorithm to automatically select thedesired formulation ie rough or smooth depending on the boundary surface propertiesBalogh et al [8] extended the approach by Parente et al [28 3] to the simulation of flowsabove complex terrains ie wind-tunnel scale 3D hill model and Askervein Hill

The present notes are organized as follows The modelling approach for the numericalsimulation of neutral ABL flows is presented by discussing the turbulence model formu-lation the different inlet profiles and the wall function Applications are presented anddiscussed for the flow over flat terrain around ground mounted bluff bodies and over hills

2 Theory

The standard kminusε model remains the most common option for the numerical simulation ofthe homogeneous ABL Such a family of models solves a transport equations for turbulentkinetic energy k and for turbulent dissipation rate ε

part

part(ρk) +

part

partxi(ρkui) =

part

partxj

[(micro+

microtσk

)partk

partxj

]+Gk minusGb minus ρεminus YM (1)

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21 Inlet conditions and turbulence model 2 THEORY

part

part(ρε) +

part

partxi(ρεui) =

part

partxj

[(micro+

microtσε

)partε

partxj

]+ Cε1

ε

k(Gk + Cε3Gb)minus Cε2ρ

ε2

k (2)

In Equations (1)-(2) ui is the ith velocity component ρ is the density Cε1 Cε2 and Cε3 aremodel constants σk and σε are the turbulent Prandtl numbers for k and ε respectively Gb

is the turbulent kinetic energy production due to buoyancy YM represents the contributionof the fluctuating dilatation in compressible turbulence to the overall dissipation rate Gk isthe generation of turbulence kinetic energy due to the mean velocity gradients calculatedfrom the mean rate-of-strain tensor Sij as

Gk = microtS2 S =

radic2SijSij Sij =

1

2

(partuipartxj

+partujpartxi

) (3)

For a steady ABL under the hypothesis of zero vertical velocity constant pressure alongvertical (z) and longitudinal (x) directions constant shear stress throughout the boundarylayer and no buoyancy effects the transport equations for turbulent kinetic energy k andturbulent dissipation rate ε simplify to

part

partz

(microtσk

partk

partz

)+Gk minus ρε = 0 (4)

part

partz

(microtσε

partε

partz

)+ Cε1Gk

ε

kminus Cε2ρ

ε2

k= 0 (5)

Gk = microt

(partu

partz

)2

(6)

The model is completed by the momentum equations which takes the form

microtpartu

partz= τw = ρu2lowast (7)

where τw is the wall shear stress and ulowast is the friction velocity

ulowast =

radicτwρ (8)

In Equations (4) (5) and (7) the laminar viscosity has been neglected with respect to theturbulent one microt expressed as

microt = ρcmicrok2

ε (9)

21 Inlet conditions and turbulence model

Fully developed inlet profiles of mean longitudinal velocity turbulent kinetic energyand dissipation rate under neutral stratification conditions are often specified followingRichards and Hoxey [4]

U =ulowastκln

(z + z0z0

)(10)

k =u2lowastradicCmicro

(11)

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2 THEORY 21 Inlet conditions and turbulence model

ε =u3lowast

κ (z + z0)(12)

where κ is the von Karman constant and z0 is the aerodynamic roughness length It canbe shown that Equations (10)-(12) are analytical solutions of the standard kminus ε model ifthe turbulent dissipation Prandtl number σvε is defined as Richards and Hoxey [4]

σε =κ2

(Cε2 minus Cε1)radicCmicro

(13)

or equivalently Parente et al [3 6] Pontiggia et al [29] if the following source term isadded to the dissipation rate equation

Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)

A weakness of the formulation presented above is the assumption of a constant value forthe turbulent kinetic energy k in Equation (11) Indeed experimental observations showa decay of k with height Leitl [7] Xie et al [25] Yang et al [5] Following this observationYang et al [5] analytically derived an alternative inlet condition for k

k =radicC1ln (z + z0) + C2 (15)

where C1 and C2 are constants determined via experimental data fitting The profile for kexpressed by Equation (15) is obtained directly as solution of the turbulent kinetic energytransport equation under the assumption of constant value for Cmicro and local equilibriumbetween production and dissipation

ε (z) =radicCmicrok

du

dz (16)

Yang et al [5] mentioned that the constant Cmicro should be correctly specified in order toensure the correct level of turbulence kinetic energy throughout the domain However thiscould be unnecessary if the effect of a non-constant k profile on the momentum equationis taken into account Gorle et al [26] generalized the expression of Cmicro as a function ofz by substituting Equations (9) and (16) into Equation (7)

microtpartu

partz= ρu2lowast rarr ρcmicro

k2

ε

partu


k2radicCmicrok

partupartz

partu

partz= ρu2lowast (17)

and then

Cmicro =u4lowastk2

(18)

Equation (18) is simply the relation proposed by Richards and Hoxey [4] inverted toensure consistency between the turbulence model and the k profile throughout the ABLdomain From the point of view of the physical interpretation the non-uniform k profileand the definition of Cmicro can be related to the large-scale turbulence present in ABL flows

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which can vary significantly with height Bottema [30] indicated the relevance of large-scale turbulence to several RANS models pointing out the necessity for case and locationdependent model constants

Using the k inlet profile by Yang et al [5] together with Equations (10) and (12) for uand ε and employing Equation (18) for Cmicro does not allow to close the system of Equations(4)-(7) with the definition of an appropriate expression for σε Only an approximatesolution Gorle et al [26] can be found using the constant value of Cmicro obtained at thewall adjacent cell In alternative the functional form of Cmicro ((18)) by introducing an anadditional source term for the k transport equation Parente et al [3] in addition to theone expressed by Eq 8 for the ε transport equation

Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)

As a consequence an arbitrary set of inlet conditions including the ones by Yang et al[5] can be adopted at the inlet boundary ensuring their conservation throughout thecomputational domain

An alternative approach is that of repeating the exercise by Yang et al [5] consideringthe functional variation of Cmicro ((18)) In particular assuming local equilibrium betweenturbulence production and dissipation Equation (16) Equation (4) becomes

part

partz

(microtσk

partk

partz

)= 0 (20)

Substituting Equations (9) (16) and (18) into Equation (20) we get

part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2

k2radicu4lowastk2k dudz

σk

partk

partz

Employing the analytical expression of the inlet velocity profile dudz

= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)

By integrating Equation (22) the following general solution for turbulent kinetic energyprofile is obtained

k (z) = C1ln (z + z0) + C2 (23)

which differs from Equation (15) since the square root operator disappears Similarlyto Equation (15) C1 and C2 are constants determined by fitting the equations to themeasured profile of k For what concerns the profile of turbulent dissipation rate the

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Table 1 Inlet conditions and turbulence model formulation

Inlet conditionsU = ulowast

κln(z+z0z0

)Equation (10)

k (z) = C1ln (z + z0) + C2 Equation (23)

ε = u3lowastκ(z+z0)

Equation (12)

Turbulence Modelmicrot = ρcmicro

k2

εEquation (9)

Sε (z) = ρu4lowast(z+z0)

2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)

equilibrium assumption (Equation (16)) and the relation for Cmicro (Equation (18)) ensurethat Equation (12) remains valid The full set of inlet conditions the turbulence modelformulation and the wall function implementation are summarized in Table 1 The set ofinlet boundary conditions provided by Equations (10) (23) and (12) for velocity turbulentkinetic energy and dissipation rate respectively represents a consistent extension of theformulation proposed by Richards and Hoxey [4] to the case of a non-constant turbulentkinetic energy profile Indeed if Equation (23) for and Equation (18) for Cmicro are used thetransport equation for the turbulent dissipation rate is identically satisfied by the sourceterm Sε (z) (Equation (14)) which is independent of the specific form of the inlet profileIn fact the equilibrium assumption and the generalization of Cmicro make the first term ofEquation (5) (which is the only one affected by the functional variation of k) universaland equal to

part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)

It should be noted that when employing the novel turbulent kinetic energy profile (Equa-tion (23)) the source term in Equation (19) reduces to zero Finally it can be observedthat assuming a constant profile for k ie C1 = 0 in Equation (23) the proposed ap-proach reduces to a formulation equivalent to the one proposed by Richards and Hoxey[4] with the difference that the proper value of Cmicro is automatically selected via Equation(18)

The profile proposed by Yang et al [5] requires the availability of experimental data todetermine the parameters C1 and C2 of Equation (23) This is not always guaranteed es-pecially for full-scale measurements In this case semi-empirical parameterizations avail-able in the literature Brost and Wyngaard [10] can be applied for the turbulent quantitiesprovided the following expressions for the mean squared fluctuating velocity components

languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

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22 Wall treatment 2 THEORY

langwprime2rang

u2lowast= 17minus z

h(27)

where h is the ABL height For neutral stratification conditions the value of h can bededuced from the following relation Bechmann [31]

hfcu2lowastasymp 033 (28)

where a typical mid-latitude value for the Coriolis parameter fc = 10minus4 can be con-sidered The variation of turbulent kinetic energy with height can be then expressedas

k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)

As stated earlier the use of a specific inlet profile for k does not result in inconsistencies inthe turbulence model formulation as long as the source term for turbulent kinetic energyis employed (Equation (19)) Such a term automatically vanishes when Equation (23) fork is employed

22 Wall treatment

Due to the importance of the surface roughness and the high Reynolds numbers associatedwith ABL flow the use of wall functions is generally required for near-wall modellingThe use of wall functions is particularly important for those regions of the computationaldomain where the actual elements are not modeled explicitly but in terms of roughness(Figure 1) The wall functions in CFD codes are generally based on the universal law ofthe wall that can be modified to account for the near-surface roughness The universallaw of the wall for a smooth surface is plotted in Figure 2 using the black solid lineusing the dimensionless variables u+ = Uulowast and y+ = ulowastyν where U is the mean velocitytangential to the wall u is a wall-function friction velocity and ν is the kinematic viscosityThe near-wall region consists of three main parts the laminar layer or linear sublayerthe buffer layer and the logarithmic layer In the linear sublayer the laminar law holds(u+ = y+) while in the log layer the logarithmic law is valid The standard smooth lawof the wall takes the form

