User:Reynolds15/sandbox

Reynolds Stress Equation Model
Reynolds stress equation model (RSM), also known as second order or second moment closure model is the most complex classical turbulence model. Several shortcomings of k-epsilon turbulence model were observed when it was attempted to predict flows with complex strain fields or substantial body forces. Under those conditions the individual Reynolds stresses were not found to be accurate while using formula $$-\rho u_{i}^\prime u_{j}^\prime$$ = $$\mu t\left (\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{i}}{\partial x_{i}}\right )-\frac{2}{3}\rho k\delta_{ij}$$ = $$2 \mu t E_{ij}-\frac{2}{3}\rho k \delta_{ij}$$

The equation for the transport of kinematic Reynolds stress $$R_{ij}=u_{i}^\prime u_{j}^\prime=-\tau _{ij}/\rho$$ is

$$\frac{DR_{ij}}{Dt} = D_{ij}+ P_{ij}+ \Pi_{ij}+ \Omega_{ij}- \varepsilon_{ij}$$

Rate of change of $$R_{ij}$$ + Transport of $$R_{ij}$$ by convection = Transport of $$R_{ij}$$ by diffusion + Rate of production of $$R_{ij}$$ + Transport of $$R_{ij}$$ due to turbulent pressure-strain interactions + Transport of $$R_{ij}$$ due to rotation + Rate of dissipation of $$R_{ij}$$.

The six partial differential equations above represent six independent Reynolds stresses. The models that we need to solve the above equation are derived from the work of Launder, Rodi and Reece (1975).

Production Term
The Production term that is used in CFD computations with Reynolds stress transport equations is

$$P_{ij}$$ =$$ -\left (R_{im}\frac{\partial U_{j}}{\partial x_{m}}+R_{jm}\frac{\partial U_{i}}{\partial x_{m}}\right )$$

Pressure-strain interactions
Pressure-strain interactions affect the Reynolds stresses by two different physical processes: pressure fluctuations due to eddies interacting with one another and pressure fluctuation of an eddy with a region of different mean velocity. This redistributes energy among normal Reynolds stresses and thus makes them more isotropic. It also reduces the Reynolds shear stresses.

It is observed that the wall effect increases the anisotropy of normal Reynolds stresses and decreases Reynolds shear stresses. A comprehensive model that takes into account these effects was given by Launder and Rodi (1975).

Dissipation Term
The modelling of dissipation rate $$\epsilon_{\rm ij}$$ assumes that the small dissipative eddies are isotropic. This term affects only the normal Reynolds stresses.

$$\epsilon_{\rm ij}$$ = $$2/3\epsilon\delta_{ij}$$

where $$\epsilon$$ is dissipation rate of turbulent kinetic energy, and $$\delta_{ij}$$ = 1 when i = j and 0 when i ≠ j

Diffusion Term
The modelling of diffusion term $$D_{ij}$$ is based on the assumption that the rate of transport of Reynolds stresses by diffusion is proportional to the gradients of Reynolds stresses. The simplest form of $$D_{ij}$$ that is followed by commercial CFD codes is

$$D_{ij}$$ = $$\frac{\partial}{\partial x_{m}}\left (\frac{v_{t}}{\sigma_{k}}\frac{\partial R_{ij}}{\partial x_{m}}\right )$$ = $$div\left (\frac{v_{t}}{\sigma_{k}}\nabla(R_{ij})\right )$$ where $$\upsilon_{t}$$ = $$C_{\mu} \frac{k^2}{\epsilon}$$, $$\sigma_{k}$$ = 1.0 and $$C_{\mu}$$ = 0.9

Pressure-Strain Correlation Term
The Pressure-Strain Correlation Term promotes isotropy of the turbulence by redistributing energy amongst the normal Reynolds stresses.The pressure-strain interactions is the most important term to model correctly. Their effect on Reynolds stresses is caused by pressure fluctuations due to interaction of eddies with each other and pressure fluctuations due to interaction of an eddy with region of flow having different mean velocity. The correction term is given as

$$\Pi_{ij}=-C_{1}\frac{\epsilon}{k}\left (R_{ij}-\frac{2}{3}k\delta_{ij}\right )-C_{2}\left (P_{ij}-\frac{2}{3}P\delta_{ij}\right )$$

Rotational Term
The rotational term is given as

$$\Omega_{ij}=-2\omega_{k}\left (R_{jm}e_{ikm}+R_{im}e_{jkm}\right )$$ here$$\omega_{k}$$ is the rotation vector, $$e_{ijk}$$=1 if i,j,k are in cyclic order and are different,$$e_{ijk}$$=-1 if i,j,k are in anti-cyclic order and are different and $$e_{ijk}$$=0 in case any two indices are same.

Advantages of RSM
1)Compared with k-ε model, it is simple because of the use of an isotropic eddy viscosity.

2)It is the most general of all turbulence models and work reasonably well for a large number of engineering flows.

3)It requires only the initial and/or boundary conditions to be supplied.

4)Since the production terms need not be modelled, it can selectively damp the stresses due to buoyancy, curvature effects etc.

Disadvantages of RSM
1)It requires very large computing costs.

2)It is not very widely validated as the k-ε model and mixing length models.

3)Due to identical problems with the ε-equation modelling, it performs just as poorly as the k-ε model in some problems.

4)Because of being isotropic, it is not good in predicting normal stresses and is unable to account for irrotational strains.