25.2 ReNormalisation Group k-e Model

ReNormalisation Group (RNG) methods provide a general set of rules which allow physical problems to be expressed in terms of equations governing the large-scale, long-time behaviour of the system. This ``coarse-graining'' embodied in the RNG approach has been applied to areas as diverse as high-energy particle physics, critical phenomena and areas of fluid dynamics such as turbulence and combustion. The RNG method can be applied to any scale invariant phenomenon which has no externally imposed characteristic length and time scale.

One of the most important features of high Reynolds number turbulent flows is their large range of both space and time scales. The size of eddies in high Reynolds number, turbulent flows range from the energy - containing eddies of size L, the integral scale, down to the dissipation-scale eddies of size L/R^3/4, where R is a microscale Reynolds number. To accurately solve the three dimensional Navier-Stokes equation for such a turbulent flow would require storage of order O(R^9/4). As the time scale of turbulent eddies is of order O(R^3/4) the amount of computational work is order O(R³). If R is large both the number of operations and the storage requirements are prohibitively large.

High Reynolds number, turbulent flow can be characterised by three ranges of spatial scales

1.

The energy spectrum is strongly anisotropic as well as being non universal for wave numbers k=O($\pi/L$). In this range the integral scale is affected by both the geometry of the flow and the physiochemical processes taking place.

2.

At smaller scales, the inertial range of small scale eddies, the velocity fluctuation spectrum, E(k), is approximately given by the Kolmogorov law

$$E(k) = C_k \overline{\epsilon}^{2/3} k^{-5/3}$$ (25.28)

where C_k is the Kolmogorov constant and $\overline{\epsilon}$ is the energy dissipation per unit volume.

3.

For wave numbers in the range k>O(K_d) the eddies have very low energy due to viscous dissipation. In this range the energy spectrum decreases exponentially with k.

Once the inertial range of eddies have been expressed in a quantitively accurate manner, the RNG methods can be applied to obtain coarse-grained equations for the turbulence related variables.

The range of scales of the effective excitation in turbulence lie between the low, energy containing wavenumber k₀ = 2$\pi/L$ and the high wavenumber viscous cutoff $\Lambda$. The RNG method removes a narrow band of modes near $\Lambda$ by expressing them in terms of lower modes in the range $k_0 < k < \Lambda \exp^{-l}$. Having removed this band of modes the equations of motions for the remaining modes is a modified system of Navier-Stokes equations in which the eddy viscosity, force and non-linear coupling have all been affected. In turn, further modes are iteratively removed from the dynamics. Thus the RNG method produces a form of the Navier-Stokes equation which allows the computation on coarser grids.

The current implementation of the RNG model uses the same equation as the standard k-e model for the turbulent viscosity with the exception that the constant $C_\mu$ has an adjusted value. A comparison of the values of a number of constants used in the standard and RNG variants of the k-e model are presented in the table below.

|t:===:t|	Standard k-e	RNG k-e
||-|-|-|| C $_{1\epsilon}$	1.44	1.42
||-|-|-|| C $_{2\epsilon}$	1.92	1.68
||-|-|-|| $\sigma_k$	1.0	0.7194
||-|-|-|| $\sigma_\epsilon$	1.314	0.7194
||-|-|-|| C$_\mu$	0.09	0.0845
|b:===:b|

In addition to the above changes in the values of the constants there is an extra source term in the dissipation rate equation which now becomes

$$\begin{split} \dxdt{\epsilon}{t} + \text{div}(\rho \underlin... ...} - C_{2\epsilon} \rho \frac{\epsilon^2}{k} - R & \end{split}$$ (25.29)

where the extra term, R, is given by

$$R \; = \; 2 \, \nu_{lam} \, G_{ij} \, \overline{\dxdt{u_l}{x_i} \dxdt{u_l}{x_j}}$$ (25.30)

In the RNG k-$\epsilon$ model the value of R is calculated using the formula

$$R \; = \; \frac{C_\mu \eta^3 (1 - \eta/\eta_0)}{1+\beta \eta^3} \rho \, \frac{\epsilon^2}{k}$$ (25.31)

where $\eta$ = $\sqrt{G} k / \epsilon$, G = 2G_ijG_ij, and the constants $\eta_0$ and $\beta$ take the values 4.8 and 0.012 respectively. This term is combined with the

$$C_{2\epsilon} \rho \frac{\epsilon^2}{k}$$

dissipation source term so that the constant $C_{2\epsilon}$ is replaced with

$$C_{2\epsilon} \; + \; \frac{C_\mu \eta^3 (1 - \eta/\eta_0)}{1+\beta \eta^3}$$

This modified constant is used in the linearisation equation described for the standard k-e model.