mathematical modeling of turbulent flows · mathematical modeling of turbulent flows prof. dr.-ing....

Faculty of Mechanical Engineering and Marine Technology

Chair of Modelling and Simulation

Mathematical Modeling ofTurbulent Flows

Prof. Dr.-Ing. habil. Nikolai Kornev

Rostock2013

Contents

1 Main Equations of Fluid Dynamics 131.1 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Classification of forces acting in a fluid. . . . . . . . . . . . . . 14

1.2.1 Body forces . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 Surface forces. . . . . . . . . . . . . . . . . . . . . . . . 151.2.3 Properties of surface forces. . . . . . . . . . . . . . . . 15

1.3 Navier Stokes Equation . . . . . . . . . . . . . . . . . . . . . . 17

2 Physics of turbulence 232.1 Definition of the turbulence . . . . . . . . . . . . . . . . . . . 232.2 Vortex dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Vorticity transport equation . . . . . . . . . . . . . . . 232.2.2 Vorticity and vortices . . . . . . . . . . . . . . . . . . . 242.2.3 Vortex amplification as an important mechanism of the

turbulence generation . . . . . . . . . . . . . . . . . . . 262.2.4 Vortex reconnection . . . . . . . . . . . . . . . . . . . . 282.2.5 Richardson poem (1922) . . . . . . . . . . . . . . . . . 292.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Experimental observations . . . . . . . . . . . . . . . . . . . . 302.3.1 Laminar- turbulent transition in pipe. Experiment of

Reynolds. . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.2 Laminar- turbulent transition and turbulence in jets. . 332.3.3 Laminar- turbulent transition in wall bounded flows. . 362.3.4 Uneven distribution of the vorticity in the turbulent

flows at large Reynolds number . . . . . . . . . . . . . 372.3.5 Distribution of the averaged velocity in the turbulent

boundary layer . . . . . . . . . . . . . . . . . . . . . . 42

3 Basic definitions of the statistical theory of turbulence 493.1 Reynolds averaging . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Isotropic and homogeneous turbulence . . . . . . . . . . . . . 50

3

3.3 Correlation function. Integral length. . . . . . . . . . . . . . . 503.3.1 Some relations in isotropic turbulence . . . . . . . . . . 533.3.2 Taylor microscale λ . . . . . . . . . . . . . . . . . . . . 553.3.3 Correlation functions in the Fourier space . . . . . . . 563.3.4 Spectral density of the kinetic energy . . . . . . . . . . 57

3.4 Structure functions . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Probability density function . . . . . . . . . . . . . . . 573.4.2 Structure function . . . . . . . . . . . . . . . . . . . . 58

4 Kolmogorov theory K41 614.1 Physical background . . . . . . . . . . . . . . . . . . . . . . . 614.2 Dissipation rate . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Kolmogorov hypotheses . . . . . . . . . . . . . . . . . . . . . . 644.4 Three different scale ranges of turbulent flow . . . . . . . . . . 664.5 Classification of methods for calculation of turbulent flows. . . 694.6 Limitation of K-41. Kolmogorov theory K-62 . . . . . . . . . . 69

5 Reynolds Averaged Navier Stokes Equation (RANS) 73

6 Reynolds Stress Model (RSM) 796.1 Derivation of the RSM Equations . . . . . . . . . . . . . . . . 79

6.1.1 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.1.2 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.1.3 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.1.4 Analysis of terms . . . . . . . . . . . . . . . . . . . . . 82

7 Equations of the k - ε Model 857.1 Derivation of the k-Equation . . . . . . . . . . . . . . . . . . . 85

7.1.1 Closure of terms of k equation . . . . . . . . . . . . . . 867.1.2 Derivation of the ε-Equation . . . . . . . . . . . . . . . 87

8 Large Eddy Simulation (LES) 898.1 LES filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.1.1 Properties of filtering . . . . . . . . . . . . . . . . . . . 908.2 LES equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.3 Smagorinsky model . . . . . . . . . . . . . . . . . . . . . . . . 92

9 Subgrid Stress (SGS) Models 959.1 Model of Germano (Dynamic Smagorinsky Model) . . . . . . . 959.2 Scale similarity models . . . . . . . . . . . . . . . . . . . . . . 979.3 Mixed similarity models . . . . . . . . . . . . . . . . . . . . . 98

9.3.1 A-posteriori and a-priori tests . . . . . . . . . . . . . . 100

4

10 Hybrid URANS-LES methods 10310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.2 Detached Eddy Simulation (DES) . . . . . . . . . . . . . . . . 10410.3 Description of the hybrid model proposed in Rostock . . . . . 10710.4 Sample of hybrid method application for the tanker KVLCC2 110

10.4.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . 113

5

List of Tables

8.1 Properties of large and small scale motions . . . . . . . . . . . 918.2 Advantages and disadvantages of the Smagorinsky model . . . 94

10.1 Results of the resistance prediction using different methods.CR is the resistance coefficient, CP is the pressure resistanceand CF is the friction resistance. . . . . . . . . . . . . . . . . 113

7

List of Figures

1.1 Body and surface forces acting on the liquid element. . . . . . 141.2 Forces acting on the liquid element. . . . . . . . . . . . . . . . 161.3 Stresses acting on the liquid cube with sizes a. . . . . . . . . . 17

2.1 Vorticity and vortices . . . . . . . . . . . . . . . . . . . . . . . 252.2 Tornado . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Vortices in two-dimensional and three dimensional cases. . . . 262.4 Velocities induced by vortices. Three dimensional curvilinear

vortices induce self induced velocities. . . . . . . . . . . . . . . 262.5 Illustration of the vortex folding. . . . . . . . . . . . . . . . . 282.6 Scenario of vortex amplification . . . . . . . . . . . . . . . . . 282.7 Scenario of vortex reconnection . . . . . . . . . . . . . . . . . 292.8 Sample of the vortex reconnection of tip vortices behind an

airplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.9 Most outstanding results in turbulence research according to

[1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.10 Sketch of the Reynolds experiment. . . . . . . . . . . . . . . . 322.11 Development of instability during the laminarturbulent tran-

sition in the circular pipe (taken from [1]). . . . . . . . . . . . 322.12 Development of instability in the jet (taken from [1]). . . . . . 332.13 Development of instability in the free jet. . . . . . . . . . . . . 342.14 Development of instability in the free jet. . . . . . . . . . . . . 342.15 Vortex structures in a free jet in a far field. . . . . . . . . . . . 342.16 Vortex structures in a free jet with acoustic impact. . . . . . . 352.17 Vortex structures in a confined jet mixer flow. . . . . . . . . . 362.18 Fine vortex structures in a confined jet mixer flow. PLIF mea-

surements by Valery Zhdanov (LTT Rostock). Spatial resolu-tion is 31µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.19 Scenario of laminar turbulent transition in the boundary layeron a flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.20 Streaks visualized by hydrogen bubbles in the boundary layeron a flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9

2.21 Conceptual model of the organization of the turbulence closeto the wall proposed by Adrian et al. (2000). . . . . . . . . . . 39

2.22 Sketch of the flow. 1- knee bend of nozzle, 2- plate for dampingof vortices shed from knee bend 1, 3- outer tube, 4- supportplates, 5- nozzle, 6- test section, 7- water box. . . . . . . . . . 39

2.23 Snapshot of the field ω2z/ < ω2

z > within the measurementwindow in jet mixer. The averaged < ω2

z > was 1.19s−2 and0.459s−2 at, respectively, x/D = 1 and 7. . . . . . . . . . . . . 40

2.24 Snapshot of the field (uxux + uyuy)/ < uxux + uyuy > withinthe measurement window in jet mixer. The averaged < uxux+uyuy > was 0.011(m/s)2 and 0.0024(m/s)2 at, respectively,x/D = 1 and 7. . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.25 Ratios εk and Ek depending on k/N . . . . . . . . . . . . . . . 41

2.26 Influence of the laser thickness on εk. . . . . . . . . . . . . . . 41

2.27 Probability density function of radius of structures of the fieldω2z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.28 Probability density function of the axis ratio a−b√ab

of structures

of the field ω2z . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.29 Condtioned probability density function of the axis ratio a−b√ab

(R >

m∆) of structures of the field ω2z . . . . . . . . . . . . . . . . . 43

2.30 Vertical distribution of the velocity ux at three different timeinstants in boundary layer. . . . . . . . . . . . . . . . . . . . . 44

2.31 Illustration of the Prandtl derivation. . . . . . . . . . . . . . . 45

2.32 Structure of the velocity distribution in the turbulent bound-ary layer. U+ = ux/uτ . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Autocorrealtion function coefficient for scalar fluctuation atthree different points A,B and C across the jet mixer. . . . . 51

3.2 Distribution of the integral length of the scalar field along thejet mixer centerline. . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Autocorrelation functions in free jet flow. . . . . . . . . . . . . 53

3.4 Illustrations of velocities used in calculations of the longitudi-nal f and transversal g autocorrelations. . . . . . . . . . . . . 54

3.5 Illustration of the autocorrelation functions f and g and Taylormicroscales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Kurtosis of the structure function for the concentration of thescalar field obtained in the jet mixer. . . . . . . . . . . . . . . 59

10

4.1 Andrey Kolmogorov was a mathematician, preeminent in the20th century, who advanced various scientific fields (amongthem probability theory, topology, intuitionistic logic, turbu-lence, classical mechanics and computational complexity). . . . 62

4.2 Illustration of the vortex cascado . . . . . . . . . . . . . . . . 62

4.3 Turbulent vortices revealed in DNS calculations performed byIsazawa et al. (2007) . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Distribution of the Kolmogorov scale along the centerline ofthe jet mixer and free jet. The dissipation rate ε is calculatedfrom the k−ε model and the experimental estimatin of Millerand Dimotakis (1991) ε = 48(U3

d/d)((x− x0)/d)−4 . . . . . . . 66

4.5 Three typical scale ranges in the turbulent flow at high Reynoldsnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6 Three typical ranges of the energy density spectrum in theturbulent flow at high Reynolds number. 1- energy containingrange, 2- inertial subrange, 3- dissipation range. . . . . . . . . 67

4.7 Experimental confirmation of the Kolmogorov law. The com-pensated energy spectrum for different flows. . . . . . . . . . . 68

4.8 Experimental confirmation of the Kolmogorov law for the con-centration fluctuations in the jet mixer. Measurements of theLTT Rostock. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.9 Three main methods of turbulent flows modelling. . . . . . . . 70

4.10 Vortex structures resolved by different models . . . . . . . . . 70

4.11 Power of the structure function. Experiments versus predic-tion of Kolmogorov and Obukhov . . . . . . . . . . . . . . . . 71

8.1 Different filtering functions used in LES . . . . . . . . . . . . 90

9.1 Illustrations for derivation of the scale similarity model . . . . 98

10.1 Zones of the Detached Eddy Simulation . . . . . . . . . . . . . 105

10.2 Squires K.D., Detached-eddy simulation: current status andperspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10.3 Squires K.D., Detached-eddy simulation: current status andperspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10.4 The division of the computational domain into the URANS(dark) and LES (light) regions at one time instant. . . . . . . 108

10.5 The cell parameters. . . . . . . . . . . . . . . . . . . . . . . . 113

10.6 The mean axial velocity field ux/u0 in the propeller plane com-puted with different models (right) vs. measurements (left). . 115

11

10.7 Circumferential distribution of the mean axial velocity field inthe propeller plane. 4 — k-ω-SST-SAS, © — DSM+V2F,solid line — KRISO experiments for the specified r/R. . . . . 115

10.8 Resolved turbulent kinetic energy k = ρ/2(u′xu′x + u′yu

′y +

u′zu′z)/u

20 multiplied with 103 in the propeller plane. Numerics

(right-half of each figure) versus measurement (left-half). . . 11610.9 Positions of probe points. R is the propeller radius. . . . . . . 11610.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

12

Chapter 1

Main Equations of FluidDynamics

1.1 Continuity equation

We consider the case of uniform density distribution ρ = const. The con-tinuity equation has the following physical meaning: The amount of liquidflowing into the volume U with the surface S is equal to the amount of liquidflowing out. Mathematically it can be expressed in form:∫

S

~u~nds = 0 (1.1)

Expressing the scalar product ~u~n∫S

(ux cos(nx) + uy cos(ny) + uz cos(nz)

)ds = 0.

and the Gauss theorem we get∫U

(∂ux∂x

+∂uy∂y

+∂uz∂z

)dU = 0

Since the integration volume U is arbitrary, the integral is zero only if

∂ux∂x

+∂uy∂y

+∂uz∂z

= 0 (1.2)

In the tensor form the continuity equation reads:

13

∂ui∂xi

= 0 (1.3)

1.2 Classification of forces acting in a fluid.

The inner forces acting in a fluid are subdivided into the body forces andsurface forces (Fig. 8.1).

Figure 1.1: Body and surface forces acting on the liquid element.

1.2.1 Body forces

Let ∆~f be a total body force (integral of ~f) acting on the volume ∆U . Letus introduce the strength of the body force as limit of the ratio of the forceto the volume:

~F = lim∆U→0

∆~f

ρ∆U(1.4)

14

which has the unit kgms2

m3

kg1m3 = ms−2. Typical body forces are gravitational,

electrostatic or electromagnetic forces. For instance, we have the followingrelations for the gravitational forces:

∆~f = ρg∆U~k (1.5)

where ∆~f is the gravitational force acting on a particle with volume ∆U . Thestrength of the gravitational force is equal to the gravitational acceleration:

~F = lim∆U→0

(−ρg∆U~k

ρ∆U) = −g~k (1.6)

The body forces are acting at each point of fluid in the whole domain.

1.2.2 Surface forces.

The body forces are acting at each point at the boundary of the fluid element.Usually they are shear and normal stresses. The strength of surface forces isdetermined as

~pn = lim∆S→0

∆~Pn∆S

(1.7)

with the unit kgms2

1m2 = kg

ms2. A substantial feature of the surface force is the

dependence of ~pn on the orientation of the surface ∆S.

The surface forces are very important because they act on the body fromthe side of liquid and determine the forces ~R arising on bodies moving in thefluid:

~R =

∫S

~pndS

~M =

∫S

(~r × ~pn)dS

(1.8)

1.2.3 Properties of surface forces.

Let us consider a liquid element in form of the tetrahedron (Fig. 1.2).Its motion is described by the 2nd law of Newton:

ρ∆Ud~u

dt= ρ∆U ~F + ~pn∆S − ~px∆Sx − ~py∆Sy − ~pz∆Sz (1.9)

15

Figure 1.2: Forces acting on the liquid element.

Dividing r.h.s and l.h.s. by the surface of inclined face ∆S results in:

ρ∆U

∆S

(d~u

dt− ~F

)= ~pn − ~px

∆Sx∆S− ~py

∆Sy∆S− ~pz

∆Sz∆S

(1.10)

Let us find the limit of (1.10) at ∆S → 0:

lim∆S→0

∆U

∆S= 0, lim

∆S→0

∆Sx∆S

= cos(nx), (1.11)

lim∆S→0

∆Sy∆S

= cos(ny), lim∆S→0

∆Sz∆S

= cos(nz) (1.12)

Substitution of (1.11) and (1.12) into (1.10) results in the following relationbetween ~pn and ~px, ~py, ~pz:

~pn = ~px cos(nx) + ~py cos(ny) + ~pz cos(nz) (1.13)

Let us write the surface forces through components:

~px =~ipxx +~jτxy + ~kτxz

~py =~iτyx +~jpyy + ~kτyz

~pz =~iτzx +~jτzy + ~kpzz

16

Here τij are shear stress (for instance τ12 = τxy), whereas pii are normalstress (for instance p11 = pxx). From moment equations (see Fig. 1.3) one canobtain the symmetry condition for shear stresses: τzya−τyza = 0⇒ τzy = τyz,and generally:

τij = τji (1.14)

Figure 1.3: Stresses acting on the liquid cube with sizes a.

