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4.1.3 Reynolds-flux model equations

Mean Enthalpy Equation via H1

PHOENICS provides for the solution of the enthalpy equation in terms of the enthalpy H1 or the temperature TEM1. For high-Reynolds-number flows, the H1 equation takes the following form when the Reynolds stress model is activated:

(rho*H1),t + (rho*Uj*H1),j = - (rho*uih),j + Sh ... (3.1)

where uih are the Reynolds enthalpy fluxes and Sh is the enthalpy source term.

The form of Sh differs depending on whether H1 represents the static or stagnation enthalpy. If H1 is the stagnation enthalpy, then:

Sh = -P,t .... (3.2)

If H1 is the static enthalpy, then:

Sh = -P,t + ui*P,i + 2*Tt:D .... (3.3)

where D is the deformation tensor given by:

D = 0.5*(Ui,j + Uj,i ) .... (3.4)

and Tt is the turbulent stress tensor, which at present is approximated by:

Tt = 2.*rho*ENUT*D .... (3.5)

Here, ENUT is a representative eddy viscosity defined by :

ENUT = CS*KE**2/EP .... (3.6)

where CS is an empirical coefficient defined earlier.

Mean Enthalpy Equation via TEM1

The TEM1 equation takes the following form when the Reynolds stress model is activated:

(rho*Cp*TEM1),t + (rho*Uj*Cp*TEM1),j = -(rho*Cp*uit),j + Sh .... (3.7)

where uit are the Reynolds temperature fluxes, and the source terms Sh have already been defined above.

Mean Scalar Equations

The scalar concentration equation takes the following form when the Reynolds stress model is activated:

(rho*C),t + (rho*Uj*C),j = - (rho*uic),j .... (3.8)

where uic are the Reynolds scalar fluxes.

Gradient-diffusion model

When heat and/or mass transfer is activated with the RSTM, the default is that the Reynolds fluxes are represented by a simple gradient-diffusion model:

-uih = {ENUT/PRT(H1)}*H1,i .... (3.9)

-uit = {ENUT/PRT(TEM1)}*TEM1,i .... (3.10)

-uic = {ENUT/PRT(C1)}*C,i .... (3.11)

where ENUT is given by:

ENUT = CMUCD*KE**2/EP .... (3.12)

where CMUCD=0.065 in wall-bounded flows and 0.115 in free turbulent flows.

Generalised-gradient-diffusion model

If the generalized gradient-diffusion model is activated, the Reynolds fluxes are computed from:

-uih = CT*uiuk*{KE/EP}*H1,k .... (3.13)

-uit = CT*uiuk*{KE/EP}*TEM1,k .... (3.14)

-uic = CT*uiuk*{KE/EP}*C1,k .... (3.15)

where the coefficient CT=0.3.

Reynolds-flux transport equations

If the full transport model is activated, the turbulent Reynolds fluxes uit ( in what follows uit is used synomously with uih and uic ) are obtained from the following modelled transport equation:

(rho*uit),t + (rho*Uk*uit),k = Diff(uit) + rho*(Pit + Rit) .... (3.16)

where Diff(uit) represents diffusive transport, and is modelled by:

Diff(uit) = (rho*CST*ukuk*KE/EP*(uit),k),k .... (3.17)

wherein CST is an empirical constant. No viscous dissipation term appears in equation (3.16) because this term is zero in locally isotropic turbulence at high Reynolds numbers.

The production term Pit needs no approximation and is defined by:

Pit = Pit1 + Pit2 .... (3.18)


Pit1 = -ukui*T,k .... (3.19)

Pit2 = -ukt*Ui,k .... (3.20)

The pressure-scrambling term Rit is the counterpart of the pressure- strain term in the uiuj equation and generally, it acts to reduce uit. It is modelled as the sum of two terms:

Rit = Rit1 + Rit2 .... (3.21)

the two contributions being associated respectively with purely turbulence interactions and interactions between mean strain and fluctuating quantities. Following Gibson and Launder [1978], these contributions are modelled as follows:

Rit1 = - C1T*uit*EP/KE .... (3.22)

Rit2 = C2T*Pit2 .... (3.23)

where C1T and C2T are empirical constants.

At present there appears to be no well tested closure model for obtaining the Reynolds heat-flux field with the SSG model. In PHOENICS, these fluxes are obtained from the foregoing closure model but with modified empirical constants so as to return satisfactory heat-flux levels in local equilibrium.

Wall-Reflection Terms

The wall corrections to the pressure-scrambling terms are modelled using the proposal of Gibson and Launder [1978], as follows:

Rit = - C1TW*ukt*ni*nk*f*EP/KE .... (3.24)

where C1TW is an empirical constant.

The empirical constants

The empirical constants employed in the Reynolds-flux equations differ depending on which pressure-strain model is employed, as follows: