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3.5 Sub-group 1.5, in which four differential equations are used


3.5.1 TWO-SCALE KE-EP TURBULENCE MODEL

Contents

  1. Introduction
  2. Description of the model
  3. Boundary conditions
  4. Activation of the model
  5. Sources of further information

1. Introduction


The dissipation rate EP in the KE-EP model can be regarded as the rate at which energy is being transferred across the spectrum from large to small eddies.

The standard KE-EP model assumes spectral equilibrium, which implies that, once turbulent energy is generated at the low-wave-number end of the spectrum (large eddies), it is dissipated immediately at the same point at the high-wave-number end (small eddies).

In general, this is not the case, because there is a vast size disparity between those eddies in which turbulence production takes place, and the eddies in which turbulence dissipation occurs.

In some flows there is an appreciable time lag between the turbulent production and dissipation processes, during which time the large- scale turbulence is continually being broken down into finer and finer scales.

Hanjalic and co-workers ( Hanjalic and Launder [1978], Hanjalic et al [1979] ) proposed a two-scale model in which the turbulence- energy spectrum is divided into two parts, roughly at the wave number above which no mean-strain production occurs.

The first part is termed the 'production' region and the second part the 'transfer' region. Spectral equilibrium is assumed between the transfer region and the region in which turbulence is dissipated.

The total turbulent energy, KE, is assumed to be divided between the production region (KP) and the transfer region (KT).

Two transport equations are employed to describe the rate of change of turbulence energy associated with each of the two regions.

The closure of these equations is accomplished by defining EP as the rate of energy transfer out of the production region, so that EP serves as a sink in KP and as a source of KT, while the dissipation rate ET defines the sink of KT.

The assumption of spectral equilibrium between the transfer and dissipation regions means that ET is the dissipation rate.

Hence, four turbulence parameters, KP, KT, EP and ET are used to characterise the production and dissipation processes.

Successful applications of the foregoing two-scale simplified split- spectrum model have been reported by Hanjalic et al [1978,1980], Fabris et al [1981] and Chen [1986].

A generalisation of the model for a multiple split-spectrum case has been reported by Schiestel [1983, 1987].

The two-scale KE-EP model provided in PHOENICS is also based on a simplified split-spectrum, but it employs the proposal of Kim and Chen [1989] for variable partitioning of the turbulent kinetic- energy spectrum.

This model is based on the work of Hanjalic et al, but differs significantly from it in the details of the modelling.

The main feature of this model is that it does not employ a fixed ratio of KP/KT to partition the turbulent kinetic-energy spectrum; instead, variable partitioning is used in such a way that the partition is moved towards the high-wave-number end when production is high and towards the low-wave-number end when production vanishes

The location of the partition ( the ratio KP/KT ) is determined as a part of the solution, and the method causes the effective eddy viscosity coefficient to decrease when production is high and to increase when production vanishes.

The advantage of the two-scale KE-EP model lies in its capability to model the cascade process of turbulent kinetic energy and its capability to resolve the details of complex turbulent flows ( such as separating and reattaching flows ) better than the standard KE-EP model ( see Kim and Chen [1989] and Kim [1990b, 1991] ).

A low-Reynolds-number extension of the model has been reported by Kim [1990a], but this has yet to be incorporated into PHOENICS.

2. Description of the model

In this model the total turbulence energy, KE, is divided equally between the production range and transfer range, thus KE is given by

KE = KP + KT (2.1)

where KP is the turbulent kinetic energy of eddies in the production range and KT is the energy of eddies in the dissipation range.

For high turbulent Reynolds numbers, the transport equations for the turbulent kinetic energies are:

(RHO*KP),t + (RHO*U,i*KP - {RHO*ENUT/PRT(KP)}*KP,i ),i =

RHO*(Pk - EP) (2.2)

(RHO*KT),t + (RHO*U,i*KT - {RHO*ENUT/PRT(KT)}*KT,i ),i =

RHO*(EP - ET) (2.3)

wherein; EP is the transfer rate of turbulent kinetic energy from the production range to the dissipation range; ET is the dissipation rate; RHO is the fluid density; ENUT is the turbulent viscosity; PK is the volumetric production rate of turbulent kinetic energy; and PRT(KP) and PRT(KT) are constant coeffcients.