The stress matrix is symmetric and contains 6 unknown elements: pxx τxy τxzτxy pyy τyzτxz τyz pzz

(1.15)

1.3 Navier Stokes Equation

Applying the Newton second law to the small fluid element dU with thesurface dS and using the body and surface forces we get:∫

U

d~u

dtρdU =

∫U

~FρdU +

∫S

~pndS (1.16)

The property of the surface force can be rewritten with the Gauss theoremin the following form:

17

∫S

~pndS =

∫S

(~px cos(nx) + ~py cos(ny) + ~pz cos(nz)) dS

=

∫U

(∂~px∂x

+∂~py∂y

+∂~pz∂z

)dU

The second law (1.16) takes the form:∫U

d~u

dtρdU =

∫U

~FρdU +

∫U

(∂~px∂x

+∂~py∂y

+∂~pz∂z

)dU =

∫U

[d~u

dtρ− ρ~F −

(∂~px∂x

+∂~py∂y

+∂~pz∂z

)]dU = 0

Since the volume dU is arbitrary, the r.h.s. in the last formulae is zero onlyif:

d~u

dt= ~F +

1

ρ

(∂~px∂x

+∂~py∂y

+∂~pz∂z

)(1.17)

The stresses in (1.17) are not known. They can be found from the generalizedNewton hypothesis: pxx τxy τxz

τxy pyy τyzτxz τyz pzz

= −

p 0 00 p 00 0 p

+ 2µSij (1.18)

where p is the pressure,

S11 = Sxx = ∂ux∂x

; S12 = Sxy = 12

(∂ux∂y

+ ∂uy∂x

); S13 = Sxz = 1

2

(∂ux∂z

+ ∂uz∂x

)

S21 = S12, S22 = Syy = ∂uy∂y, S23 = Syz = 1

2

(∂uy∂z

+ ∂uz∂y

)

S31 = S13, S32 = S23, S33 = Szz = ∂uz∂z

The liquids obeying (1.18) are referred to as the Newtonian liquids.The normal stresses can be expressed through the pressure p:

18

pxx = −p+ 2µ∂ux∂x, pyy = −p+ 2µ∂uy

∂y, pzz = −p+ 2µ∂uz

∂z

The sum of three normal stresses doesn’t depend on the choice of the coor-dinate system and is equal to the pressure taken with sign minus:

pxx + pyy + pzz3

= −p (1.19)

The last expression is the definition of the pressure in the viscous flow: Thepressure is the sum of three normal stresses taken with the sign minus. Sub-stitution of the Newton hypothesis (1.18) into (1.17) gives (using the firstequation as a sample):

ρduxdt

= ρFx + ∂∂x

(− p+ 2µ∂ux

∂x

)+ ∂

∂y

(µ

(∂uy∂x

+ ∂ux∂y

))

+ ∂∂z

(µ

(∂ux∂z

+ ∂uz∂x

))=

= %Fx − ∂p∂x

+ µ

(∂2ux∂x2

+ ∂2ux∂y2

+ ∂2ux∂z2

)+

+µ ∂∂x

(∂ux∂x

+ ∂uy∂y

+ ∂uz∂z

)

The last term in the last formula is zero because of the continuity equation.Doing similar transformation with resting two equations in y and z direc-tions, one can obtain the following equation, referred to as the Navier-Stokesequation:

d~u

dt= ~F − 1

%∇p+ ν∆~u (1.20)

The full or material substantial derivative of the velocity vector d~udt

is theacceleration of the fluid particle. It consists of two parts: local accelerationand convective acceleration:

d~udt

=∂~u

∂t︸︷︷︸+ux∂~u

∂x+ uy

∂~u

∂y+ uz

∂~u

∂z︸︷︷︸local convective

acceleration acceleration

19

The local acceleration is due to the change of the velocity in time. It ispresent even if the particle is at the rest. The convective acceleration isdue to particle motion in a nonuniform velocity field. The Navier-StokesEquation in tensor form is:

∂ui∂t

+ uj∂ui∂xj

= Fi −1

ρ

∂p

∂xi+ ν

∂

∂xj

(∂

∂xjui

)(1.21)

Using the continuity equation (1.3) the convective term can be written in theconservative form:

uj∂ui∂xj

=∂

∂xj

(uiuj

)(1.22)

Finally,the Navier Stokes in the tensor form is:

∂ui∂t

+∂

∂xj(uiuj) = Fi −

1

ρ

∂p

∂xi+ ν

∂

∂xj

(∂

∂xjui

)(1.23)

The Navier Stokes equation together with the continuity equation (1.3) isthe closed system of partial differential equations. Four unknowns velocitycomponents ux, uy, uz and pressure p are found from four equations. Theequation due to presence of the term ∂

∂xj(uiuj) is nonlinear.

The boundary conditions are enforced for velocity components and pressureat the boundary of the computational domain. The no slip condition ux =uy = uz = 0 is enforced at the solid body boundary. The boundary conditionfor the pressure at the body surface can directly be derived from the NavierStokes equation. For instance, if y = 0 corresponds to the wall, the NavierStokes Equation takes the form at the boundary:

∂p

∂x= %Fx + µ

∂2ux∂y2

∂p

∂y= %Fy + µ

∂2uy∂y2

∂p

∂z= %Fz + µ

∂2uz∂y2

Very often the last term in the last formulae is neglected because secondspatial derivatives of the velocity are not known at the wall boundary.

20

Till now, the existence of the solution of Navier Stokes has been not proven bymathematicians. Also, it is not clear whether the solution is smooth or allowssingularity. The Clay Mathematics Institute has called the Navier–Stokesexistence and smoothness problems one of the seven most important openproblems in mathematics and has offered one million dollar prize for itssolution.

21

Chapter 2

Physics of turbulence

2.1 Definition of the turbulence

Flow motions are subdivided into laminar flows and turbulent ones. Theword ”Laminar” in Greek means layer. The fluid particles move orderlyin layers without intense lateral mixing. The disruption between layers isabsent. On the contrary the turbulent flow is very chaotic with strong eddiesand intense mixing across the flow.Turbulent motion is the three dimensional unsteady flow motion with• chaotical trajectories of fluid particles,• fluctuations of the velocity and• strong mixingarisen at large Re numbers due to unstable vortex dynamics.

2.2 Vortex dynamics

The vortex dynamics is the key to understand what happens in the turbulentflow.

2.2.1 Vorticity transport equation

The vector calculus relation reads:

1

2∇(A ·A) = A× (∇×A) + (A∇)A (2.1)

Taking u = A we get:

1

2∇(u · u) = u× ω + (u∇)u (2.2)

23

where ω = ∇×u is the vorticity. Here we used the identity ∇×(12∇(u·u)) =

0. Application of the curl operator to (2.3) results in

∇×((u∇)u) = −∇×(u×ω) = −u(∇ω)+ω(∇u)−(ω∇)u+(u∇)ω (2.3)

Both vectors u and ω satisfy the continuity equation, i.e. ∇ω = 0 and∇u = 0:

∇× ((u∇)u) = −(ω∇)u + (u∇)ω (2.4)

Let us apply the curl operator to the Navier Stokes equation

∇× (∂u

∂t+ (u∇)u) = ∇× (−1

ρ∇p+ ν∆u) (2.5)

∂ω

∂t+∇× ((u∇)u) = ν∆(∇× u) = ν∆ω (2.6)

Substituting (2.4) into (2.6) results in

∂ω

∂t+ (u∇)ω = (ω∇)u + ν∆ω (2.7)

Dω

Dt= (ω∇)u + ν∆ω (2.8)

The equation (2.8) is the vorticity transport equation.

2.2.2 Vorticity and vortices

The vortices are main players in turbulent flows. Here we would like toemphasize the difference between the vorticity and vortices. The vorticityis the curl of the velocity ω = ∇ × u. The vorticity is usually not zero inviscous flows especially in areas close to the walls. Speaking about vorticeswe bear in mind the concentrated structures of the vorticity field ω = ∇×u.The difference between the vorticity and vortices is illustrated in Fig. 2.1.The boundary layer is the flow area with strong but smoothly distributedvorticity (Fig. 2.1a). Due to instabilities, that will be discussed later, theconcentrated vortex structures arise in the smooth vorticity field (Fig. 2.1b).A famous sample of concentrated vortex structures is the tornado (Fig. 2.2).

The vorticity is solenoidal:

∇ω = ∇(∇× u) = 0 (2.9)

The consequence of the condition (2.9) is:• All vortex lines, defined as the lines which are tangential to the vorticity

vector ω × dl = 0, are closed in the three dimensional case (Figure 2.3).

24

The velocity induced by vorticity ω occupied the volume U are calculatedfrom the Biot- Savart law:

u(x, t) =1

4π

∫U

ω × (x− r)

|x− r|3(2.10)

The velocities induced by two dimensional and three dimensional vortexstructures are shown in Fig. 2.4. An important fact is the appearance ofself induced velocities on curvilinear three dimensional vortex structures.They are responsible for leapfrog vortex ring motion (http://www.lemos.uni-rostock.de/galerie/). The self induced velocities is the reason for convectiveinstability of three dimensional vortex structures.

Figure 2.1: Vorticity and vortices

Figure 2.2: Tornado

25

Figure 2.3: Vortices in two-dimensional and three dimensional cases.

Figure 2.4: Velocities induced by vortices. Three dimensional curvilinearvortices induce self induced velocities.

2.2.3 Vortex amplification as an important mechanismof the turbulence generation

As mentioned above the vortices are main players during the laminar- tur-bulent transition. The vorticity can be essentially intensified (amplified) dueto action of neighboring vortices or even due to self induction. The reason

26

for that can be explained by analysis of the vorticity transport equation

Dω

Dt= (ω∇)u + ν∆ω (2.11)

The r.h.s. of (2.11) contains two terms. The first term (ω∇)u is responsiblefor the rotation of the vorticity vector ω and enlargement or reduction ofits magnitude |ω|. The second diffusion term results in spreading of thevorticity in the space. The term (ω∇)u is responsible for the amplificationof the vorticity.

The effect of the amplification can easily be understood if we considerthe vortex with vector aligned along the x-axis ωx > 0. If such a vortexis in the fluid stretching area ∂ux

∂x> 0, the term ωx

∂ux∂x

> 0 is positive. Asa result, Dωx

Dt> 0 is positive, what leads to the increase of the vorticity

ωx. As shown analytically by Novikov [2] for a simple model problem, thevortex strength of vortons structures can increase exponentially up to theinfinity without viscosity effects. The vorticity growth caused by inviscidamplification term is counterbalanced by the diffusion term. Two terms onr.h.s. of (2.11) compete with each other. In the inviscid fluid the circulationof the vortex core is constant Γ =

∫SωxdS = const. Increase of ωx results

in the decrease of the cross section S. The vortex becomes thinner. Thediffusion acts against and makes the vortex thicker. In some flow regionsthe amplification can be stronger that diffusion. The thin vortex losses thestability and is folded.

As shown by by Chorin [3] and [4] the folding is necessary mechanismpreventing the exponential growth of vorticity. If the amplification is toostrong, the vorticity goes to infinity and the energy is not kept constant.Chorin [3] notes that ”as the vortices stretch, their cross-section decreasesand the energy associated with them would increase unless they arrangedthemselves in such a way that their velocity canceled. The foldings achievessuch cancelation”. This could be easily explained using a simple sample. Ifwe have just one straight infinite vortex it induces the velocity in the planeperpendicular to its axis. If this vortex is tangled the vortex pieces withdifferent vorticity direction are approaching close each to other cancelingtheir induction (see Fig. 2.5). Chorin [4] explicitly specifies typical scales offolding: ” the inertial range 1 properties are due to the appearance of foldedvortex tubes, which behave on large scales as self-avoiding walks, and onsmall scales contain a large number of folds (=hairpins) that are needed tosatisfy the constraint of energy conservation”.

Summarizing all effects mentioned above one can imagine the followingscenario (see Fig. 2.6). Let the vorticity at a certain point of the flow grows.

1 this term will be introduced in the next chapter

27

Figure 2.5: Illustration of the vortex folding.

In the real physical process folding prevents the growth of the kinetic energyand increases the canceling effect of viscosity. Then the vortex structuresbreaks down into small structures due to reconnection mechanism describedin the next subsection.

Figure 2.6: Scenario of vortex amplification

2.2.4 Vortex reconnection

Let us consider the vortex ring (see Figure (2.7)). Due to convective insta-bility or influence of neighboring vortices the vortex ring is deformed. Dueto self induction two opposite sides of the ring are merged. As soon as twoelements with opposite vorticity sign are approaching each to other, theystart to cancel each other by mutual diffusion. The vortices disappear inthe area of the contact. Two small vortices are created from one big vortex.Each small vortex ring breaks then down into smaller vortices and so on.

28

Figure 2.7: Scenario of vortex reconnection

The energy of small vortices is equal to the energy of the big vortex witha small loss caused by the dissipation. This fact is formulated in the sen-tence, which is common in the turbulence theory: The energy is transferredfrom large scale vortices to small scales vortices. The reconnection processcan be observed on macroscales. The decay, break up of tip vortices behindthe airplane proceeds according to the same scenario (see Fig. (2.8)). Thereconnection process is, perhaps, the main mechanism of vortex cascade inthe turbulent flows, i.e. transformation of big vortices into small ones. Thereconnection can also lead to enlargement of small vortices if two rings ap-proach each to other as shown in Fig. 2.7 (see red circle). In this case theenergy of small vortices turns into the energy of the big vortex. This processis referred to as the energy back scattering. Statistically, the direct energyflux sufficiently surpasses the backward one.

2.2.5 Richardson poem (1922)

The vortex turbulence cascade means that large eddies break down to formsmall eddies as turbulence cascades from large scales to small ones. This ideawas formulated in the famous poem by Richardson (1922):

Big whorls have little whorls,Which feed on their velocity;

29

Figure 2.8: Sample of the vortex reconnection of tip vortices behind an air-plane.

And little whorls have lessor whorls,And so on to viscosity (in the molecular sense).

2.2.6 Summary

Vortex arise in the fluid due to viscosity effects. They experience instabilityand amplification. Diffusion counteracts the amplification. If the Reynoldsnumber is large, the vortex structures are strong and concentrated. Theamplification can dominate at some fluid region over the diffusion. Thevortex instability is not damped by viscosity. The flow becomes stochasticdue to mutual interaction of unstable vortices. The big vortices break downinto small ones by means of vortex reconnection. The vortex instabilityprocess is identified as the turbulence.

2.3 Experimental observations

Different regimes of the fluid motion were revealed very early, perhaps, inantique times. Much later, Leonardo da Vinci recognized two states of thefluid motion and introduced the term ”la turbolenza”. Arkady Tsinoberin his book ”An informal introduction to turbulence” [1] presented mostoutstanding results in turbulence research in chronological order (Fig. 2.9).

30

Figure 2.9: Most outstanding results in turbulence research according to [1].

2.3.1 Laminar- turbulent transition in pipe. Experi-ment of Reynolds.

Quantitative study of turbulence was started by Osborne Reynolds (1842-1912) who performed in 1883 very famous experiment shown in Fig. 2.10.The water flows from the vessel A to the pipe B. The ink injected into thepipe B with the local flow velocity is not mixed in transversal direction andkeeps its identity if the flow velocity is small (Fig. 2.10, right). The flowunder such a condition is laminar. As soon as the flow velocity increases dueto water level raise in the vessel A, the ink jet loses the stability and is mixedwith surrounding water (Fig. 2.10, right). The flow becomes turbulent. Inkjet development at different flow velocities in the circular pipe is shown inFig. 2.11. The great merit of Reynolds lies in the fact, that he in contrast tohis predecessors quantified the laminar turbulent transition. He showed thatthe transition in pipes occurs if the Reynolds number Re = UbD/ν exceedsthe threshold around ∼ 2400. Here Ub is the bulk velocity in pipe determinedas the ratio of the flow rate to the pipe cross section, i.e. Ub = Q/(πD2/4),D is the pipe diameter and ν is the kinematic viscosity coefficient (ν ∼10−6m2s−1 for water and ν ∼ 10−6m2s−1 for air). Later, it was shown thatthe transition strongly depends on the perturbations presented in the flow.The experimental setup of Reynolds has been preserved at the University ofManchester in UK. The experiments done nowadays shown that the laminar-

31

Figure 2.10: Sketch of the Reynolds experiment.

Figure 2.11: Development of instability during the laminarturbulent tran-sition in the circular pipe (taken from [1]).

turbulent transition Reynolds number is less than that documented originallyby Reynolds. The reason is, presumably, the building vibration and noisecaused by traffic which was not in time of Reynolds. If the perturbations areeliminated the transition can be delayed up to Re ∼ 40000...50000.

32

2.3.2 Laminar- turbulent transition and turbulence injets.

The jet flows experiences also laminar turbulent transition shown in Fig. 2.12.Obviously, the flow can be fully or only partially turbulent. Close to thenozzle the flow is laminar. The instability is developed downstream. Theshear flow at the jet boundary is the area of rapid velocity change fromthe jet velocity to zero outside of the jet. The shear flow experiences theso called Kelvin Helmholtz instability (Fig. 2.13) resulting in formation ofconcentrated vortices which are approximately circular. It happens close tojet nozzle at x/D < 1 (see Fig. 2.14), where x is the distance from the nozzleand D is the nozzle diameter.

