# Reynolds stress model (RSM)

## Introduction

The Reynolds Stress Models (RSM), also known as the Reynolds Stress Transport Models, are higher level turbulence closures and represent the most complete classical turbulence model. The method of closure employed is usually called a Second Order Closure. This modelling approach originates from the works by Chou (1945) and Rotta (1951). In Reynolds Stress Models, the eddy viscosity approach is avoided and the individual components of the Reynolds stress tensor are directly computed. These models rely on the exact Reynolds stress transport equation. They are able to account for complex interactions in turbulent flow fields, such as the directional effects of the Reynolds stresses.

## Shortcomings of Eddy-viscosity based models

Eddy-viscosity based models like the $k-\epsilon$ and the $k - \omega$ models have significant shortcomings in complex, real-life turbulent flows that are often encountered in engineering applications. In flows with high degrees of anisotropy, significant streamline curvature, flow separation, flows with zones of re-circulation or flows influenced by mean rotational effects, the performance of such models is unsatisfactory (Craft (1996)).

Eddy viscosity based closures cannot account for the limiting states of turbulence. In decaying turbulent flows, it has been observed that the state of turbulence rapidly approaches an isotropic state with equipartition of the turbulent kinetic energy among the components of the Reynolds stresses. Eddy-viscosity based models can never replicate this return to isotropy behaviour. Eddy-viscosity based models cannot replicate the behaviour of turbulent flows in the Rapid Distortion limit, where the turbulent flow essentially behaves as an elastic medium (instead of viscous).

Reynolds Stress Models have an clear advantage over lower-order models for such flows where the transport of terms like the Reynolds stress or heat flux, play important roles. This is because the physics of the transport of second-moment terms can be built into the second-order models.

## 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: $\frac{\partial}{\partial t}\left(\rho \overline{u'_iu'_j}\right) + \frac{\partial}{\partial x_{k}}\left(\rho u_{k} \overline{u'_iu'_j}\right) = - \frac{\partial}{\partial x_k}\left[\rho \overline{u'_iu'_ju'_k} + \overline{p'\left(\delta_{kj}u'_i + \delta_{ik}u'_j\right)}\right]$ $+ \frac{\partial}{\partial x_k}\left[{\mu \frac{\partial}{\partial x_k}\left(\overline{u'_iu'_j}\right)}\right] - \rho\left(\overline{u'_iu'_k}\frac{\partial u_j}{\partial x_k}+\overline{u'_ju'_k}\frac{\partial u_i}{\partial x_k}\right)$ $+ \overline{p'\left(\frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i}\right)} - 2\mu\overline{\frac{\partial u'_i}{\partial x_k} \frac{\partial u'_j}{\partial x_k}}$ $-2\rho\Omega_k\left(\overline{u'_ju'_m}\epsilon_{ikm} + \overline{u'_iu'_m}\epsilon_{jkm}\right)$

or

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

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, $\phi_{ij}$ 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 modelling. However, $D_{T,ij}$, $\phi_{ij}$, and $\epsilon_{ij}$ have to be modelled for closing the equations. The fidelity of the Reynolds stress model depends on the accuracy of the models for the turbulent transport, the pressure-strain correlation and the dissipation terms.

### 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.

### Johansson & Hallbäck

Johansson, A. V. and Hallbäck, M. (1994), "Modelling of rapid pressure—strain in Reynolds-stress closures", Journal of Fluid Mechanics, Vol. 269, pp. 143-168.

### Mishra & Girimaji

Mishra, A. A. and Girimaji, S. S. (2017), "Toward approximating non-local dynamics in single-point pressure–strain correlation closures", Journal of Fluid Mechanics, Vol. 811, pp. 168-188.