Upulowast

=ln (y+)

κ+B (30)

where the integration constant B = 50ndash54 The effect of roughness is modeled byintroducing a shift in the intercept ∆B

(k+S) Moving the integration constant inside the

logarithm the following expression is obtained for the logarithmic law for a rough wall

Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)

where κ is the von Karman constant E is the wall function constant (E = 9 minus 97935The function ∆B

(k+S) depends of the dimensionless roughness height k+S = ulowastkSν

and measures the departure of the wall velocity from smooth conditions It can assumedifferent forms depending on the equivalent sand grain roughness values Nikuradze [19]Cebeci and Bradshaw [18] In particular when k+S gt 90

VKI - 11 -

2 THEORY 22 Wall treatment

Figure 1 Computational domain with building models for CFD simulation of ABL flowsand indication of different parts in the domain for roughness modelling Blocken et al [1]

Figure 2 Law of the wall for smooth and sand-grain roughened surfaces as a function ofthe dimensionless sand-grain roughness height k+S Blocken et al [1]

VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)

which givesUpulowast

=1

κln

(Ey+

CSk+S

) (33)

where Cs is a roughness constant By comparing Equations (10) and (33) it becomesclear that the two treatment are inconsistent and they may lead to discrepancies in theprediction of the near wall velocity if a proper selection of the roughness constants isnot performed To avoid that Blocken et al [1] proposed a first order match betweenthe rough wall of the wall and the velocity inlet profile at the first cell centroid zp toappropriately select CS

Ey+

CSk+S

=zp + z0z0

rarr CS =E ulowastzp

νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)

In Equation (34) a common requirement of ABL simulations has been made explicitnamely that the distance zp between the centroid of the wall-adjacent cell and the wallshould be larger than the sand-equivalent roughness ks of the terrain This requirementis generally translated into an upper limit for ks taken equal to zp zp ge kS Blockenet al [1] Different implementations of Equation (33) can be found in commercial CFDcodes Currently the value of CS can be freely set in the commercial software StarCCM+In Ansys CFX a fixed value for the roughness constant is adopted CS = 03 whereasFLUENT allows defining custom profiles for CS through user defined functions (UDF)However even when the value of the velocity at the first cell matches the one provided byEquation (10) the standard rough wall function suffers from two main drawbacks Firstit poses strong limitations on the maximum size of the wall adjacent cell the maximumallowable value for CS being limited by the wall function constant E In fact at the firstcell centroid zp Equations (33) gives

Upulowast

=1

κln

(E

CS

)

taking kS = zp This implies that CS cannot be higher than the value of the parameterE Moreover the standard wall function does not imply any direct effect of the roughnessproperties on the turbulence quantities at the wall This can be shown by taking thesquared derivative of the velocity profile (proportional to the production of k using thesand-grain law of the wall and the ABL velocity inlet profile

(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)

To overcome these drawbacks an alternative boundary condition could be imple-mented with the same functional form of the Richards and Hoxey [4] inlet profiles To

VKI - 13 -


that purpose velocity turbulent dissipation rate and turbulent kinetic energy productionare specified as follows Richards and Hoxey [4]

Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w

ρκC025micro k05 (z + z0)

(39)

Richards and Hoxey [4] proposed a different expression for Gk by integrating the pro-

duction term over the first cell height ie Gk = 12zp

acute 2zp0

Gkdz Such formulation was

found to produce a peak in the k profile close to the wall Parente et al [3 6] and theevaluation of Gk at the cell centroid may be preferable (Equation (39)) Figure 3 showsthe turbulent kinetic energy at the inlet and outlet sections of an empty domain wherefully developed profiles of velocity and turbulence are specified at the domain inlet Thetest case is presented to show how the proposed wall function formulation (green dots)provides the best agreement (deviation below 4) between the profile specified at theinlet and the one retrieved at the outlet By averaging both Gk and ε (blue short-dashes)the deviation at the wall is about 15 but it remains higher than the error obtainedusing the cell values of Gk and ε Averaging Gk over the first cell while keeping the cellvalue for ε (red dashes) leads to an overestimation of the turbulent kinetic energy at thewall of about 100 Richards and Norris [32] have shown that such a peak in the profilefor k can also be avoided by changing the discretization of the production term to ensureequilibrium between production and dissipation

Parente et al [3 6]proposed an implementation of the rough wall function whichpreserves the the form of the universal law of the wall through the introduction of a newwall function constant and non-dimensional wall distance

Upulowast

=1

κln(Ey+

)(40)

with

y+ =ulowast (z + z0)

νE =

ν

z0ulowast (41)

The non-dimensional distance y+ is simply a y+ shifted by the aerodynamic roughnesswhereas the new wall function constant E depends on the roughness characteristics of thesurface In Equation (40) the friction velocity ulowast is not kept constant in the longitudinaldirection but calculated locally as ulowast = C025

micro k05 The present approach removes thedrawbacks of the standard wall function without limiting its flexibility In particular itis easily extendable to mixed rough and smooth surface through a redefinition of the lawof the wall constants Moreover it allows full flexibility from the point of view of meshgeneration as the wall function parameters do not impose any limitation on the first cellheight A schematic of the rough wall implementation is shown in Figure 4

VKI - 14 -


Figure 3 Turbulent kinetic energy profiles at the inlet and outlet sections of an emptycomputational domain (see Figure 2b) Dashes cell value for turbulent dissipation rateand turbulent kinetic energy averaged over the first cell Short dashes cell value forturbulent dissipation rate and kinetic energy Dots turbulent dissipation rate and kineticenergy averaged over the first cell

Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w

C025micro k05 (z + z0)

Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall

Richards and Hoxey (1993) Implementation(Fluent OpenFOAM Fine Open)

centroid

UP Gk ϵP

Figure 4 Rough law of the wall implementation

VKI - 15 -

2 THEORY23 Generalization of the ABL model for the case of obstacles immersed in the flow

23 Generalization of the ABL model for the case of obstaclesimmersed in the flow

The proposed methodologies for neutral ABL flows is valid when the ABL is undisturbedWhen an obstacle is immersed in the flow field or for the flow on more complex terrains theturbulence model formulation (including the definition of the turbulence model parameterCmicro (Equation (18)) and the turbulent dissipation rate source term source term Sε (z)(Equation (14))Pare no longer valid since the governing equations are no longer given byEquations (4)-(7)

Gorle et al [26]proposed a modification of the turbulence model parameters Cmicro andσvε in a building influence area (BIA) defined following Beranek [2] as a half sphere withradius r = 16H and centered 016H downstream of the building where H is the buildingheight Parente et al [3] further investigated the effect of the BIA size and shape bylimiting the BIA area to the region downstream of the building The investigated config-urations indicates as PS1 and PS2 respectively are shown in Figure for the flow arounda bluff-body Parente et al [6] proposed an approach for the automatic determination ofthe BIA to allow a gradual transition of the turbulence model parameters from the for-mulation proposed for the undisturbed ABL to one more suitable for wake flow regions Alocal deviation from the undisturbed ABL conditions is introduced which automaticallyidentifies the extent of the flow region affected by the obstacle The relative deviation δof the actual local velocity profile u with respect to the inlet logarithmic one uABL istaken as blending parameter between the two formulations

δ = min

[(uminus uABLuABL

) 1

](42)

The coefficient Cmicro and the source term Sε (z) are then weighted as a function of δ toallow a gradual transition between different flow regions

φ = δαφwake + (1minus δα)φABL = φwake + (1minus δα) (φABL minus φwake) (43)

where wake and ABL indicate the wake and undisturbed ABL values for the turbulencemodel parameters If the standard k minus ε is employed in the wake as proposed by Gorleet al [26] Parente et al [3 6] Balogh et al [8] then Sεwake = 0 and Cmicrowake = 009The parameter α determines the shape of the transition function between the different

Figure 5 Configurations PS1 (a) and PS2 (b) for the definition of the building influencearea (BIA)

VKI - 16 -

23 Generalization of the ABL model for the case of obstacles immersed in the flow2 THEORY

formulations Different values of α have been tested by Parente et al [6] Balogh et al [8]showing that quadratic blending is the most appropriate choice Figure 6 shows the varia-tion of the model parameter Cmicro for the ABL model for the prescribed switch approach andfor an undisturbed ABL (top) using a prescribed region Beranek [2] (middle) and usingan automatic switching function (bottom) To improve the performances of the standardk minus ε model in wakes different corrections may be considered such as the ones proposedby Kato and Launder [33] Yap [34] Tsuchiya et al [35] to improve the simulation ofseparation and reattachment regions as reported in Parente et al [3 6] Balogh et al [8]The Kato-Launder (KL) approach is based on the reformulation of the production termwritten as

GkKL = microtSΩ (44)

Ω =radic

2ΩijΩij Ωij =1

2

(partuipartxjminus partujpartxi

)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)

where S is the strain tensor (symmetric part of the velocity gradient tensor) and Ω is thevorticity (antisymmetric part of the velocity gradient tensor) The approach proposed byYap is a correction for separated flows it corrects the turbulent kinetic energy predictionin the separated regions through an additive source term in the ε transport equationdefined as

Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)

where le = Cminus075micro κdw and dw is the nearest wall distance Following Launder [36] the

Yap correction is applied together with the Kato-Launder approach The modificationproposed by Tsuchiya et al [35] (usually reported as MMK model in the literature)differs from the standard k minus ε model for the evaluation of the eddy viscosity microt In thisapproach the turbulent viscosity if modified with a multiplier expressed as a constrainedratio between the vorticity and the modulus of the rate of strain tensor

microt MMK = FMMKρCmicrok2

εFMMK = min

(Ω

S 1

) (46)

In alternative to linear closures nonlinear eddy viscosity models the Reynolds stressrelation can be used to improve the predictions in wake regions For instance a quadraticclosure can be employed [37]

Rij = uiuj =2

3kδij minus νtSij + c1Cmicro

k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3

ε2(ΩikSjk + ΩjkSik) + c3Cmicro

k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)

where Sij = Uij + Uji and Ωij = UijminusUji according to Craft et al [37]