The corresponding transport equations for the energy transfer rate and the dissipation rate are given by:

(RHO*EP),t + (RHO*U,i*EP - {RHO*ENUT/PRT(EP)}*EP,i ),i =

RHO*(CP1*Pk*Pk/KP + CP2*Pk*EP/KP - CP3*EP*EP/KP)

(2.4)

(RHO*ET),t + (RHO*U,i*ET - {RHO*ENUT/PRT(ET)}*ET,i ),i =

RHO*(CT1*EP*EP/KT + CT2*EP*ET/KT - CT3*ET*ET/KT)

(2.5)

where PRT(EP), PRT(ET), CP1, CP2, CP3, CT1, CT2 and CT3 are constant model coefficients. The CP1*PK*PK/KP and CT1*EP*EP/KT terms can be interpreted as variable energy transfer functions. The former term increases the energy transfer rate when production is high, and the second term increases the dissipation rate when the energy transfer rate is high.

The model constants are given as: PRT(KP)=0.75, PRT(EP)=1.15, PRT(KT)=0.75, PRT(ET)=1.15, CP1=0.21, CP2=1.24, CP3=1.84, CT1=0.29, CT2=1.28 and CT3=1.66.

The eddy viscosity is computed from:

ENUT = CMUCDF*KE**2/EP = CMUCD*KE**2/ET (2.6)

where CMUCD=CMUCDF*ET/EP is the effective eddy-viscosity coefficient and CMUCDF=0.09. For turbulent flows in equilibrium, Pk=ET and ET=EP so that CMUCDF=CMUCD. In this case the location of the partition is in the high-wave-number region. When Pk > EP > ET, CMUCD <CMUCDF and the partition is located in a higher wave-number region than that of the equilibrium case. When the production vanishes, KP < KT, EP CMUCDF and the partition is located in the low- wave-number region. For arbitrary ratios of Pk/ET, the partition is determined as part of the solution.

3. Boundary Conditions

Wall boundary conditions


The model may be used in combination with equilibrium (GRND2) or non-equilibrium (GRND3) wall functions ( see the Encyclopaedia entry WALL Functions ). For equilibrium wall functions, the following boundary conditions are applied for the turbulence variables:

KP = KE/(1.+BETA) (3.1)

KT = BETA*KE/(1.+BETA) (3.2)

EP = (CMUCD)**0.75*KE**1.5/(k*Y) (3.3)

ET = EP (3.4)

Here KE= UTAU**2/SQRT(CMUCD), UTAU is the resultant friction velocity ( = SQRT(TAUW/RHO) ), TAUW is the wall shear stress, Y is the normal distance of the first grid point from the wall, k is von Karman's constant, and BETA is given by:

BETA = K**2/(PRT(EP)*SQRT(CMUCD)/(CP3-CP1-CP2) - 1. (3.5)

which yields a value of BETA=0.25.

For the non-equilibrium wall functions, the treatment is similar to that employed for the standard KE-EP model, with KP and KT determined from their respective balance equations and for simplicity it is assumed that EP=ET.

Inlet conditions


At mass-inflow boundaries, the inlet values of KE and ET are usually unknown, and one needs to take guidance from experimental data for similar flows. The simplest practice is to assume uniform values of KE and ET computed from:

KE = (I *U)**2; ET = (CMUCDF)**0.75*KE**1.5/LM (3.6)

where U is the bulk inlet velocity, I is the turbulent intensity (typically in the range 0.01 .lt.I.lt. 0.05) and LM ~ 0.1H, where H is a characteristic inlet dimension, say the hydraulic radius of the inlet pipe.

The turbulence may be assumed to be in equilibrium ( production ~ dissipation ), so that the inlet values of KP, KT and EP are given by equations (3.1), (3.2) and (3.4) respectively, with BETA=0.25.

Free-stream conditions


At free (entrainment) boundaries, where a fixed-pressure condition is employed, it is necessary to prescribe free stream values for KP, KT, EP and ET. If the ambient stream is assumed to be free of turbulence, then KE and ET can be set to negligibly small values. However, it is advisable to ensure that these values are such that ENUT ~ ENUL and that KT/KP=N and ET/EP=N+1 where N=1.5 as given by the decay of homogeneous free-stream turbulence. Thus, for a given free-stream value of KE, one uses:

EP=CMUCDF*KE**2/ENUL ; KP=KE/(N+1) ; KT=N*KE/(N+1) (3.7)

wherein the free-stream value of ENUT has been taken as ENUL.