The Kelvin Helmholtz vortices experience then the pairing (see Fig. 2.14).One vortex overtakes the neighbor vortex creating a pair. This processhas an inviscid convective nature and can be explained thinking back tothe famous leapfrog motion of two vortex rings. In the inviscid flow theleapfrog motion is running as long as the convective instability destroysthe vortices. One vortex runs the next down, its radius decreases whereasthe speed increases. The radius of the next vortex increases, the speeddecreases. The first ring moves through the second one. The process isthen repeated. The movie illustrating this process can be downloaded fromhttp://www.lemos.uni-rostock.de/en/gallery/.

The paired vortices experience the azimuthal instability and takes thecrude ring form. Later they are destroyed downstream in the region 1 <x/D < 6. In far field at large x/D the vortex structures look like a tree withbranches oriented against the main flow direction (see Fig. 2.15).

Figure 2.12: Development of instability in the jet (taken from [1]).

Creation of vortex rings is the reason for the jet noise. The noise pro-duced, for instance, by jet propulsors of airplanes is the action of these vor-tices. The vortices play a significant positive role in jet mixers widely used infood industry, chemical engineering, etc. That is why, one of the most per-

33

Figure 2.13: Development of instability in the free jet.

Figure 2.14: Development of instability in the free jet.

Figure 2.15: Vortex structures in a free jet in a far field.

34

spective way to reduce the noise or to increase mixing is the manipulationof vortices arising behind the jet nozzle. To increase the mixing it is neces-sary to strengthen the Kelvin Helmholtz vortices. To decrease the noise thevortices should be broken down into small ones. Fig. 2.16 shows the effectof the acoustic impact on jet. The original vortices (lower picture) are splitinto small ones (upper picture).

Figure 2.16: Vortex structures in a free jet with acoustic impact.

High resolution laser diagnostics methods LIF and PIV allow to get a deepinside into the structure of the turbulent flow. Fig. 2.17 shows the structureof the confined jet mixer flow displayed by Planar Laser Induced Fluores-cence (PLIF) Method. The macrostructure obtained with low resolution isshown in the upper Figure. A small window with sizes 2.08mm × 2.72mmwas selected for high resolution PLIF measurements. The vortex microstruc-tures are presented in the lower Figure. A very important observation is thepresence of fine vortex structures. A gallery of such vortices obtained at dif-ferent distances x/D, where D is the diameter of the closing pipe, is given inFig. 2.18. The smallest vortices are the so called Kolmogorov vortices whichare considered in the next chapter.

35

Figure 2.17: Vortex structures in a confined jet mixer flow.

2.3.3 Laminar- turbulent transition in wall boundedflows.

The boundary layer on a plate is the thin layer of rapid change of the velocityfrom zero to 99.5% of the incident flow. A possible scenario of the laminarturbulent transition in the boundary layer on a flat plate is shown in Fig. 2.19First, the transversal vortices are generated in the boundary layer due to theso called Tollmien- Schlichting instability. They experience the secondaryinstability and form downstream the lambda structures. The latter interacteach with other and experience the tertiary instability. They loss originalregular form and become stochastic. An important feature of the turbulentboundary layer is the presence of streaks (strips) of the low velocity fluidregions (see Fig. 2.20). They arise due to induction of lambda structures

36

Figure 2.18: Fine vortex structures in a confined jet mixer flow. PLIF mea-surements by Valery Zhdanov (LTT Rostock). Spatial resolution is 31µm.

schematically shown in Fig. 2.21.

2.3.4 Uneven distribution of the vorticity in the tur-bulent flows at large Reynolds number

In this subsection we analyze high resolved PIV measurements data obtainedby Valery Zhdanov at the Institute of Technical Thermodynamics of theRostock University. The flow is the turbulent axisymmetric jet developing ina coflow confined by a pipe of diameter D = 50mm and length 5000mm. Aschematic of this flow system is given in Fig. 2.22. Medium in both flows iswater. The inner tube had diameter d = 10mm and the length 600mm chosenfrom the condition that perturbations caused by the knee bend are suppressednear the nozzle exit. The test section of the mixer was installed in a Perspexrectangular box filled with water to reduce refraction effects. More detailedinformation about the hydrodynamic channel can be found in [? ]. Since theReynolds number based on the jet exit velocity Ud is Red = dUd/ν = 104 thejet can be considered as a fully- developed turbulent jet. PIV measurementswere performed within the window 3.232mm× 2.407mm with pixel distanceof ∆ = 68.8µm. The laser thickness estimated as ∼ 40µm is very thin.The measurement window was located on the centerline of the jet mixer atthe distances x/D = 1 and 7 from the nozzle. The vorticity was calculatedusing the central differential scheme (CDS). The snapshots of the vorticity

37

Figure 2.19: Scenario of laminar turbulent transition in the boundary layeron a flat plate.

Figure 2.20: Streaks visualized by hydrogen bubbles in the boundary layeron a flat plate.

component squared ω2z/ < ω2

z >, where <> stands for quantity averaged overthe window, is shown in Fig. 2.23. Strong uneven distribution of ω2

z pointedclearly out, that the vorticity is concentrated in a relatively small numberof spots or vortices. This character of the distribution is typical for bothinitial development of the confined jet at x/D = 1.0 with weak anisotropy(R11 = 0.09, R22 = 0.073) and in the region of its strong decay at x/D = 7.0where the flow is almost isotropic (R11 = 0.036, R22 = 0.035). Distributionfor the doubled two dimensional turbulent kinetic energy uxux +uyuy, whereui are fluctuations, is shown in Fig. 2.24 for the same time instants.

Strong concentration of vorticity is especially obvious in Fig. 2.25. Thecells are sorted in order of descend of ω2

z , i.e. the first cell has the maximumvalue of ω2

z and the last one with the number N has the minimum value.

The ratio εk =k∑i=1

ω2zi/

N∑i=1

ω2zi shows the contribution of k cells to the total

amount Ω =N∑i=1

ω2zi. The ratio along the horizontal axis shows the fraction

38

Figure 2.21: Conceptual model of the organization of the turbulence close tothe wall proposed by Adrian et al. (2000).

of cells containing εk. As seen from this figure, the dependence εk(k/N)is strongly nonlinear and reaches the saturation very quickly. Five percentof cells contains around fifty percent of the total Ω, twenty percent of cellscontains more than eighty percent, sixty percent of cells contains less thanfive percent of Ω. Therefore, the number of active cells and, respectively,number of active vortices are very small. Note that the background vorticityobtained as the average over the whole measurement window is of order of∼ 10−3 although the maximum and minimum values of a few dozens.

Figure 2.22: Sketch of the flow. 1- knee bend of nozzle, 2- plate for dampingof vortices shed from knee bend 1, 3- outer tube, 4- support plates, 5- nozzle,6- test section, 7- water box.

39

Figure 2.23: Snapshot of the field ω2z/ < ω2

z > within the measurementwindow in jet mixer. The averaged < ω2

z > was 1.19s−2 and 0.459s−2 at,respectively, x/D = 1 and 7.

Figure 2.24: Snapshot of the field (uxux + uyuy)/ < uxux + uyuy > withinthe measurement window in jet mixer. The averaged < uxux + uyuy > was0.011(m/s)2 and 0.0024(m/s)2 at, respectively, x/D = 1 and 7.

40

The distribution of Ek =k∑i=1

u2i /

N∑i=1

u2i is less nonlinear indicating the fact

that the distribution of the energy is more uniform than that of ω2z . The field

of the kinetic energy contains less spots as clearly seen from the comparisonof Figures 2.23 and 2.24. The dependence Ek(k/N) is nonlinear only atthe beginning and then becomes almost linear. The reason of more uniformdistribution is that the energy is an integral quantity, whereas ωz is thelocal one. The contribution to E is carried out not only by vortices locatedat adjacent cells within the measurement plane but also by all vortices ofthe volume including vortices located outside of the measurement window.Uneven character of the εk(k/N) distribution, revealed above, is strengthened

Figure 2.25: Ratios εk and Ek depending on k/N .

when the laser thickness of PIV measurements is decreased from ∼ 400µmto ∼ 40µm (Fig. 2.26).

Figure 2.26: Influence of the laser thickness on εk.

The vortices are inclined to the measurement window at different angles β.The trace of vortices on the measurement plane is ω sin β. One can assume

41

that the maximum ω2z corresponds to vortices which are perpendicular to

the measurement plane (β = π/2). Already visual analysis of Fig. 2.23suggests that the strongest vortices are approximately axisymmetric. Weapply the algorithm proposed in [? ] to detect the vortex structures in thefield of ω2

z using two dimensional linear approximation of ω2z . Note that the

linear approximation is consistent with CDS applied for the calculation ofωz. Fig. 2.27 shows the probability density function of the structures of thefield ω2

z . The most frequent structures have radius around ∼ 2.5∆.The p.d.f of the ratio a−b√

abindicating the circularity of the vortex cross

section is given in Fig. 2.28. As seen from Fig. 2.28 the circular cross sectioncorresponding to a = b is the most frequent case. A large fraction of vorticesis inclined to the measurement plane. Even if they are axisymmetric theirintersection with measurement plane is not circular. It means that the truenumber of circular vortices is sufficiently larger than these corresponding toa−b√ab

= 0 in Fig. 2.28. Analysis of the circularity should be done with care

because the peak of a−b√ab

= 0 can be just due to a low resolution. Indeed,if the real size of vortex is smaller than the cell size ∆, being identified atany node of the grid, it occupies four adjacent cells. In our algorithm such avortex is identified as the circle with the radius of R = ∆. The circularity ofvortices with the radius equal to ∆ is indefinable. To exclude their influencewe calculated conditioned p.d.f. of a−b√

abat R > m∆ shown in Fig. (2.29). As

clearly seen the peak- like character of p.d.f. in vicinity of a−b√ab

= 0 is kepteven at m = 4. Taking the fact into account, that the most frequent vorticeshave according to Fig. 2.27m ≈ 2.5, and results in Fig. 2.29, one can concludethat the axisymmetric approximation of fine vortices can be considered asquite appropriate. Increase of the order of spline approximation of ω2

z field upto three doesn’t change the qualitative conclusions drawn from the bilinearapproximation.

2.3.5 Distribution of the averaged velocity in the tur-bulent boundary layer

A remarkable feature of the turbulent boundary flow is the presence of threetypical velocity distribution regions. The instantaneous velocity distributionscan be quite different (s. Fig. 2.32). However, the averaged velocity hastypical distribution close to the wall regardless of the flow type.

Let us introduce the following designations:• y is the distance from the wall,• τw is the stress at the wall,

• uτ =√

τwρ

is the friction velocity,

42

Figure 2.27: Probability density function of radius of structures of the fieldω2z

Figure 2.28: Probability density function of the axis ratio a−b√ab

of structures

of the field ω2z

Figure 2.29: Condtioned probability density function of the axis ratioa−b√ab

(R > m∆) of structures of the field ω2z .

• y+ = uτyν

is the dimensionless wall distance.

The stress in wall turbulence flow can be considered as the sum of the

43

Figure 2.30: Vertical distribution of the velocity ux at three different timeinstants in boundary layer.

laminar and turbulent stresses:

τ = τl + τt (2.12)

Close to the wall the turbulent fluctuations are weak. The laminar stressτl dominates over the turbulent one τt, i.e. τ ∼ τl. We consider the thinboundary layer, i.e. the stress is approximately equal to the wall stress τw:

τ ≈ τw (2.13)

Applying the Newton hypothesis (1.18) to the two dimensional wall boundedflow, one gets from (2.13)

τw = ρνduxdy

(2.14)

oruxuτ

= y+ + C (2.15)

From the condition at the wall ux = 0 the unknown constant C is zero, i.e.

uxuτ

= y+ (2.16)

Close to the wall the velocity increases linearly ux ∼ y. This law confirmedin measurements is valid in the range 0 < y+ < 5. This region is reffered toas the viscous sublayer.Far from the wall the laminar stresses are smaller than the turbulent onesτ ∼ τt. The turbulent stress τt can be found from the Prandtl mixing length

44

model. The instantaneous velocities are presented as the sum of averaged uand fluctuation u

′parts:

ux = ux + u′

x, uy = u′

y, uy = 0 (2.17)

The averaged velocities are defined as

ux = limT→∞

T∫0

ux(t)dt (2.18)

The turbulent stress τ12 = τxy according to the definition is

τ12 = ρu′xu′y (2.19)

Figure 2.31: Illustration of the Prandtl derivation.

Prandtl proposed in 1925 a very simple algebraic relation for u′xu′y using

ideas from the kinetic gas theory developed by Boltzmann. Let us considerthe fluid particle at the distance y from the wall. Let the particle velocitybe equal to the averaged velocity at y: ux. Due to some perturbations theparticle jumps from the position y to the position y+ lx and attains the fluidlayer with the averaged velocity ux + dux

dylx. Since the particle has velocity

ux, the velocity at the point y + dy is changed. Obviously, this change is−dux

dylx, or: √

u′2x =duxdy

lx (2.20)

45

Root square of the averaged squared pulsation in vertical direction is writtenin a similar form: √

u′2y =duxdy

ly (2.21)

Introducing the correlation coefficient Rxy = u′xu′y/(√

u′2y

√u′2y)

and using

(2.20) and (2.21) one gets:

|τ12| = |τt| = ρRxylxly

(duxdy

)2

= ρl2(duxdy

)2

(2.22)

where l2 = Rxylxly is the mixing length of Prandtl. The mixing lengthis determined from empirical data. For instance, the length for the wallbounded flow is

l = κy (2.23)

κ is the first constant of the turbulence, or the constant of Karman. It isequal to 0.41. Van Driest proposed the modification of (2.23) to take thewall damping effect into account:

l = κy(

1− e−yuτνA

)(2.24)

where A is the Van Driest constant, which is equal to 26 or 27. In shear flowsl = Const · δ(x), where δ is the shear layer thickness.

We consider again the thin boundary layer, i.e. the stress is approximatelyequal to the wall stress τw = τ12. Using (2.22) and (2.23) we get

τw = ρl2(duxdy

)2

(2.25)

duxdy

=1

l

√τwρ

=uτκy

(2.26)

The differential equation (2.26) reads

uxuτ

=1

κln y+ + C (2.27)

The constant C is approximately equal to 5.2. The region (2.27) is referredto as the logarithmic region which takes place within 30 < y+ < 300. Theregion 5 < y+ < 30 between the viscous and the logarithmic regions is calledthe buffer layer. The region at y+ > 300 is the wake region. The results ofthe analysis are summarized in Fig. 2.22.

46

Figure 2.32: Structure of the velocity distribution in the turbulent boundarylayer. U+ = ux/uτ

47

Chapter 3

Basic definitions of thestatistical theory of turbulence

3.1 Reynolds averaging

Reynolds proposed to represent any stochastic quantity in turbulent flow asthe sum of its averaged part and fluctuation. For instance, this representationapplied for velocity components reads

ux = ux + u′x; uy = uy + u′y; uz = uz + u′z; (3.1)

The Reynolds averaged velocities are

ux = limT→∞

1

T

T∫0

uxdt; uy = limT→∞

1

T

T∫0

uydt;

uz = limT→∞

1

T

T∫0

uzdt

(3.2)

The averaged fluctuation is zero

u′x,y,z = 0

The root of the averaged square of fluctuations is called root mean square,or r.m.s.. The quantity averaged twice is equal to quantity averaged onceu = u.

If the turbulence process is statistically unsteady (for instance r.m.s ischanged in time), the definition of the Reynolds averaging (3.2) is not ap-plicable and should be extended using the concept of ensemble averaging.

49

Within the ensemble averaging the stochastic process is repeated N timesfrom the initial state. The turbulent quantity u is measured at a certaintime t∗ N times. The ensemble averaged quantity is then:

u(t∗) = limN→∞

1

N

N∑i=1

ui(t∗)

3.2 Isotropic and homogeneous turbulence

The turbulence is isotropic if r.m.s of all three velocity fluctuations are equal

u′2x = u′2y = u′2z (3.3)

The turbulence parameters are invariant with respect to the rotation of thereference system. The turbulence is homogeneous in some fluid volume ifall statistical parameters are the same for all points in this volume, i.e.u′2x,y,z(~x) = u′2x,y,z(~x+ ~r). This equality can be written for all statistical mo-ments. The turbulence parameters are invariant with respect to the transla-tion of the reference system.