VKI - 17 -


Figure 6 Variation of the turbulence model parameter Cmicro for an undisturbed ABL (top)using a prescribed region Beranek [2] (middle) and using an automatic switching function(bottom)

VKI - 18 -

3 APPLICATIONS

3 Applications

The methodologies for the simulation of the ABL are tested and presented in this sectionThe first objective is that of verifying that the proposed approaches can provide thedesired uniformity of velocity and turbulent quantities for an empty domain case Thenthe simulation over more complex terrains and around obstacles will be considered

31 Empty fetch at wind-tunnel and full scale

The first test case is a wind tunnel scale ABL Leitl [7] fully characterized in terms ofvelocity and turbulence intensity measurements investigated by several authors Theresults presented here refer to Parente et al [3 6] The computational domain consistsof a two-dimensional box of 4 m length and 1 m height with a grid of 400x71 cellsThe domain (Figure 7a) is discretized with a grid uniform in the longitudinal directionand stretched in the vertical one to have the centre point of the wall adjacent cell at aheight of 00025 m At the inlet boundary profiles of velocity turbulent kinetic energyand dissipation rate are specified The inlet conditions proposed by Richards and Hoxey[4] Yang et al [5] and Parente et al [6] are tested for turbulent kinetic energy (Table2) and the turbulence model parameters are modified accordingly A pressure outletcondition is applied for the outlet section Both standard and modified formulations forthe wall function are applied at the lower boundary whereas a constant stress is appliedto the upper boundary following the recommendation of Richards and Hoxey [4] Thesecond test case is the full scale ABL adopted for the blind-test exercise on the CFDmodelling of wind loading on the full-scale Silsoe cube Richards et al [38] Results fromParente et al [3] are presented for a 2D domain (Figure 7b) of 5000 m length and 500m height with a grid of 500x50 cells The generated grid is uniform in the longitudinaldirection and stretched in the vertical direction to have the centre point of the walladjacent cell at a height of 1 m The velocity profile is taken from the blind-test exerciseRichards et al [38] whereas the k profile is specified using both the relations proposedby Richards and Hoxey [4] and the one obtained using the semi-empirical correlation byBrost and Wyngaard [10] As for the wind tunnel scale test case a constant stress isapplied to the upper boundary while only the modified wall treatment is used to model

Table 2 Fitting parameters for velocity and turbulent kinetic energy inlet profiles accord-ing to Yang et al [5] Parente et al [6] and turbulent model parameters

Parameter Richards and Hoxey [4] Yang et al [5] Parente et al [6]ulowast 0374z0 000075C1 0 -011 -004

C2u4lowastCmicro

053 052

Cmicro 009 u4lowastk2

Skρulowastκσk

part[(z+z0) partkpartz ]partz

-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -

3 APPLICATIONS 31 Empty fetch at wind-tunnel and full scale

Figure 7 Computational domain and main boundary conditions applied for the numericalsimulation of the unperturbed ABL at wind tunnel (a) and full (b) scale

the rough terrain properties following the conclusions of the results obtained from thewind-tunnel scale simulations The presented results were replicated using both FLUENTand OpenFOAM codes with a double precision pressure based solver The standarddiscretization scheme is applied to pressure while second order schemes are adopted formomentum and turbulence quantities and the SIMPLE algorithm is chosen for pressure-velocity coupling The simulations are considered converged when the residuals level outresulting in a decrease of at least six orders of magnitude

Figure 8 shows the profiles of velocity turbulent kinetic energy and dissipation rate atthe inlet and outlet section of the computational domain in Figure 7 for the inlet condi-tions specified by Richards and Hoxey [4] (Figure 8a-c) and Yang et al [5] (Figure 8d-f)For the case of a constant profile for turbulent kinetic energy Richards and Hoxey [4] theresults obtained with the standard wall function with a constant Cs defined according toEquation (18) are compared to those provided by the modified law of the wall Parenteet al [3] The turbulence model parameters used in combination with the different sets ofboundary conditions are summarized in Parle 2 The results (Figure 8a-c) indicate thatthe two approaches are comparable however the modified wall approach better preservesthe homogeneity of velocity and turbulence throughout the domain In particular Figure8b-c confirms that the modification of Cs only affects the velocity profile whereas theturbulent dissipation rate at the wall is overestimated as a result of omitting the aero-dynamic roughness in the denominator of Equation (38) Consequently the turbulentkinetic energy at the wall adjacent cell is slightly underestimated Therefore the formu-lation of the wall function alone already has a non-negligible impact on the simulationresults for what concerns the reproduction of the turbulent quantities For the case of anon constant k profile the comparison between the profiles of velocity turbulent kineticenergy and dissipation rate at the inlet and outlet sections of the domain indicate that theproposed source terms and the implemented wall function ensure the desired longitudinalhomogeneity The highest differences are observed for the k profile but they are below4 in all cases Figure 9 shows the evolution of the wall shear stress along the longitu-dinal coordinate for the two cases of constant Richards and Hoxey [4] and variable inletk profiles Yang et al [5] It can be remarked that the modified wall function produces a

VKI - 20 -

31 Empty fetch at wind-tunnel and full scale 3 APPLICATIONS

Figure 8 Profiles of velocity turbulent kinetic energy and turbulent dissipation rate atinlet and outlet section of the computational domain (Figure 7) obtained when applyinginlet conditions given by Equations 10-12 (a-c) and Equations 1015 and 12 (d-f) STDWF = Standard Wall Function MOD WF = Modified Wall Function [3]

VKI - 21 -


constant and realistic wall shear stress after a short adaptation length The wind tunnelanalysis is completed by the assessment of the approach proposed by Parente et al [6]Figure 10 presents shows the comparison between inlet and outlet profiles of velocity tur-bulent kinetic energy and turbulent dissipation rate when the approach by Parente et al[6] is employed It can be observed how the overall modelling approach is able to ensurelongitudinal and vertical homogeneity for the simulated boundary layer The highest dif-ferences are observed for the k profile however these are safely below 2 in all casesThe results shown in Figure 10 a-c are further confirmed by Figure 10d which shows the

non-dimensional velocity gradient Φ =(κzulowast

partupartz

) which allows assessing the deviation of the

simulated velocity profile from the ideal logarithmic one Indeed the Φ profile obtainedat the outlet section of the domain closely follows the theoretical non-dimensional velocitygradient It can be concluded that the new set of boundary conditions together with therough wall function formulation and the modification of the turbulence model provide acomprehensive and consistent modelling approach for the neutral ABL Figure 11 showsthe profiles of velocity turbulent kinetic energy and dissipation rate at the inlet and outletsection of the full-scale computational domain shown in Figure 7b obtained when apply-ing inlet conditions by Richards and Hoxey [4] (a-c) and Brost and Wyngaard [10] (d-f)Table 3 shows the fitting parameters for velocity and turbulent kinetic energy as well asthe turbulent model parameters for the two sets of inlet conditions employed Similarlyto the wind-tunnel scale it can be concluded that the modelling approach summarized inTable 3 can be successfully applied also to the numerical simulation of full scale unper-turbed ABL flows As far as the longitudinal velocity component is concerned maximumdeviations of about 3 are observed for both sets of inlet conditions tested The turbulentkinetic energy profile specified at the inlet section is also very well maintained throughoutthe computational domain maximum differences of about 5 and 3 are observed whenapplying the Richards and Hoxey [4] Brost and Wyngaard [10] boundary condition fork respectively It is also interesting to observe that the semi-empirical correlations byBrost and Wyngaard [10] result in a consistent level of turbulent kinetic energy althoughslightly higher (˜30) than the one by Richards and Hoxey [4] Finally the wall shearstress along the longitudinal coordinate (Figure 12) indicates that a constant and realisticshear stress is retrieved also for the full-scale ABL simulation

Table 3 Fitting parameters for velocity and turbulent kinetic energy inlet profiles accord-ing to Richards and Hoxey [4] Brost and Wyngaard [10] and turbulent model parameters

Parameter Richards and Hoxey [4] Brost and Wyngaard [10]ulowast [ms] 0325z0 [m] 001

k u2lowastradicCmicro

u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -

32 Flow around a ground-mounted building 3 APPLICATIONS

Figure 9 Wall shear stress as a function of the axial coordinate Solid black line theo-retical value ρu2lowast Dashed red line Richards and Hoxey [4] profile Blue dots Yang et al[5] profile [3]

32 Flow around a ground-mounted building

The simulation of the flow around a single rectangular building [7] is considered to furtherassess the modelling proposed in [3 6] The building (Figure 13) has width 01m length015m height H = 0125m and 4 source elements on the leeward building side Thecomputational domain inlet boundary is set 1m upstream of the building where ABLprofiles are measured in the wind tunnel whereas the outlet boundary is located 4mdownstream of the building As the model is symmetrical with respect to the planey = 0m only half of the domain is represented The width and height of the domain are065m and 1m respectively corresponding to the wind tunnel size A structured meshconsisting of 17 million cells (200 x 114 x 107 elements) is applied The height of theground adjacent cell is 000075m which correspond to one cell between the ground surfaceand the lower edge of the source elements The wall function formulation (Section 22)based on the use of the aerodynamic roughness [3 6] allows maintaining such resolutionin the entire computational domain Previous investigations by Gorle et al [39] employedthe standard rough wall formulation with a modification of the roughness constant CS[1] As a consequence the computational grid had to be non-conformal with coarserwall adjacent cells in the far field to allow the correct reproduction of the incomingABL without exceeding the requirement CS le E (Section 22) Hence the test case iswell suited to demonstrate the advantage of a wall function based on the aerodynamicroughness which does not require the definition of CS and kS and allows preserving thesame high near-wall resolution throughout the entire domain (Figure 14) The presenceof an obstacle immersed in the ABL flow has important modelling implications Theturbulence model formulation presented in Section 1 is valid for an unperturbed ABLAs indicated in Section 23 the application of turbulence model modifications derivedfor unperturbed ABL flows in regions of the flow affected by obstacles can negativelyaffect model predictions The problem can be alleviated by dividing the computationaldomain in two regions a region unaffected by the building where the modified turbulencemodel parameter Cmicro and the source terms Sk (if needed) and Sε are applied and a region

