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TWO-EQUATION MODELS

One-equation models typically include flow history effects, which allow them to predict separated regions more accurately. However, this feature is very limited, and the turbulent length scale is still obtained primarily through empirical means. One of the main reasons for adding a second partial differential equation to the system is to include transport effects on the length scale. Moreover, because of the additional equation, two-equation models are generally complete and can be applied to a wide range of flows. As long as the fundamental hypotheses of the model are not violated, it will perform reasonably well without any interaction from the user.

General assumptions

Most two-equation models consider that the fluctuations are isotropic. Although the flow can be locally isotropic at high Reynolds number (Kolmogorov hypothesis - 1941 [2]), the larger scales are anisotropic because of the mean strain rates. This approximation is therefore contestable, but remains acceptable for solving many engineering problems. The second major hypothesis is called the local equilibrium assumption. The convection and diffusion of turbulent kinetic energy are typically small in the near-wall region of an attached boundary layer. At any point in the flow, it is assumed that the dissipation roughly balances the production of TKE. This hypothesis greatly simplifies the modeling of the length scale.

Equation for the dissipation rate ϵ

There is a large number of two-equation models available in the literature. The most widely used is probably the k − ϵ model (Jones and Launder - 1972 [1]), which includes a second partial differential equation for the dissipation rate ϵ. Its derivation will be shown as an example in this section. Remember from the TKE equation that the dissipation rate is defined by

(1) ϵ = ν(∂u_i’)/(∂x_j)(∂u_i’)/(∂x_j)

An exact equation for ϵ can be derived by considering the following moment of the Navier-Stokes equation.

(2) 2ν(∂u_i’)/(∂x_j)(∂)/(∂x_j)[...] = 0

After considerable algebra, the exact equation for the dissipation rate is given by

(∂ϵ)/(∂t) + U_j(∂ϵ)/(∂x_j) = − 2ν(u_i, k’u_j, k’ + u_k, i’u_k, j’)(∂U_i)/(∂x_j) − 2νu_k’u_i, j(∂²U_i)/(∂x_k∂x_k)

− 2νu_i, k’u_i, m’u_k, m’ − 2ν²u_i, km’u_i, km’

(3) + (∂)/(∂x_j)⎛⎝ν(∂ϵ)/(∂x_j) − νu_j’u_i, m’u_i, m’ − 2(ν)/(ρ)p_m’u_j, m⎞⎠

where φ_i, j = ∂φ_i ⁄ ∂x_j. Once again, drastic modeling will be required to solve the equation. It is important to remember that each term is more than just a mathematical expression, and possesses a physical meaning. The first row on the right hand side of Eq.(3↑) represents the production of ϵ due to interactions between the fluctuations and the mean flow. The second line describes the destruction of ϵ due to turbulent fluctuations. Finally, the last row describes the transport of ϵ within the domain. Similarly to the turbulent kinetic energy equation, this phenomenon can be due to viscous diffusion and turbulent fluctuations.

Production:

The convection and diffusion of turbulent kinetic energy are typically small in the near-wall region of an attached boundary layer. From the TKE equation, it follows that the production of kinetic energy roughly balances the dissipation. The flow is said to be in local equilibrium, such that

(4) P_k = − u_i’u_j’(∂U_i)/(∂x_j) ~ ϵ so that P_ϵ ∝ (P_k)/(t_ch)

where t_ch is a characteristic time scale for the production of dissipation ϵ. The local equilibrium assumption implies that the dissipation equals the rate at which the turbulent kinetic energy k is being produced. From dimensional analysis, one can therefore obtain

(5) t_ch ∝ (k)/(ϵ) which leads to P_ϵ = C_ϵ₁(k)/(ϵ)P_k

with C_ϵ₁ being a closure coefficient.

Destruction:

Similarly, the destruction term can be estimated using dimensional analysis.

(6) D_ϵ ∝ (ϵ)/(t_ch) which implies that D_ϵ = C_ϵ₂(ϵ²)/(k)

where C_ϵ₂ is another closure coefficient.