theoretical manual · 1-4 fine™ 1-3.2 time averaging of navier-stokes equations in principle, the...

Theoretical ManualFINE™/Turbo v8.10

Flow Integrated Environment

- November 2012 -

N U M E R I C A L M E C H A N I C S A P P L I C A T I O N S

Theoretical ManualFINE™/Turbo v8.10

Documentation v8.10c

NUMECA InternationalChaussée de la Hulpe, 189

1170 BrusselsBelgium

Tel: +32 2 647.83.11Fax: +32 2 647.93.98

Web: http://www.numeca.com

N U M E R I C A L M E C H A N I C S A P P L I C A T I O N S

Contents

CHAPTER 1: Euler and Navier-Stokes Equations 1-1 1-1 Overview 1-11-2 Euler Equations 1-11-3 Navier-Stokes Equations 1-2

General Navier-Stokes Equations 1-2Time Averaging of Navier-Stokes Equations 1-4Treatment of Turbulence in the RANS Equations 1-5Formulation in Rotating Frame for the Relative Velocity 1-6Formulation in Rotating Frame for the Absolute Velocity 1-7

CHAPTER 2: Fluid Models 2-1 2-1 Overview 2-12-2 Calorically Perfect Gas 2-1

Equation of State 2-1Transport Properties 2-2

2-3 Real Gas 2-3Equation of State 2-3Transport Properties 2-4

2-4 Incompressible Fluid 2-4Liquid 2-4Barotropic Liquid 2-5

2-5 Condensable Fluid 2-6Introduction 2-6Thermodynamic Tables 2-6Integration of Thermodynamic Tables in the Solver 2-7

CHAPTER 3: Turbulence Models 3-1 3-1 Overview 3-13-2 Linear Eddy Viscosity Turbulence Models 3-2

Baldwin-Lomax Model 3-2Spalart-Allmaras Model 3-3SARC Model 3-4k-ε Turbulence Models 3-5k-ω Turbulence Models 3-7v2-f Model 3-9

3-3 Nonlinear Eddy Viscosity Turbulence Models 3-10EARSM Model 3-10

3-4 Detached Eddy Simulations 3-133-5 Laminar-Transition Model 3-13

Transition Models 3-13Intermittency Distribution 3-15Modification of the turbulence models 3-16

3-6 Modelling Flow near the Wall 3-18Velocity 3-18Temperature 3-21Turbulence 3-21

FINE™ iii

Contents

CHAPTER 4: Boundary Conditions 4-1 4-1 Overview 4-14-2 Inlet Boundary Conditions 4-1

Cylindrical Inlet Boundary Conditions 4-1Cartesian Inlet Boundary Conditions 4-7Turbulent Inlet Boundary Conditions 4-8

4-3 Outlet Boundary Conditions 4-9Outlet Boundary Conditions for Subsonic Flow 4-9Outlet Boundary Conditions for Supersonic Flow 4-12Treatment of Backflow at Outlet (Radial Diffuser) 4-12

4-4 Solid Wall Boundary Conditions 4-13Euler Walls 4-13Navier-Stokes Walls 4-14

4-5 Far-field Boundary Condition 4-17External 4-17

4-6 Unsteady Inlet and Outlet Boundary Conditions 4-18user specified time function 4-18rotation of the reference system 4-18

CHAPTER 5: Numerical Schemes 5-1 5-1 Overview 5-15-2 Discretization and Solution Theory 5-1

Spatial Discretization 5-1Time Discretization: Multistage Runge-Kutta 5-7Multigrid Strategy 5-10Full Multigrid Strategy 5-13Implicit residual smoothing 5-14

5-3 Preconditioning 5-15Hakimi Preconditioning 5-16Merkle Preconditioning 5-19

5-4 Time Configuration 5-21Standard Time Accurate Solver (Non-preconditioned Formulation) 5-21Density Based Time Accurate Solver and Preconditioning 5-23

CHAPTER 6: Physical Models 6-1 6-1 Overview 6-16-2 Fluid-Particle Interaction 6-1

Trajectories Calculation - Integration Method 6-1Integration Time Step - Relaxation Time 6-2Full Coupling Carrier & Particle Flows 6-3Interaction with Turbulence 6-3

6-3 Fluid-Structure 6-4Conjugate Heat Transfer 6-4Mesh Deformation and Solver Adaptations 6-6Modal Coupling 6-10

6-4 Passive Tracers 6-11

iv FINE™

Contents

6-5 Porous Media Model 6-12Porous Media Model 6-12Thermal Heat Flux model 6-13

6-6 Cavitation Model 6-14

CHAPTER 7: Dedicated Turbomachinery Models 7-1 7-1 Overview 7-17-2 Rotor/Stator Interaction 7-1

Introduction 7-1Default Mixing Plane Approach 7-2Full Non-matching Technique for Mixing Planes 7-5Non Reflecting Boundary Conditions 7-7Frozen Rotor 7-10

7-3 Unsteady Turbomachinery Calculations 7-10Unsteady Inlet and Outlet Boundary Conditions 7-11Phase Lagged Periodicity Boundary Conditions 7-11Domain Scaling Method 7-13Phase-lagged Method 7-15Choice of the Physical Time Step 7-15

7-4 Harmonic Method 7-16Theory 7-16Rotor/Stator interaction with Harmonic Method 7-19

7-5 Clocking Effects with NLH 7-19Theory 7-19

7-6 Cooling/Bleed 7-207-7 Aeroacoustics 7-21

APPENDIX A:Output Variables A-1 A-1 Computed Variables A-1A-2 Surface Averaged Variables A-2A-3 Azimuthal Averaged Variables A-3A-4 Global Performance Output A-5

FINE™ v

Contents

vi FINE™

NOMENCLATURE

FINE™

acceleration vectorc speed of soundcp specific heat at constant pressure

cv specific heat at constant volume

e internal energy per unit massE total energy per unit volume

external force vector

vector (column matrix) of inviscid flux

vector (column matrix) of viscous fluxg gravity accelerationh enthalpy per unit massH total enthalpyk turbulent kinetic energy

mass flow rateM molecular weight; Mach numberp pressureP pitchPr Prandtl numberPrt turbulent Prandtl number

q heat fluxQ flux of variables; limiterr gas constant per unit massr, θ, z cylindrical coordinatesR universal gas constantRe Reynolds numbers entropy per unit massS surfaceST vector (column matrix) of source term

surface vectort timeT temperature

a

fe

FI

FV

m·

S

v

Nomenclature

vi

T+ dimensionless temperatureTu turbulence intensity

U vector (column matrix) of conservative variables

velocity vector with component u, v, wV specific volume; vector (column matrix) of primitive variables

relative velocity vectorx, y, z cartesian coordinates

y+ dimensionless normal distance from the wallspecific heat ratio

turbulent dissipation rate

coefficient of thermal conductivity

coefficient of dynamic viscosity

coefficient of turbulent viscosity

density

stress tensor

kinematic viscosity

turbulent kinematic viscosity

volume

angular velocity; specific dissipation rate

v u v w, ,( )

w

γ

ε

κ

μ

μt

ρ

τ

ν

νt

Ω

ω

FINE™

CHAPTER 1: Euler and Navier-Stokes Equations

FINE™

1-1 OverviewThe flow solver integrated into the FINE™/Turbo user environment is a finite volume solver for 2D and 3D flows in complex geometries, using the latest numerical developments in CFD. A structured mesh is required and complex geometries can be easily handled through a flexible multiblock meshing procedure.

This chapter describes the basic governing equations solved in the FINE™/Turbo solver:

• Euler Equations•Navier-Stokes Equations

General Navier-Stokes equationsTime averaging of Navier-Stokes equationsTreatment of turbulence in the RANS equations Formulation in rotating frame for the relative velocityFormulation in rotating frame for the absolute velocity

1-2 Euler EquationsThe general Euler equations written in a Cartesian frame can be expressed as:

(1-1)

where is the volume and S is the surface. is the vector of the conservative variables:

∂∂t---- U Ωd

Ω FI Sd⋅

S+ ST Ωd

Ω=

Ω U

1-1

Euler and Navier-Stokes Equations Navier-Stokes Equations

1-2

(1-2)

is the inviscid flux vectors:

(1-3)

where the total energy is defined by:

(1-4)

contains the source terms:

(1-5)

with , and are the component of external forces . the work performed by those

external forces: . Other source terms are possible, like gravity, depending on the selected functionalities.

1-3 Navier-Stokes Equations

1-3.1 General Navier-Stokes EquationsThe general Navier-Stokes equations written in a Cartesian frame can be expressed as:

, (1-6)

where is the volume and S is the surface, is the vector of the conservative variables:

U

ρρv1ρv2ρv3ρE

=

FI

FI

ρviρv1vi pδ1i+ρv2vi pδ2i+ρv3vi pδ3i+

ρE p+( )vi

=

E e 12---vivi+=

ST

ST

0ρfe1ρfe2ρfe3Wf

=

fe1 fe2 fe3 fe Wf

Wf ρfe v⋅=

∂∂t---- U Ωd

Ω FI Sd⋅

S FV Sd⋅

S+ + ST Ωd

Ω=

Ω U

FINE™

Navier-Stokes Equations Euler and Navier-Stokes Equations

FINE™

(1-7)

and are respectively the inviscid and viscous flux vectors:

and (1-8)

where the total energy and the heat flux components are defined by:

(1-9)

(1-10)

is the laminar thermal conductivity.

contains the source terms:

(1-11)

with , and are the component of external forces . is the work performed by those

external forces: . Other source terms are possible, like gravity, depending on the selected functionalities.

To close the Navier-Stokes equations, it is necessary to specify the constitutive laws and the defini-tion of the shear stress tensor in function of the other flow variables. Here only Newtonian fluids are considered for which the shear stress tensor is given by:

(1-12)

where is the dynamic molecular viscosity.

More informations about the constitutive laws and properties of the fluids can be referred to Chapter 2.