VKI - 23 -

3 APPLICATIONS 32 Flow around a ground-mounted building

Figure 10 Profiles of velocity (a) turbulent kinetic energy (b) turbulent dissipation rate(c) and non-dimensional velocity gradient (d) at inlet and outlet section of the computa-tional domain obtained with Equations 10 (23) and 12 [6]

VKI - 24 -


Figure 11 Profiles of velocity turbulent kinetic energy and turbulent dissipation rate atinlet and outlet section of the computational domain (Figure 7) obtained when applyinginlet conditions given by Equations 10-12 (a-c) and Equations 10 (29) and 12 (d-f)Results obtained with the wall function formulation by [3] [3]

VKI - 25 -


Figure 12 Wall shear stress as a function of the axial coordinate Solid black linetheoretical value ρu2lowast Dashed red line Richards and Hoxey [4] profile Blue dots Yanget al [5] profile

Figure 13 Building geometry and location of measurement planes for the flow around theobstacle [7]

Figure 14 Computational domain and grid for the flow around the obstacle [7]

VKI - 26 -


influenced by the building called building influence area (BIA) where other turbulencemodel formulations of the k minus ε model are used (Section 23) The results in this sectionwere obtained with Fluent 63 using the steady 3d double precision pressure based solverThe standard discretization scheme was applied to pressure while second order schemeswere adopted for momentum and turbulence quantities and the SIMPLE algorithm forpressure-velocity coupling

Results are first presented for the case of a BIA with a pre-defined size as discussedin Section 23 and shown in Figure 5 The quality of the results is assessed using the hitrate metric defined for both velocity and turbulent kinetic energy on the basis of the Navailable measurements δI

q =Nsum

i=1

δi (48)

with

δi =

1 if

∣∣∣VCFDminusVTESTVTEST

∣∣∣ le 025 or |VCFD minus VTEST | le W

0 else(49)

The hit rate indicates the fraction of N measurement points at which the CFD results arewithin 25 of the measurement data or within the uncertainty interval W of the dataFranke et al [24] The value used for the uncertainty interval W is 0012 for the non-dimensional velocity and for the non-dimensional turbulence kinetic energy The selectedvalues affect the absolute value of the hit rate but they are verified not to change theobserved model performance trends

Table 4 shows both the total hit rate for all measurement points and local hit ratesfor selected points (upstream on the side on top and downstream of the building) Inaddition to the two prescribed BIA size models (PS1 and PS2) the result presented inGorle et al [39] for settings similar to the present PS1 and the result obtained usingthe classic Richards and Hoxey [4] boundary conditions and the ABL model everywhere(indicated as UABL) The test conditions for the different cases are summarized in Table5 For velocity an improvement in the hit rate is obtained using any of the modifiedapproaches as compared to the Richards and Hoxey [4] boundary conditions Moreoverwe observe no major differences in the hit rate for uUref between the PS1 or PS2 settingsFor the turbulence kinetic energy however the PS2 setting produces a significant increaseof the hit rate (59) with respect to the PS1 settings (52) An important improvementis also observed with respect to the reference case from Gorle et al [39] which providesa hit rate of 47 and the results obtained using the Richards and Hoxey [4] boundaryconditions (51) in all regions of the flow field

Figures 15 and 16 present the contour plots for the non-dimensional velocity andturbulent kinetic energy respectively at planes y = 0 and z = 0035m The experimentalmeasurements are compared to the results obtained applying the PS1 and PS2 modelIt can be observed that the main difference between the PS1 and PS2 settings is foundin front of the building with the PS2 setting resulting in a slightly better prediction ofthe upstream separation bubble and a smaller over-prediction of the turbulent kineticenergy Both settings over-predict the size of the wake region by about 30 ( 22Hversus 17H) However a larger overestimation is observed when applying the standardmodel and the Richards and Hoxey [4] boundary conditions ( 28H versus 17H) To

VKI - 27 -


Table 4 Hit rate values for non-dimensional velocity and turbulent kinetic energy for theprescribed BIA size approach varying the turbulence model settings [3]

UUref

upstream sizetop downstream allUABL 09 075 049 066

Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059

Table 5 Test cases and corresponding model settings for the numerical simulation of theflow around a bluff-body [7] using the prescribed BIA size approach TM=TurbulenceModel

Test case Inlet conditions Wall function BIA BIA TMRH Richards and Hoxey [4] Richards and Hoxey [4] - -

Gorle et al [39] Yang et al [5] Blocken et al [1] Half sphere KLPS1 Yang et al [5] Parente et al [3] Half sphere KLPS2 Yang et al [5] Parente et al [3] Cut half sphere KL

VKI - 28 -


Figure 15 Contour plots of non-dimensional velocity on the planes y = 0 (left) andz = 0035m (right) Experimental measurements are compared to the results obtainedapplying the PS1 and PS2 model [3]

Figure 16 Contour plots of non-dimensional turbulence kinetic energy on the planesy = 0(left) and z = 0035m (right) Experimental measurements are compared to theresults obtained applying the PS1 and PS2 model [3]

VKI - 29 -


allow a more quantitative comparison between experiments and numerical simulationsthe calculated and measured non-dimensional vertical profiles of velocity and k on thesymmetry plane are shown in Figure and Figure respectively for different locationsupstream over and downstream of the building The different configurations (PS1 andPS2) used to prescribe the BIA are compared to the experimental measurements It canbe observed how the difference between the two configurations mainly affects the turbulentkinetic energy field (Figure ) in the region upstream of the building In particular theapplication of the ABL model at the axial location x = minus0075m allows significantlyreducing the overestimation of turbulent kinetic energy This effect remains at the firstlocation on top of the building (x = minus004m) On the other hand on the roof furtherdownstream (x = 0m and x = 004m) and behind of the building (x = 0105m andx = 03m) the two model predictions become comparable almost collapsing onto a singleline

The approach based on the a priori prescription of the BIA can be modified by consid-ering the approach discussed in Section 23 using the blending function given by Equation(43) to achieve a gradual transition between the different flow regions as shown in Figure6 Depending on the value of the exponent α controlling the transition between the tworegions different models can be obtained If α = 1 and α = 2 are chosen a linear and aquadratic switch are obtained indicated by the acronyms ASL and ASQ respectively Acomparison between PS and ASQ is shown in the following for the flow around the bluff-body As a reference a test case without separation between flow regions is also includedand denoted Undisturbed ABL (UABL) The test conditions for the different cases aresummarized in Table 5 The calculated and measured non-dimensional vertical profiles ofvelocity and k are shown in Figures 19 and 20 respectively for different axial locationsover and downstream of the building Results were obtained applying the UABL ASQand PS models It can be observed how the difference between the models becomes morepronounced downstream of the building in the wake region (x = 0105m and x = 03m)In particular the automatic quadratic switch (ASQ) allows reducing the under-predictionof velocity (Figure 19) and the over prediction of turbulent kinetic energy (Figure 19)downstream of the building with respect to the UABL model It can be also remarkedthat the results obtained with ASQ are very similar to those provided by the prescribedswitch model (PS) for what concerns the velocity distribution On the other hand theASQ model results in a better prediction of turbulent kinetic energy (Figure 19) in par-ticular at location x = 03m where the prescribed switch model (PS) shows a very strongover-prediction of the k values Nevertheless the peak of k at x = 0105m is better cap-tured by the PS model Finally Table 4 lists the hit rate values for the non-dimensionalvelocity and turbulent kinetic energy

Table 6 Test cases and corresponding model settings for the numerical simulation ofthe flow around a bluff-body [7] using the automatic switch approach BIA for the BIATM=Turbulence Model

Test case Inlet conditions Wall function BIA BIA TMUABL Parente et al [6] Parente et al [3] - -

PS Parente et al [6] Parente et al [3] Half sphere Kato and Launder [33]ASL ASQ Parente et al [6] Parente et al [3] Automatic Kato and Launder [33]

VKI - 30 -


Figure 17 Experimental and numerical profiles of non-dimensional velocity upstreamover and downstream of the obstacle Solid line experimental data [3] Dashes PS1configuration Short dashes PS2 configuration

Figure 18 Experimental and numerical profiles of non-dimensional turbulent kinetic en-ergy upstream over and downstream of the obstacle [3] Solid line experimental dataDashes PS1 configuration Short dashes PS2 configuration

Figure 19 Experimental and numerical profiles of non-dimensional velocity over anddownstream of the obstacle Solid line experimental data [6] Dashes UABL modelShort dashes PS model Dots ASQ model

VKI - 31 -


Figure 20 Experimental and numerical profiles of non-dimensional turbulent kinetic en-ergy over and downstream of the obstacle Solid line experimental data [6] DashesUABL model Short dashes PS model Dots ASQ model

Table 7 shows the hit rates for all investigated conditions The first remark is that thehit rates obtained applying the UABL and ASQ models are very similar In other wordsit is not possible to distinguish which of the two model settings provides the best resultsby just comparing the hit rate values This is obviously due to the global nature of the hitrate metric which does not provide any information regarding the local prediction of flowfeatures To better clarify this aspect

rdquoFigure 21 shows the local hits for the turbulent

kinetic energy field obtained applying the ASQ and UABL models respectively It isclear that the quadratic automatic switch (ASQ) provides better results in the wakewhich is confirmed by the increased number of hits However the global hit rate valuesfor the two cases are very similar due to the poorer performances of the ASQ modelat the border of the wake where the turbulent kinetic energy is over-estimated As faras the other approaches are concerned both the ASQ and the PS models optimize thereproduction of the velocity field but penalize significantly the prediction of turbulentkinetic energy To conclude the ASQ model can be regarded as the best performingoption for the present investigation In particular it ensures the best compromise forwhat concerns the prediction of velocity and turbulent kinetic energy within the BIAproviding hit rate values higher than 60 for both fields

