 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-84/4/f/solution

Since

2 p - q = 1

, the leading turbulent viscosity is linear:

ν_{T} = C_{μ} k_{0} x .

Using

U_{0} = C_{μ} p k_{0} / σ_{k}

and

1/ p = 2 - σ

gives

\frac{d ν _{T}}{d x}_{0^{+}} = C_{μ} k_{0} = σ_{k} (2 - σ) U_{0} = (2 σ_{k} - σ_{ϵ}) U_{0} .

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