Reynolds stress model (RSM)

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Introduction

The Reynold's stress model (RSM) is a higher level, elaborate turbulence model. It is usually called a Second Order Closure. This modelling approach originates from the work by [Launder (1975)]. In RSM, the eddy viscosity approach has been discarded and the Reynolds stresses are directly computed. The exact Reynolds stress transport equation accounts for the directional effects of the Reynolds stress fields.

Equations

The Reynolds stress model involves calculation of the individual Reynolds stresses, $\overline{u'_iu'_j}$ , using differential transport equations. The individual Reynolds stresses are then used to obtain closure of the Reynolds-averaged momentum equation.

The exact transport equations for the transport of the Reynolds stresses, $\overline{u'_iu'_j}$ , may be written as follows:

$\frac{\partial}{\partial t}\left(\rho \overline{u'_iu'_j}\right) + \frac{\partial}{\partial x_{k}}\left(\rho u_{k} \overline{u'_iu'_j}\right) = - \frac{\partial}{\partial x_k}\left[\rho \overline{u'_iu'_ju'_k} + \overline{p\left(\delta_{kj}u'_i + \delta_{ik}u'_j\right)}\right]$

$+ \frac{\partial}{\partial x_k}\left[{\mu \frac{\partial}{\partial x_k}\left(\overline{u'_iu'_j}\right)}\right] - \rho\left(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}\right) - \rho\beta\left(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta}\right)$

$+ \overline{p\left(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i}\right)} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}}$

$-2\rho\Omega_k\left(\overline{u'_ju'_m}\epsilon_{ikm} + \overline{u'_iu'_m}\epsilon_{jkm}\right) + S_{user}$

or

Local Time Derivate + $C_{ij}$ = $D_{T,ij}$ + $D_{L,ij}$ + $P_{ij}$ + $G_{ij}$ + $\phi_{ij}$ - $\epsilon_{ij}$ + $F_{ij}$ + User-Defined Source Term

where $C_{ij}$ is the Convection-Term, $D_{T,ij}$ equals the Turbulent Diffusion, $D_{L,ij}$ stands for the Molecular Diffusion, $P_{ij}$ is the term for Stress Production, $G_{ij}$ equals Buoyancy Production, $\phi_{ij}$ is for the Pressure Strain, $\epsilon_{ij}$ stands for the Dissipation and $F_{ij}$ is the Production by System Rotation.

Of these terms, $C_{ij}$ , $D_{L,ij}$ , $P_{ij}$ , and $F_{ij}$ do not require modeling. After all, $D_{T,ij}$ , $G_{ij}$ , $\phi_{ij}$ , and $\epsilon_{ij}$ have to be modeled for closing the equations.

Modeling Turbulent Diffusive Transport

Modeling the Pressure-Strain Term

Effects of Buoyancy on Turbulence

Modeling the Turbulence Kinetic Energy

Modeling the Dissipation Rate

Modeling the Turbulent Viscosity

Boundary Conditions for the Reynolds Stresses

Convective Heat and Mass Transfer Modeling

Model constants

The constants suggested for use in this model are as follows:

$C_s \approx 0.25, C_l \approx 0.25, C_\gamma \approx 0.25$

Model variants

Performance, applicability and limitations

Implementation issues

References

Launder, B. E., Reece, G. J. and Rodi, W. (1975), "Progress in the Development of a Reynolds-Stress Turbulent Closure.", Journal of Fluid Mechanics, Vol. 68(3), pp. 537-566.

This article is a stub, a short article which needs to be improved. You can help by expanding it.

@@ Line 11: / Line 11: @@
 <math>
-\frac{\partial}{\partial t}(\rho \overline{u'_iu'_j}) + \frac{\partial}{\partial x_{k}}(\rho u_{k} \overline{u'_iu'_j}) = - \frac{\partial}{\partial x_k}[\rho \overline{u'_iu'_ju'_k} + \overline{p(\delta_{kj}u'_i + \delta_{ik}u'_j)}] </math>
+\frac{\partial}{\partial t}\left(\rho \overline{u'_iu'_j}\right) + \frac{\partial}{\partial x_{k}}\left(\rho u_{k} \overline{u'_iu'_j}\right) = - \frac{\partial}{\partial x_k}\left[\rho \overline{u'_iu'_ju'_k} + \overline{p\left(\delta_{kj}u'_i + \delta_{ik}u'_j\right)}\right] </math>
 <math>
-  + \frac{\partial}{\partial x_k}[{\mu \frac{\partial}{\partial x_k}(\overline{u'_iu'_j})}] - \rho(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}) - \rho\beta(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta})
+  + \frac{\partial}{\partial x_k}\left[{\mu \frac{\partial}{\partial x_k}\left(\overline{u'_iu'_j}\right)}\right] - \rho\left(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}\right) - \rho\beta\left(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta}\right)
 </math>
@@ Line 21: / Line 21: @@
 <math>
-+ \overline{p(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i})} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}}
++ \overline{p\left(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i}\right)} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}}
 </math>
@@ Line 27: / Line 27: @@
 <math>
--2\rho\Omega_k(\overline{u'_ju'_m}\epsilon_{ikm} + \overline{u'_iu'_m}\epsilon_{jkm}) + S_{user}
+-2\rho\Omega_k\left(\overline{u'_ju'_m}\epsilon_{ikm} + \overline{u'_iu'_m}\epsilon_{jkm}\right) + S_{user}
 </math>
 </center>