U

ρρv1ρv2ρv3ρE

=

FI FV

FI

ρviρv1vi pδ1i+ρv2vi pδ2i+ρv3vi pδ3i+

ρE p+( )vi

= F– V

0τi1τi2τi3

qi vjτi j+

=

E e 12---vivi+=

qi κ∂

∂xi-------T=

κ

ST

ST

0ρfe1ρfe2ρfe3Wf

=

fe1 fe2 fe3 fe Wf

Wf ρfe v⋅=

τij μ∂vj∂xi-------

∂vi∂xj-------+

23--- ∇ v⋅ δij–=

μ

1-3


1-4

1-3.2 Time Averaging of Navier-Stokes Equations In principle, the Navier-Stokes equations describe both laminar and turbulent flows. However, tur-bulence is a nonlinear process with a wide range of spatial and temporal scales. The direct simula-tion of complex turbulent flows in most engineering applications is not possible and will not be for the foreseeable future.

In the context of scale modelling, the most direct approach is offered by the partitioning of the flow field into a mean and fluctuating part. This process will produce the Reynolds Averaged Navier-Stokes (RANS) equations.

The Reynolds averaged equations are derived by averaging the viscous conservation laws over a time interval T, large enough with respect to the time scales of the turbulent fluctuations, but small enough with respect to all other time-dependent effects. Time averaging is performed as follows.

The quantity A in the Navier-Stokes equations is time averaged related to the instantaneous value through:

(1-13)

where

(1-14)

is the time averaged value and is the fluctuating part. Furthermore, we have:

(1-15)

The corresponding density weighted average is defined through:

(1-16)

with

(1-17)

and

(1-18)

Density and pressure are time averaged, whereas energy, velocity components and temperature are density weighted time averaged.

The averaged form of the Navier-Stokes equations is the same as Eq. 1-6, with:

(1-19)

where the density averaged total energy is given by:

A A A'+=

A x t,( ) 1T--- A x t τ+,( ) τd

T 2⁄–

T 2⁄

=

A A'

A′ 0=

Ã ρAρ

-------=

A Ã A″+=

ρA″ 0=

U

ρρṽ1ρṽ2ρṽ3ρẼ

= FI

ρṽiρṽ1ṽi ρv1″vi″ pδ1 i+ +

ρṽ2ṽi ρv2″vi″ pδ2 i+ +

ρṽ3ṽi ρv3″vi″ pδ3 i+ +

ρẼ p+( )ṽi ρE″vi″ pvi″+ +

= FV–

0τi1τi2τi3

qi viτij+

=

FINE™


FINE™

(1-20)

k is the turbulent kinetic energy and is defined by:

(1-21)

1-3.3 Treatment of Turbulence in the RANS EquationsThe Reynolds-averaged Navier-Stokes equations lead to the introduction of the Reynolds stress ten-sor and turbulent heat diffusion term. Since these quantities are unknown, the application of the Navier-Stokes equations to the computation of turbulent flows requires the introduction of some modelizations of these unknown relations, based on theoretical considerations coupled to unavoida-ble empirical information.

For the linear eddy viscosity turbulence models, a first-order closure model, based on Boussinesq's assumption, is used for the Reynolds stress:

(1-22)

where is the turbulent eddy viscosity.

For the turbulent heat diffusion term, a gradient approximation is used:

(1-23)

where is the turbulent thermal conductivity and is connected to the turbulent eddy-viscosity

through a turbulent Prandtl number :

(1-24)

Finally, the resulting system of governing equations with the assumptions above the same as Eq. 1-6, with:

(1-25)

where the Reynolds stress and the heat flux components are given by:

(1-26)

Ẽ ẽ 12---vi

˜ vi˜ k̃+ +=

k 12--- ρvi″vi″ ρ⁄( )=

ρvi″vj″– μt xj∂∂ṽi

xi∂∂ṽj 2

3--- ∇ v⋅ δij–+

23---ρk̃δi j–=

μt

Cpρvi″T κt∂

∂xi-------T̃–=

κt μtPrt

κtμtCpPrt

-----------=

U


= FI

ρṽiρṽ1ṽi p*δ1i+

ρṽ2ṽi p*δ2i+

ρṽ3ṽi p*δ3i+

ρẼ p*+( )ṽi

= FV–

0τi1τi2τi3

qi ṽiτij+

=

τij μ μt+( )∂ṽi∂xj-------

∂ṽj∂xi------- 2

3--- ∇ v⋅ δij–+=

1-5


1-6

and (1-27)

and need to be solved by the turbulence models. More informations about the turbulence models can be referred to Chapter 3.

In contrast to the laminar case, both the static pressure and the total energy contain contributions from the turbulent kinetic energy and are defined as:

(1-28)

(1-29)

1-3.4 Formulation in Rotating Frame for the Relative VelocityIn many applications, such as turbomachinery problems, it is necessary to describe the flow behav-iour in the relative system and solve the governing equations for the relative velocity components.

The Reynolds averaged Navier-Stokes equations for the relative velocities in the rotating frame of reference become:

(1-30)

where is the component of the relative velocity . The source term vector contains con-tributions of Coriolis and centrifugal forces and is given by:

(1-31)

with is the angular velocity of the relative frame of reference.

The stress and the heat flux components are given by:

(1-32)

and

(1-33)

The static pressure and the total energy are defined as:

qi κ κt+( )∂

∂xi-------T̃=

μt κt

k

p* p 23---ρk̃+=

Ẽ ẽ 12--- ṽiṽi k̃+ +=

U

ρρw̃1ρw̃2ρw̃3ρẼ

= FI

ρw̃ip*δ1i ρw̃1w̃i+

p*δ2i ρw̃2w̃i+

p*δ3i ρw̃3w̃i+

ρẼ p*+( )w̃i

= FV–

0τi1τi2τi3

qi w̃iτi j+

=

wi xi w ST

ST

0

ρ–( ) 2ω w× ω ω r×( )×( )+[ ]

ρw ∇ 0.5ω2r2( )⋅

=

ω

τij μ μt+( )∂w̃i∂xj--------

∂w̃j∂xi-------- 2

3--- ∇ w̃⋅ δi j–+=

qi κ κt+( ) xi∂∂ T̃=

FINE™


FINE™

(1-34)

(1-35)

1-3.5 Formulation in Rotating Frame for the Absolute VelocityAlthough the governing equations for rotating systems are usually formulated in the relative system and solved for the relative velocity components, the formulation retained for ship propeller applica-tions, ventilators or wind turbine are often expressed in the relative frame of reference for the abso-lute velocity components. This formulation is different from the one generally used to solve internal turbomachinery problems, where the equations are solved for the relative velocity. The two formu-lations should lead to the same flow solution. However, experience shows that the solution can be different, especially in the far field region. For propeller problems, the formulation based on rela-tive velocities has the disadvantage that the far field relative velocity can reach high values. This induces an excess of artificial dissipation leading to a non-physical rotational flow in the far field region, this dissipation being based on the computed variables.

The Reynolds averaged Navier-Stokes equations for the absolute velocities in the rotating frame of reference become:

(1-36)

where the velocity is the xi component of the relative velocity and the velocity is the xicomponent of the absolute velocity . This formulation involves thus both the absolute and the rel-ative velocity components.

The source term vector is given by:

(1-37)

with is the angular velocity of the relative frame of reference. Other source terms are possible, like gravity, depending on the chosen functionalities.

The stress and the heat flux components are given by:

(1-38)

and

p* p 23---ρk̃+=

Ẽ ẽ 12---w̃iw̃i k̃+ +=

U


= FI

ρw̃ip*δ1i ρw̃iṽ1+

p*δ2i ρw̃iṽ2+

p*δ3i ρw̃iṽ3+

ρẼw̃i p*ṽi+

= FV–

0τi1τi2τi3

qi ṽjτij+

=

wi w viv

ST

ST

0

ρ ω ṽ×( )–0

=

ω

τij μ μt+( )∂ṽi∂xj-------

∂ṽj∂xi------- 2

3--- ∇ ṽ⋅ δij–+=

1-7


1-8

(1-39)

The static pressure and the total energy are defined as:

(1-40)

(1-41)

qi κ κt+( ) xi∂∂ T̃=

p* p 23---ρk̃+=

Ẽ ẽ 12--- ṽiṽi k̃+ +=

FINE™

CHAPTER 2: Fluid Models

FINE™

2-1 OverviewIn this chapter, details are provided about the different fluid types available in FINE™/Turbo:

• Perfect gas,•Real gas,• Incompressible fluid,•Condensable fluid.

2-2 Calorically Perfect Gas

2-2.1 Equation of StateThe perfect gas law is used as constitutive equation:

(2-1)

with the gas constant for the perfect gas under consideration:

(2-2)

with the universal gas constant and the molecular weight of the perfect gas.

is the specific heat ratio, the specific heat at constant pressure, and the specific heat at con-stant volume with:

(2-3)

p ρrT=

r

r RM----- cp cv–= =

R M

γ cp cv

γcpcv----=

2-1

Fluid Models Calorically Perfect Gas

2-2

When the gas temperature is so low that the vibrational and electronic modes are frozen, the inter-nal energy of the gas will be proportional to the temperature. The specific heats as well as are then constant. The gas is called "calorically perfect". This is the usual assumption of moderate speed aerodynamics.

The relation between and is given by

(2-4)

The values of and are user input. The static pressure is obtained from the conservative varia-bles through the following relation:

(2-5)

2-2.2 Transport Properties

2-2.2.1 The Laminar ViscosityThe laminar kinematic viscosity can be

• constant,• given in terms of a polynomial function of the temperature,• given by a profile for a certain temperature range,• varying according to Sutherland’s law.

The dynamic viscosity is then computed within the code by multiplying the kinematic viscosity by the reference density.

The Sutherland law is given by:

For

(2-6)

For

(2-7)

The viscosity in Eq. 2-6 and Eq. 2-7 is the dynamic viscosity specified in the Fluid Model page or which is obtained from the kinematic viscosity and the density both specified in the Fluid Modelpage. and are the reference temperature and the Sutherland temperature respectively, also specified in the Fluid Model page.

