Welcome to AE Resources

Converted document Converted document

ONE-EQUATION MODELS

In order to determine the eddy viscosity, an additional partial differential equation is introduced. In most cases, this equation solves for the specific turbulence kinetic energy k = (1)/(2)u_i’u_i’.

The turbulence kinetic energy (TKE) equation

Combining the Navier-Stokes equation for u_i with the RANS equation for U_i and multiplying the result by the fluctuation u_i’ yields the turbulence kinetic energy equation.

(1) (Dk)/(Dt) = (∂)/(∂x_j)⎛⎝ν(∂k)/(∂x_j) − (1)/(2)u_i’u_i’u_j’ − (1)/(ρ)u_jp’⎞⎠ − u_i’u_j’(∂U_i)/(∂x_j) − ν(∂u_i’)/(∂x_k)(∂u_i’)/(∂x_k)

(i) (ii) (iii) (iv)

Eq.(1↑) looks pretty complex, but can be explained in simple terms. It is physically equivalent to

(2) rate of change of TKE (i) = Transport (ii) + Production (iii) − Dissipation (iv)

The transport term can be decomposed in two parts. First, the molecular motion is responsible for diffusing the TKE in the flow. In addition, the turbulent fluctuations in pressure and velocity transport energy from one point to another. This term is in the form of a gradient in space of statistical averages. It is therefore equal to zero if those averages are independent of location in space. The production term represents the TKE that an eddy will gain from the mean flow per unit time. It has a minus sign in front of it, but most of the time, P = − u_i’u_j’S_i, j > 0 so that the TKE is increased. Eventually, the dissipation ϵ = 2νs_i, js_i, j > 0 tends to reduce the TKE in the flow. It represents the amount of TKE that is being converted to heat at the smallest scales. The energy dissipated is taken from the intermediate scales, ultimately governed by the large scales.

Modeling of the TKE equation

The new equation for the turbulent kinetic energy k introduced a large number of unknowns, leading to the familiar closure problem. In order to resolve the equation, some terms will therefore need to be modeled, based on physical intuition and experimental data.

Production:

As discussed previously, one-equation models are typically based on the Boussinesq hypothesis. From a dimensional perspective, the eddy viscosity can be defined by ν_T ~ k^1 ⁄ 2l, where l is some turbulence length scale. This formulation, very similar to the mixing length model, allows non zero ν_T in locations where the mean strain rate is zero. However, it remains an isotropic relation and still assumes that the time scale of turbulence is proportional to that of the mean flow. These approximations are not completely accurate, but provide reasonable results for many engineering problems

Dissipation:

The kinetic energy is dissipated at the small scales, but the rate at which the energy is dissipated is controlled by the large scales. This rate is therefore imposed by an inviscid mechanism. It follows that:

(3) ϵ ~ (k^3 ⁄ 2)/(l)

Turbulent transport:

This term is relatively complex and requires drastic approximations. It is generally modeled by assuming a gradient diffusion mechanism. In theory, there is no equivalence for the pressure term, which is therefore included into the transport model. This hypothesis is acceptable, since DNS results by Mansour, kim and Moin (1988) have shown that this term was small for incompressible flows [4]. The transport equation is

(4) (1)/(2)u_i’u_i’u_j’ + (1)/(ρ)u_j’p’ = − (ν_T)/(σ_k)(∂k)/(∂x_j)

where σ_k is a closure coefficient sometime called turbulent Prandtl number. Indeed, this parameter modifies the magnitude of the turbulence transport term throughout the flow. Eventually, the modeled turbulent kinetic energy equation is obtained:

(5) (Dk)/(Dt) = (∂)/(∂x_j)⎡⎣⎛⎝ν + (ν_T)/(σ_k)⎞⎠(∂k)/(∂x_j)⎤⎦ + ⎡⎣ν_T⎛⎝(∂U_i)/(∂x_j) + (∂U_j)/(∂x_i)⎞⎠ − (2)/(3)kδ_ij⎤⎦(∂U_i)/(∂x_j) − C_D(k^3 ⁄ 2)/(l)

where ν_T = C_μk^1 ⁄ 2l and C_D, C_μ and σ_k are closure coefficients. To summarize, this model has three closure coefficients and a closure function (the length scale). Emmons (1954) [2] and Glushko (1966) [3] implemented this model with reasonable success using the values C_D ~ 0.08, C_μ = 1 and σ_k = 1. Their length scale function was very similar to the traditional mixing length initially postulated by Prandtl.