Finally to improve the predictions in the wake a non-linear closure by Craft et al[37] (Section 23) has been tested [11] and benchmarked against the results provided bythe k minus ε model using the Kato-Launder correction as summarized in Table 8 It can beobserved that the mean stream-wise velocity and turbulent kinetic energy hit rates do notimprove much with respect to the ASQ model For linear models the increase in turbulentkinetic energy hit rates had the consequence of worsening flow field results In the presentcase it appears that an improvement in both direction is achieved The approach has thepotential of providing satisfactory results but the transition between the two flow regionsappears not to be currently optimal It can be observed that a sharper transition (α = 5)improves the prediction of the velocity field as well as of turbulent kinetic energy bothupstream and downstream of the building On the building top the increase of α hasthe effect of worsening the predictions suggesting that a multi-zone approach should beprobably pursue This is currently under investigation

VKI - 32 -


Table 7 Hit rate values for non-dimensional velocity and turbulent kinetic energy for theautomatic switch approach varying the turbulence model settings [6]

UUref

upstream sizetop downstream allUABL 087 075 045 063ASL 087 075 049 065ASQ 085 077 045 062PS1 09 079 058 071

kU2ref


Figure 21 Local hit rates for the non-dimensional turbulent kinetic energy applying theASQ model (left) and the UABL model (right) [6]

VKI - 33 -

3 APPLICATIONS 33 Flow over complex terrains wind-tunnel and full-scale hills


UUref

upstream sizetop downstream allASQ 085 077 045 062

Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061

33 Flow over complex terrains wind-tunnel and full-scale hills

In this section the comprehensive modelling approach for ABL flows is assessed on com-plex terrains such rough hills The approach by Parente et al [6] was tested againstthe measurements obtained in the thermally stratified wind tunnel of The University ofTokyo using three-dimensional laser doppler anemometry [40] The study was reportedby Balogh et al [8]The model is an axisymmetric hill whose shape is given by

h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else

with the radius at the hill base of rmax = 042m and with the height at the hill-tophmax = 02m The hill model was positioned 2m downstream of the test section inletof the 6 x 22 x 18m wind tunnel The computational domain contains the entire testsection with the origin set x = 0 y = 0 The hill and wind tunnel floor are modeledas rough walls with the same aerodynamic roughness of z0 = 000122m while smoothwall boundary conditions are applied on the ceiling and side walls At the downwindside a pressure outlet boundary condition was used The fitting parameters for velocityand turbulent kinetic energy [6] inlet profiles are shown in Table 9 The computationalmesh is composed of 200 x 87 x 60 cells resulting in 1044 million hexahedral elementsrefined horizontally at the near-field of the hill Numerical simulations were performedwith Fluent and OpenFOAM solver [8] To allow a meaningful comparison between Flu-ent and OpenFOAM numerical settings were selected to be as similar as possible (see [8]for more details) Figure 22 shows the measured and computed velocity and turbulentkinetic energy profiles along the longitudinal direction provided by the comprehensiveapproach [6] with a blending exponent α = 3 for the automatic switch Such a value ofα was chosen as it provided the best compromise between velocity and turbulent kineticenergy predictions It can be observed that OpenFOAM simulations always present alarge underestimation of k downstream of the hill whereas Fluent provides calculatedvalues more in agreement with the measurements However this effect is due to the largeoverestimation of the size of the separation bubble by Fluent which results in erroneousvelocity predictions but turbulent kinetic energy levels closer to the measured ones A

VKI - 34 -

33 Flow over complex terrains wind-tunnel and full-scale hills 3 APPLICATIONS

Table 9 Fitting parameters for velocity and turbulent kinetic energy inlet profiles accord-ing to Parente et al [6] and turbulent model parameters for the 3D hill simulation

Parameter ValueExpressionulowast [ms] 00923z0 [m] 000122

C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

Figure 22 Stream-wise velocity (top) and turbulent kinetic energy profiles (bottom) inthe symmetry plane against measurements obtained on the 3D hill at laboratory scaleusing Fluent and OpenFOAM [8]

VKI - 35 -


comparison of the wall shear stress as a function of the non-dimensional longitudinal coor-dinate is given in Figure 23 where the circles denote the measurements The comparisonshows that the simulations are in fair accordance with the observations in the separatedregion whereas an important deviation can be remarked at the first measurement locationat the top of the hill The latter can be caused by the difference between the roughnesselements mounted on the flat part of the wind tunnel and the hill surface The wall shearstress obtained by Fluent better reproduces the experimental data especially at higherdistances from the hilltop Table 10 reports the hit rate values for both the velocity andturbulent kinetic energy prediction with OpenFOAM and Fluent It can be observed thatthe modification of the turbulence model within the wake does not have a significant im-pact on the results Instead the use of MMK results in slightly lower values of hit rates inmost cases (with the exception of the it rate of k for Fluent simulations) Moreover thenumerical hit rate values consistently show significant differences between OpenFOAMand Fluent concerning the reproduction of the separation bubble Table 11 shows thenormalized errors in the prediction of the separation point εSP and wake length εWL forthe 3D hill using OpenFOAM and Fluent Results show that when the comprehensiveapproach is applied in OpenFOAM both the location of the separation point and thesize of the wake show a fairly good agreement with the measurements regardless of theturbulence model applied in the wake region as indicated by the error values listed inTable 11 Fluent results show on the other hand much larger discrepancies The causesfor that which may be related to the limited access to the source code granted n Fluentare still under investigation

In practical atmospheric applications the surface is generally not as regular as in theprevious example therefore simulations were performed on more complex geometry tovalidate the approach and possibly propose modifications In particular the full scalemeasurements obtained over the Askervein hill [41 42] were chosen to this purpose Thisis a popular case study for validating CFD models for ABL simulations [43 44 45 46 47]The Askervein hill has a nearly elliptical form with major and minor axis of 2000m and1000m The height of the hill is 116m and its slopes range from 12 to 25 Thesurrounding area is flat at the upwind side of the hill and it is hilly at the downwindside For describing the surface coverage the aerodynamic roughness was taken as z0 =00353m based on the data measured at the reference mast The computational domain(Figure 24) has dimensions 6000 x 6000 x 1000m with an origin located at the hill top

Figure 23 Simulated wall shear stress along the symmetry of the domain against theoret-ical values extracted from the inlet profile (Inlet τw) and against values extracted frommeasured profiles (meas)[8]

VKI - 36 -


Table 10 Hit rate values for non-dimensional velocity and turbulent kinetic energy forthe 3D hill simulation varying the turbulence model settings [8]

OpenFOAMWake model Velocity Kinetic energy

STD 081 049MMK 079 047

FluentWake model Velocity Kinetic energy

STD 071 047MMK 070 052

Table 11 Normalized errors in the prediction of the separation point εSP and wakelength εWL for the 3D hill Negative and positive values sign the under- and overestima-tion respectively[8]

OpenFOAMWake model εSP [] εWL []

STD -6 -13MMK -9 -5

FluentWake model εSP [] εWL []

STD -26 -23MMK -29 32

VKI - 37 -


(HT) Similarly to a previous study [47] the mesh was generated with 97 x 111 x 30hexahedral elements using 15 m maximum resolution on the hill and 1m for the first cellheight The refinement the grid at the hill is shown in Figure 24 Measurements for inletconditions are obtained at almost neutral condition stable wind direction and relativelyhigh wind speed [48] The wind direction was 210 degrees which determines the meshorientation Namely the inlet is fixed on the western side of the computational domain andthe outlet is the eastern side On its southern and northern side the symmetry boundarycondition are defined On the top of the domain the uniform values corresponding tothe fitted profiles are imposed as summarized in Table 12 The measurement campaignwas carried out using sonic anemometers at 10 meters above the surface in every mustalong a given line (denoted as line-A in Figure 24) and at the hill top thus the velocitycomponents such as the turbulent fluctuations for the three directions are available forthis locations The comparison between computed and measured horizontal profiles isshown in Figure 25 The horizontal component (Uh) vertical velocity (W ) and turbulentkinetic energy are presented and the RMS of the velocity is used to characterize theuncertainty The agreement between measured and computed velocity can be consideredsufficiently satisfactory Although the approach overestimates the horizontal velocity atthe far upwind of the hill summit (194 for OpenFOAM and 121 for Fluent) theprediction at the top of the hill (Figures 25 and 26) and along the downwind of thehill is fairly good (underestimation of 21 for OpenFOAM and 55 for Fluent) Asfor turbulent kinetic energy the results obtained with the comprehensive approach aresignificantly better than the ones given by the original approach (resulting in hit ratesbelow 30) The differences between the CFD solvers are significant OpenFOAM resultsare better for velocity whereas Fluent provides a more accurate prediction of turbulentkinetic energy Table 13 shows the hit rate of the horizontal velocity and the turbulentkinetic energy for the OpenFOAM and Fluent codes A hit-rate value gt 99 for velocityis obtained for the OpenFOAM simulation using the comprehensive approach with theMMK correction in the wake region with a corresponding hit rate of about 44 forturbulent kinetic energy A comparable HR value for k is obtained only using FluentHowever for this case the hit-rate value for velocity is below 99 ie 81


VKI - 38 -


Table 12 Fitting parameters for velocity and turbulent kinetic energy inlet profiles accord-ing to Parente et al [6] and turbulent model parameters for the Askervein hill simulation


C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

Figure 25 Comparison of simulated and measured horizontal and vertical stream veloc-ity (Uh and W ) and turbulent kinetic energy (k) along line-A using the comprehensiveapproach [6] with α = 3 [8]

Figure 26 Comparison of simulated and measured vertical profiles (U and k) at the hillsummit [8]

VKI - 39 -

4 INFLUENCE OF STABILITY CLASSES

Table 13 Hit rate values for non-dimensional velocity and turbulent kinetic energy for theAskervein hill simulation varying the turbulence model settings within the wake region[8]


STD 094 031MMK gt099 044


STD 081 044MMK 081 044

4 Influence of stability classes

The models proposed above (Section 2) are derived under the hypothesis of neutral strat-ification ie when the the heat flux from the ground is equal to zero Recently [29]attempts have been made to extend ABL models for non-neutral conditions From theMonin-Obukhov similarity theory the universal profiles of velocity and temperature canbe expressed as