γ

r cp

r γ 1–γ

-----------cp=

γ cp

p γ 1–( ) ρE ρw( )2

2ρ--------------–=

T∞ 120K>

μ T( ) μ 120( ) T120---------= T 120K≤

μ T( ) μ∞T

T∞------ 1.5 T∞ Tsuth+

T Tsuth+------------------------ = T 120K≥

T∞ 120K≤

μ T( ) μ∞T

T∞------= T 120K≤

μ T( ) μ 120( ) T120---------

1.5 120 Tsuth+T Tsuth+

-------------------------- = T 120K≥

μ∞

T∞ Tsuth

FINE™

Real Gas Fluid Models

FINE™

2-2.2.2 The Heat ConductivityThe laminar heat conductivity can be

• constant,• given in terms of a polynomial function of the temperature,• given by a profile for a certain temperature range,• specified through the Prandtl number.

In case the Prandtl number Pr is specified, the laminar thermal conductivity is obtained as:

(2-8)

The turbulent viscosity is calculated iteratively using one of the turbulence models discussed in Chapter 3. The turbulent conductivity is obtained from the turbulent viscosity and a turbulent Prandtl number whose value can be controlled through the expert parameter PRT (default: 1.0):

(2-9)

2-3 Real Gas

2-3.1 Equation of StateFor real gases or more precisely thermally perfect gases, and depend on the temperature T. The corresponding values can be entered either through a polynomial approximation or through a profile.

The thermally perfect gas model is based on two equations:

• the perfect gas relation, still valid:

(2-10)

• the enthalpy equation:

(2-11)

where T0 is a reference temperature. In practice T0 is set equal to the minimum temperature of the temperature range specified in the Fluid Model page.

These two equations lead to:

(2-12)

κμcpPr--------=

κtμtcpPrt----------=

cp γ

p ρrT ρ RM-----T= =

h cp TdT0

T

e pρ---+ cv Td

T0

T

rT+= = =

rT cp cv–( ) TdT0

T

=

2-3

Fluid Models Incompressible Fluid

2-4

In the case of a real gas with varying specific heats one can notice that it is not possible to respect the 2 above equations with a constant value of r, unless the difference (cp-cv) does not vary with temperature.

Two solutions are proposed in the FINE™/Turbo solver in order to model real gases:

•Constant r model. The user specifies the gas constant r (R). Then cp(T) or γ(T) is provided and

respectively γ is calculated as or cp is calculated as .

• Perfect gas with compressibility factor. It is used when the user specifies cp(T) and γ(T) to define the specific heat law. The equation of state is modified to include a compressibility fac-tor Z:

(2-13)

(2-14)

where T0 corresponds to the new TMIN when from the two ranges, [TMING,TMAXG] for γ(T) and [TMINC,TMAXC] for cp(T), the smaller range is taken: TMIN=MIN(TMING,TMINC) and TMAX=MAX(TMAXG,TMAXC). Then the range is expanded by (TMAX-TMIN)/2 on the lower and upper side. This results in respectively a lowered TMIN and a higher TMAX, and a two times wider range. cp(T) and γ(T) are constant (zero order extrapolation) outside their respective ranges.

In such case, in Eq. 2-14, the variable "ZrT" computed from user data (cp(T) and γ(T)) is directly used in Eq. 2-13 to define the pressure from the density.

2-3.2 Transport PropertiesThe theory of the real gas transport properties is the same as that of the perfect gas. The details can be seen in Chapter 2-2.2.

2-4 Incompressible Fluid

2-4.1 Liquid

2-4.1.1 Equation of StateThe density is user input:

(2-15)

This implies a decoupling of the mass and momentum conservation equations from the energy con-servation equation, since no term in the mass and momentum equations depends on the tempera-ture.

γcp

cp r–-------------= cp

r

1 1γ---–

------------=

p ZρrT Zρ RM-----T= =

Z

cp cv–( ) TdT0

T

rT-----------------------------------------=

ρ Cst=

FINE™

Incompressible Fluid Fluid Models

FINE™

2-4.1.2 Transport PropertiesThe theory of the liquid transport properties is the same as that of the perfect gas. The details can be seen in Chapter 2-2.2.

2-4.2 Barotropic Liquid

2-4.2.1 Equation of StateThe density is a user defined function of the pressure:

(2-16)

The corresponding values can be entered either through a polynomial approximation or through a profile.

The particularity of the barotropic liquid formulation is that although the density is not constant, a decoupling of the energy equation is still adopted, exactly as in the full incompressible formulation.

Modification of the treatment of the energy equation

A modification of the treatment of the energy is required in order to account for the density varia-tions. The temperature equation for any type of fluid can be written as:

(2-17)

where εv is the viscous dissipation term and V is the specific volume (inverse of density). Since for a barotropic fluid there is a unique relation between the pressure and the density the first term on the right hand side of the above equation vanishes and the temperature for a barotropic fluid becomes:

(2-18)

In addition, since from classical thermodynamics it is known that

(2-19)

the specific heat at constant volume depends only on the temperature . Note that for an incompressible fluid the specific heats at constant pressure and at constant volume are equal ( ).

The modification of the formulation of the energy for a barotropic fluid can be derived from the above relations. For a barotropic fluid:

(2-20)

The barotropic relation can be integrated from a reference state to finally obtain:

ρ f p( )=

ρcvdTdt------ T ∂p

∂T------

V∇ V⋅ – ∇ κ∇T

⋅ εv+ +=

ρcvdTdt------ ∇ κ∇T

⋅ εv+=

∂cv∂V-------- T ∂

2p∂T2--------

V

=

cv cv T( )=

cv cp=

de cvdT T∂p∂T------

Vp–

dV+ cvdT pdV–= =

2-5

Fluid Models Condensable Fluid

2-6

(2-21)

Compared to an incompressible formulation the energy depends on both pressure and temperature.

2-4.2.2 Transport PropertiesThe theory of the barotropic liquid transport properties is the same as that of the perfect gas. The details can be seen in Chapter 2-2.2.

2-5 Condensable Fluid

2-5.1 IntroductionThe aim of the condensable fluid module is modelling the real thermodynamic properties of the fluid by means of interpolation of the variables from dedicated tables.

The module can be used for a single-phase fluid whose properties are too complex to be treated with a perfect or real gas model. It can also be used in order to treat thermodynamic conditions that are close to the saturation line. Note that the model can be used on the liquid or on the vapour side of the saturation curve. In case the thermodynamic state lies inside the two-phase region a homoge-neous equilibrium two-phase mixture of vapour and liquid is considered. The hypothesis of an equi-librium mixture is however not valid if the dryness (wetness) fraction exceeds 20%. The module can not be used above these fractions, as it completely ignores evaporation-condensation phenom-ena.

For a real fluid the equation of state may be a complicated and usually implicit expression, generat-ing unacceptable computational overhead if the corresponding equations are explicitly introduced in the solver. Similarly when an equilibrium mixture of vapour and droplets (wet steam for instance) is considered as a single fluid, the calculation from the saturation properties of thermody-namic variables must be done iteratively.

The approach that has been adopted in the FINE™/Turbo solver consists of using a series of ther-modynamic tables, one table being required each time a thermodynamic variable must be deduced from two other ones. This implies the creation of many tables as input, but presents the advantage that no iterative inversion of the tables is done in the solver, with as a consequence a very small additional CPU time. This additional time corresponds to the one that is needed by the bilinear (or bicubic) interpolation procedures through the tables. In order to optimize the efficiency of these interpolations, the input tables are always built on basis of a Cartesian mesh in the plane of the input variables (V1,V2) (N1 and N2 values for the variables V1 and V2 respectively).

2-5.2 Thermodynamic TablesTables are generated based on TabGen, which are based on various sets of well-known equations as described below:

• Vander Waals equation.• Benedict Webb Rubin equation• Starling modified Benedict Webb Rubin equation (usually used for hydrocarbons),

e p T,( ) e0 cv Td

T0

T

p

ρ2-----∂ρ

∂p------ pd

p0

p

+ +=

FINE™

Condensable Fluid Fluid Models

FINE™

• 32 terms modified Benedict-Webb-Rubin equation• Equation of state derived from the Helmoltz free energy equation (as for water tables that are

based on the release 1997 IAPWS)

Single tables are generally used, which means that the tables cover both single-phase and two-phase regions. A bilinear interpolation approach is well adapted to these tables, providing an adequate smoothing of the saturation region. Bicubic interpolation techniques have also been implemented and tested. They provide a higher accuracy for a given number of data points but tend to create spu-rious oscillations in the saturation region. These oscillations can be weakened by using more points.

Five categories of tables are used by the solver:

1. Basic tables: p(e,ρ) and T(e,ρ) or ρ(h,p) and T(h,p)

For non-preconditioned equations, the numerical scheme is based on a formulation of the equations on basis of the energy and the density. The basic tables are used in order to update the pressure and the temperature after each update of the formulation variables. For preconditioned equations, the basic variables are enthalpy and pressure, they are used in order to update the density and tempera-ture.

2. Entropy tables: p(h,s), ρ(h,s), s(h,p), h(s,p)

The entropy tables are used at the inlet/outlet boundary conditions, and are required in order to cal-culate the total (static) thermodynamic conditions from the static (total) ones.

3. The (p,T) tables: e(p,T) and ρ(p,T)

These tables are required in order to allow the use of the inlet boundary conditions based on total pressure and temperature. They are also required by the turbomachinery initial solution procedure.

.4. The viscosity and conductivity tables: μ(e,ρ) and κ(e,ρ)

The viscosity and conductivity can also be interpolated from tables. If these tables are not present the other laws available within the user interface can be used.

5. The saturation table

This table contains the liquid and vapour values of all thermodynamic variables along the saturation line. It is only required if the user activates the dryness fraction output and/or if the inlet boundary condition based on total enthalpy and dryness fraction is selected.

2-5.3 Integration of Thermodynamic Tables in the SolverProvided that all the input tables are present, the full functionality of the basic solver is accessible. All boundary conditions are available, as well as the initial solution procedures. The numerical schemes are not different, making use of the same acceleration techniques (multigrid, local time stepping, residual smoothing) providing robust and fast convergence to steady state.