3.3 Correlation function. Integral length.

The product of two fluctuations is the correlation. The product of two fluc-tuations at two point separated by the distance ~r is the correlation function:

Rij(~x,~r) = u′i(~x)u′j(~x+ ~r) (3.4)

If i = j the correlation function Rii is reffered as to the autocorrelationfunction. In homogeneous turbulence Rii depends only on the separation:

Rij(~r) = u′i(~x)u′j(~x+ ~r) (3.5)

The coefficient of the autocorrelation function is changed between zero andone:

ρii(~x,~r) =u′i(~x)u′i(~x+ ~r)

u′2i (~x)(3.6)

A sample of the autocorrealtion function coefficient for scalar fluctuation f ′

ρf (~x,~r) =f ′(~x)f ′(~x+ ~r)

f ′2(~x)(3.7)

50

Figure 3.1: Autocorrealtion function coefficient for scalar fluctuation at threedifferent points A,B and C across the jet mixer.

at three different points A,B and C across the jet mixer is shown in Fig.3.1. In measurements presented in Fig. 3.1 the scalar f is the concentrationof the dye injected from the nozzle (see Fig. 3.1, low picture, right). Thechange of the function has a certain physical meaning. Let us consider theautocorrelation function with respect to the point C (blue line):

ρf (rC , r) =f ′(rC)f ′(rC + r)

f ′2(rC)(3.8)

where r is the radial coordinate across the pipe. The ρf (rC , rA) is negative.It means the increase of the quantity f at the point C (f ′(rC) > 0) is followedby the decrease of this quantity at the point A (f ′(rA) < 0) . This is true instatistical sense, i.e. the most probable consequence of the increase f(rC) isthe decrease of f(rA).

The correlation function and autocorrelation function can be written notonly for spatial separation but also for separation in time. For example, theautocorrelation temporal function of the ui fluctuation is

ρii(~x, τ) =u′i(~x, t)u

′i(~x, t+ τ)

u′2i (~x, t)(3.9)

51

Figure 3.2: Distribution of the integral length of the scalar field along the jetmixer centerline.

The integral of the spatial autocorrelation functions

Lij(~x) =

∞∫0

ρii(~x, xj)dxj (3.10)

is the integral length. The integral lengths are estimations of the size of thelargest vortex in flow. A sample of the integral length of the scalar field falong the jet mixer centerline (Fig. 3.1, right)

Lf (x) =

D/2∫−D/2

ρf (r)dr (3.11)

is shown in fig. 3.2, where ρf is the autocorrelation function across the jetmixer, d is the nozzle diameter, D is the diameter of the closing pipe. Lfis the estimation of the largest structure of the scalar field (in this case, thesize of the spot of colored liquid injected from the nozzle). The integral ofthe temporal autocorrelation functions

Ti(~x) =

∞∫0

ρii(~x, τ)dτ (3.12)

is the integral time length. Coefficients of the autocorrelation function of theaxial velocity fluctuations for the free jet are presented in Fig. 3.3. The line 1corresponds to the autocorrelation function calculated along the jet boundary

52

Figure 3.3: Autocorrelation functions in free jet flow.

line (shown by the blue line in Fig. 3.3, right). The line 2 corresponds to theautocorrelation function calculated along the jet axis. Oscillating characterof the dependency Ruu(∆x/d) indicates the presence of vortex structuresarising at the jet boundary and attaining the jet axis. The distance betweenzero points is roughly the vortex size.

3.3.1 Some relations in isotropic turbulence

In the isotropic turbulence u′2 = u′l(x)u′l(x) = u′t(x)u′t(x) = ... the autocorre-lation function can be represented in the form [5]

Rij = u2

(f − gr2

rirj + gδij

)(3.13)

where

f(r) =u′l(x)u′l(x+ r)

u′l(x)u′l(x), (3.14)

is the autocorrelation of the longitudinal velocity calculated in longitudinaldirection. For instance, the autocorrelation function of the ux fluctuationcalculated in x direction,

f(r) =u′x(x)u′x(x+ r)

u′x(x)u′x(x), (3.15)

or the autocorrelation function of the uy fluctuation calculated in y direction:

f(r) =u′y(y)u′y(y + r)

u′y(y)u′y(y), (3.16)

53

Figure 3.4: Illustrations of velocities used in calculations of the longitudinalf and transversal g autocorrelations.

The function f(r) is the same in both cases (3.15) and (3.16). The autocor-relaton function is calculated for transversal velocities along any direction

g(r) =u′t(x)u′t(x+ r)

u′t(x)u′t(x)(3.17)

For instance, the autocorrelation function of the ux fluctuation calculated iny direction,

g(r) =u′x(y)u′x(y + r)

u′x(y)u′x(y), (3.18)

or the autocorrelation function of the uy fluctuation calculated in x direction:

g(r) =u′y(x)u′y(x+ r)

u′y(x)u′y(x), (3.19)

The function g(r) is the same in both cases (3.18) and (3.19). The velocitiescomponents used in the previous definitions are illustrated in Fig. 3.4. ui isthe velocity pulsation vector, uil is its projection on the direction connectingtwo points (i.e. longitudinal direction), uit is its projection on the transversaldirection. Products like uituit are the correlations between points 1 and 2.

The following relations are valid between g and f in the isotropic homo-geneous turbulence (see [5]):

g = f +1

2r∂f

∂r(3.20)

Typical form of f and g is shown in Fig. 3.5. The change of the sign of gfunction is due to the continuity equation of the velocity field. The integrallength calculated using f is twice as large as that calculated using g.

54

Figure 3.5: Illustration of the autocorrelation functions f and g and Taylormicroscales.

3.3.2 Taylor microscale λ

Until Kolmogorov derived his estimations for vortices in 1941, it has beenthought that the minimum vortices arising in the turbulent flow have sizesestimated by Taylor. Let us consider the parabola fitted to the autocorre-lation function at point r = 0. The parabola intersects the horizontal axisat a certain point. The coordinate of this point λ is the scale introduced byTaylor and called as the Taylor microscale. The Taylor microscale can becalculated through the second derivative of the autocorrelation function atr = 0. The Taylor series of f in the vicinity of the point r = 0 is

f(r) = 1 +1

2

∂2f

∂r2(0)r2 + O(r4) (3.21)

The parabola fitted to the curve f(r) at r = 0 intersects the horizontal axisat the point:

λf =

√− 2

∂2f∂r2

(0)(3.22)

Similar relations can be derived for the transversal autocorrelation

λg =

√− 2

∂2g∂r2

(0)(3.23)

Today the Taylor microscale λ is still in use in turbulent research althoughit has no physical meaning. Very popular is the Reynolds number based onthe Taylor microscale

Re = u′λ/ν (3.24)

which characterizes the state of the turbulence in the flow.

55

3.3.3 Correlation functions in the Fourier space

Any continuous function can be represented in the Fourier space:

f(~r, t) =1

8π3

∞∫−∞

f(~k, t)ei~k~rd~k (3.25)

The new function f(~k, t) is then known as the Fourier transform and/orthe frequency spectrum of the function f . The Fourier transform is also areversible operation:

f(~k, t) =

∞∫−∞

f(~r, t)e−i~k~rd~r (3.26)

The Fourier transformation can be also written for the correlation function:

Rij(~r) =

∞∫−∞

Φij(~k)ei~k~rd~k, Φij(~k) =

1

8π3

∞∫−∞

Rij(~r)e−i~k~rd~r (3.27)

Very often one uses one dimensional correlation functions defined as

Θij(k1) =1

2π

∞∫−∞

Rij(r1, 0, 0)e−ik1r1dr1 =

∞∫−∞

∞∫−∞

Φij(k1, k2, k3)dk2dk3 (3.28)

Proof of the formula (3.28) The inverse Fourier transform of the functionRij(r1, 0, 0):

Θij(k1, 0, 0) =1

2π

∞∫−∞

Rij(r1, 0, 0)eik1r1dr1 (3.29)

The inverse transformation reads:

Rij(r1, 0, 0) =

∞∫−∞

Θij(k1, 0, 0)e−ik1r1dr1 (3.30)

The general definition is

Rij(r1, r2, r3) =

∞∫−∞

∞∫−∞

∞∫−∞

Φij(k1, k2, k3)ei~k~rdk1dk2dk3 (3.31)

56

From the last formula we have:

Rij(r1, 0, 0) =

∞∫−∞

( ∞∫−∞

∞∫−∞

Φij(k1, k2, k3)dk2dk3

)eik1r1dk1 (3.32)

Comparison of (3.32) with (3.30) results in the desired formula

Θij(k1, 0, 0) =

∞∫−∞

∞∫−∞

Φij(k1, k2, k3)dk2dk3 (3.33)

3.3.4 Spectral density of the kinetic energy

According to the definition of the correlation function

Rij(~r) = ui(~x)uj(~x+ ~r) (3.34)

the total kinetic energy is

TKE =1

2Rii(0) =

1

2

∞∫−∞

Φii(~k)d~k =

∞∫0

Φii(~k)d~k (3.35)

The quantity

E(k) =

∫|~k|

Φii(~k)d~k (3.36)

is the spectral density of the kinetic energy. Physically it is the energy onthe sphere k =

√k2

1 + k22 + k2

3 in the Fourier space. The total kinetic energyis then:

TKE =

∞∫0

E(k)dk (3.37)

3.4 Structure functions

3.4.1 Probability density function

We take the definitions from Wikipedia: In probability theory, a probabilitydensity function (pdf), or density of a continuous random variable, is a func-tion that describes the relative likelihood for this random variable to take

57

on a given value. The probability for the random variable to fall within aparticular region is given by the integral of this variables density over theregion. The probability density function is nonnegative everywhere, and itsintegral over the entire space is equal to one.

3.4.2 Structure function

Kolmogorov introduced the structure function of q order for any stochasticfunction. For instance, for the longitudinal velocity along the longitudinaldirection (see Fig. 3.5) it reads:

Sq(l) = 〈(u2l − u1l)q〉 (3.38)

The standard deviation squared is then σ2 = S2. If the p.d.f. of the structurefunction is described by the Gaussian function

p.d.f.(x) =1√2πσ

e−(x−µ)2

2σ2 (3.39)

where µ is the mean value of the stochastic value, the turbulence is Gaussian.In reality, the most of the turbulence parameters are not Gaussian. Thedeviations from the Gaussian turbulence is characterized by the KurtosisKurt and Skewness Sk. The Kurtosis

Kurt =〈(u2l − u1l)

4〉(〈(u2l − u1l)2〉)2

(3.40)

is three for the Gaussian turbulence. Big values of the Kurtosis means thatthe p.d.f. distribution of the structure function S1(l) = 〈u2l − u1l〉 is veryflat. The Kurtosis is also often called as flatness. If Kurtosis for small l ∼ 0is big, it means that the field of the stochastic field is very intermittent. Bigdifferences are possible even the separation between two points l is small.The p.d.f. function has long tails in this case. A sample of Kurtosis for thescalar structure function S(x) = f(x + r) − f(x) is given in Fig.3.6. Theskewness

Sk =〈(u2l − u1l)

3〉(〈(u2l − u1l)2〉)3/2

(3.41)

is zero for the Gaussian process. For the isotropic turbulence the Skewnessof the derivative

Sk =〈(∂ul∂l

)3〉[〈(∂ul∂l

)2〉]3/2

(3.42)

58

Figure 3.6: Kurtosis of the structure function for the concentration of thescalar field obtained in the jet mixer.

is −0.5. Physically it means that negative values of the derivative ∂ul∂l

aremore probable than positive ones. Please prove that the skewness (3.42) isthe skewness of the structure functions of the first order S1(l) = 〈u2l − u1l〉calculated at l→ 0.

59

Chapter 4

Kolmogorov theory K41

4.1 Physical background

One of the most outstanding results in turbulence theory was obtained byKolmogorov in 1941. The Kolmogorov theory known as K41 is based thehypothesis of local isotropy of the turbulent motion at small scales. Thephysical model behind the Kolmogorov theory is the vortex cascado illus-trated in Fig. 4.2. Big vortices with scales L (corresponds to the wave num-bers π/L in the Fourier space) break up to small ones, which in turn splitinto even smaller and so on up to the smalles vortices with the scale η. Oneof the most important vortex break up mechanisms is the vortex reconnec-tion described above. The energy is transferred from big vortices to smallones almost without the loss. The massive dissipation ε takes place at smallvortices referred to as the dissipative or the Kolmogorov vortices. The realturbulent vortices are similar to these calculated by Isazawa et al. (Fig. 4.3).Vortices are displayed at three different time instants. The upper picturesare obtained from the lower ones by filtering out the hight frequencies. Asseen big vortices are revealed in low frequency simulation. If the resolutionis increased more and more small scale vortex filaments appears on the placeof big smooth vortices. Thus, the most important physical processes duringthe vortex break up are:

• Transfer energy from large scales to small ones and

• Dissipation of the energy in small vortices.

61

Figure 4.1: Andrey Kolmogorov was a mathematician, preeminent in the20th century, who advanced various scientific fields (among them probabil-ity theory, topology, intuitionistic logic, turbulence, classical mechanics andcomputational complexity).

Figure 4.2: Illustration of the vortex cascado

62

Figure 4.3: Turbulent vortices revealed in DNS calculations performed byIsazawa et al. (2007)

4.2 Dissipation rate

Two parameters, which are of importance during the cascado process, arethe kinematic viscosity ν and the dissipation rate ε. The energy dissipationrate per unit mass of a turbulent fluid is given by

ε =ν

2

∑i,j

(∂u′i∂xj

+∂u′j∂xi

)2= 2νsijsij (4.1)

where sij = 12(∂u′i∂xj

+∂u′j∂xi

) is the fluctuating rate of strain:

sij = Sij − Sij =1

2(∂ui∂xj

+∂uj∂xi

)− 1

2(∂ui∂xj

+∂uj∂xi

) (4.2)

The dissipation ε is a random function of the coordinates and time, whichfluctuates together with the field u(x, t). In what follows we consider themean dissipation rate ε designating it as ε. The energy dissipated by thesmall vortices is generated by large scale vortices. The energy production isdefined as

P = u′iu′j

∂ui∂xj

(4.3)

Based on the dimension analysis, Prandtl and Kolmogorov proposed theestimation of the integral length of the turbulent flow

63

L ∼ k3/2

ε(4.4)

The formula (4.4) is valid for very high Reynolds numbers for the turbulencebeing in the equilibrium, i.e. the production of the turbulence is compensatedby its dissipation, i.e. P = ε.

4.3 Kolmogorov hypotheses

The basis of the Kolmogorov theory are three hypotheses, which are supposedto be valid for high Reynolds numbers Ret = vL

ν, where v =

√TKE =

√k

is the characteristic fluctuation velocity.

The Kolmogorov hypothesis of local isotropy reads:

At sufficiently high Reynolds number Ret, the small-scale turbulent motions(l lEI) are statistically isotropic

Here lEI is the lengthscale as the demarcation between the anisotropic largeeddies and the isotropic small eddies. Kolmogorov argued that all informa-tion about the geometry of the large eddies - determined by the mean flowfield and boundary conditions - is also lost. Directional information at smallscales is lost. With the other words, the direction of the vorticity vector ωof small turbulent vortices is uniformly distributed over the sphere. As aconsequence, the statistics of the small-scale motions are in a sense universal- similar in every high-Reynolds-number turbulent flow (see [6]).

The Kolmogorov first similarity hypothesis reads:

In every turbulent flow at sufficiently high Reynolds number Ret, the statisticsof the small-scale motions (l < lEI) have a universal form that is uniquelydetermined by ν and ε.

For this range of scales we can introduce characteristic size η, characteristicvelocity uη and characteristic time τη which depend only on two parametersν and ε:

η = ναηεβν , uη = ναuεβu , τη = νατ εβτ (4.5)

The analysis of dimension allows one to derive the following dependences:

64

η =

(ν3

ε

)1/4

,

uη = (νε)1/4,

τη = (ν/ε)1/2

(4.6)

Here η is the scale of the smallest dissipative vortices (Kolmogorov scale),uη the characteristic velocity of turning of Kolmogorov vortices,τη characteristic turn over time of Kolmogorov vortices.Using expressions

ε ≈ k3/2

Lu ≈ k1/2 (4.7)

some useful estimations can be derived from (4.6):

η/L ≈ (Ret)−3/4,

uη/u = (Ret)−1/4,

τη/T = (Ret)−1/2

(4.8)

Very remarkable is the first formula defining the ratio between the smallestand largest vortices in the flow. If L is, say one meter, and the fluctu-ation 1m/s, the turbulent Reynolds number in water is Ret = 106. TheKolmogorov scale is in this case 32000 as less as the flow macroscale L. Es-timations of the Kolmogorov scale in the jet mixer with nozzle diameter ofd = 1cm and closing pipe of D = 5cm diameter is shown in Fig. 4.4.