U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)

where the Monin-Obukhov length L = u2lowastTwκgTlowast

is an estimate of the height where theturbulent dissipation due to the buoyancy is comparable with the shear stress productionof turbulence Tw is the surface temperature qw the surface heat flux g the gravitationalacceleration module and Φm is a function that depends on z and L For neutral and stablestratification Φm = 1 + 5 z

L However for neutral stratification the heat flux from the

ground is zero and the Monin-Obukhov length is infinite the parameter Φm tends to 1thus simplifying the form of the inlet profiles In fact the friction temperature Tlowast tendsto zero and temperature is constant along z In stable conditions Φm is different than 1and the temperature profile is not homogeneous In such conditions the following profilesfor k and ε become [29]


radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)

Similarly to what discussed for the neutral ABL (Section 2) two source terms need to beadded to the kminus ε turbulence model to ensure that the inlet profile satisfy the transport

VKI - 40 -


equations

Sk (z) = minuspart[(micro+ microT

σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)

The implementation of such approach has been recently performed in OpenFOAM [49]Figure 27 shows the profiles of (a) velocity (b) turbulent kinetic energy (c) turbulentdissipation rate (d) turbulent viscosity and e) temperature at the inlet and outlet sectionof a 2D computational domain (60m high and 400m long ) as well as the (f) shear stressat the domain wall The inlet conditions are taken from [9] Results indicate that themodification of the turbulence model by means of the source terms given by Equations(55) and (56) is able to provide the desired uniformity of velocity and turbulent quantitiesthroughout the domain

VKI - 41 -


(a) (b)

(c) (d)

(e) (f)

Figure 27 Profiles of (a) velocity (b) turbulent kinetic energy (c) turbulent dissipationrate (d) turbulent viscosity and e) temperature at the inlet and outlet section of a 2Dcomputational domain (60m high and 400m long ) and (f) shear stress at the wall Inletconditions taken from [9]

VKI - 42 -

REFERENCES REFERENCES

References

[1] B Blocken T Stathopoulos J Carmeliet CFD simulation of the atmosphericboundary layer wall function problems Atmospheric Environment 41 (2007) 238ndash252

[2] W Beranek General rules for the determination of wind environment Proceedingsof the 5th international conference on wind engineering (Fort Collins Colorado USA1979) 225ndash234

[3] A Parente C Gorle J van Beeck C Benocci Improved k-epsilon model and wallfunction formulation for the rans simulation of abl flows Journal of Wind Engineeringand Industrial Aerodynamics 99 (2011) 267ndash278

[4] P J Richards R P Hoxey Appropriate boundary conditions for computationalwind engineering models using the k-epsilon turbulence model Journal of Wind En-gineering and Industrial Aerodynamics 46-47 (1993) 145 ndash 153

[5] Y Yang M Gu S Chen X Jin New inflow boundary conditions for modelling theneutral equilibrium atmospheric boundary layer in computational wind engineeringJournal of Wind Engineering and Industrial Aerodynamics 97 (2) (2009) 88 ndash 95

[6] A Parente C Gorle J van Beeck C Benocci A comprehensive modelling approachfor the neutral atmospheric boundary layer Consistent inflow conditions wall func-tion and turbulence model Boundary-Layer Meteorology 140 (2011) 411 ndash 428

[7] B Leitl Cedval at hamburg universityURL httpwwwmiuni-hamburgdecedval

[8] M Balogh A Parente C Benocci Rans simulation of abl flow over complex terrainsapplying an enhanced k-epsilon model and wall function formulation Implementationand comparison for fluent and openfoam Journal of Wind Engineering and IndustrialAerodynamics 104 - 106 (2012) 360 ndash 368

[9] M Magnusson A S Smedman Influence of atmospheric stability on wind turbinewakes Wind Engineering 18 (1994) 139 ndash 151

[10] R A Brost J C Wyngaard A model study of the stably stratified planetary bound-ary layer Journal of the Atmospheric Sciences 35 (8) (1978) 1427ndash1440

[11] A Rakai Simulation of flow around a building-shaped obstacle with openfoam Techrep von Karman Institute for Fluid Dynamics (2012)

[12] K B Shah J H Ferziger A fluid mechanicians view of wind engineering Large eddysimulation of flow past a cubic obstacle Journal of Wind Engineering and IndustrialAerodynamics 67 (1997) 211ndash224

[13] H C Lim T Thomas I P Castro Flow around a cube in a turbulent boundarylayer Les and experiment Journal of Wind Engineering and Industrial Aerodynam-ics 97 (2) (2009) 96ndash109

VKI - 43 -


[14] Z-T Xie I P Castro Les and rans for turbulent flow over arrays of wall-mountedobstacles Flow Turbulence and Combustion 76 (3) (2006) 291ndash312

[15] Z-T Xie I P Castro Large-eddy simulation for flow and dispersion in urban streetsAtmospheric Environment 43 (13) (2009) 2174ndash2185

[16] A Dejoan J Santiago A Martilli F Martin A Pinelli Comparison between large-eddy simulation and reynolds-averaged navierndashstokes computations for the must fieldexperiment part ii Effects of incident wind angle deviation on the mean flow andplume dispersion Boundary-Layer Meteorology 135 (1) (2010) 133ndash150

[17] W Rodi Comparison of les and rans calculations of the flow around bluff bodiesJournal of Wind Engineering and Industrial Aerodynamics 69-71 (0) (1997) 55ndash 75 ltcetitlegtProceedings of the 3rd International Colloqium on Bluff BodyAerodynamics and ApplicationsltcetitlegtURL httpwwwsciencedirectcomsciencearticlepii

S0167610597001475

[18] T Cebeci P Bradshaw Momentum Transfer in Boundary Layers Hemisphere Pub-lishing Corporation Washington 1977

[19] J Nikuradze Stromungsgesetze in rauhen rohren Forschungsheft 361 (1933)

[20] A Riddle D Carruthers A Sharpe C McHugh J Stocker Comparisons betweenfluent and adms for atmospheric dispersion modelling Atmospheric Environment38 (7) (2004) 1029 ndash 1038

[21] J Franke C Hirsch A G Jensen H W Krus M Schatzmann P S WestburyS D Miles J A Wisse N G Wright Recommendations on the use of cfd inwind engineering in C A C14 (Ed) Proceedings of the International Conferenceon Urban Wind Engineering and Building Aerodynamics von Karman InstituteRhode-Saint-Genese Belgium 2004 pp C11ndashC111

[22] T S B Blocken J Carmeliet Cfd evaluation of wind speed conditions in passagesbetween parallel buildings - effect of wall-function roughness modifications for theatmospheric boundary layer flow Journal of Wind Engineering and Industrial Aero-dynamics 95 (2007) 941ndash962

[23] D Hargreaves N Wright On the use of the k-epsilon model in commercial cfd soft-ware to model the neutral atmospheric boundary layer Journal of Wind Engineeringand Industrial Aerodynamics 95 (5) (2007) 355 ndash 369

[24] J Franke A Hellsten K H Schlunzen B Carissimo The cost 732 best practiceguideline for cfd simulation of flows in the urban environment a summary Interna-tional Journal of Environment and Pollution 44 (1-4) (2011) 419ndash427URL httpwwwingentaconnectcomcontentindijep201100000044

F0040001art00047

[25] Z Xie P R Voke P Hayden A G Robins Large-eddy simulation of turbulent flowover a rough surface Boundary-Layer Meteorology 111 (3) (2004) 417ndash440

VKI - 44 -


[26] C Gorle J van Beeck P Rambaud G V Tendeloo Cfd modelling of small particledispersion The influence of the turbulence kinetic energy in the atmosphericboundary layer Atmospheric Environment 43 (3) (2009) 673 ndash 681URL httpwwwsciencedirectcomsciencearticlepii

S1352231008009084

[27] A Parente C Benocci On the rans simulation of neutral abl flows in Proceed-ings of the Fifth International Symposium on Computational Wind Engineering(CWE2010) Chapel Hill North Carolina USA 2010 pp 1ndash8

[28] A Parente C Gorle J V Beck C Benocci Rans simulation of abl flows applicationof advanced wall boundary conditions to configurations with mixed rough and smoothsurfaces in Proceedings of the Fifth International Symposium on ComputationalWind Engineering (CWE2010) Chapel Hill North Carolina USA 2010 pp 1ndash8

[29] M Pontiggia M Derudi V Busini R Rota Hazardous gas dispersion A cfd modelaccounting for atmospheric stability classes Journal of Hazardous Materials 171 (1 -3) (2009) 739 ndash 747

[30] M Bottema Turbulence closure model constants and the problems of inactive atmo-spheric turbulence Journal of Wind Engineering and Industrial Aerodynamics 67-68(1997) 897 ndash 908

[31] A Bechmann Large-eddy simulation of atmospheric flow over complex terrain PhDthesis Technical University of Denmark (2006)

[32] P J Richards S E Norris Appropriate boundary conditions for computational windengineering models revisited Journal of Wind Engineering and Industrial Aerody-namics 99 (2011) 257ndash266

[33] M Kato B E Launder The modelling of turbulent flow around stationary andvibrating square cylinders in Proceedings of 9th symposium on turbulence andshear flows Kyoto Japan 1993 pp 1ndash6

[34] C J Yap Turbulent heat and momentum transfer in recirculating and impingingflows PhD thesis University of Manchester (1987)

[35] M Tsuchiya S Murakami A Mochida K Kondo Y Ishida Development of anew k-epsilon model for flow and pressure fields around bluff body Journal of WindEngineering and Industrial Aerodynamics 67-68 (0) (1997) 169 ndash 182

[36] B E Launder On the computation of convective heat transfer in complex turbulentflows Journal of Heat Transfer 110 (4b) (1988) 1112ndash1128

[37] T J Craft B E Launder K Suga Development and application of a cubic eddyvis-cosity model of turbulence International Journal of Heat and Fluid Flow 17 (1995)108ndash115

