Reynolds stress model (RSM)

(Difference between revisions)
 Revision as of 21:45, 16 May 2006 (view source) (→Introduction)← Older edit Revision as of 10:13, 4 October 2006 (view source) (→Equations)Newer edit → Line 5: Line 5: == Equations == == 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 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: + + $+ \underbrace{\frac{\partial}{\partial t}(\rho \overline{u'_iu'_j})} + \frac{\partial}{\partial x_{k}}(\rho u_{k} \overline{u'_iu'_j}) = - \frac{\partial}{\partial x_k}[\rho \overline{u'_iu'_ju'_k} + \overline{p(\delta_{kj}u'_i + \delta_{ik}u'_j)}]$ + + + $+ + \frac{\partial}{\partial x_k}[{\mu \frac{\partial}{\partial x_k}(\overline{u'_iu'_j})}] - \rho(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}) - \rho\beta(g_i\overline{u'_j\theta}+g_j\overline{u'_i\theta}) +$ + + + + $+ + \overline{p(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i})} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}} +$ + + + + $+ -2\rho\Omega_k(\overline{u'_ju'_m}\epsilon_{ikm} + \overline{u'_iu'_m}\epsilon_{jkm}) + S_{user} +$ +
+ + 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== + + + {{Stub}} + [[Category:Turbulence models]] == Model constants == == Model constants ==

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:

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

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

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

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

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.

Model constants

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

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

References

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