The momentum equations take the following form when the Reynolds- stress model is activated:
(rho*Ui),t + (rho*Uj*Ui),j = -P,i -(rho*uiuj),j + Bi ... (2.1)
where Bi represents any body forces, uiuj are the Reynolds stresses, and the viscous terms have been neglected in comparison to turbulent transport.
Physically, the Reynolds stresses, multipled by rho, represent the transport of momentum due to the turbulent fluctuating motion. Thus, -rho*uiuj is the transport of xi-momentum in the direction xj ( or vice versa); it acts as a stress on the fluid and is therefore called turbulent or Reynolds stress.
The Reynolds stresses are obtained the following modelled transport equation:
(rho*uiuj),t + (rho*Uk*uiuj),k = Diff(uiuj)
+ rho*(Pij + Rij - Eij)...(2.2)
Diff(uiuj) represents diffusive transport, and it is modelled by using a simplified form of the Daly-Harlow [1970] model:
Diff(uiuj) = (rho*CS*ukuk*{KE/EP}*(uiuj),k),k... (2.3)
wherein CS is an empirical constant, KE is the turbulent kinetic energy and EP is the dissipation of KE.
The stress production term Pij needs no approximation and is defined by:
Pij = -(uiuk*Uj,k + ujuk*Ui,k) .... (2.4)
Eij represents the viscous destruction of uiuj, and is modelled by assuming local isotropy. Thus:
Eij = 2*EP*dij/3 .... (2.5)
where dij is the Kronecker delta. Consequently, the dissipation term is zero for shear stresses; and for the normal stresses the same amount of energy is dissipated in each component of KE.
The pressure-strain term Rij acts to redistribute energy among the normal stresses and to reduce shear stresses. This term, which tends to make the turbulence more isotropic, is modelled as the sum of two contributions, as follows:
Rij = Rij1 + Rij2 .... (2.6)
where Rij1 is the non-linear turbulent part, and Rij2 is the mean- strain (or 'rapid') part.
Rij is modelled by using the IP closure model of Launder et al [1975]:
Rij1 = - C1*(uiuj-2*dij/3)*EP/KE .... (2.7)
Rij2 = - C2*(Pij - dij*Pkk/3) .... (2.8)
where C1 and C2 are empirical constants.
The IPY model of Younis [1984] also uses (2.7) and (2.8) for Rij, but with different values of C1 and C2.
The QI closure model of Launder et al [1975] uses (2.7) for Rij1, and the following model for Rij2:
Rij2 = - alfa*(Pij - dij*Pkk/3) - beta*(Ui,j+Uj,i)*KE
- gama*(Dij - dij*Pkk/3) ..... (2.9)
where
Dij = - (uiuk*Uk,j + ujuk*Uk,i) .... (2.10)
and alfa, beta and gama are empirical coefficients defined by:
alfa = (C2+8)/11 , beta = (8*C2-2)/11 ,
gama = (30*C2-2)/55 ..... (2.11)
The SSG model models the complete pressure-strain term as:
Rij = - (C1*EP + C1S*Pkk)*bij + C2*EP*(bik*bij - bmn*bmn*dij/3)
+ [C3 - C3S*(bmn*bmn)**0.5]*KE*Sij
+ C4*KE*(bik*Sjk + bjk*Sik - 2*bmn*Smn*dij/3)
+ C5*KE*(bik*Wjk + bjk*Wik) ..... (2.12)
where bij, Sij and Wij are, respectively, the Reynolds-stress anisotropy, the mean rate of strain and the mean vorticity tensors
bij = 0.5*uiuj/KE - dij/3 ; Sij = 0.5*(Ui,j + Uj,i)
Wij = 0.5*(Ui,j - Uj,i) .... (2.13)
This model is quadratic in the Reynolds stresses, and does not require any wall-reflection terms.
The IP, IPY and QI pressure-strain models do not allow for the influence of wall-reflected pressure fluctuations on the redistrib- ution of the stresses. In near-wall turbulence, this effect causes the level of fluctuating velocity normal to the wall to be much damped, while that parallel to the main flow to be enhanced relative to free-shear flow. This so-called 'wall-reflection' effect is accounted for in PHOENICS by adding the following wall-correction terms to the pressure-strain model ( see Rodi [1980] ):
Rij1w = C1W*(ukum*nk*nm*dij-1.5*ukui*nk*nj-1.5*ukuj*nk*ni)*f*EP/KE....(2.14)
Rij2w = C2W*(Rkm2*nk*nm*dij-1.5*Rij2*nk*nj-1.5*Rkj2*nk*ni)*f....(2.15)
where: C1W and C2W are empirical constants; nk is the unit vector normal to a wall, and f is the wall-damping function.
The wall-damping function is computed from:
f = CW*L/Xn ..... (2.16)
where: Xn is the normal distance from a wall; CW=(Xn/L) at the near- wall grid point; and L is the turbulence lengthscale given by:
L = CD*KE**1.5/EP .....(2.17)
where CD is an empirical constant. The value chosen for CW ensures that f is unity in near-wall turbulence.
The turbulence energy dissipation rate is computed from the following modelled transport equation:
(rho*EP),t + (rho*Ui*EP),i = Diff(EP) + rho*(0.5*C1E*Pkk
-C2E*EP)*EP/KE .... (2.18)
where Diff(EP) represents diffusive transport, and is modelled by:
Diff(EP) = (CEP*rho*KE/EP*ukuk*EP,k),k ...... (2.19)
In the foregoing CEP, C1E and C2E are empirical constants.
The empirical constants
The empirical constants employed in the RSTM differ depending on which pressure-strain model is employed, as follows:
| IPM | IPY | QIM | SSG | |
| CEP | 0.18 | 0.15 | 0.15 | 0.183 |
| C1E | 1.45 | 1.40 | 1.44 | 1.44 |
| C2E | 1.90 | 1.80 | 1.90 | 1.83 |
| CS | 0.22 | 0.22 | 0.21 | 0.21 |
| C1 | 1.80 | 3.00 | 1.50 | 3.4 |
| C2 | 0.60 | 0.30 | 0.40 | 4.2 |
| C1W | 0.50 | 0.75 | 0.50 | - |
| C2W | 0.30 | 0.50 | 0.06 | - |
| C1S | - | - | - | 1.8 |
| C3 | - | - | - | 0.8 |
| C3S | - | - | - | 1.3 |
| C4 | - | - | - | 1.25 |
| C5 | - | - | - | 0.4 |
The empirical constant CD=(CMUCD)**0.75 where CMUCD=0.065.
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