The Kolmogorov second similarity hypothesis reads:

In every turbulent flow at sufficiently high Reynolds number, the statistics ofthe motions of scale l in the range L l η have a universal form that isuniquely determined by ε, independent of ν.

This range is called as the inertial subrange. Since the vortices of this rangeare much larger than Kolmogorov vortices, we can assume that their Reynoldsnumbers lul/ν are large and their motion is little affected by the viscosity.The energy density depends on the wave number k and the dissipation rateε

E(k) = εαkβ (4.9)

65

Figure 4.4: Distribution of the Kolmogorov scale along the centerline ofthe jet mixer and free jet. The dissipation rate ε is calculated from thek − ε model and the experimental estimatin of Miller and Dimotakis (1991)ε = 48(U3

d/d)((x− x0)/d)−4

The analysis of dimension leads to the Kolmogorov law

E(k) = αε2/3k−5/3 (4.10)

where α ≈ 1.5 is the constant.

4.4 Three different scale ranges of turbulent

flow

Three different ranges can be distinguished in the spectrum of scales in thefull developed turbulence at high Reynolds numbers Ret (Fig.4.5):

• Energy containing range at l > lEI (according to Pope [6], lEI ≈ 16L).

Within this range the kinetic energy of turbulence is generated and bigturbulent eddies are created.

• Inertial subrange at lDI > l > lEI (according to Pope [6], lDI ≈ 60η).Within this subrange the energy is transferred along the scales towardsdissipative vortices without any significant loss, i.e. ε ∼ 0. The energydensity obeys the Kolmogorov law (4.10).

• Dissipation range l < lDI . The dissipation of the energy of big vorticesoccurs within the dissipation range.

66

The inertial subrange and dissipation range belong to the universal equilib-rium range. Three corresponding ranges can be distinguished in the distri-bution of the energy density over the wave numbers k Fig. 4.6. The presenceof the inertial and dissipation subranges was confirmed in numerous exper-imental measurements performed after development of the K41 theory (seeFig. 4.7, 4.8).

Figure 4.5: Three typical scale ranges in the turbulent flow at high Reynoldsnumbers

Figure 4.6: Three typical ranges of the energy density spectrum in the tur-bulent flow at high Reynolds number. 1- energy containing range, 2- inertialsubrange, 3- dissipation range.

67

Figure 4.7: Experimental confirmation of the Kolmogorov law. The compen-sated energy spectrum for different flows.

68

Figure 4.8: Experimental confirmation of the Kolmogorov law for the con-centration fluctuations in the jet mixer. Measurements of the LTT Rostock.

4.5 Classification of methods for calculation

of turbulent flows.

The energy spectrum Fig. 4.6 is used to classify three main methods of tur-bulent flows modelling (Fig. 4.9). The most general strategy is the DirectNumerical Simulation (DNS). Within the DNS the whole spectrum of tur-bulent structures is modelled starting from the biggest vortices of the energycontaining range till the smallest dissipative Kolmogorov vortices. The LargeEddy Simulation (LES) models the energy containing vortices and a fractionof vortices corresponding to the inertial subrange. The effect of remain-ing vortices is considered using different approximation models. Since smallvortices are universal, the models are also supposed to be universal. TheReynolds averaged Navier Stokes (RANS) models are dealing with the largevortices corresponding to the energy containing range. The effect of othervortices is taken by different semi- empiric models which are not universal.

4.6 Limitation of K-41. Kolmogorov theory

K-62

The strongest and simultaneously the most questionable assumption of theKolmogorov-41 is: Dissipation rate is an universal constant for each turbulentflow. Already in 1942, during a scientific seminar the Nobel price laureate

69

Figure 4.9: Three main methods of turbulent flows modelling.

Figure 4.10: Vortex structures resolved by different models

Landau noted, that the dissipation rate is a stochastic function, it is notconstant. We consider the consequences of the neglect of this fact.According to the Kolmogorov - Obukhov law the structure function of theq-th order

Sq(l) = 〈(u2l − u1l)q〉 (4.11)

has the following asymptotic behaviour at small l

Sq(l) ∼ (εl)q ∼ (l)ζq (4.12)

Fig. 4.11 shows that the predictions of Kolmogorov and Obukhov deviatefrom measurement data. The reason of the discrepancy is the physical phe-

70

nomenon called the intermittency. The intermittency is caused by the pres-ence of laminar spots in every turbulent flows even at very high Reynoldsnumbers.

Figure 4.11: Power of the structure function. Experiments versus predictionof Kolmogorov and Obukhov

After the deviation between the K-41 and measurement was documented,Kolmogorov tried to improve his theory. New Kolmogorov theory called asK-62 was published in 1962.New theory is based on two following assumptions:

• Assumption 1:

Sq(l) =< δνql >∼< εq/3l > lq/3,

ζq =q

3+ τq/3 < εql >∼ lτq

(4.13)

• Assumption 2:

P (εl) = ce(lnε−a)2

2σl2 a = lnε σ2

l = A+ µln(L/l)

τq =µ

2q(1− q) ςq =

q

3+

µ

18q(3− q) < ε2

l >∼ l−µ(4.14)

Unfortunately, various experiments showed later that the second assumptionis proved to be wrong.

71

Chapter 5

Reynolds Averaged NavierStokes Equation (RANS)

According to the Reynolds averaging each fluctuating quantity is representedas the sum of the averaged value and its fluctuation:

ux = ux + u′x; uy = uy + u′y; uz = uz + u′z (5.1)

where the averaged part is defined as:

ux =1

T

T∫0

uxdt; uy =1

T

T∫0

uydt; uz =1

T

T∫0

uzdt (5.2)

The Reynolds averaging has the following properties:

• averaged fluctuation is zero:

f ′ = 0 (5.3)

• double averaged quantity is equal to once averaged one:

¯f = f (5.4)

• averaged sum is equal to the sum of averaged:

f + g = f + g (5.5)

• operators of averaging and differentiation commutate:

∂f

∂t=∂f

∂t,

∂f

∂x=∂f

∂x(5.6)

73

• averaged product of two fluctuating quantities is not zero:

f ′g′ 6= 0, fg = f g (5.7)

• averaged product of any averaged quantity and fluctuation is zero:

f g′ = f g′ = 0 (5.8)

The starting point of the derivation of the RANS equation is the originalNavier Stokes (NS) equation:

∂ui∂t

+ uj∂ui∂xj

= Fi +1

ρ

∂τji∂xj

(5.9)

Here we use the summation convention of Einstein:

uj∂ui∂xj

= u1∂ui∂x1

+ u2∂ui∂x2

+ u3∂ui∂x3

(5.10)

The NS equation is supplied with the continuity equation, which for the caseof incompressible flow takes the form:

∂ui∂xi

=∂u1

∂x1

+∂u2

∂x2

+∂u3

∂x3

= 0 (5.11)

Using the continuity equation the convective term is written in the conser-vative form:

uj∂ui∂xj

= uj∂ui∂xj

+ ui∂uj∂xj

=∂(uiuj)

∂xj(5.12)

With (5.12) the NS equation reads

∂ui∂t

+∂(uiuj)

∂xj= Fi +

1

ρ

∂τji∂xj

(5.13)

This equation is valid for fluctuating quantities represented in Reynoldsform (5.1)

∂(ui + u′i)

∂t+∂(ui + u′i)(uj + u′j)

∂xj= Fi + F ′ +

1

ρ

∂(τji + τ ′ji)

∂xj(5.14)

Both r.h.s. and l.h.s. are averaged:

∂(ui + u′i)

∂t+∂(ui + u′i)(uj + u′j)

∂xj= Fi + F ′ +

1

ρ

∂(τji + τ ′ji)

∂xj(5.15)

74

Utilization of Reynolds averaging properties results in:

∂ui∂t

+∂(uiuj + u′iu

′j)

∂xj= Fi +

1

ρ

∂τji∂xj

(5.16)

Writing the term u′iu′j on the r.h.s we obtain the Reynolds averaged Navier

Stokes equation (RANS). Its unsteady version is called as the unsteadyReynolds averaged NS equation (URANS):

ρ∂ui∂t

+ ρ∂uiuj∂xj

= ρFi +∂

∂xj(τji − %u′iu′j) (5.17)

There are two important features of URANS in comparison with NS:

• The URANS is written for the averaged quantities, whereas the NS forinstantaneous ones,

• The URANS has additional term on the r.h.s. −%u′iu′j which is calledthe Reynolds stress Rij.

Generally the Reynolds stress is the matrix with nine terms:

Rij

∣∣∣∣∣∣−ρu′xu′x −ρu′xu′y −ρu′xu′z−ρu′xu′y −ρu′yu′y −ρu′yu′z−ρu′xu′z −ρu′yu′z −ρu′zu′z

∣∣∣∣∣∣ (5.18)

Due to symmetry condtitions the number of unknown stresses is six. Theterm −%u′iu′j is called the stress since it has the same appearance in NSequation as the laminar stress:

τji = ρν

(∂ui∂xj

+∂uj∂xi

)− pδij (5.19)

Laminar stress appears due to viscosity effects whereas the Reynolds stressis caused by flow fluctuations. Now the system of four fluid equations (threeURANS+ continuity) has ten unknowns: three averaged velocity compo-nents ui, averaged pressure p and six Reynolds stresses. The system offluid dynamics is not closed. Additional relations are necessary to expressthe Reynolds stresses through the velocities and pressure. This problem ofdetermination of Reynolds stresses is called as the closure problem of theturbulence. A huge amounts of closure models was developed within theframework of URANS methodology. As shown above the URANS methodsmodels the biggest vortices of the flow. The resting scales, which are filteredby the Reynolds avergaing out, are big enough and not universal. This is

75

the reason why the URANS models are not universal. Most of them are ofsemi empirical character. They are based on submodels with few constantsselected for simple canonical flows. Non universality of closure models is thebiggest weakness of URANS turbulence modelling.The majority of URANS models used in engineering are based on the Boussi-nesq hypothesis which is the formal extension of the Newton hypothesis toturbulent flows. Boussinesq proposed to express the Reynolds stress through

the strain rate tensor Sij = 12

(∂uj∂xi

+ ∂ui∂xj

)in the form of the Newton hy-

pothesis with the only difference that instead of the kinematic viscosity νthe turbulent viscosity νt is used

−ρu′iu′j = ρνt

(∂uj∂xi

+∂ui∂xj

)− 2

3ρδijk (5.20)

In the simplest form for flow along the plate with ux(y) and uy = uz = 0 theformula (5.20) reads

−ρu′xu′y = ρνtduxdy

(5.21)

The last term in (5.20) is introduced to keep the consistency. Indeed thesum of three diagonal terms of the Reynolds matrix is equal to the turbulentkinetic energy k = 1

2u′iu′i. Without this term the sum of r.h.s of (5.20)

would result in the sum of the diagonal terms of the strain rate matrixS11 + S22 + S33 which is zero due to the continuity equation. It would bewrong result because k 6= 0. The turbulent closures (5.20) are referred to asthe isotropic because the coefficient νt is equal for all matrix elements Rij.While the kinematic viscosity depends on the liquid, the turbulent kinematicviscosity depends on the turbulent state of the flow. According to estima-tion of Landau the ratio of the kinematic viscosity to the turbulent one isproportional to the ratio of the Reynolds number to that corresponding tothe transition for this type of flow

νt/ν ∼ Re/Recrit (5.22)

The URANS closure models are subdivided into algebraic and differentialones. The most prominent model amount the algebraic models is the Prandtlmodel described above. The disadvantages of the algebraic models are:

• they are good only for the simplest flow,

• not suitable for 3D flows,

76

• not suitable for separation flows,

• turbulent viscosity depends on averaged values of velocities,

• do not consider the flow history.

These disadvantages can be overcome using differential models which aresubdividided into one, two and multi equation models. One and two equa-tion models are usually isotropic based on the Boussinesq approach (5.20).Among the one equation models the most modern and efficient is the modelof Spalart Allmares (SA model) written for the modified kinematic turbulentviscosity υ. The equation for υ reads

∂υ

∂t+ uj

∂υ

∂xj= Cb1ξυ − Cw1fw

(υ

d

)2

+1

σ

∂

∂xk

((ν + υ)

∂υ

∂xk

)+Cb2σ

dυ

dxk

dυ

dxk(5.23)

where

Cb1 = 0, 1355, Cb2 = 0, 622, Cν1 = 7, 1, σ = 2/3,

Cw1 =Cb1k2

+1 + Cb2σ

, Cw2 = 0, 3, Cw3 = 2, 0, k = 0, 41(5.24)

fν1 =χ3

χ3 + C3ν1

, fν2 =χ

1 + χfν1

, fw = g

[1 + C6

w3

g6 + C6w3

]1/6

,

χ = υν, g = r + Cw2(r6 − r), r =υ

ξk2d2

(5.25)

ξ = S +υ

k2d2fν2, S =

√2ΩijΩij, Ωij =

1

2

(∂ui∂xj− ∂uj∂xi

)(5.26)

Within the more advanced k− ε model the turbulent kinetic energy and thedissipation rate are calculated from the transport equations:

∂k

∂t+ uj

∂k

∂xj=

∂

∂xj

((ν +

νtσk

)∂k

∂xj

)+ τij

∂ui∂xj− ε

∂ε

∂t+ uj

∂ε

∂xj=

∂

∂xj

[(ν +

νtσε

)∂ε

∂xj

]+Cε1ε

kτij∂ui∂xj− Cε2ε

2

k

(5.27)

If k and ε are known the turbulent viscosity νt can be found from the dimen-sion analysis, applied to the dissipation rate

77

ε ≈ k3/2/L (5.28)

turbulent kinematic viscosity

νt = Cµ√kL (5.29)

Here Cµ = 0.09 is the empirical constant. Substitution of (5.28) into (5.29)leads to the sought relation:

νt = Cµ√kL = Cµ

k2

ε(5.30)

If νt is known, the Reynolds stresses can be determined from the Boussinesqapproach (5.20).

