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Determination of the closure coefficients

In the theory section, it was mentionned that the closure coefficients were obtained empirically. In this example, the calibration process will be explained in more details for three canonical flows.

Shear flow in local-equilibrium

From dimensional analysis, remember that the turbulent viscosity ν_T was given by

(1) ν_T = C_μ(k²)/(ϵ)

As a first example, the closure coefficient C_μ will be extracted from experimental data. The production of kinetic energy in the flow is

(2) P = − u_i’u_j’ (∂U_i)/(∂x_j)≃((u_i’u_j’)²)/(ν_T)

from the Boussinesq assumption. Introducing Eq.(1↑) into Eq.(2↑), the following equation is obtained.

(3) P = (ϵ(u_i’u_j’)²)/(C_μk²) or (|u_i’u_j’|)/(k) = C^1 ⁄ 2_μ⎛⎝(P)/(ϵ)⎞⎠^1 ⁄ 2

For shear layers in local-equilibrium (production P = dissipation ϵ), experimental results [2] suggest that uv ⁄ k≃0.3. Hence, it directly follows from Eq.(3↑) that C_μ = 0.09.

Decaying isotropic turbulence (grid turbulence)

The modeled equation for the TKE and dissipation of TKE are given in the theory section by

(4) (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)

and

(5) (∂ϵ)/(∂t) + U_j(∂ϵ)/(∂x_j) = (∂)/(∂x_j)⎡⎣⎛⎝ν + (ν_T)/(σ_ϵ)⎞⎠(∂ϵ)/(∂x_j)⎤⎦ − C_ϵ₁(k)/(ϵ)u_i’u_j’(∂U_i)/(∂x_j) − C_ϵ₂(ϵ²)/(k)

In the case of decaying isotropic turbulence, there is no mean velocity gradient in the flow and no transport phenomenon. Eq.(4↑) and Eq.(5↑) are respectively reduced to

(6) (dk)/(dt) = − ϵ and (dϵ)/(dt) = − C_ϵ₂(ϵ²)/(k)

It appears that only the closure coefficient C_ϵ₂ plays a role in this flow. According to experimental data [1], the turbulent kinetic energy decays as a power of time.

(7) k ∝ t^− n so that ϵ = − (dk)/(dt) ∝ nt^{− (n + 1)} and (dϵ)/(dt) ∝ − n(n + 1)t^{− (n + 2)}

On the other hand, the modeled dissipation equation (Eq.(6↑)) becomes

(8) (dϵ)/(dt) ∝ − C_ϵ₂n²t^{− (n + 2)}

In order to match these two expressions, it is required that n(n + 1) = n²C_ϵ₂, or C_ϵ₂ = 1 + 1 ⁄ n. Most experimental data [1] suggest that n≃1.3, so that C_ϵ₂≃1.77. However, this value somehow depends on the initial and boundary conditions of the experiment. In the standard k − ϵ model, C_ϵ₂ = 1.92 was chosen, which represents a value of roughly 1.1 for n.