[38] P J Richards A D Quinn S Parker A 6m cube in an atmospheric boundary layerflow part 2 computational solutions Wind and Structures 5 (2) (2002) 177ndash192

VKI - 45 -


[39] C Gorle J Beeck P Rambaud Dispersion in the wake of a rectangular buildingValidation of two reynolds-averaged navierndashstokes modelling approaches Boundary-Layer Meteorology 137 (2010) 115ndash133URL httpdxdoiorg101007s10546-010-9521-0

[40] T Takahashi S Kato S Murakami R Ooka M F Yassin R Kono Wind tunneltests of effects of atmospheric stability on turbulent flow over a three-dimensionalhill Journal of Wind Engineering and Industrial Aerodynamics 93 (2) (2005) 155 ndash169

[41] P A Taylor H W Teunissen Askervein rsquo82 Report on the septemberoctober 1982experiment to study boundary layer flow over askervein Technical Report MSRS-83-8 Meteorological Services Research Branch Atmospheric Environment Service (1983)

[42] P A Taylor H W Teunissen The askervein hill project Report on the septem-beroctober 1983 main field experiment Technical Report MSRB-84-6 Meteorolog-ical Services Research Branch Atmospheric Environment Service (1985)

[43] G D Raithby G D Stubley P A Taylor The askervein hill project A finitecontrol volume prediction on three-dimensional flows over the hill Boundary-LayerMeteorology 39 (1987) 107ndash132

[44] H Kim V Patel Test of turbulence models for wind flow over terrain with separationand recirculation Boundary-Layer Meteorology 94 (2000) 5ndash21

[45] F A Castro J M L M Palma A S Lopes Simulation of the askervein flow part 1Reynolds averaged navier stokes equations (k-epsilon turbulence model) Boundary-Layer Meteorology 107 (3) (20003) 501ndash530

[46] F A Castro J M L M Palma A S Lopes Simulation of the askervein flow part2 Large-eddy simulations Boundary-Layer Meteorology 125 (1) 85ndash108

[47] C A V Rodrigues Analysis of the atmospheric boundary layer flow over mountain-ous terrain Masterrsquos thesis von Karman Institute for Fluid Dynamics (2005)

[48] R Mickle N Cook A Hoff N Jensen J Salmon P Taylor G Tetzlaff H Teunis-sen The askervein hill project Vertical profiles of wind and turbulence Boundary-Layer Meteorology 43 (1988) 143ndash169

[49] B Nagy Simulation of stratified atmospheric flows with openfoam Tech rep vonKarman Institute for Fluid Dynamics (2012)

VKI - 46 -

Introduction

Theory

Inlet conditions and turbulence model

Wall treatment

Generalization of the ABL model for the case of obstacles immersed in the flow

Applications

Empty fetch at wind-tunnel and full scale

Flow around a ground-mounted building

Flow over complex terrains wind-tunnel and full-scale hills

Influence of stability classes


List of Figures

1 Computational domain with building models for CFD simulation of ABLflows and indication of different parts in the domain for roughness modellingBlocken et al [1] 12

2 Law of the wall for smooth and sand-grain roughened surfaces as a functionof the dimensionless sand-grain roughness height k+S Blocken et al [1] 12

3 Turbulent kinetic energy profiles at the inlet and outlet sections of an emptycomputational domain (see Figure 2b) Dashes cell value for turbulentdissipation rate and turbulent kinetic energy averaged over the first cellShort dashes cell value for turbulent dissipation rate and kinetic energyDots turbulent dissipation rate and kinetic energy averaged over the firstcell 15

4 Rough law of the wall implementation 15

5 Configurations PS1 (a) and PS2 (b) for the definition of the building influ-ence area (BIA) 16

6 Variation of the turbulence model parameter Cmicro for an undisturbed ABL(top) using a prescribed region Beranek [2] (middle) and using an auto-matic switching function (bottom) 18

7 Computational domain and main boundary conditions applied for the nu-merical simulation of the unperturbed ABL at wind tunnel (a) and full (b)scale 20

8 Profiles of velocity turbulent kinetic energy and turbulent dissipation rateat inlet and outlet section of the computational domain (Figure 7) ob-tained when applying inlet conditions given by Equations 10-12 (a-c) andEquations 1015 and 12 (d-f) STD WF = Standard Wall Function MODWF = Modified Wall Function [3] 21

9 Wall shear stress as a function of the axial coordinate Solid black linetheoretical value ρu2lowast Dashed red line Richards and Hoxey [4] profileBlue dots Yang et al [5] profile [3] 23

10 Profiles of velocity (a) turbulent kinetic energy (b) turbulent dissipationrate (c) and non-dimensional velocity gradient (d) at inlet and outlet sectionof the computational domain obtained with Equations 10 (23) and 12 [6] 24

11 Profiles of velocity turbulent kinetic energy and turbulent dissipation rateat inlet and outlet section of the computational domain (Figure 7) ob-tained when applying inlet conditions given by Equations 10-12 (a-c) andEquations 10 (29) and 12 (d-f) Results obtained with the wall functionformulation by [3] [3] 25

12 Wall shear stress as a function of the axial coordinate Solid black linetheoretical value ρu2lowast Dashed red line Richards and Hoxey [4] profileBlue dots Yang et al [5] profile 26


14 Computational domain and grid for the flow around the obstacle [7] 26

VKI - 2 -















VKI - 3 -


List of Tables














VKI - 4 -

1 INTRODUCTION

1 Introduction


VKI - 5 -

2 THEORY



2 Theory


part

part(ρk) +

part

partxi(ρkui) =

part

partxj

[(micro+

microtσk

)partk

partxj


VKI - 6 -


part

part(ρε) +

part

partxi(ρεui) =

part

partxj

[(micro+

microtσε

)partε

partxj

]+ Cε1

ε


ε2

k (2)



Gk = microtS2 S =

radic2SijSij Sij =

1

2

(partuipartxj

+partujpartxi

) (3)


part

partz

(microtσk

partk

partz


part

partz

(microtσε

partε

partz

)+ Cε1Gk

ε

kminus Cε2ρ

ε2

k= 0 (5)

Gk = microt

(partu

partz

)2

(6)


microtpartu



ulowast =

radicτwρ (8)


microt = ρcmicrok2

ε (9)



U =ulowastκln

(z + z0z0

)(10)


(11)

VKI - 7 -


ε =u3lowast

κ (z + z0)(12)


σε =κ2


(13)


Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)


k =radicC1ln (z + z0) + C2 (15)



du

dz (16)


microtpartu


k2

ε

partu


k2radicCmicrok

partupartz

partu


and then

Cmicro =u4lowastk2

(18)


VKI - 8 -




Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications



















VKI - 3 -


List of Tables














VKI - 4 -

1 INTRODUCTION

1 Introduction


VKI - 5 -

2 THEORY



2 Theory


part

part(ρk) +

part

partxi(ρkui) =

part

partxj

[(micro+

microtσk

)partk

partxj


VKI - 6 -


part

part(ρε) +

part

partxi(ρεui) =

part

partxj

[(micro+

microtσε

)partε

partxj

]+ Cε1

ε


ε2

k (2)



Gk = microtS2 S =

radic2SijSij Sij =

1

2

(partuipartxj

+partujpartxi

) (3)


part

partz

(microtσk

partk

partz


part

partz

(microtσε

partε

partz

)+ Cε1Gk

ε

kminus Cε2ρ

ε2

k= 0 (5)

Gk = microt

(partu

partz

)2

(6)


microtpartu



ulowast =

radicτwρ (8)


microt = ρcmicrok2

ε (9)



U =ulowastκln

(z + z0z0

)(10)


(11)

VKI - 7 -


ε =u3lowast

κ (z + z0)(12)


σε =κ2


(13)


Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)


k =radicC1ln (z + z0) + C2 (15)



du

dz (16)


microtpartu


k2

ε

partu


k2radicCmicrok

partupartz

partu


and then

Cmicro =u4lowastk2

(18)


VKI - 8 -




Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications






List of Tables














VKI - 4 -

1 INTRODUCTION

1 Introduction


VKI - 5 -

2 THEORY



2 Theory


part

part(ρk) +

part

partxi(ρkui) =

part

partxj

[(micro+

microtσk

)partk

partxj


VKI - 6 -


part

part(ρε) +

part

partxi(ρεui) =

part

partxj

[(micro+

microtσε

)partε

partxj

]+ Cε1

ε


ε2

k (2)



Gk = microtS2 S =

radic2SijSij Sij =

1

2

(partuipartxj

+partujpartxi

) (3)


part

partz

(microtσk

partk

partz


part

partz

(microtσε

partε

partz

)+ Cε1Gk

ε

kminus Cε2ρ

ε2

k= 0 (5)

Gk = microt

(partu

partz

)2

(6)


microtpartu



ulowast =

radicτwρ (8)


microt = ρcmicrok2

ε (9)



U =ulowastκln

(z + z0z0

)(10)


(11)

VKI - 7 -


ε =u3lowast

κ (z + z0)(12)


σε =κ2


(13)


Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)


k =radicC1ln (z + z0) + C2 (15)



du

dz (16)


microtpartu


k2

ε

partu


k2radicCmicrok

partupartz

partu


and then

Cmicro =u4lowastk2

(18)


VKI - 8 -




Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications





1 INTRODUCTION

1 Introduction


VKI - 5 -

2 THEORY



2 Theory


part

part(ρk) +

part

partxi(ρkui) =

part

partxj

[(micro+

microtσk

)partk

partxj


VKI - 6 -


part

part(ρε) +

part

partxi(ρεui) =

part

partxj

[(micro+

microtσε

)partε

partxj

]+ Cε1

ε


ε2

k (2)



Gk = microtS2 S =

radic2SijSij Sij =

1

2

(partuipartxj

+partujpartxi

) (3)


part

partz

(microtσk

partk

partz


part

partz

(microtσε

partε

partz

)+ Cε1Gk

ε

kminus Cε2ρ

ε2

k= 0 (5)

Gk = microt

(partu

partz

)2

(6)


microtpartu



ulowast =

radicτwρ (8)


microt = ρcmicrok2

ε (9)



U =ulowastκln

(z + z0z0

)(10)