The evaluation of the advective, viscous and artificial dissipation fluxes involves the pressure in the momentum and energy equations and the temperature in the energy equation. For a perfect gas both are easily deduced from the equation of state, the conversion being so rapid that only the pressure and the density are stored and the other variables recalculated from these whenever they are needed. In the condensable fluid module the density, the pressure, the temperature and the energy are stored in order to limit the number of interpolations. Only one interpolation is made in order to deduce the new values of pressure and temperature after the update of the density and the energy.

The speed of sound is also required. It enters the numerical solution process through the spectral radius in the time step and in the residual smoothing and artificial dissipation coefficients. It can either be interpolated from a table or computed from the partial derivatives of the p(e,ρ) table:

2-7

Fluid Models Condensable Fluid

2-8

(2-22)

The Baldwin-Lomax, Spalart-Allmaras and turbulence models can be used. The turbulent conductivity calculation is based on the usual relation:

(2-23)

The specific heat at constant pressure can either be constant (the value being provided in the user interface) or deduced from the specific table.

c2 ∂p∂ρ------

e

pρ2----- ∂p

∂e------

ρ+=

κ ε–

κtμtPrt-------cp=

FINE™

CHAPTER 3: Turbulence Models

FINE™

3-1 OverviewThis chapter provides details of the different turbulence models, laminar-transition model and the wall functions available in FINE™/Turbo:

• Linear Eddy Viscosity Turbulence models:—Algebraic model:

Baldwin-Lomax— 1-equation model:

Spalart-AllmarasSpalart-Allmaras (Extended Wall Function)SARC

— 2-equation models:k-ε (Standard wall function)k-ε (Extended wall function)k-ε (Low Re Chien)k-ε (Low Re Yang-Shih)k-ε (Low Re Launder-Sharma)k-ω (Wilcox)Shear Stress Transport (SST)SST (Extended wall function)

— 4-equation model:

v2-f (code friendly)•Nonlinear Eddy Viscosity Turbulence models:

— 2-equation models:EARSMEARSM (Extended wall function)

•Detached Eddy Simulations (DES)• Laminar-Transition Model

3-1

Turbulence Models Linear Eddy Viscosity Turbulence Models

3-2

•Modelling Flow near the Wall

3-2 Linear Eddy Viscosity Turbulence Models

3-2.1 Baldwin-Lomax ModelThe Baldwin-Lomax algebraic turbulence model, Baldwin & Lomax (1978), is a two layer model where the turbulent viscosity in the inner layer is determined by using the Prandtl mixing length model, and the turbulent viscosity in the outer layer is determined from the mean flow and a length scale. The strain-rate parameter in the Prandtl mixing length model is taken to be the magnitude of the vorticity.

The influence on the mean flow equations through the turbulent kinetic energy is neglected.

The turbulent viscosity is given by

(3-1)

where is the normal distance to the wall, and is the smallest value of at which the inner and outer viscosity is equal.

The inner viscosity is

(3-2)

where

(3-3)

(3-4)

and

(3-5)

where w is the subscript for wall quantities and is the Kronecker symbol.

The outer viscosity is

(3-6)

where is the smaller of

(3-7)

μtμt( )i n nc≤,

μt( )0 n nc>,

=

n nc n

μt( )i ρl2 ω=

l κn 1 e y–+ A+⁄–( )=

y+ρwτwμw

----------------

n=

ωi εi jk xk∂∂uj=

εijk

μt( )0 KCcpρFwakeFKleb n( )=

Fwake

nmaxFmax

and Cwknmax u2 v2 w2+ +( )max u2 v2 w2+ +( )min–2/Fmax

FINE™

Linear Eddy Viscosity Turbulence Models Turbulence Models

FINE™

The term is the value of n corresponding to the maximum value of , , where

(3-8)

and

(3-9)

The constants used are hard coded and equal to: , , , ,

, .The turbulent Prandtl number, needed to calculate the turbulent con-ductivity, is accessible through the expert parameter PRT (see Chapter 2-2.2.2).

3-2.2 Spalart-Allmaras ModelThe Spalart-Allmaras turbulence model is a one equation turbulence model which can be consid-ered as a bridge between the algebraic model of Baldwin-Lomax and the two equation models. The Spalart-Allmaras model has become quite popular in the last years because of its robustness and its ability to treat complex flows. The main advantage of the Spalart-Allmaras model when compared to the one of Baldwin-Lomax is that the turbulent eddy viscosity field is always continuous. Its advantage over the k-ε model is mainly its robustness and the lower additional CPU and memory usage.

The principle of this turbulence model is based on the resolution of an additional transport equation for the eddy viscosity. The equation contains an advective, a diffusive and a source term and is implemented in a non conservative manner. The implementation is based on the papers of Spalart and Allmaras (1992) with the improvements described in Ashford and Powell (1996) in order to

avoid negative values for the production term ( in Eq. 3-15).

The turbulent viscosity is given by

(3-10)

where is the turbulent working variable and a function defined by

(3-11)

with is the ratio between the working variable and the molecular viscosity ,

(3-12)

The turbulent working variable obeys the transport equation

(3-13)

where is the velocity vector, ST the source term and , constants.

The source term includes a production term and a destruction term:

nmax F Fmax

F n( ) n ω 1 e n+/A+––( )=

FKleb 1 5.5 nCKleb/nmax( )6+

1–=

A+ 26= Cwk 1= Ccp 1.6= κ 0.41=

Ckleb 0.3= K 0.0168=

S̃

νt ν̃fv1=

ν̃ fv1

fv1χ3

χ3 cv13+

-------------------=

χ ν̃ ν

χ ν̃ν---=

ν̃∂t∂

------ v ν̃∇⋅+ 1σ--- ∇ ν 1 cb2+( )ν̃+( ) ν̃∇[ ] cb2ν̃ ν̃Δ–⋅{ } ST+=

v σ cb2

3-3


3-4

(3-14)

where

(3-15)

(3-16)

The production term P is constructed with the following functions:

; (3-17)

; (3-18)

where d is the distance to the closest wall and S the magnitude of vorticity.

In the destruction term (Eq. 3-16), the function is

(3-19)

with

, (3-20)

The constants arising in the model are

, , , ,

, , ,

3-2.3 SARC ModelThe SARC model results from modifications brought to the Spalart-Allmaras (S-A) model, which have been introduced by Spalart and Shur (1997) and Shur et al. (2000), in order to take into account rotation and curvature effects.

The only difference of the SARC model with the original Spalart-Allmaras model is that the pro-duction term in the Spalart-Allmaras eddy viscosity transport equation is multiplied by the follow-ing function:

(3-21)

where cr1, cr2, and cr3 are model constants.

The variables and are derived from the strain rate and the vorticity tensors:

ST ν̃P ν̃( ) ν̃D ν̃( )–=

ν̃P ν̃( ) cb1Sν̃˜

=

ν̃D ν̃( ) cw1fwν̃d---

2=

S̃ Sfv3ν̃

κ2d2-----------fv2+=

fv21

1 χ cv2⁄+( )3

-------------------------------= fv31 χfv1+( ) 1 fv2–( )

χ--------------------------------------------=

fw

fw g1 cw3

6+

g6 cw36+

-------------------

16---

=

g r cw2 r6 r–( )+= r ν̃

S̃κ2d2--------------=

cw1 cb1 κ2⁄ 1 cb2+( ) σ⁄+= cw2 0.3= cw3 2= cv1 7.1= cv2 5=

cb1 0.1355= cb2 0.622= κ 0.41= σ 2 3⁄=

fr1 r∗ r̃,( ) 1 cr1+( )2r∗

1 r∗+-------------- 1 cr3arctg cr2 r̃( )–[ ] cr1–=

r∗ r̃

FINE™


FINE™

, (3-22)

where the strain rate and the vorticity tensors are defined as

(3-23)

(3-24)

and the remaining variables derived from those two tensors:

and and D2=0.5(S2+ ) (3-25)

In general, the modification will improve cases where rotation and curvature effects are important. In the current implementation, the temporal derivative (unsteady term) of the strain tensor is disre-garded.

3-2.4 k-ε Turbulence Models

3-2.4.1 General Formulation

In the turbulence model two additional transport equations for the turbulent kinetic energy, , and the turbulent dissipation rate, , are solved. In the FINE™/Turbo solver, 4 linear models are currently used.

In the following, the trace of the tensor X will be written {X}These two additional equations can be put in the following general form:

(3-26)

(3-27)

where is the mean strain tensor and the turbulent Reynolds stress tensor.

The variable is the modified dissipation rate

(3-28)

and the turbulent viscosity is given by the following relation

(3-29)

3-2.4.2 Linear ModelsIn the linear models, the turbulent Reynolds stress tensor is related to the mean strain tensor in a lin-ear way.

r̃ 2ωikSjkDSijDt

---------- εimnSjn εjmnSin+( )Ωm+ D4⁄= r∗ S

ω----=

Sij12---

∂ui∂xj-------

∂uj∂xi-------+

=

ωij12---

∂ui∂xj-------

∂uj∂xi-------– 2εmjiΩm+

=

S2 2SijSij= ω2 2ωijωij= ω

2

k ε– kε

ρk∂t∂

--------- ρwk μμtσk-----+ k∇–

∇•+ ρw″ w″⊗ S

– ρε–=

ρε̃∂t∂

--------- ρwε̃ μμtσε-----+ ε̃∇–

∇•+ 1T--- Cε1f1 ρw″ w″⊗ S

Cε2f2ρε̃+

–= E+

S ρw″ w″⊗–

ε̃

ε̃ ε D–=

μt

μt ρCμfμkT=

3-5


3-6

(3-30)

The implemented linear models are:

- Chien, low Reynolds number k-ε model (Chien, 1982).

- Extended wall functions (Hakimi, 1997) = Standard high Reynolds (Launder & Spalding, 1974) if the expert parameter INEWKE is set to 0.

- Launder-Sharma, low Reynolds number k-ε model (Launder & Sharma, 1974).

- Yang-Shih, low Reynolds number k-ε model (Yang & Shih, 1993).

