# Reynolds stress model (RSM)

(Difference between revisions)
 Revision as of 10:13, 4 October 2006 (view source) (→Equations)← 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 → (15 intermediate revisions not shown) Line 1: Line 1: == Introduction == == 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 [[#Referencec|[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. + 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 [[#References|[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 == == 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, $\overline{u'_iu'_j}$ , 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: The exact transport equations for the transport of the Reynolds stresses, $\overline{u'_iu'_j}$ , may be written as follows: +
$[itex] - \underbrace{\frac{\partial}{\partial t}(\rho \overline{u'_iu'_j})} + \frac{\partial}{\partial x_{k}}(\rho u_{k} \overline{u'_iu'_j}) = - \frac{\partial}{\partial x_k}[\rho \overline{u'_iu'_ju'_k} + \overline{p(\delta_{kj}u'_i + \delta_{ik}u'_j)}]$ + \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] - + $[itex] - + \frac{\partial}{\partial x_k}[{\mu \frac{\partial}{\partial x_k}(\overline{u'_iu'_j})}] - \rho(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}) - \rho\beta(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta}) + + \frac{\partial}{\partial x_k}\left[{\mu \frac{\partial}{\partial x_k}\left(\overline{u'_iu'_j}\right)}\right] - \rho\left(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}\right) - \rho\beta\left(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta}\right)$ [/itex] - - $[itex] - + \overline{p(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i})} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}} + + \overline{p'\left(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i}\right)} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}}$ [/itex] - - $[itex] - -2\rho\Omega_k(\overline{u'_ju'_m}\epsilon_{ikm} + \overline{u'_iu'_m}\epsilon_{jkm}) + S_{user} + -2\rho\Omega_k\left(\overline{u'_ju'_m}\epsilon_{ikm} + \overline{u'_iu'_m}\epsilon_{jkm}\right) + S_{user}$ [/itex]
Line 34: Line 29:
- 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 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 43: Line 36: 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. 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 Turbulent Diffusive Transport=== - ==Modeling the Pressure-Strain Term== + ===Modeling the Pressure-Strain Term=== - ==Effects of Buoyancy on Turbulence== + ===Effects of Buoyancy on Turbulence=== - ==Modeling the Turbulence Kinetic Energy== + ===Modeling the Turbulence Kinetic Energy=== - ==Modeling the Dissipation Rate== + ===Modeling the Dissipation Rate=== - ==Modeling the Turbulent Viscosity== + ===Modeling the Turbulent Viscosity=== - ==Boundary Conditions for the Reynolds Stresses== + ===Boundary Conditions for the Reynolds Stresses=== - ==Convective Heat and Mass Transfer Modeling== + ===Convective Heat and Mass Transfer Modeling=== + ===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. - {{Stub}} + Rotta's model describes the linear return-to-isotropy behavior of a - [[Category:Turbulence models]] + 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 == == Model constants == Line 70: 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 == == References == == References == - {{reference-paper|year=1975|author=Launder, B. E., Reece, G. J. and Rodi, W.|title=Progress in the Development of Reynolds Stress Turbulent Closure|rest=Journal of Fluid Mechanics, Vol. 68, pp. 537-566}} + {{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}} + + {{Stub}} + [[Category:Turbulence models]]

## 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.