(11)

VKI - 7 -


ε =u3lowast

κ (z + z0)(12)


σε =κ2


(13)


Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)


k =radicC1ln (z + z0) + C2 (15)



du

dz (16)


microtpartu


k2

ε

partu


k2radicCmicrok

partupartz

partu


and then

Cmicro =u4lowastk2

(18)


VKI - 8 -




Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications





2 THEORY



2 Theory


part

part(ρk) +

part

partxi(ρkui) =

part

partxj

[(micro+

microtσk

)partk

partxj


VKI - 6 -


part

part(ρε) +

part

partxi(ρεui) =

part

partxj

[(micro+

microtσε

)partε

partxj

]+ Cε1

ε


ε2

k (2)



Gk = microtS2 S =

radic2SijSij Sij =

1

2

(partuipartxj

+partujpartxi

) (3)


part

partz

(microtσk

partk

partz


part

partz

(microtσε

partε

partz

)+ Cε1Gk

ε

kminus Cε2ρ

ε2

k= 0 (5)

Gk = microt

(partu

partz

)2

(6)


microtpartu



ulowast =

radicτwρ (8)


microt = ρcmicrok2

ε (9)



U =ulowastκln

(z + z0z0

)(10)


(11)

VKI - 7 -


ε =u3lowast

κ (z + z0)(12)


σε =κ2


(13)


Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)


k =radicC1ln (z + z0) + C2 (15)



du

dz (16)


microtpartu


k2

ε

partu


k2radicCmicrok

partupartz

partu


and then

Cmicro =u4lowastk2

(18)


VKI - 8 -




Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications






part

part(ρε) +

part

partxi(ρεui) =

part

partxj

[(micro+

microtσε

)partε

partxj

]+ Cε1

ε


ε2

k (2)



Gk = microtS2 S =

radic2SijSij Sij =

1

2

(partuipartxj

+partujpartxi

) (3)


part

partz

(microtσk

partk

partz


part

partz

(microtσε

partε

partz

)+ Cε1Gk

ε

kminus Cε2ρ

ε2

k= 0 (5)

Gk = microt

(partu

partz

)2

(6)


microtpartu



ulowast =

radicτwρ (8)


microt = ρcmicrok2

ε (9)



U =ulowastκln

(z + z0z0

)(10)


(11)

VKI - 7 -


ε =u3lowast

κ (z + z0)(12)


σε =κ2


(13)


Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)


k =radicC1ln (z + z0) + C2 (15)



du

dz (16)


microtpartu


k2

ε

partu


k2radicCmicrok

partupartz

partu


and then

Cmicro =u4lowastk2

(18)


VKI - 8 -




Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications






ε =u3lowast

κ (z + z0)(12)


σε =κ2


(13)


Sε (z) =ρu4lowast

(z + z0)2

((Cε2 minus Cε1)

radicCmicro

κ2minus 1

σε

) (14)


k =radicC1ln (z + z0) + C2 (15)



du

dz (16)


microtpartu


k2

ε

partu


k2radicCmicrok

partupartz

partu


and then

Cmicro =u4lowastk2

(18)


VKI - 8 -




Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








Sk (z) =ρulowastκ

σk

part[(z + z0)

partkpartz

]

partz (19)



part

partz

(microtσk

partk

partz

)= 0 (20)


part

partz

(ρcmicro

k2

ε

σk

partk

partz

)=

part

partz

ρcmicro

k2radicCmicrok

dudz

σk

partk

partz

=

part

partz

ρu4lowastk2


σk

partk

partz


= ulowastκ

1(z+z0)

(Equation

(10))

part

partz

(ρu2lowastσk

dudz

partk

partz

)=

part

partz

(ρu2lowast

σkulowastκ

1(z+z0)

partk

partz

)=

part

partz

(ρulowastκ

σk(z + z0)

partk

partz

)= 0 (21)

which gives

(z + z0)partk

partz= const (22)


k (z) = C1ln (z + z0) + C2 (23)


VKI - 9 -




κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








κln(z+z0z0

)Equation (10)



Equation (12)


k2

εEquation (9)


2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)Equation (14)

Cmicro = u4lowastk2

Equation (18)


part

partz

ρcmicro

k2radicCmicrok

dudz

σε

partradicCmicrok

dudz

partz

=

part

partz

ρ

radicu4lowastk2k

σεulowast

κ(z+z0)

partradic

u4lowastk2

kulowastκ(z+z0)

partz

=

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)

part

partz

(minusρu

4lowast

σε

1

(z + z0)

)=ρu4lowastσε

1

(z + z0)2 (24)



languprime2rang

u2lowast= 5minus 4

z

h(25)

langvprime2rang

u2lowast= 2minus z

h(26)

VKI - 10 -


langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications






langwprime2rang

u2lowast= 17minus z

h(27)




k (z) =1

2

(languprime2rang

+langvprime2rang

+langwprime2rang)

=u2lowast2

(87minus 6

z

h

)(29)


22 Wall treatment


Upulowast

=ln (y+)

κ+B (30)




Upulowast

=1

κln(Ey+

)minus∆B

(k+S) (31)




VKI - 11 -




VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








VKI - 12 -


∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications






∆B(k+S)

=1

κln(CSk

+S

)(32)


=1

κln

(Ey+

CSk+S

) (33)


Ey+

CSk+S

=zp + z0z0


νz0

ulowastkSν

(z0 + zp)sim Ez0

kSsim Ez0

zp(34)


Upulowast

=1

κln

(E

CS

)


(partU

partz

)2

wf std

=

[part

partz

(ulowastκln

(Ez

CSkS

))]2=(ulowastκz

)2(35)

(partU

partz

)2

in ABL

=

[part

partz

(ulowastκln

(z + z0z0

))]2=

(ulowast

κ (z + z0)

)2

(36)


VKI - 13 -



Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







Up =ulowastκln

(z + z0z0

)(37)

εp =C075micro k15

κ (z + z0)(38)

Gk =τ 2w


(39)



acute 2zp0




Upulowast

=1

κln(Ey+

)(40)

with


νE =

ν

z0ulowast (41)



VKI - 14 -



Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







Up =u

ln

z + z0

z0

p =C075

micro k15

(z + z0)

Gk =2w


Up

u=

1

lneEey+

ey+ =u (z + z0)

eE =

z0u

ey+ = y+ eE = E

Rough wall

Smooth wall


centroid

UP Gk ϵP


VKI - 15 -





δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications









δ = min

[(uminus uABLuABL

) 1

](42)





VKI - 16 -




Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








Ω =radic

2ΩijΩij Ωij =1

2


)

S =radic

2SijSij Sij =1

2

(partuipartxj

+partujpartxi

)


Sε Y AP = 083ε2

k

(k15

εleminus 1

)(k15

εle

)2

(45)




εFMMK = min

(Ω

S 1

) (46)


Rij = uiuj =2


k3

ε2

(SikSjk minus

1

3SklSklδij

)+

+ c2Cmicrok3


k3

ε2

(ΩikΩjk-

1

3ΩklΩklδij

)(47)


VKI - 17 -



VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







VKI - 18 -

3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications





3 APPLICATIONS

3 Applications






C2u4lowastCmicro

053 052


Skρulowastκσk


-

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 19 -





VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications









VKI - 20 -



VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







VKI - 21 -




partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








partupartz






u2lowast2

(87minus 6 z

h

)


Sk - ρulowastκσk


Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)

VKI - 22 -





VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications









VKI - 23 -



VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







VKI - 24 -



VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







VKI - 25 -





VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications









VKI - 26 -




q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








q =Nsum

i=1

δi (48)

with

δi =

1 if



0 else(49)




VKI - 27 -



UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







UUref


Gorle et al [39] 087 081 059 071PS1 089 078 056 07PS2 09 079 058 071

kU2ref


Gorle et al [39] 054 031 049 047PS1 054 04 054 052PS2 058 054 062 059




VKI - 28 -




VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








VKI - 29 -







VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications











VKI - 30 -





VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications









VKI - 31 -







VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications











VKI - 32 -



UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







UUref


kU2ref



VKI - 33 -



UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications







UUref


Craft α = 2 090 076 050 066Craft α = 5 091 076 052 068

kU2ref


Craft α = 2 062 040 061 057Craft α = 5 063 038 066 061



h (r) =

hmax

12

(1 + cos 2πr

rmax

)if r lt rmax

0 else


VKI - 34 -




C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








C1 [m2s2] -00053C2 [m2s2] 0050

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)


VKI - 35 -





VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications









VKI - 36 -




STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications








STD 081 049MMK 079 047


STD 071 047MMK 070 052



STD -6 -13MMK -9 -5


STD -26 -23MMK -29 32

VKI - 37 -




VKI - 38 -




C1 [m2s2] -0351C2 [m2s2] 261

Cmicrou4lowastk2

Sk -

Sερu4lowast

(z+z0)2

((Cε2minusCε1)

radicCmicro

κ2minus 1

σε

)



VKI - 39 -




STD 094 031MMK gt099 044


STD 081 044MMK 081 044



U =ulowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

](50)

T = TW +Tlowastκ

[ln

(z

z0

)+ φm

( zL

)minus 1

]minus g

cp(z minus z0) (51)






radicΦe

(zL

)

Φm

(zL

) (52)

ε =u3lowastκz

Φe (53)

with

Φe = 1 + 4z

L (54)


VKI - 40 -


equations


σk

)partkpartz

]

partz(55)

Sε (z) =ρu4lowast

(z + z0)2

[(Cε2 minus Cε1)

radicCmicro

κ2Φe

radicΦm

Φe

minus 1

σε

(2

Φm

minus 1

Φ2m

+TlowastκT

)] (56)


VKI - 41 -


(a) (b)

(c) (d)

(e) (f)


VKI - 42 -


References














VKI - 43 -






S0167610597001475








F0040001art00047


VKI - 44 -



S1352231008009084













VKI - 45 -













VKI - 46 -

Introduction

Theory


Wall treatment


Applications





cfd inflow conditions, wall functions and turbulence models for flows around obstacles

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