The constants or functions Cμ, Cε1, Cε2, σk, σε, fμ,f1, f2, D, E and T are model dependent and are

defined in the Table 3-1 where and

TABLE 3-1 Coefficients of the k-ε modelsk-ε Model

Chien Standard high-Re

Launder & Sharma Yang & Shih

Cμ 0.09 0.09 0.09 0.09

Cε1 1.35 1.44 1.44 1.44

Cε2 1.80 1.92 1.92 1.92

σk 1.0 1.0 1.0 1.0

σε 1.3 1.3 1.3 1.3

fμ 1.0

x Rey

A 1.5 10-4

a 1

B 5 10-7

b 3

C 10-10

c 5

d 0.5

f1 1.0 1.0 1.0 1.0

f2 1.0 1.0

T

D 0. 0.

ρ– w″ w″⊗( )i j 2μt Sij23--- ∇w( )δij–

23---ρkδij–=

Sij12---

xj∂∂w̃i

xi∂∂w̃j+

=

Retk2

νε------= Rey

k0.5yν

-----------=

1 e 0.0115y+–– e

3.4–1 Ret( ) 50⁄+( )

2---------------------------------------

1 e Axa Bxb Cxc+ +( )––[ ]

d

1 0.22eRet

2 36⁄–– 1 0.3e

Ret2–

–

k ε̃⁄ k ε⁄ k ε̃⁄ k ε⁄ ν ε⁄( )0.5+

2νk y2⁄ 2ν k∇( )2

FINE™


FINE™

The constants Cμ, Cε1, Cε2, σk, σε can however be changed by the user (respectively the expert param-eters CMU, CE1, CE2, SIGK, SIGE) in the Computation Steering/Control variables page. The turbulent Prandtl number, needed to calculate the turbulent conductivity, is also an expert parameter (PRT).

Two variants of the high Reynolds k-ε with wall functions exist: •If the expert parameter INEWKE=10 (default), the wall functions are derived from DNS (Direct Numerical Simulation) curve fitting. The k-ε model is in that case derived from the Yang-Shih model. This new model, named extended wall functions, allows to obtain accurate results on fine mesh contrary to the standard high Reynolds k-ε model.

•if the expert parameter INEWKE=0, standard wall functions are applied and the y+

value of the near wall cell should be greater than 20.

3-2.5 k-ω Turbulence Models

3-2.5.1 Wilcox k-ω ModelThe turbulence model is a widely tested two-equation eddy viscosity model with integration to wall. This model was originated at about the same time and was developed in parallel with the

model as an alternative to define the eddy viscosity function. In the turbulence model two additional transport equations for the turbulent kinetic energy, , and the specific dissipation rate, , are solved.

The Wilcox model has been proven to be superior in numerical stability to the model primarily in the viscous sublayer near the wall. This model does not require explicit wall-damping functions as the model and other two-equation models due to the large values of in the wall regions. The numerical wall boundary conditions require the specification of the distance from the wall to the first point off the wall. In the logarithmic region, the model gives good agreement with experimental results for mild adverse pressure gradient flows.

In this model, the turbulent kinematic viscosity is defined as a function of the turbulent kinetic energy, , and the specific dissipation rate of the turbulent frequency, , as follows:

(3-31)

The two transport model equations for the k and scalar turbulence scales are defined below.

E 0.

kw 0./

DNS0. 0.

εw 0./

DNS0.

TABLE 3-1 Coefficients of the k-ε modelsk-ε Model

Chien Standard high-Re

Launder & Sharma Yang & Shih

2με̃e 0.5y+––

y2----------------------------- 2νμt S∇•( )

2 νμt S∇•( )2

uτ2 Cμ

0.5⁄

uτ3 κy( )⁄

2ν k∇( )2

k ω–

k ε– k ω–k

ω

k ω– k ε–

k ε– ω

k ω

νtkω----=

ω

3-7


3-8

(3-32)

(3-33)

where is the production rate of turbulence. The model constants are defined as:

, , , ,

3-2.5.2 Shear-Stress Transport (SST) ModelThe worst shortcoming of the Wilcox turbulent model is that its results are extremely sensi-tive to the small free stream value of in free-shear layer and adverse-pressure-gradient boundary layer flows. In order to solve this problem, a blended model has been proposed by Menter (1994) that allows to use Wilcox model near solid walls and the standard model, in a formulation, near boundary layer edges and in free-shear layers, as called BSL model. The BSL model uses the definition of the turbulence viscosity derived from the original model.

In order to blend the and the model, the latter is transformed into a formulation. The differences between this formulation and the original model are that an additional cross-diffusion term appears in the -equation and that the modelling constants are different. The origi-nal model is then multiplied by a function and the transformed model by the func-

tion and the corresponding equations of each model are added together. The function is designed to be a value of one in the half inner part of the boundary layer (where the model behaves like the original model) and decreases to vanish far from the wall.

The Shear-Stress Transport model (SST) is identical to the BSL model except that the values of the model constants and the definition of the turbulence viscosity. The SST model modifies the turbulence viscosity function to improve the prediction of separated flows and to avoid invariably overestimated Reynolds stresses with the and models in adverse pressure gradients. This modification is done on the basis of a Townsend-Lighthill design for the specification of turbu-lent eddy viscosity. Two-equation models generally under-predict the retardation and separation of the boundary layer due to adverse pressure gradients. This is a serious deficiency, leading to an underestimation of the effects of viscous-inviscid interaction which generally results in too optimis-tic performance estimates for aerodynamic bodies. The reason of this deficiency is that two-equa-tion models do not account for the important effects of transport of the turbulent stresses. The Johnson-King model (1985) has demonstrated that significantly improved results can be obtained with algebraic models by modelling the transport of the shear stress as being proportional to that of the turbulent kinetic energy. A similar effect is achieved in the present model by a modification in the formulation of the eddy viscosity using a blending function in boundary layer flows.

The turbulent kinematic viscosity is defined as the following function of the turbulent kinetic energy, k, and specific dissipation rate of turbulent frequency, :

(3-34)

where

DkDt------- Pk β′ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- αωk

----Pk β′ω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

+=

Pk

β' 0.09= α 5 9⁄= β 3 40⁄= σk 0.85= σω 0.5=

k ω–ω

k ω– k ε– k ω–k ω–

k ω– k ω–

k ω– k ε– k ω–k ω–

ωk ω– F1 k ε–

1 F1– F1

k ω–

k ω– k ε–

F2

ω

νta1k

max a1ω 2SF2,( )----------------------------------------------=

a1 0.31=

FINE™


FINE™

In turbulent boundary layers, the maximum value of the eddy viscosity is limited by forcing the tur-bulent shear stress to be bounded by the turbulent kinetic energy times . The auxiliary function,

, is defined as a function of wall distance d as:

(3-35)

The two transport equations of the model are defined below with a blend function for the model

coefficients of the original and model equations.

(3-36)

(3-37)

where is the production rate of turbulence. The last term on the right hand side of Eq. 3-37 is a cross diffusion term that only active outside of the boundary layer.

The constant . The variables , , , denoted with the generic symbol are

defined by blending the coefficients of the original model, denoted as , with those of the

transformed model, denoted as :

(3-38)

with the coefficients of the original models defined as:

• Inner model constants , , ,

•Outer model constants , , ,

The auxiliary blending function , designed to blend the model coefficients of the original

model in boundary layer zones with the transformed model in free-shear layer and free-stream zones, is defined as:

(3-39)

with

3-2.6 v2-f ModelThough they produces relatively good results for a number of flow aspects, the Low-Reynolds k-εturbulence models use damping functions that correct the improper asymptotic behaviour of the eddy-viscosity formulation close to solid walls. Insofar as the objective of this damping effect is to represent the kinematic blocking by the wall, Durbin (1991) suggests to use the turbulent stress nor-mal to streamlines instead of the turbulent kinetic energy in the definition of the turbulent viscosity. Therefore, he introduces a new v2-f turbulence model that does not require any damping function: v2 represents the turbulent stress normal to streamlines and f is a redistribution function.

a1F2

F2 max2 k

0.09dω------------------ 500ν

ωd2------------,

2

tanh=

F1ω ε

DkDt------- Pk β∗ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- γωk

----Pk βω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

2 1 F1–( )σω21ω---- ∂k

∂xj-------∂ω

∂xj-------+ +=

Pk

β∗ 0.09= γ β σk σω φ

k ω– φ1k ε– φ2

φ F1φ1 1 F1–( )φ2–=

γ1 0.5532= β1 0.075= σk1 0.85= σω1 0.5=

γ2 0.4403= β2 0.0828= σk2 1= σω2 0.856=

F1 k ω–

k ε–

F1 min maxk

β∗ωd-------------- 500ν

ωd2------------,

4ρσω2k

CDkωd2

--------------------, 4

tanh=

CDkω max 2ρσω21ω---- ∂k

∂xj-------∂ω

∂xj------- 1.0e 20–,

=

3-9

Turbulence Models Nonlinear Eddy Viscosity Turbulence Models

3-10

The present code-friendly variant of v2-f turbulence model derived from the one described in Lien and Kalitzin (2001). It is based on the high-Reynolds k-ε model with two additional equations for the turbulent stress normal to streamlines and the redistribution function:

(3-40)

(3-41)

(3-42)

(3-43)

The eddy viscosity is computed as where the turbulent time (T) and length (L) scales are defined by:

(3-44)

(3-45)

The terms that include the strain-rate magnitude (S) are linked to the realizability constraint intro-duce by Durbin (1995) in order to overcome a too high production close to stagnation points. The turbulent time and length scales are also bounded by the Kolgomorov scales.

The parameter used in this model and available through expert parameter in the Control Variables

page under Expert Mode are: Cμ = 0.22, σk = 1, σε = 1.3, Cε1 = , Cε2 = 1.9,

C1 = 1.4, C2 = 0.3, CL = 0.23, Cη = 70.

3-3 Nonlinear Eddy Viscosity Turbulence Models

3-3.1 EARSM ModelExplicit Algebraic RSM (EARSM) can be considered as a subset of nonlinear constitutive relations in which a part of the higher-order description of physical processes on the RSM-level is transferred into the two-equation modeling level. As a result, EARSM is much less demanding than RSM from the computational standpoint and, at the same time, is capable of reproducing some important fea-tures of turbulence (e.g. its anisotropy in the normal stresses), which are beyond the capabilities of linear eddy viscosity turbulence models.