78

Chapter 6

Reynolds Stress Model (RSM)

6.1 Derivation of the RSM Equations

6.1.1 Step 1

The k − th Navier-Stokes equation

∂uk∂t

+∂

∂xj(ujuk) = −1

ρ

∂p

∂xk+

1

ρ

∂

∂xjτjk (6.1)

is multiplied with the velocity component ui

ui

(∂uk∂t

+∂

∂xj(ujuk)

)= ui

(− 1

ρ

∂p

∂xk+

1

ρ

∂

∂xjτjk

)(6.2)

The i− th equation is multiplied with k − th velocity component:

uk

(∂ui∂t

+∂

∂xj(ujui)

)= uk

(− 1

ρ

∂p

∂xi+

1

ρ

∂

∂xjτjk

)(6.3)

Resulting equations (6.2) and (6.3) are then sumed:

∂(ρuiuk)

∂t+∂(ρuiukuj)

∂xj= −ui

∂p

∂xk− uk

∂p

∂xi+ ui

∂τjk∂xj

+ uk∂τji∂xj

(6.4)

Substitution of Reynolds decomposition

ui = ui + u′i, p = p+ p′, τji = τji + τ ′ji (6.5)

into the equation (6.4) results in

79

∂

∂t(ρuiuk) +

∂

∂t(ρu′iu

′k) +

∂

∂xj

(ρuiujuk + ρuiu′ju

′k +

+ρuju′iu′k + ρuku′iu

′j + ρu′iu

′ju′k

)= −ui

∂p

∂xk− u′i

∂p′

∂xk

− uk∂p

∂xi− u′k

∂p′

∂xi+ ui

∂τ ′jk∂xj

+ u′i∂τ ′jk∂xj

+ uk∂τji∂xj

+ u′k∂τ ′ji∂xj

(6.6)

6.1.2 Step 2

The k − th Reynolds averaged Navier Stokes equation

∂ρuk∂t

+∂

∂xj(ρujuk) = − ∂p

∂xk+

∂

∂xj(τjk − ρu′ju′k) (6.7)

is multiplied with the i− th component of averaged velocity

ui

(∂ρuk∂t

+∂

∂xj(ρujuk)

)= ui

(− ∂p

∂xk+

∂

∂xj(τjk − ρu′ju′k)

)(6.8)

Again the i−th Reynolds averaged Navier Stokes equation is multiplied withthe k − th component of averaged velocity

uk

(∂ρui∂t

+∂

∂xj(ρujui)

)= uk

(− ∂p

∂xi+

∂

∂xj(τjk − ρu′ju′i)

)(6.9)

The sum of two last equations reads

∂(ρuiuk)

∂t+∂(ρuiukuj)

∂xj= −ui

∂p

∂xk−uk

∂p

∂xi+ui

∂(τjk − ρu′ju′k)∂xj

+uk∂(τji − ρu′ju′i)

∂xj(6.10)

6.1.3 Step 3

Subtracting the last equation from (6.6) results in

∂

∂t(ρu′iu

′k) +

∂

∂xj(ρuju′iu

′k) +

∂

∂xj(ρu′iu

′ju′k) = −u′i

∂p′

∂xk− u′k

∂p′

∂xi+

+ u′i∂τ ′jk∂xj

+ u′k∂τ ′ji∂xj− ρu′ju′k

∂ui∂xj− ρu′ju′i

∂uk∂xj

(6.11)

80

Using identities (please prove them)

−u′i∂p′

∂xk− u′k

∂p′

∂xi= p′

(∂u′i∂xk

+∂u′k∂xi

)−[δjk

∂

∂xj(u′ip

′) + δij∂

∂xj(u′kp

′)

](6.12)

u′i∂τjk∂xj

+ u′k∂τji∂xj

= µ

(u′i∂2u′k∂x2

j

+ u′k∂2u′i∂x2

j

)= µ

∂2

∂x2j

u′iu′k − 2µ

∂u′i∂xj

∂u′k∂xj

(6.13)

we get the Reynolds stress model equation

∂

∂t(ρu′iu

′k) +

∂

∂xj(ρuju′iu

′k) +

∂

∂xj(ρu′iu

′ju′k) =

p′(∂u′i∂xk

+∂u′k∂xi

)−[δjk

∂

∂xj(u′ip

′) + δij∂

∂xj(u′kp

′)

]µ∂2

∂x2j

u′iu′k − 2µ

∂u′i∂xj

∂u′k∂xj

−ρu′ju′k∂ui∂xj− ρu′ju′i

∂uk∂xj

which can be written in a compact form

∂

∂t(u′iu

′k) + uj

∂

∂xj(u′iu

′k) =

∂

∂xjDjk +Rik + Pik − εik (6.14)

The physical meaning of terms on the r.h.s is as follows

Dik = ν∂(u′iu

′k)

∂xj− u′iu′ju′k +

1

ρ(δjku′i + δiju′k)p

′ → Diffusion (6.15)

Rik =1

ρ

(∂u′i∂xk

+∂u′k∂xi

)p′ → Re distribution (energy exchange) (6.16)

Pik = −u′ju′k∂ui∂xj− u′ju′i

∂uk∂xj

→ Generation (6.17)

εik = 2ν∂u′i∂xj

∂u′k∂xj

→ Dissipation (6.18)

81

6.1.4 Analysis of terms

The diffusion of energy

Dik = ν∂(u′iu

′k)

∂xj− u′iu′ju′k +

1


′ (6.19)

is due to

• molecular diffusion, described by the term:

ν∂(u′iu

′k)

∂xj(6.20)

• turbulent diffusion, described by the term:

−u′iu′ju′k (6.21)

• turbulent diffusion caused by correlation between pressure and velocityfluctuations

1


′ (6.22)

The two last terms are unclosed. Here we face with the famous problem notedfirst by Friedmann and Keller (1924): Effort to derive the equations for thesecond order moments u′iu

′k results in the necessity of determination of new

unclosed terms including third order moments u′iu′ju′k. Using the method pro-

posed by Friedmann and Keller in 1924 it is possible to derive equations formoments of arbitrary order. However, the equation for the m− th order willcontain unclosed moments of the m+1− th order. Impossibility of obtainingof a closed system of equations for a finite number of moments, known as theFriedmann-Keller problem) is a direct consequence of the nonlinearity of theNavier Stokes equations.

The also unclosed term

Rik =1

ρ

(∂u′i∂xk

+∂u′k∂xi

)p′ (6.23)

describes the redistribution of the energy between different tensor compo-nents u′iu

′k caused by correlation between the stresses and pressure fluctua-

tions.

The term

82

Pik = −u′ju′k∂ui∂xj− u′ju′i

∂uk∂xj

(6.24)

is responsible for the energy generation, i.e. the transport of the energytransfer from averaged (mean) flow to oscillating flow (fluctuations). And,finally,

εik = 2ν∂u′i∂xj

∂u′k∂xj

(6.25)

is the dissipation. This unclosed term is responsible for the transformationof the turbulent kinetic energy into the inner energy of the flow.RSM model based on equations (6.14) is used to determine the Reynoldsstresses from the transport equations. It is not based on the Boussinesqapproach and takes the anisotropy of stresses into account. This model isthe best one among RANS models.

83

Chapter 7

Equations of the k - ε Model

7.1 Derivation of the k-Equation

According to definition

k = u′ku′k/2 (7.1)

Assuming i = k in the Reynolds stress model equations (6.14)

∂

∂t(u′iu

′k) + uj

∂

∂xj(u′iu

′k) =

∂

∂xjDik +Rik + Pik − εik (7.2)

ans summing equations for k=1,2 and 3 we obtain the transport equationsfor the total kinetic energy k:

∂k

∂t+ uj

∂k

∂xj=

∂

∂xjDs + P − εS (7.3)

where

DS = ν∂k

∂xj− 1

ρδjku′kp

′ − u′jk′, k′ = u′ku′k/2 Diffusion (7.4)

P = −u′ju′k∂uk∂xj

Generation (7.5)

εS = ν∂u′k∂xj

∂u′k∂xj

Dissipation (Pseudodissipation) (7.6)

The relation between the true and pseudodissipation is

85

ε =ν

2

(∂u′k∂xj

+∂u′j∂xk

)2

≈ εS +∂

∂xjν∂

∂xku′ju

′k (7.7)

For large Reynolds numbers the true dissipation and the pseudodissipationare equal.

ε ≈ εS (7.8)

More precise analysis shows that

εεS ≈ Re−1t (7.9)

where

Ret =√kL/ν (7.10)

7.1.1 Closure of terms of k equation

Two unknown terms in the diffusion

DS = ν∂k

∂xj− 1

ρδjku′kp

′ − u′jk′ (7.11)

are determined by the gradient assumption

−1

ρδjku′kp

′ − u′jk′ =νtσk

∂k

∂xj(7.12)

where σk is an empirical constant. Dissipation is determined by the energycontaining motion using the formula of Prandtl- Kolmogorov

εs = CDk3/2/L (7.13)

The Reynolds stresses are seeking in form proposed by Boussinesq:

−ρu′iu′j = ρνt

(∂uj∂xi

+∂ui∂xj

)− 2

3ρδijk (7.14)

Substitution of all these approximations into the equation (7.3) results in thek-Equation

∂k

∂t+ uj

∂k

∂xj=

∂

∂xj

[(ν +

νtσk

)∂k

∂xj

]+ νt

(∂uj∂xi

+∂ui∂xj

)∂uj∂xi−CD

k3/2

L(7.15)

86

7.1.2 Derivation of the ε-Equation

The Navier Stokes equation

∂uk∂t

+ uj∂uk∂xj

= −1

ρ

∂p

∂xk+

1

ρ

∂τjk∂xj

(7.16)

is differentiated and multiplied with the derivative∂u′i∂xk

∂

∂xk

∂ui∂t

+ uj∂ui∂xj

= −1

ρ

∂p

∂xi+

1

ρ

∂τji∂xj

∂u′i∂xk

(7.17)

This gives:

∂εs∂t

+ uj∂εs∂xj

=∂

∂xjDε + Pε − εε (7.18)

where

Dε = ν∂εs∂xj− u′jε′s − 2

ν

ρδij∂u′j∂xk

∂p′

∂xk(7.19)

Pε = −2νu′j∂u′i∂xk

∂2ui∂xj∂xk

− 2ν

(∂u′i∂xk

∂u′j∂xk

∂ui∂xj

+∂u′i∂xj

∂u′i∂xk

∂uj∂xj

)− 2ν

∂u′i∂xj

∂u′j∂xk

∂u′i∂xk

(7.20)

εε = 2ν2 ∂2u′i∂xj∂xk

∂2u′i∂xj∂xk

, ε′s = ν∂u′i∂xk

∂u′i∂xk

(7.21)

The terms on the r.h.s. were approximated according to the following for-mula:

∂

∂xjDε =

∂

∂xj

[(ν +

νTσε

)∂ε

∂xj

];Pε =

Cε1ε

kτij∂ui∂xj

; εε =Cε2ε

2

k(7.22)

Constants are taken from planar jet and mixing layer:

Cε1 = 1.44, Cε2 = 1.92, σk = 1, σε = 1.3 (7.23)

Hereby the full closed system of the k − ε model reads:

87

∂k

∂t+ uj

∂k

∂xj=

∂

∂xj

((ν +

νtσk

)∂k

∂xj

)+ τij

∂ui∂xj− ε

∂ε

∂t+ uj

∂ε

∂xj=

∂

∂xj

[(ν +

νtσε

)∂ε

∂xj

]+Cε1ε

kτij∂ui∂xj− Cε2ε

2

k

(7.24)

Under assumption that the generation of the turbulent energy equals to theits dissipation (the turbulence is in equilibrium, turbulent scales are in theinertial range) Kolmogorov and Prandtl derived the relation between thekinetic energy, the dissipation rate and the integral lengths L:

ε ≈ k3/2

L(7.25)

From the dimension analysis

νt = Cµ√kL (7.26)

follows

νt = Cµ√kL = Cµ

k2

ε(7.27)

As soon as k and ε are known the turbulent kinematic viscosity νt is computedfrom (7.27) and Reynolds stresses can be calculated from the Boussinesqhypothesis and then substituted into the Reynolds averaged Navier Stokesequations. The problem is mathematically closed.

The k− ε model is the classical approach, which is very accurate at large Renumbers. At small Re number, for instance close to the wall, the approxima-tions used in derivation of k− ε model equations are not valid. To overcomethis disadvantage various low Reynolds k − ε models were proposed.

88

Chapter 8

Large Eddy Simulation (LES)

8.1 LES filtering

Within the LES all vortices are subdivided into large resolved vortices andfine subgrid vortices. The border between vortices should lie within the in-ertial range. The separation of fine scale motions (small fine vortices) fromlarge ones is done using the spatial filtering. Let ϕ be any stochastic functionwhich is represented as the sum of filtered part and fluctuation:

Where the filtered part is defined as

ϕ = ϕ+ ϕ′

ϕ(~x, t) =

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

ϕ(~x− ~s, t)F (~s)d~s

Here F(s) is the filtering function, satisfying the condition

∞∫−∞

∞∫−∞

∞∫−∞

F (~s)d~s = 1

Three different filtering functions shown in Fig. 1: ideal filter, Gauss filterand top hat filter. Ideal filter is applied in Fourier space. High frequenciesare cut off. Low frequencies are simulated directly. The top hat filter is somekind of smoothing applied in physical space. The simplest case is smoothingover three neighboring points

ϕi =1

b(ϕi−1 + aϕi + ϕi+1)

where b = 2 + a.

89

8.1.1 Properties of filtering

The spatial filtering and Reynolds averaging are both filtering operations.LES spatial filtering has properties which differ from these of Reynolds av-eraging. First, the spatial averaged quantity is not zero. Double filtering isnot equal once filtering.

ϕ′ 6= 0, ˜ϕ 6= ϕ,

Both conditions are compatible because

ϕ′ = ϕ− ϕ = ϕ− ˜ϕ 6= 0

Figure 8.1: Different filtering functions used in LES

Other important properties are similar to these of RANS averaging:

90

• Averaged sum of two quantities is equal to the sum of averaged quan-tities:

ϕ+ g = ϕ+ g

• Filtering operator commutes with the differentiation operator

∂ϕ

∂t=∂ϕ

∂t

∂ϕ

∂xj=

∂ϕ

∂xj

A very important relation which is the consequence of these properties is

ϕφ = ϕφ+ ϕφ′ + φϕ′ + ϕ′φ′

In the case of Reynolds averaging only the first and the last terms remain.The properties of large and small scale motions are shown in the table 8.1.

Large scale motion Small scale motionGenerated by mean flow Generated by large scale structuresDepends on the flow geometry Universalregular StochasticDeterministic description Stochastic descriptionHeterogeneous HomogeneousAnisotrop IsotropExists long time Exists short timeDiffusive DissipativeModelling is complicated easy to model

Table 8.1: Properties of large and small scale motions

A very important conclusion from this table is the fact that the small scalemotion is universal. Therefore one can expect that the models describing thesmall scale motion in contrast to RANS models are also universal.

8.2 LES equations

The governing equations of LES are derived from the Navier Stokes equation

∂

∂t(ρui) +

∂

∂xj(ρuiuj) =

∂

∂xj

[ρν

(∂uj∂xi

+∂ui∂xj

)]− ∂p

∂xi+ ρgi (8.1)

Application of the filter operation to 8.1 results in

91

∂

∂t(ρui) +

∂

∂xj(ρuiuj) =

∂

∂xj

[ρν

(∂uj∂xi

+∂ui∂xj

)]− ∂p

∂xi+ ρgi

Averaged sum is equal to the sum of averaged terms:

∂

∂t(ρui) +

∂

∂xj(ρuiuj) =

∂

∂xj

[ρν∂ui∂xj

]− ∂p

∂xi+ ρgi (8.2)

Introducing the term

τSGSij = uiuj − uiuj

The equation 8.2 is rewritten in the final form

∂

∂t(ρui) +

∂

∂xj(ρuiuj) =

∂

∂xj

[ρν∂ui∂xj− ρτSGSij

]− ∂p

∂xi+ ρgi

The term τSGSij = uiuj − uiuj is the subgrid stress (SGS) which considersthe effect of small fine vortices on large scale motion directly resolved on thegrid.

8.3 Smagorinsky model

Note that the fine scale vortices are not resolved. They filtered out by thefiltering operation. The effect of these vortices is taken by the term τSGSij

into account. Since the small vortices are not modeled, the subgrid stress arecalculated using phenomenological models. The most recent phenomenolog-ical model was proposed by Smagorinsky in 1963. The Smagorinsky modelis just the extension of the Boussinesq approach

τij −1

3τkkδij ≈ −νt2Sij,

Smagorinsky introduced the subgrid viscosity νSGS instead of the turbulentkinematic viscosity

τSGSij − 1

3τSGSkk δij ≈ −νSGS2Sij,

Expression for the subgrid viscosity was obtained by Smagorinsky with theuse of idea taken from the Prandtl mixing length theory. According toPrandtl, the turbulent kinematic viscosity is proportional to the mixinglength squared and the velocity gradient close to the wall

92

νt = l2|duxdy|

According to Smagorinsky, the subgrid viscosity is proportional to the mag-nitude of the strain rate tensor Sij and to a certain length ls squared

νSGS = l2S|Sij|, |Sij| =√

2SijSij

Where

Sij =1

2

(∂ui∂xi

+∂uj∂xi

)The length lS is assumed to be proportional to the mesh size

lS = CS∆

where CS is the constant of Smagorinsky.

The Smagorinsky constant was estimated first by Lilly. The main assumptionof the Lilly analysis is the balance between generation

Pr = −τijSij = 2νtSijSij = νt|Sij|2

and dissipation of the turbulent kinetic energy

ε = P = νt|Sij|2 = l2S|Sij|3 (8.3)

Lilly estimated the strain rate tensor magnitude for Kolmogorov spectrum

S2 ≈ 7Cε2/3∆−4/3

Substitution of the last formula into 8.3 results in:

lS =∆

(7C)3/4

(S2

3/2

S3

)Assuming additionally that S2

3/2

≈ S3, the length lS and the Smagorinskyconstant are expressed through the Kolmogorov constant C = 1.5:

CS =lS∆

=1

(7C)3/4≈ 0.17

The Smagorinsky constant 0.17 is derived analytically with a few strongassumptions. The experience shows that numerical results agree with mea-surements much better if a reduced value of the Smagorinsky constant is

93

used. Common values are 0.065 and 0.1.