4. Activation of the model

The 2-scale KE-EP model is selected by TURMOD(TSKEMO) which is equivalent to the following PIL commands:


         STORE(KE); SOLVE(EP,KT,ET,KP)

         OUTPUT(KE,Y,N,N,Y,Y,Y);PRT(KP)=0.75;PRT(EP)=1.15

         PRT(KT)=0.75;PRT(ET)=1.15;GENK=T;IENUTA=7;

         ENUT=GRND5;EL1=GRND4

         PATCH(TSKESO,PHASEM,1,NX,1,NY,1,NZ,1,1)

         COVAL(TSKESO,KP,GRND5,GRND5); COVAL(TSKESO,EP,GRND5,GRND5)

         COVAL(TSKESO,KT,GRND5,GRND5); COVAL(TSKESO,ET,GRND5,GRND5)

         VARMIN(KE)=1.E-10;VARMIN(KT)=1.E-10;VARMIN(KP)=1.E-10

         VARMIN(ET)=1.E-10

When STORE(KTKP,ETEP) appears in the Q1 file, the ratios KT/KP and ET/EP are written to the PHI ( and RESULT ) file, so that they may be viewed via PHOTON and AUTOPLOT.

The FORTRAN coding sequences for the 2-scale model may be found in Subroutine GXTSKE of the file GXKE.FOR, and also in the file GXWALL.FOR.

The WALL and CONPOR commands create the required COVAL settings automatically, as follows:

COVAL(WALLN,KP,GRND2,GRND2); COVAL(WALLN,KT,GRND2,GRND2) COVAL(WALLN,ET,GRND2,GRND2); COVAL(WALLN,EP,GRND2,GRND2)

or


     COVAL(WALLN,KP,GRND3,GRND3)

     COVAL(WALLN,ET,GRND3,GRND3); COVAL(WALLN,EP,GRND3,GRND3)

if WALLCO=GRND3.

Finally, a number of Q1's may be found in the advanced-turbulence- models input library which demonstrate the use of the model.

5. Sources of further information

C.P.Chen, 'Multiple-scale turbulence model in confined swirling- jet predictions', AIAA J., Vol.24, pp1717, (1986).

G.Fabris, P.T.Harsha and R.B.Edelman, 'Multiple-scale turbulence modelling of boundary-layer flows for scramjet applications', NASA-CR-3433, (1981).

S.W.Kim and C.P.Chen, 'A multi-time-scale turbulence model based on variable partitioning of the turbulent kinetic energy spectrum', Numerical Heat Transfer, Part B, Vol.16, pp193, (1989).

S.W.Kim, 'Near-wall turbulence model and its application to fully- developed turbulent channel and pipe flows', Numerical Heat Transfer, Part B, Vol.17, pp101, (1990a).

S.W.Kim, 'Numerical investigation of separated transonic turbulent flows with a multiple-time-scale turbulence model', Numerical Heat Transfer, Part A, Vol.18, pp149, (1990b).

S.W.Kim, 'Calculation of divergent channel flows with a multiple- time-scale turbulence model', AIAA J., Vol.29, pp547, (1991).

K.Hanjalic and B.E.Launder,'Turbulent transport modelling of separating and reattaching shear flows', Mech.Eng.Rept. TF/78/9, University of California, Davis, USA, (1978).

K.Hanjalic, B.E.Launder and R.Schiestel,'Multiple time-scale concept in turbulent transport modelling', In Turbulent Shear Flows II, Springer Verlag, p36, (1980).

R.Schiestel, 'Multiple-scale concept in turbulence modelling, II Reynolds stresses and turbulent heat fluxes of a passive scalar, algebraic modelling and simplified model using Boussinesq Hypothesis', J.Mech.Theor.Appl., Vol.2, pp601, (1983).

R.Schiestel, 'Multiple-time-scale modelling of turbulent flows in one-point closure', Phys.Fluids, Vol.30, pp722, (1987).

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