DkDt------- Pk xj∂

∂ ννtσk-----+

∂k∂xj-------

+=

DεDt-------

Cε1Pk Cε2ε–T

--------------------------------xj∂∂ ν

νtσε-----+

∂ε∂xj-------

+=

Dv2

Dt--------- kf 6v2εk

--–xj∂∂ ν

νtσk-----+

∂v2

∂xj--------

+=

L2∇2f f–C1 1–( )

T-------------------- v

2

k---- 2

3---– C2

Pkk

----- 5T---v

2

k----––+=

νt Cμv2T=

T min max kε-- 6 ν

ε---,

0.6k6Cμv

2S----------------------,=

L CLmax mink3 2⁄

ε--------- k

3 2⁄

6Cμv2S

----------------------,

Cην3 4⁄

ε1 4⁄----------,=

1.4 1 0.05 kv2----

+

FINE™

Nonlinear Eddy Viscosity Turbulence Models Turbulence Models

FINE™

The EARSM model implemented in FINE™/Turbo is the simplified baseline explicit algebraic Reynolds stress model proposed by Menter et al. (2009), which allows the inclusion of anisotropic effects into the turbulence model.

In the EARSM model, the Reynolds-stress tensor is expressed using an effective eddy-viscosity

formulation including a corrective extra-anisotropy tensor :

(3-46)

where the effective turbulence eddy viscosity is defined as:

(3-47)

and the extra anisotropy tensor terms are defined as:

(3-48)

All the terms and coefficient of Eq. 3-47 and Eq. 3-48 are defined in the following equations as:

aijex( )

ρvi'vj'– μt xj∂∂vi

xi∂∂vj 2

3--- ∇ v⋅ δi j–+

23---ρkδij– ρkaij

ex( )–=

μt12---β1ρkτ–=

aijex( ) β3T3 i j, β4T4 ij, β6T6 i j,+ +=

T3 ij, ΩikΩkj13---ΩijΩjiδi j–=

T4 ij, SikΩkj ΩikSkj–=

T6 ij, SikΩklΩlj ΩikΩklSlj23---SikΩkjΩjiδij Ωi jΩjiSij––+=

β1NQ----–=

β32SikΩkjΩji

NQ1---------------------------–=

β41Q----–=

β6NQ1------–=

QN2 2ΩijΩj i–

A1-------------------------------=

Q1Q6---- 2N2 Ωi jΩji–( )=

N C'194--- 2β∗SijSji+=

Sijτ2---

xj∂∂vi

xi∂∂vj 2

3---δi j xk∂

∂vk–+

=

Ωi jτ2---

xj∂∂vi

xi∂∂vj–

=

3-11

Turbulence Models Nonlinear Eddy Viscosity Turbulence Models

3-12

The turbulence time scale, , is

(3-49)

The and equations write:

(3-50)

(3-51)

where the production term is

(3-52)

and the turbulent kinematic viscosity is:

(3-53)

The variable and the constant . The variables , , denoted with the

generic symbol are defined as:

(3-54)

with the coefficients:

• Inner model constants , ,

•Outer model constants , ,

The auxiliary blending function , designed to blend the model coefficients of the original

model in boundary layer zones with the transformed model in free-shear layer and free-stream zones, is defined as:

(3-55)

with

C'194--- C1 1–( )=

C1 1.8=

A1 1.245=

β∗ 0.09=

τ

τ max 1β∗ω---------- 6 ν

β∗kω-------------,

=

k ω

DkDt------- Pk β∗ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- γωk

----Pk βω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

2 1 F1–( )σω21ω---- ∂k

∂xj-------∂ω

∂xj-------+ +=

Pk v'iv'j∂ui∂xj-------–=

νtkω----=

γ ββ∗------

σωκ2

β∗-------------–= κ 0.41= β σk σω

φ

φ F1φ1 1 F1–( )φ2–=

β1 0.075= σk1 0.5= σω1 0.5=

β2 0.0828= σk2 1= σω2 0.856=

F1 k ω–

k ε–

F1 min maxk

β∗ωd-------------- 500ν

ωd2------------,

4ρσω2k

CDkωd2

--------------------, 4

tanh=

CDkω max 2ρσω21ω---- ∂k

∂xj-------∂ω

∂xj------- 1.0e 20–,

=

FINE™

Detached Eddy Simulations Turbulence Models

FINE™

3-4 Detached Eddy SimulationsWith steady RANS unable to meet the increasing demand for very detailed analysis of three-dimen-sional flows and the continuing advance in computing development on one side, the fact that, on the other side, for the foreseeable future, DNS and LES remain far too expensive for today’s engineer-ing applications, methods hybridizing LES and RANS such as Detached Eddy Simulation, origi-nally proposed by Spalart et al (1997), have appeared.

The model was proposed by Schur et al. (1999), based on Spalart-Allmaras model (1994).

Turbulent length scale (distance to the wall) is replaced by DES length scale .

(3-56)

where

is a measure of the width of the filter related to the calculation. is a calibration constant.

Recommended value for is 0.65. This model is often called DES97.

3-5 Laminar-Transition ModelThe boundary layer which develops on the surface of a solid body starts as a laminar layer but becomes turbulent over a relatively short distance known as the transition region. Four models are available in FINE™/Turbo to take into account the transition onset:

• Fully turbulent• Fully laminar• Forced transition•Abu-Ghannam and Shaw Model

The transition models can be used only with the Spalart-Allmaras, Yang-Shih k-epsilon, Wilcox k-omega and SST turbulence models.

3-5.1 Transition ModelsThe transition models consist of introducing a so-called intermittency, Γ, defined as the fraction of time during which the flow over any point on a surface is turbulent. It should be zero in the laminar boundary layer and one in fully developed turbulent boundary layer.

3-5.1.1 Fully TurbulentWhen the Fully Turbulent model is selected, the intermittency, Γ is set to 1 on the whole blade suction and/or pressure sides (fully developed turbulent boundary layer).

3-5.1.2 Fully LaminarWhen the Fully Laminar model is selected, the intermittency, Γ is set to 0 on the whole blade suc-tion and/or pressure sides (laminar boundary layer).

dW lDES

lDES min dW CDESΔ,( )=

Δ max Δx Δy Δz, ,( )=

Δ CDESCDES

3-13

Turbulence Models Laminar-Transition Model

3-14

3-5.1.3 Forced TransitionThe first step is to have a description of the blades in the FINE™/Turbo solver to be able to position the transition line. Therefore AutoGrid™ has been adapted to output the position of the leading and trailing edges and the topological patches composing the blade surfaces.

This last information is read by FINE™/Turbo and transmitted to the flow solver. A dialog box has been created to allow the positioning the transition line on the blade surfaces (Figure 3.5.1-1).

In Forced Transition mode, each transition line is positioned through a point on the hub and a point on the shroud. By default, these points are defined by a percentage of the arc length (Figure 3.5.1-1). However, a special treatment is also available where the transition points are defined by a percentage of the axial chord (only available for axial machines). This option can be activated via the expert parameter INTERL that should be set to 1 (default value = 0).

FIGURE 3.5.1-1 Example of the calculation of a transition point on a blade suction side as a percentage of the chord length.

The intermittency, Γ, is then computed on each point of the blade surface (it is set to 0 upstream of the transition line and tends towards 1 downstream). Each point of the computational domain has the same intermittency as the one at the closest point on the blade surface. However a special treat-ment is required close to the trailing edge and in tip gaps.

3-5.1.4 Abu-Ghannam & Shaw Model (AGS)The location of transition could be computed from the flow solution by using empirical relations related to external parameters. Here the correlations obtained by Abu-Ghannam and Shaw [1980] and derived from experimental data from transition on a flat plate with pressure gradients are used. According to these authors transition starts at a momentum thickness Reynolds number:

FINE™

Laminar-Transition Model Turbulence Models

FINE™

(3-57)

λθ is a dimensionless pressure gradient defined as:

(3-58)

where Ue is the velocity at the edge of the boundary layer, s is the streamwise distance from the leading edge, and θ denotes the momentum thickness in the laminar region.

τ is the free stream turbulence level (in %). For the Spalart-Allmaras turbulence model, τ is com-puted based on the upstream (inlet or rotor/stator) eddy viscosity and assuming a turbulent length scale of about 10% of the reference length scale:

(3-59)

For the Yang-Shih k-ε turbulence model, τ is computed by using the following definition:

(3-60)

The function F(λθ ) depends on the sign of the pressure gradient:

for adverse pressure gradient (λθ < 0)

for a favourable pressure gradient (λθ > 0)

Therefore, according to these relations, transition is promoted in adverse pressure gradient whereas it is retarded in favourable pressure gradient.

The range of application of the AGS correlation is 0.1 > λθ > -0.1, and for a free stream turbulent level ranging from 0.3 to 10%.

3-5.2 Intermittency Distribution

3-5.2.1 Spalart-Allmaras model:The intermittency function is defined on each point of the blade surface and at each iteration. It is extended to the computational domain by a simple orthogonal extension (i.e., each point of the computational domain has the same intermittency than the closest point on the blade surface).

Either using the Forced Transition option or the Abu-Ghannam and Shaw (AGS) Model, the stream-wise evolution of intermittency should be defined. It could be either a binary field or a smoother field. In the current implementation those two options are available. It is controlled by the expert parameter INTERI. With its default value (INTERI = 0) the intermittency is 0 before transi-tion and 1 after the transition onset. If INTERI = 1, intermittency is defined following the relation of Dhawan and Narasimha [1958]:

(3-61)

(3-62)

Rθs 163 F λθ( )F λθ( )6.91

--------------τ– exp+=

λθθ2

ν-----

dUeds

---------=

τ 86νt

ULref-------------=

τ 100U--------- 2

3---k=

F λθ( ) 6.91 12.75λθ 63.64 λθ( )2+ +=

F λθ( ) 6.91 2.48λθ 12.27 λθ( )2–+=

Γ 1 0.412ξ2–( )exp–=

ξ 1λ---Max s st 0,–( )=

3-15

Turbulence Models Laminar-Transition Model

3-16

where st is the position of the transition onset and s is the current position on the arc from leading edge to trailing edge. λ is the characteristic extent of the transition region and is determined from the correlation:

(3-63)

Close to the trailing edge the intermittency is set to 1. This is necessary in order to allow a turbulent wake to be generated downstream of the blade. This special treatment is controlled by the expert parameter FTURBT that is set to 0.95 by default (i.e., only the last 5% of a blade side will have an intermittency of 1). In order to get a tri-dimensional distribution of the intermittency, the intermit-tency of a cell is the one of the closest cell on a solid surface (hub, shroud, and tip gap are fully tur-bulent and have an intermittency equal to unity).