Advantages and disdvantages of the Smagorinsky model are summarized inthe table 8.2.

Advantages DisadvantagesSimple Laminar flow is not modelledLow computational costs Constant of Smagorinsky is constant in time and

spaceStable Actually, the constant is chosen arbitrarily

depending on the problem under considerationGood accuracy in ideal Sensible to gridconditions

Purely dissipativDamping of pulsation is too strong

Table 8.2: Advantages and disadvantages of the Smagorinsky model

We complete this section with a very important comment:

• the LES models are consistent when the resolution increases, i.e. ∆→ 0.

Indeed, if the resolution is increased, ∆ → 0, the SGS stresses disappear.The LES equation is passed to the original Navier Stokes equations. TheLES simulation becomes the DNS simulation if ∆→ 0. On the contrary, theURANS simulation is not consistent when ∆ → 0. The Reynolds stressesdon’t disappear if the resolution is increased ∆→ 0. The turbulence is thentwice resolved.

94

Chapter 9

Subgrid Stress (SGS) Models

The classical model of Smagorinsky with the parameter CS being constantfor the whole computational domain is proved to be very diffusive. Germanoproposed to calculate the Smagorinsky constant CS being variable both inspace and in time, i.e. CS = CS(x, t). The constant is determined using thedynamic procedure which is then referred to as the Dynamic SmagorinskyModel (DSM).

9.1 Model of Germano (Dynamic Smagorin-

sky Model)

According to the definition the subgrid stress is

τSGSij = uiuj − uiuj (9.1)

Germano introduces the double filtering or the test filtering designated asau = û. Here the tilde symbol means the first filtering with filter width ∆whereas the hat symbol stands for the second filtering with filter width ∼ 2∆.The symbol means the resulting double filtering. Using the definition (9.1)we can write

T testij =_uiuj −

aui

auj = uiuj − ûi ûj (9.2)

Filtration of (9.1) results in


Subtracting (9.3) from (9.2) yields

95

T testij − τSGSij = uiuj − ûi ûj (9.4)

We suppose that the double filter width is small. Therefore the Smagorinskymodel is valid for both stresses τSGSij and T testij :

τSGSij − 1

3τSGSkk δij = −2

(C∆s ∆)2|Sij|

(Sij)

= 2CmSGSij ,

T testij −1

3T testkk δij = −2

(C∆s ∆)2| ˆSij|

( ˆSij)

= 2Cmtestij ,

(9.5)

The application of the double filter to τSGSij gives:

τSGSij − 1

3τSGSkk δij = 2CmSGS

ij (9.6)

where C = C2S. Here we supposed that the filtered product of the constant

C with mij is equal to the product of filtered mij with the same constant

Cm ≈ Cm

We introduce the tensor Lij which is equal to the difference between testfilter and once filtered original SGS stress:

Lij = T testij − τSGSij = uiuj − ûi ûj (9.7)

Using this designation we get from (9.5) and (9.6)

2CMij = Lij −1

3Lkkδij (9.8)

where

Mij = mtestij − mSGS

ij

The system (9.8) is overdefined (six equations for one unknown coefficientC). To get an unique solution we multiply both the l.h.s. and r.h.s of (9.8)with the tensor Sij. The final result for C is

C =LijSij

2MijSij(9.9)

Use of (9.11) is problematic since the denumerator MijSij can become zero.To overcome this difficulty Lilly proposed to determine the constant from thecondition of the minimum residual of the equation (9.8):

96

Q =

(Lij −

1

3Lkkδij − 2CMij

)2

→ min (9.10)

The minimum is attained at the point with zero derivative of the functionalQ on the parameter C:

∂Q

∂C= −4Mij

(Lij −

1

3Lkkδij − 2CMij

)= 0 (9.11)

It follows directly from (9.11):

C =MijLij − 1

3LkkδijMij

2MijMij

=MijLij

2MijMij

(9.12)

since δijMij = 0. The solution (9.12) corresponds to the minimum of Q(C)since the second derivative ∂2Q/∂C2 is positive at this point

∂2Q

∂C2= 8MijMij > 0 (9.13)

Theoretically the constant C can become negative. The case C < 0 andνSGS can be considered as the energy backscattering. However, this leads tostrong numerical instability. That is why the dynamic constant is limitedfrom below:

C = max

MijLij

2MijMij

, 0

≥ 0 (9.14)

The subgrid kinematic viscosity is always positive

νSGS = C∆2|Sij| ≥ 0

9.2 Scale similarity models

Despite the fact that diffusion of the classic Smagorinsky model was sub-stantially reduced by the dynamic choice of the Smagorinsky constant, theDynamic Smagorinsky model remains very diffusive. This disadvantage wasovercome within the similarity models. The main point of the similaritymodel is the assumption that the statistical properties of the once filteredfield ui are identical to these of the double filtered field ˜ui. It is the caseif the filter width is small. The difference between once and double filteredvelocities is negligible, i.e. different scale motions are similar.

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Figure 9.1: Illustrations for derivation of the scale similarity model

Let us consider uj as the original (unfiltered) field. ˜uj is the filtered fieldand uj − ˜uj is the pulsation (see Fig. 9.1). Then from the definition of thesubgrid stress one obtains

τSGSij = ˜uiuj − ˜ui ˜uj (9.15)

The formula (9.15) is the scale similarity model proposed by Bardina et al. [7].As seen the SGS stress can be calculated directly from the resolved field ui.

9.3 Mixed similarity models

The experience shows that diffusion produced by the scale similarity mo-del (9.15) is too low. The numerical calculations are often unstable usingthis model. Taking the fact into account, that the diffusion of the Smagorin-sky model is too high, it was decided to combine the Samgorinsky and scalesimilarity models to get the proper diffusion. The advantages and disadvan-tages of both models are summarized as follows

• Dynamic Smagorinsky Model (DSM): energy dissipation is overesti-mated (drawback), energy backscattering is not reproduced (draw-back),

• Scale similarity model: energy backscattering is reproduced (advan-tage), energy dissipation is underestimated (drawback).

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The Idea is to combine models to strengthen the advantages and to over-come disadvantages of both models. The hybrid model called as the mixedsimilarity model is written as

τ rij =(˜uiuj − ˜ui ˜uj

)− 2(C∆S ∆)2|S|Sij (9.16)

The mixed model can be derived in a more formal way. For that the velocitydecomposition into filtered and pulsation parts:

uiuj = ˜(ui + u′i)(uj + u′j) = ˜uiuj + u′iuj + u′jui + u′iu′j (9.17)

is substituted into the SGS stress expression:


Finally we have Leonard’s formulation of the mixed model:

τSGSij = Lij + Cij +Rij (9.19)

whereLij = ˜uiuj − uiuj is the Leonard stress

Cij = ˜uiu′j + ˜uju′i is the Cross-stress

Rij = u′iu′j is the Reynolds stress

The sum of the cross and Reynolds stresses is calculated via the Smagorinskymodel with the dynamically determined constant CS

Cij +Rij = −2(C∆S ∆)2|S|Sij (9.20)

A substantial disadvantage of this formulation is the fact that the Leonardstress does not satisfy the Galilean invariance condition. The Galilean in-variance is the independence of basic formula of mechanics on the speed ofthe reference system.

The classical definition of the SGS stresses possesses the Galilean invariance.Indeed, let V be the speed of the reference system. The velocity relative tothe reference system is

~W = ~u+ ~V (9.21)

The SGS stress does not depend on the reference system speed:

99

WiWj − WiWj = ˜(ui + Vi + u′i)(uj + Vj + u′j)− ˜(uiVi + u′i)˜(uj + Vj + u′j) =

= ˜(ui + u′i)(uj + u′j)− ˜(ui + u′i)˜(uj + u′j) = uiuj − uiuj

(9.22)

On the contrary, the Leonard stress is not Galilean invariant:

˜WiWj − WiWj = ˜uiuj − uiuj − Viu′j − Vju′i (9.23)

Germano proposed an alternative formulation

τSGSij = L0ij + C0

ij +R0ij (9.24)

where all stresses are Galilean invariant:

L0ij = ˜uiuj − ˜ui ˜uj is the Leonard stress

C0ij = ˜uiu′j + u′iuj − ˜uiu

′j − u′i ˜uj is the Cross stress

R0ij = u′iu

′j − u′iu′j is the Reynolds stress

C0ij +R0

ij = −2(C∆S ∆)2|S|Sij (9.25)

Exercise: Prove the following facts:

• equivalence of subgrid stresses computed from formulations of Ger-mano (9.24) and the original one proposed by Leonard (9.19),

• Galilean invariance of stresses in Germano‘s formulation (9.24).

9.3.1 A-posteriori and a-priori tests

Two tests are used to verify LES models. The comon way is the a-posterioritest. The LES simulation is performed and then the flow parameters obtainedfrom the simulation are compared with these from measurement. Dependingon comparison results the conclusion about quality of LES models is drawn.The disadvantage of such approach is that the LES results are affected bymodelling errors, errors of approximation of differential operators and round-ing errors. In a-priori test they can not be separated.

100

Direct test of quality of subgrid stresses is the a-priori test. First, the subgridstress is calculated at each time instant from the definition

τSGSdefij = uiuj − uiuj (9.26)

Then the subgrid stress is computed again at each time instant from anymodel, say Smagorinsky one

τSGSmodij − 1

3τSGSkk δij = −2(CS∆)2|Sij|Sij (9.27)

The subgrid stresses τSGSdefij and τSGSmodij averaged in time are comparedeach with other. If

τSGSdefij ≈ τSGSmodij

the SGS model is accurate.

A big difficulty of a-priori tests is the determination of velocities ui. Forthat it is necessary first to obtain the unfiltered velocities ui with spatial andtemporal resolutions compared with the Kolmogorov scales. At present thisis a big challenge to measure three components of velocity in a volume withhigh spatial and temporal resolutions. The Particle Image Velocimetry (PIV)measurements are mostly planar measurements within a two dimensionalwindow. Direct Numerical Simulation data are often used as the source fora-priori tests. Three components of velocity in a volume, obtained from DNS,are filtered and utilized for the test. However, it should be noted that DNSsimulation is restricted by relatively low Reynolds numbers, whereas mainlaws of LES are valid for high Re numbers.

101

Chapter 10

Hybrid URANS-LES methods

10.1 Introduction

As discussed above, the most promising approach to resolve the flow un-steadiness is the Large Eddy Simulation (LES), which is already widely usedfor research purposes. Typical Reynolds numbers in engineering are verylarge even at model scales. The grid resolution necessary for a pure LESis so huge that it makes the direct application of LES impossible (see Sec.10.4). A practical solution of this problem is the use of hybrid URANS-LESmethods, where the near body flow region is treated using URANS and farflow regions are treated with LES.

According to Peng [23] the hybrid techniques can be subdivided into flowmatching and turbulence matching methods. Within the flow matchingmethods the interface between URANS and LES is explicitly defined. LESfiltered equations are solved in the LES region, whereas URANS equationsare solved in the URANS domain. The flow parameters (velocities, kineticenergy) are matched at the interface between the URANS and the LES re-gions. Among the most important contributions to the development of flowmatching methods we mention the works of (Davidson, Dalstroem [13]; Ter-racol [29]; Jakirlic et. al. [16]; Temmerman et. al. [28]) and others. Aserious weakness of this approach is the development of robust procedures toset the URANS-LES interface for complicated flow geometries. Within theframework of the turbulence matching method an universal transport equa-tion is solved in the whole computational domain. The stress terms in thisequation are treated in different ways in LES and URANS domains. Thereare various procedures to distinguish between LES and URANS cells. Themost popular hybrid method is Detached Eddy Simulation (DES) proposed

103

by (Spalart et al.[26]). The original version of this method is based on theclassic Smagorinsky LES model and the Spalart-Allmaras (SA) URANS ap-proach. SA is used close to the wall, whereas LES in the rest part of theflow. The switching between the two techniques is smooth and occurs in a”gray” subdomain. There are two major improvements of DES, developedrecently. The first one, DDES (Delayed DES), has been proposed to detectthe boundary layers and to prolong the RANS mode, even if the wall-parallelgrid spacing would normally activate the DES limiter (Spalart [27]). The sec-ond one, IDDES (Improved DDES), allows one to solve the problems withmodelled-stress depletion and log-layer mismatch. For the details see thereview (Spalart [27]). In spite of a wide application area DES has seriousprinciple limitations thoroughly analyzed by (Menter, Egorov [21]). Otherversions of the turbulence matching methods using different blending func-tions to switch the solution between LES and URANS modes were proposedby (Peng [23]; Davidson, Billson [12]; Abe, Miyata [8]) and others.

A very critical point of the turbulence matching methods is the transitionfrom the time (or ensemble) averaged smooth URANS flow to the oscillat-ing LES flow, see (Menter, Egorov, 2005). The oscillations have to appearwithin a short flow domain in a ”gray zone” between LES and URANS. Ex-perience shows that it is extremely difficult to provide a smooth transitionof the turbulent kinetic energy passing from the URANS to LES domain. Toovercome this problem (Schlueter et al. [25]) and (Benerafa et al. [11]) usedan additional forcing term in the Navier Stokes equation artificially enhanc-ing fluctuations in the gray zone. However, the problem of smooth solutiontransition from URANS to LES still remains as the main challenge for theturbulence matching methods.

10.2 Detached Eddy Simulation (DES)

The most popular hybrid method -detached eddy simulation- was proposedin 1997 by Spalart et al. [26]. The principle of DES is illustrated in Fig. 10.1.Close to the body the solution is calculated using the URANS mode. Farfrom the wall the LES equations are solved. The grey zone between URANSand LES is the mixed solution.

The classical version of the DES approach is based on the Spalart Almaras(SA) model formulated with respect to the modified turbulent viscosity ν =

104

Figure 10.1: Zones of the Detached Eddy Simulation

νt/fν1. The transport equation for ν reads

∂ν

∂t+ vj

∂ν

∂xj= Cb1Sν︸︷︷︸

Generation

−Cw1fw

(ν

d

)2

︸︷︷︸Destruction

+

+1

σ

∂

∂xk

((ν + ν)

∂ν

∂xk

)+Cb2σ

∂ν

∂xk

∂ν

∂xk︸︷︷︸Diffusion

(10.1)

where

Cb1 = 0.1355, Cb2 = 0.622, Cν1 = 7.1,

σ = 2/3, Cw1 =Cb1κ2

+1 + Cb2σ

,

Cw2 = 0.3, Cw3 = 2.0, κ = 0.41,

fν1 =χ3

χ3 + C3ν1

, fν2 = 1− χ

1 + χfν1

,

fw = g

[1 + C6

w3

g6 + C6w3

]1/6

, χ = ν/ν,

g = r + Cw2(r6 − r), r =ν

Sκ2d2,

S = S +ν

κ2d2fν2, S =

√2ΩijΩij,

Ωij =1

2

(∂νi∂xj− ∂νj∂xi

)105

Here d is the distance from the wall. The physical sense of different termsis illustrated in (10.1). Far from the wall the generation and the distructionterms are approaching each to other and the turbulence attains the equilib-rium state:

Cb1Sν︸︷︷︸Generation

−Cw1fw

(ν

d

)2

︸︷︷︸Destruction

∼ 0

The kinematic viscosity is the calculated from the formula

ν ∼ Cb1Cw1

Sd2

which is similar to the Smagorinsky one:

νt = l2S|Sij|, |Sij| =√

2SijSij

lS = CS∆

DES inventors proposed to use the following expression for d:

d = mind, CDES∆, ∆ = max∆x,∆y,∆z

where CDES ≈ 1.3 is the DES constant. Now the main idea of the DESbecomes obvious:

• At small wall distance d < CDES∆ the Spalart Almaras URANS model isactive

• At large wall distance d > CDES∆ the Spalart Almaras URANS model issmoothly passed into the Smagorinsky model.

Samples of DES applications are presented in Fig. 10.2 and 10.3.

Despite of the wide application Detached Eddy Simulation technique is notfree of disadvantages. Menter [21] notes: The essential concern with DESis that it does not continuously change from RANS to LES under grid re-finement. In order for LES structures to appear, the grid spacing and timestep have to be refined beyond a case-dependent critical limit. In addition, asufficiently large instability mechanism has to be present to allow the rapidformation of turbulent structures in regions where the DES limiter is acti-vated. If one of the two, or both requirements are violated, the resultingmodel is undefined and the outcome is largely unpredictable.

