# Reynolds stress model (RSM)

(Difference between revisions)
 Revision as of 01:43, 24 February 2009 (view source) (Change p into p' in the transport and pressure strain terms)← Older edit Revision as of 09:05, 12 August 2010 (view source)Peter (Talk | contribs) (added model variants LRR and SSG with references)Newer edit → Line 4: Line 4: == Equations == == Equations == - The Reynolds stress model involves calculation of the individual Reynolds stresses, $\rho\overline{u'_iu'_j}$ , using differential transport equations. The individual Reynolds stresses are then used to obtain closure of the Reynolds-averaged momentum equation. The Reynolds stress model involves calculation of the individual Reynolds stresses, $\rho\overline{u'_iu'_j}$ , using differential transport equations. The individual Reynolds stresses are then used to obtain closure of the Reynolds-averaged momentum equation. Line 13: Line 12: $[itex] \frac{\partial}{\partial t}\left(\rho \overline{u'_iu'_j}\right) + \frac{\partial}{\partial x_{k}}\left(\rho u_{k} \overline{u'_iu'_j}\right) = - \frac{\partial}{\partial x_k}\left[\rho \overline{u'_iu'_ju'_k} + \overline{p'\left(\delta_{kj}u'_i + \delta_{ik}u'_j\right)}\right]$ \frac{\partial}{\partial 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] [/itex] - $[itex] + \frac{\partial}{\partial x_k}\left[{\mu \frac{\partial}{\partial x_k}\left(\overline{u'_iu'_j}\right)}\right] - \rho\left(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}\right) - \rho\beta\left(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta}\right) + \frac{\partial}{\partial x_k}\left[{\mu \frac{\partial}{\partial x_k}\left(\overline{u'_iu'_j}\right)}\right] - \rho\left(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}\right) - \rho\beta\left(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta}\right)$ [/itex] - - $[itex] + \overline{p'\left(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i}\right)} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}} + \overline{p'\left(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i}\right)} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}}$ [/itex] - - $[itex] Line 35: Line 29: - Local Time Derivate + [itex]C_{ij}$ = $D_{T,ij}$ + $D_{L,ij}$ + $P_{ij}$ + $G_{ij}$ + $\phi_{ij}$ - $\epsilon_{ij}$ + $F_{ij}$ + User-Defined Source Term 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 -
Line 91: Line 83: == Model variants == == Model variants == + + ===LRR, Launder-Reece-Rodi=== + + {{reference-paper|year=1975|author=Launder, B. E., Reece, G. J. and Rodi, W.|title=Progress in the Development of a Reynolds-Stress Turbulent Closure.|rest=Journal of Fluid Mechanics, Vol. 68(3), pp. 537-566}} + + ===SSG, Speziale-Sarkar-Gatski=== + + {{reference-paper|year=1991|author=Speziale, C.G., Sarkar, S., Gatski, T.B.|title=Modeling the Pressure-Strain Correlation of Turbulence: an Invariant Dynamical Systems Approach|rest=Journal of Fluid Mechanics, Vol. 227, pp. 245-272}} + == Performance, applicability and limitations == == Performance, applicability and limitations == == Implementation issues == == Implementation issues ==

## 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, $\rho\overline{u'_iu'_j}$ , 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.

### Return-to-isotropy models

For an anisotropic turbulence, the Reynolds stress tensor, $\rho\overline{u'_iu'_j}$ , is usually anisotropic. The second and third invariances of the Reynolds-stress anisotropic tensor $b_{ij}$ are nontrivial, where $b_{ij}=\overline{u'_iu'_j}/2k-\delta_{ij} /3$ and $k = \overline{u'_iu'_i}/2$. It is naturally to suppose that the anisotropy of the Reynolds-stress tensor results from the anisotropy of turbulent production, dissipation, transport, pressure-stain-rate, and the viscous diffusive tensors. The Reynolds-stress tensor returns to isotropy when the anisotropy of these turbulent components return to isotropy. Such a correlation is described by the Reynolds stress transport equation. Based on these consideration, a number of turbulent models, such as Rotta's model and Lumley's return-to-isotropy model, have been established.

Rotta's model describes the linear return-to-isotropy behavior of a low Reynolds number homogenous turbulence in which the turbulent production, transport, and rapid pressure-strain-rate are negligible. The turbulence dissipation and slow pressure-strain-rate

are preponderant. Under these cirsumstance, Rotta suggested
$\frac{d b_{ij}}{dt}=-(C_{R}-1) \frac{\varepsilon}{k} b_{ij}$
. Here, $C_{R}$ is called the Rotta constant.

## 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

### LRR, Launder-Reece-Rodi

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.

### SSG, Speziale-Sarkar-Gatski

Speziale, C.G., Sarkar, S., Gatski, T.B. (1991), "Modeling the Pressure-Strain Correlation of Turbulence: an Invariant Dynamical Systems Approach", Journal of Fluid Mechanics, Vol. 227, pp. 245-272.

## 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.