3-5.2.2 Yang-Shih k-ε, Wilcox k-ω and SST models:The intermittency function is defined on each point of the blade surface and at each iteration. Then it is extended to the computational domain. This is done by comparing the distance to blade with the momentum thickness θ computed by Thwaites relation and multiplied by expert parameter BLFACT (default value is 5). If the distance to blade is smaller than (BLFACT*θ) the intermittency of the current cell will be set to the intermittency of the closest point on the blade. Otherwise the intermittency will be set to unity. Near the leading edge the distance on which the intermittency is applied is computed in the same way but the θ value is taken at the position corresponding to the expert parameter FTRAST to avoid insignificant distance.

The definition of the stream-wise evolution of intermittency is the same as that used by the Spalart-Allmaras turbulence model.

3-5.3 Modification of the turbulence models

3-5.3.1 The Spalart-Allmaras turbulence model:The intermittency is used to multiply the turbulence production term in the Spalart-Allmaras turbu-lence model in Eq. 3-14. This gives:

(3-64)

where is the production term. Free-stream wakes (with only a weak production term) will not be affected by the intermittency. A laminar boundary layer will be preserved upstream of the transition location and turbulence could freely develop thereafter.

3-5.3.2 The Yang-Shih k-ε turbulence model:In the original Yang-Shih k-ε turbulence model (Eq. 3-26, Eq. 3-27), three terms (the turbulent kinetic energy production term, the turbulent kinetic dissipation term and the dissipation rate pro-duction term) need to be modified. Three modifications have been implemented. This is controlled by the expert parameter KETRAN:

When KETRAN is equal to 1:

(3-65)

Reλ 9Rest0.75=

S̃ ΓS̃=

S̃

ρk∂t∂

--------- ρwk( )∇•+ Γ μμtσk-----+ k∇

∇• ρw″ w″⊗ S– ρε–=

FINE™

Laminar-Transition Model Turbulence Models

FINE™

(3-66)


(3-67)

(3-68)


(3-69)

(3-70)

3-5.3.3 The Wilcox k-ω turbulence model:In the original Wilcox k-ω turbulence model (Eq. 3-32, Eq. 3-33), three terms (the turbulent kinetic energy production term, the turbulent kinetic dissipation term and the specific dissipation rate pro-duction term) need to be modified. Three modifications have been implemented. This is controlled by the expert parameter KETRAN:


(3-71)

(3-72)


(3-73)

(3-74)


(3-75)

(3-76)

ρε̃∂t∂

--------- ρwε̃( )∇•+ μμtσε-----+ ε̃∇

∇• 1T--- Cε1f1 ρw″ w″⊗ S

Cε2f2ρε̃+

–= E+

ρk∂t∂

--------- ρwk( )∇•+ Γ μμtσk-----+ k∇

∇• ρw″ w″⊗ S– Γρε–=

ρε̃∂t∂

--------- ρwε̃( )∇•+ μμtσε-----+ ε̃∇

∇• 1T--- Cε1f1 ρw″ w″⊗ S

Cε2f2ρε̃+

–= E+

ρk∂t∂

--------- ρwk( )∇•+ Γ μμtσk-----+ k∇

∇• ρw″ w″⊗ S– ρε–=

ρε̃∂t∂

--------- ρwε̃( )∇•+ Γ μμtσε-----+ ε̃∇

∇• 1T--- Cε1f1 ρw″ w″⊗ S

Cε2f2ρε̃+

–= E+

DkDt------- ΓPk β′ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- αωk

----Pk β′ω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

+=

DkDt------- ΓPk Γβ′ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- αωk

----Pk β′ω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

+=

DkDt------- ΓPk β′ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- Γαωk

----Pk β′ω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

+=

3-17

Turbulence Models Modelling Flow near the Wall

3-18

3-5.3.4 The SST turbulence model:In the original SST turbulence model (Eq. 3-36, Eq. 3-37), three terms (the turbulent kinetic energy production term, the turbulent kinetic dissipation term and the specific dissipation rate production term) need to be modified. Three modifications have been implemented. This is controlled by the expert parameter KETRAN:


(3-77)

(3-78)


(3-79)

(3-80)


(3-81)

(3-82)

3-6 Modelling Flow near the Wall The wall functions are used to mimic the presence of the walls by reflecting the effect of the steep, non-linear variations of the flow properties through the turbulent boundary layer. They define the shear stress and the heat flux on the cell faces lying on solid boundaries as well as the values of the turbulent variables in the vicinity of the wall.

For the models that do not use wall functions, the profile of the boundary layer is directly inferred from the input values at the wall. Consequently, particular attention must be paid to the grid refine-ment in this region in order to capture the viscous sublayer. A typical value is given by at the first node from the wall.

3-6.1 Velocity

3-6.1.1 Smooth wallsThe wall function in the viscous sublayer is given by:

DkDt------- ΓPk β∗ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- γωk

----Pk βω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

2 1 F1–( )σω21ω---- ∂k

∂xj-------∂ω

∂xj-------+ +=

DkDt------- ΓPk Γβ( )∗ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- γωk

----Pk βω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

2 1 F1–( )σω21ω---- ∂k

∂xj-------∂ω

∂xj-------+ +=

DkDt------- ΓPk β∗ωk–

∂∂xj------- ν σkνt+( )

∂k∂xj-------

+=

DωDt-------- Γγωk

----Pk βω2– ∂

∂xj------- ν σωνt+( )

∂ω∂xj-------

2 1 F1–( )σω21ω---- ∂k

∂xj-------∂ω

∂xj-------+ +=

1 y+ 10<

Modelling Flow near the Wall Turbulence Models

FINE™

(3-83)

with , the dimensionless normal distance from the walls and , the wall fric-

tion velocity directly related to the wall shear stress.

While in the turbulent layer, the wall function becomes:

(3-84)

where κ is the Von Karman constant (default value 0.41) and Bo another constant (default value 5.36). Both κ and Bo can be modified by the user in the Boundary Condition page.

3-6.1.2 Rough walls:

a) Old Method (IROUGH=0)The wall function is given by (no viscous sublayer considered):

(3-85)

where κ is the Von Karman constant (default value 0.41). For rough walls, the constant Bo is usu-

ally zero (when ks+ > 70 with ). Otherwise the value of Bo is evolving as presented in

below graph.

k0 and d0 are constants asked for each rough wall. The constant k0 is the equivalent roughness height:

(3-86)

where ks is the average roughness height.

The constant d0 is known as the height of the zero displacement plane, defined by:

if Bo=0 (3-87)

uuτ----- y+=

y+yuτν

--------= uτ τwall ρ⁄=

uuτ----- 1

κ--- y+ln Bo+=

uuτ----- 1

κ---

y d0–k0

--------------ln Bo+=

ks+ uτks

ν----------=

s+s+

k0ks30------=

u k0 d0+( ) 0=

3-19


3-20

For small roughness elements, . For high roughness elements, the flow behaves as if the wall was located at a distance d0 from the real wall position.

b) New Method (IROUGH=1)Using the old method, seen above, the user has to give the constant Bo as input parameters. How-ever, the value of the constant Bo is depending on the value of the wall friction velocity . And the value of Bo will also correct the wall friction velocity . The user has to iterate this procedure and converge to the desired value of Bo. This is not straightforward and a new approach has been intro-duced in the flow solver in order to simplify the computations with roughness.

For the new method, the wall function is given by (no viscous sublayer considered):

(3-88)

where κ is the Von Karman constant (default value 0.41) and Bo another constant (default value 5.36). Both κ and Bo can be modified by the user in the Boundary Condition page.

McKeon et al. (2004) have shown the is a function of with , as shown in

Figure 3.6.1-2. The function has a special behaviour in the range . Below =3, the

flow is considered as smooth and there is no modification of the classical log-law ( = 0). For

>70, the flow is considered as fully rough and shows logarithmic increase with increasing

.

FIGURE 3.6.1-2Roughness function

A function is used to interpolate in the flow solver, seeing the Rough function fitting curve in Figure 3.6.1-2:

(3-89)

d0 0=

uτuτ

uuτ----- 1

κ---

y d0–ν

--------------uτ ln Bo U+Δ–+=

U+Δ ks+ ks

+ uτksν

----------=

ks+ 3 70–[ ]∈ ks

+

U+Δ

ks+ U+Δ

ks+

U+Δ

U+Δ 1κ--- ks

+ Bo*– ea1ks

+ a2+( )+ln=

FINE™


FINE™

where a1=-0.7649, a2=1.9 and is a constant.

The new method has the advantage that no more iteration has to be done on the constant Bo.

3-6.2 TemperatureSimilarly, for isothermal walls, the following laws are considered in the code:

in the viscous sublayer (3-90)

where Pr is the laminar Prandtl number and:

in the turbulent layer (3-91)

where .

The heat flux at the wall is defined as:

(3-92)

A complete description of the available thermal boundary conditions at walls is provided in section 4-4.

The "law of the wall" distributions of velocity, temperature and other variables are assumed to prevail across the boundary layer. They are imposed at the node of a single grid cell for which the user is advised to check that . As a result, if the first

cell node from the wall is placed too close, calculation is conducted in the viscous sub-layer and the law is no longer valid. Similarly, if the first node is placed too far away, then a discrepancy may exist between the profiles and their assumed shape.

3-6.3 Turbulence

3-6.3.1 Standard Wall function for k-ε modelThe values of k and ε in the first inner cell are imposed based on the value of the wall friction veloc-ity :

(3-93)

(3-94)

where Cμ = 0.09 and κ is the Von Karman constant (default value 0.41).