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Figure 10.2: Squires K.D., Detached-eddy simulation: current status andperspectives

Figure 10.3: Squires K.D., Detached-eddy simulation: current status andperspectives

10.3 Description of the hybrid model proposed

in Rostock

Our hybrid model is based on the observation that the basic transport equa-tions have the same form in LES and RANS

∂ui∂t

+∂(uiuj)

∂xj= −∂p

∗

∂xi+∂(τ lij + τ tij)

∂xj, (10.2)

but the interpretation of the overline differs. In LES it means filtering, butin RANS it stands for the Reynolds, or ensemble, averaging. Here we usedthe standard notation of p∗ for the pseudo-pressure, and τ lij and τ tij for thelaminar and turbulent stresses respectively. Note that the turbulent stressesare calculated in different ways in LES and URANS regions.

107

Figure 10.4: The division of the computational domain into the URANS(dark) and LES (light) regions at one time instant.

The computational domain in our model is dynamically (i.e. at each timestep) divided into the LES and URANS regions. A cell of the mesh belongsto one or the other region depending on the relation between the integrallength scale L and the extended LES filter ∆ according to the following rule:

if L > ∆ then the cell is in the LES region,

if L < ∆ then the cell is in the URANS region.(10.3)

The integral length scale is calculated from the known formula of Kolmogorovand Prandtl with the correction factor 0.168 taken from [24]

L = Ck3/2

ε, (10.4)

where k is the turbulent kinetic energy and ε is the dissipation rate. Theconstant C is C ∼ 0.168 close to the wall y/δ < 0.2, C ∼ 0.35 at 0.2 < y/δ <0.7 and C ∼ 1.0 in the outer area of the bounder layer y/δ > 0.7, where δ isthe boundary layer thickness. L varies from one time step to another, whichresults in varying decomposition of the computational domain into the LESand URANS regions. The extended LES filter is computed as

∆ =√d2

max + δ2, (10.5)

where dmax is the maximal length of the cell edges dmax = max(dx, dy, dz)and δ = (the cell volume)1/3 is the common filter width used in LES. Thischoice ensures that very flat cells in the boundary layer (for which δ ≈ 0but dmax > 0) are treated correctly. ∆ depends only on the mesh and it isprecomputed only once before the main computation.

108

The LES and URANS regions are shown in Fig. 10.4. The URANS region islocated close to the ship surface and plays the role of a dynamic wall func-tion. In areas of bilge vortices formation, the boundary layer is sheddingfrom the hull and penetrates into the outer flow part. Since the boundarylayer is a fine scale flow the procedure (10.3) recognizes the bilge vortexformation zones as URANS ones. There is a technical issue concerning thecells which are far from the ship hull and where both k and ε are small,so large numerical errors are introduced into the integral length scale com-puted according to Eq. (10.4). To avoid an irregular distribution of URANSand LES zones, the general rule (10.3) of the domain decomposition is cor-rected in such a way that the LES region is switched to URANS one if kis getting less than some threshold. This procedure has no influence on theship flow parameters since it is used far from the area of the primary interest.

We have performed several calculations with different combinations of LESand URANS models to find the most efficient one for the problem underconsideration. Among the models we used in our computations are the linearand nonlinear k-ε, k-ω SST and kεv2f URANS models combined with thesimple and dynamic Smagorinsky as well as with the dynamic mixed LESclosure models. The experience shows that the most satisfactory results areobtained using the URANS approach based on the kεv2f turbulent modelof [14] and LES approach based on the Smagorinsky dynamic model. Theturbulent stresses τ tij are calculated from the Boussinesq approximation usingthe concept of the turbulent viscosity. The only difference between LESand URANS is the definition of the kinematic viscosity. Within LES it isconsidered as the subgrid viscosity and calculated according to the dynamicmodel of Smagorinsky:

νSGS = cDδ2|Sij|, Sij =

1

2

(∂uj∂xi

+∂ui∂xj

), (10.6)

where Sij is the strain velocity tensor and cD is the dynamic constant. In theURANS region the viscosity is calculated from the turbulent model of [14]:

νt = min

(0.09

k2

ε, 0.22v2Tt

), (10.7)

where v2 is the wall normal component of the stresses and Tt is the turbulenttime Tt = max(k/ε, 6

√ν/ε).

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10.4 Sample of hybrid method application for

the tanker KVLCC2

The hybrid model described in the previous section has been implementedin the open-source code OpenFOAM based on the finite volume method.The doubled model of the KRISO tanker KVLCC2 (Kim et al.[18]) withthe scale 1/58 has been chosen for our investigations since it is a well-knownbenchmark which is widely used in the shipbuilding community (Goetheburgworkshop [15]). The model has length of 5.517 m, breadth of 1 m, draught of0.359 m and block coefficient of 0.8098. The computations have been carriedout on two unstructured 3D-grids. The coarse grid contains 1.2 × 106 cells,the fine one — 1.8 × 106 cells. Both grids are coarse in the foreship area(y+

1 ≈ 10). The grid with 1.2 × 106 cells has y+1 ≈ 2 − 6 in the wall region

of aftership. The grid with 1.8 × 106 cells was additionally refined in theaftership area and has y+

1 ≈ 0.1− 4 in the wall region. This grids were gen-erated by the Ship Model Basin Potsdam (SVA Potsdam) and proved to bean appropriate grid for RANS calculations. The computations on the bothgrids have been carried out with fixed maximal Courant number of 0.6. Thetime step was about 0.0008 s for the coarse grid and 0.0005 s for the fine one.

For the space discretization, the central difference scheme is used for all termsin the momentum equation whereas the Crank-Nicholson scheme is used forthe time discretizations. Steady RANS solutions are used to initialize theflow in the computational domain. The time-averaged solutions has beenobtained when the resolved flow reached a statistically steady state (usuallyit requires the ship way of 3−4 lengths). A typical time period for statisticalaveraging takes about 40−50 seconds, which corresponds to 8−10 lengths ofthe ship way. Study of the wake has been performed for the constant velocityof 1.047 m/s corresponding to the Reynolds number of Re = 4.6× 106. TheFroude number Fn = 0.142 is small which makes it possible to neglect thewater surface deformation effects.

Estimations of the resolution necessary for a pure shipLES

The precise determination of the necessary LES resolution is quite difficult.Estimations presented below are based on the idea that about 80% of theturbulent kinetic energy should be directly resolved and the rest is modeledin a properly resolved LES simulation. Implementation of this idea impliesthe knowledge of the Kolmogorov η and the integral length L scales which are

110

used to draw the typical spectra of the full developed turbulence E(k). Thewave number k∗ separating the resolved and modeled turbulence is foundfrom the condition ∫∞

k∗E(k)dk∫∞

0E(k)dk

≈ 0.2. (10.8)

The maximum possible cell size is then ∆max = 2π/k∗. The scales L and ηare found from the known expression η = (ν3/ε)1/4 and Eq. (10.4), wherethe kinetic energy k and the dissipation rate ε are taken from RANS simula-tions using k-ε linear model. The ratio λ = ∆max/η is then used as the scaleparameter for grid generation. Both lengths vary in space which makes thegrid generation procedure very complicated. To roughly estimate the size ofthe grid we assume that λ is constant. We performed different calculationsdetermining λ at the two following points: i) the point where L/η is maximalin the boundary layer and ii) the point in the propeller disk where the vor-ticity ~ω is maximal (region of the concentrated vortex structure). The latteris dictated by the wish to resolve the most intensive vortex flow structureswhich have the strongest influence on the propeller operation. Since LES ap-plication is required in the ship stern area only this part of the computationalvolume has been meshed. It covers the boundary layer of the stern regionstarting from the end of the parallel midship section. The thickness of themeshed region has been constant and equal to the maximum boundary layerthickness at the stern δBL. The grid for a pure LES is generated using thefollowing algorithm. The minimum Kolmogorov length ηmin is determinedin the near wall region. The cell sizes in x and z directions along the wallare calculated by multiplication of ηmin with the scale parameter λ. Thesesizes remain constant for all cells row in y direction which is normal to theship surface (see Fig. 10.5). The cells have at least two equal sizes which isdesirable from the point of view of LES accuracy. The choice of the size in ydirection is dictated by proper resolution of the boundary layer. Close to thewall this size is chosen from the condition ∆w = min(yw, ηmin). Since yw ischosen as the ordinate where y+ = 1 the first nodes lay deeply in the viscoussublayer. The size in y direction at the upper border of the boundary layer isequal to ∆∞ = ληδ, where ηδ is the Kolmogorov scale at y = δBL. A simplegrading is used in y direction between ∆w and ∆∞.

Results of the estimations are as follows: the required grid size ranges from∼ 5 M to ∼ 25 M for Re = 2.8 × 106, and from ∼ 7 M to ∼ 60 M forRe = 5.8 × 106. The results vary depending on the value of λ in use, sothey should be considered as very rough estimations. Together with simi-

111

lar estimations for the nonlinear k-ε model these results show that the LESgrid should have the order of tens of millions of nodes. Nowadays, the com-putations with hundred millions and even with a few billions of nodes arebecoming available in the research community. However, a numerical studyof engineering problems implies usually many computations which have to beperformed within a reasonable time with moderate computational resources.In this sense, the results of the present subsection clearly demonstrate thatthe pure LES is impossible for ship applications so far. To verify that theresolution estimation procedure we used gives meaningful results, it has beenapplied for turbulent boundary layer (TBL) benchmark. We found from me-thodical calculations that the pure LES with 1M cells is quite accurate forprediction of the velocity distribution, TBL thickness, TBL displacementthickness and the wall shear stress. The estimation procedure presentedabove predicted the necessary resolution around 0.5M. Therefore, the esti-mations presented for a ship model are rather lower bound for the resolutionrequired for a pure LES.

In the CFD community one can observe tendency to use pure LES withoutpaying any attention to resolution problems. Very often LES is running ontypical RANS grids. In fact, such computations can give correct results if theflow structures to be captured are large enough and exist for a long time. Insome cases modeling of such structures does not require detailed resolutionof boundary layers and a thorough treatment of separation regions. As anexample one can mention flows around bluff bodies with predefined separa-tion lines like ship superstructures. Application of underresolved LES forwell streamlined hulls should be considered with a great care. First of all,one should not forget that the basic LES subgrid models are derived underthe assumption that at least the inertial turbulent subrange is resolved. Sec-ond, underresolution of wall region leads to a very inaccurate modeling of theboundary layer, prediction of the separation and overall ship resistance. It isclearly illustrated in the Table 10.1. The resistance obtained from underre-solved LES using the wall function of (Werner, Wengle [30]) is less than half ofthe measured one and that obtained from RANS. Obviously, the applicationof modern turbulence LES models, more advanced than RANS models, doesnot improve but even makes the results much worse with the same space res-olution. The change from RANS to LES should definitely be followed by theincrease of the resolution which results in a drastic increase of the computa-tional costs. These facts underline necessity of further development towardshybrid methodology. Although in (Alin et al. [9]) it has been shown thatthe accuracy of the resistance prediction using pure LES at a very moderateresolution with y+ ∼ 30 can be improved using special wall functions, the

112

d

xy

z

l h· min

l hmin·

l h· d

y=1+

Figure 10.5: The cell parameters.

most universal way for the present, to our opinion, is application of hybridmethods.

10.4.1 Validation

CR CP CFKRISO Exp. 4.11× 10−3 15% 85%RANS kεv2f 4.00× 10−3 16% 84%k-ω SST SAS 3.80× 10−3 18% 82%Underresolved LES 1.70× 10−3 81% 19%Hybrid RANS LES 4.07× 10−3 17% 83%

Table 10.1: Results of the resistance prediction using different methods. CR isthe resistance coefficient, CP is the pressure resistance and CF is the frictionresistance.

Before we start to analyze unsteady effects, we show that our hybrid methodpredicts averaged flows with the accuracy not worse than that of RANS.Table 10.1 confirms that the hybrid method works well for ship resistanceprediction. Both the overall resistance and the resistance components ratioare in a good agreement with the KRISO measurements (Kim et al. [18])and RANS. Disagreement between LES results and measurements is due tocoarse resolution of the boundary layer.

The axial mean velocity field in the propeller plane for the KVLCC2 shown

113

in Fig. 10.6(a) is compared with the experimental data of KRISO (Lee et al.2003). The axial velocity ux is normalized to the ship model velocity u0 andthe coordinates are normalized to the length between perpendiculars of theship model Lpp. The mean velocity field is very similar to the experimentalone. The lines of the constant velocity have the typical form and reflect theformation of a large longitudinal bilge vortex in the propeller disk. The sec-ond longitudinal vortex is formed near the water plane, but it has a muchsmaller strength compared to the bilge one. More detailed comparison isgiven in Fig. 10.7. The axial mean velocity in circumferential direction atthe propeller radii r/R = 0.7 and r/R = 1 is compared with KRISO experi-mental data and simulations using the k-ω SST-SAS and hybrid models. Theagreement in the ranges 0 < θ < 40 and 90 < θ < 180 can be considered asquite satisfactory. The results for the range 40 < θ < 90 require additionalnumerical investigations and measurements. However, bearing in mind a bigscattering of experimental data in this range, the observed discrepancy isnot necessary the sign of modeling weakness. Fig. 10.8 shows the resolvedturbulent kinetic energy normalized to u2

0 for both computational grids incomparison with experimental data of KRISO. Topologically, the isolines aresimilar to those of the axial mean velocity shown in the Fig. 10.6(a). Refine-ment of the grid leads to a significant improvement of the numerical results.The position of the area with the strongest fluctuations and the magnitudesof these fluctuations are reproduced better when the resolution increases. Ac-cording to (Lee at al.[20]), the uncertainty of the measured TKE is ∼ 12%,so our results are quite satisfactory. The OpenFOAM implementation ofthe detached eddy simulation approach DDES failed to predict the averagedvelocity field properly, see Fig. 10.6(b). Advanced URANS technique k-ωSST-SAS provides quite satisfactory results, see Fig. 10.6(c) and Fig. 10.7.

114

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.01

-0.03

-0.05

-0.07

0.9

0.80.7

0.60.5

0.4

0.3

0.40.3

0.4

0.9

0.80.7

0.60.5

0.4

0.3

0.3

YLpp

ZL

pp

(a) Hybrid URANS-LES.

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.01

-0.03

-0.05

-0.07

0.9

0.80.7

0.60.5

0.4

0.3

0.40.3

0.9

0.8

0.70.6

0.4

0.5

0.3

0.40.3

YLpp

ZL

pp

(b) SA-DDES.

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.01

-0.03

-0.05

-0.07

0.9

0.80.7

0.60.5

0.4

0.3

0.40.3

0.9

0.8

0.70.6

0.40.5

0.3

0.40.3

YLpp

ZL

pp

(c) k-ω-SST-SAS.

Figure 10.6: The mean axial velocity field ux/u0 in the propeller plane com-puted with different models (right) vs. measurements (left).

éééééééé

éé

éééééé

éé

éé

éééééééééééééé

ééé

óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó

óóó

uxu0

Θ

0 50 100 1500

0.2

0.4

0.6

0.8

(a) r/R = 0.7

ééééééééééééé

é

é

ééé

ééééééééééééééééééé

óóó

óóóóóóóóó

óó

óóóóóóóóóóóóóóóóóóóó

óóó

0°

90°

180°

uxu0

Θ

0 50 100 1500

0.2

0.4

0.6

0.8

(b) r/R = 1.0

Figure 10.7: Circumferential distribution of the mean axial velocity field inthe propeller plane. 4 — k-ω-SST-SAS, © — DSM+V2F, solid line —KRISO experiments for the specified r/R.

115

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.01

-0.03

-0.05

-0.07

24

46

24

610

14

8

2

2

44

4

6

6

10

10

YLpp

ZL

pp

(a) Computational grid 1.2× 106 cells.

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.01

-0.03

-0.05

-0.07

24

46

24

610

14

8

6

24 6

14

8

1014

YLpp

ZL

pp

(b) Computational grid 1.8× 106 cells.

Figure 10.8: Resolved turbulent kinetic energy k = ρ/2(u′xu′x + u′yu

′y +

u′zu′z)/u

20 multiplied with 103 in the propeller plane. Numerics (right-half

of each figure) versus measurement (left-half).

-0.06 -0.04 -0.02 0

-0.01

-0.03

-0.05

-0.07YLpp

ZL

pp

123456

1 0 0

2 0.0036 0.23

3 0.0073 0.47

4 0.0110 0.70

5 0.0150 0.94

6 0.0180 1.20

YLpp rR

Figure 10.9: Positions of probe points. R is the propeller radius.

116

Figure 10.10:

117

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