3-6.3.2 Extended Wall Function for k-ε modelFor the extended wall functions model, the wall functions for k and ε are fitted with polynomials to the DNS data.

Bo* 8.5 Bo–=

T+ Pr y+=

T+Prtκ

------- y+

13.2---------- 13.2Pr+ln=

T+Tw T–

Tτ----------------=

qw ρcpuτTτ=

20 y+ 100< <

uτ

kuτ

2

Cμ-----------=

εuτ

3

κy------=

3-21


3-22

3-6.3.3 Extended Wall Function for Spalart-Allmaras model

The turbulent working variable in the first inner cell is imposed based on the value of the wall friction velocity :

(3-95)

with , and κ is the Von Karman constant (default value 0.41).

This relation can be rewritten as:

(3-96)

and holds true in the viscous layer and in the log layer.

3-6.3.4 Extended Wall Function for SST modelThe value of k in the first inner cell is obtained from the tabulated values already used for the k-εmodel. ω in the first inner cell can be imposed based on the value of the wall friction velocity ,

in the sublayer (3-97)

in the log layer (3-98)

where β1=0.075, Cμ=0.09 and κ is the Von Karman constant (default value 0.41).

In the intermediate region, is approximated with the form of interpolation between the viscous and log layer value:

(3-99)

with

References:

Ashford G.A., and Powell K.G., 1996, "An unstructured grid generation and adaptative solution technique for high-Reynolds number compressible flow", VKI (Von Karman Institute) Lecture Series 1996-06.

Baldwin B., Lomax H., 1978, "Thin Layer Approximation and Algebraic Model for Separated Tur-bulent Flows", AIAA-78-257.

Chien K.Y., 1982, "Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model", AIAA J., Vol. 20, No. 1.

Durbin P.A., 1991, "Near-wall turbulence closure modeling without damping function", Theor. Comput. Fluid Dynamics, Vol. 3, No. 1, pp 1-13.

Durbin P.A., 1995, "Separated flow computations with the k-e-v2 model". AIAA J., Vol. 33, No. 4, pp 659-664.

ν̃uτ

ν̃+

κy+=

ν̃+ ν̃

ν---= y+

yuτν

--------=

ν̃ κyuτ=

uτ

ωvis+ 6

β1 y+( )

2-------------------=

ωlog+ 1

κ Cμy+

--------------------=

ω+

ω+ ωvis+( )

2ωlog

+( )2

+=

ω+ ων

uτ2

-------=

FINE™


FINE™

Jameson A., Schmidt W., Turkel E., 1981, "Numerical Solutions of the Euler Equations by Finite Volume Methods using Runge-Kutta Time-Stepping Schemes", AIAA-81-1259.

Jameson A., 1995, "Positive schemes and shocks modelling for compressible flows", International Journal for Numerical Methods in Fluids, Vol 20, pp 773-776.

Johonson J. and King L., 1985, "A mathematically simple turbulence closure model for attached and separated turbulent boundary layers", AIAA Journal, vol.23, pp.1684-1692.

Kalitzin G., Medic G., Iaccarion G., Durbin P., 2005, "Near-Wall Behavior of RANS Turbulence Models and Implications for Wall Functions", Journal of Computational Physics, Vol.204, pp265-291.

Launder B.E. and Sharma B.I., 1974, "Application of the Energy-Dissipation Model of turbulence to the Calculation of Flow Near a Spinning Disc", Letters in Heat and Mass Transfer, Vol. 1, pp. 131-138.

Launder B.E. and Spalding D.B., 1974, "The numerical computation of turbulent flow", Comput. Methods App. Mech. Eng., vol. 3, pp. 269-289.

Lien F.S. and Kalitzin G., 2001, "Computations of transonic flow with the v2-f turbulence model", International Journal of Heat and Fluid Flow 22, 53-61.

McKeon B.J., Li J., Jiang W., Morrison J.F., and Smits A.J., 2004, "Further observations on the mean velocity distribution in fully developed pipe flow", J. Fluid Mech., Vol.501:135-147.

Mehdizadeh O., et al, 2012, "Applications of EARSM turbulence models to internal flows", Pro-ceedings of ASME Turbo Expo 2012, GT2012-68886, Copenhagen, Denmark.

Menter F., 1992, "Influence of freestream values on turbulence model predictions", AIAA J., Vol. 30, No. 6, pp1657-1659.

Menter F., 1993, "Zonal two-equation turbulence model for aerodynamic flows", in AIAA 24th Fluid Dynamics Conference, AIAA paper 93-2906, (Orlando, FL).

Menter F., 1992, "Performences of popular turbulence models for attached and separated adverse pressure gradient flows", AIAA Journal, vol.30, pp.2066-2072.

Menter F., 1994, "Two-equation eddy viscosity turbulence models for engineering applications", AIAA Journal, vol.32, pp.1299-1310.

Menter F., et al, 2009, "Explicit algebraic Reynolds stress models for anisotropic wall-bounded flows", in EUCASS – 3rd European Conference for Aero-Space Sciences, July 6-9th, Versailles.

Nichols R.H., 1990, "A Two-Equation Model for Compressible Flows", AIAA paper 90-0494.

Piquet F., 1999, "Turbulent flow: Models and Physics", Berlin, Heilderberg, New York: Springer-Verlag.

Sarkar S., Erlebacher G., Hussaini M.Y., Kreiss H.O., 1991, "The analysis and modelling of dilata-tional terms in compressible turbulence", J Fluid Mech, Vol.227, pp.473-493.

Shur M. L., Strelets M. K., Travin A. K., Spalart P. R., 2000, "Turbulence Modeling in Rotating and Curved Channels: Assessing the Spalart-Shur Correction", AIAA Journal, Vol 38:784-792.

Spalart P.R. and Allmaras S. R., 1992, "A one equation turbulence model for aerodynamic flows", AIAA 92-0439.

Spalart P.R. and Shur M. L., 1997, "On the sensitisation of turbulence models to rotation and curva-ture", Aerospace Science and Technology, No 5:297-302.

Vandromme D., 1991, "Turbulence Modelling for Compressible Flows and Implementation in Navier-Stokes Solvers", VKI Lecture Series 1991-02.

k ω–

k ω–

3-23


3-24

Wilcox D., 1988, "Reassessment of the scale-determining equation for advanced turbulence mod-els", AIAA Journal, vol.26, pp.1299-1310.

Wilcox D., 1993, "Turbulence model for CFD", tech. rep., DCW Industries, Inc., 5354 Palm Drive, La Canada, Calif.

Yang Z. and Shih T.H., 1993, "A k-e model for turbulence and transitional boundary layer", Near-Wall Turbulent Flows, R.M.C. So., C.G. Speziale and B.E. Launder(Editors), Elsevier-Science Pub-lishers B. V., pp. 165-175.

Zhang H.S., So R.M.C., Speziale C.G., Lai Y.G., 1991, "A Near-Wall Model for Compressible Tur-bulent Flow", ICASE Report 91-82.

FINE™

CHAPTER 4: Boundary Conditions

FINE™

4-1 OverviewThis chapter describes all the theories related to the boundary conditions available in the FINE™/Turbo solver:

• Inlet Boundary Conditions•Outlet Boundary Conditions• Solid Wall Boundary Conditions• Far-field Boundary Condition•Unsteady Inlet and Outlet Boundary Conditions

4-2 Inlet Boundary Conditions

4-2.1 Cylindrical Inlet Boundary Conditions

4-2.1.1 Static Quantities Imposed (Subsonic)The complete absolute velocity vector followed by the static temperature on the boundary are spec-ified. Several possibilities exist:

• the magnitude of the velocity and the flow angles α and γ are specified:

(4-1)

• the magnitude of the velocity and other flow angles δ and ε are specified:

αvθvz-----atan=

γvrvz----atan=

4-1

Boundary Conditions Inlet Boundary Conditions

4-2

(4-2)

with the meridional velocity ( ). This option has been designed to allow radial inlet with zero axial velocities,• the cylindrical velocity components are specified.

The static pressure is extrapolated from the interior. Using subscript 0 for the boundary, and sub-script 1 for the first internal cell one has:

(4-3)

which allows to calculate all the primitive variables on the boundary.

These conditions can also be used for incompressible flows.

4-2.1.2 Total Quantities Imposed (Subsonic)Several variants are available based on the specification of flow angles or velocity components.

a) Absolute flow angles from axial direction:The user specifies two flow angles (of the absolute flow) at the boundary, α and γ, which are defined by Eq. 4-1 and given in radians.

The absolute total pressure and the absolute total temperature on the inlet boundary are also speci-fied and imposed.

Three variants are available depending on the value that is extrapolated.

• The module of the absolute velocity vector is extrapolated from the interior field. Using sub-script 0 for the boundary, and subscript 1 for the first internal cell:

(4-4)

where v represents the absolute velocity.

• the axial velocity component is extrapolated:

(4-5)

Note that it is assumed that the angular velocity vector is in the z-direction,

(4-6)

which makes this direction the axial one. Also note that, as a result of Eq. 4-6 the absolute and relative axial velocity component are equal.• the mass flow is extrapolated, assuming that the meridional component of the velocity is the

one contributing to the mass flow.

δvrvm-----acos=

εvtvm-----atan=

vm vm vr2 vz

2+=

p0 p1=

v0 v1=

vz( )0 vz( )1=

ω ω 1z⋅=

FINE™

Inlet Boundary Conditions Boundary Conditions

FINE™

(4-7)

with (4-8)

This boundary condition is specially dedicated to radial turbomachinery for which a problem of mass flow conservation can arise.

b) Relative flow angles from axial directionFor a stator calculation it may be convenient to impose the velocity angles of the relative flow at the exit of the upstream rotor. In case of rotors the user may also prefer to impose relative boundary conditions. This procedure is only available in case total conditions and flow angles are imposed. In addition to that the extrapolation of vz must be selected. Note also that both flow angles and total conditions are then treated in the relative mode.

The adopted approach consists of imposing boundary conditions, which are still defined in the absolute frame of reference, but which vary from one iteration to

theoretical manual · 1-4 fine™ 1-3.2 time averaging of navier-stokes equations in principle, the...

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