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non dimensionalisation eddy viscosity, mixing length and turbulent shear stress

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Old   January 30, 2019, 09:20
Default non dimensionalisation eddy viscosity, mixing length and turbulent shear stress
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The velocity profile is defined as law of the wall is defined as u^+ = f(y^+), where u^+ = \frac{\bar{u}}{v^*}; y^+ = \frac{yv^*}{v} and v^*=\sqrt{\frac{\tau_w}{\rho}}.

How would one then non dimensionalize the eddy viscosity v_t and turbulent sheer stress \tau_{turb} / \rho = -\overline{u^{'}v^{'}} as a function of y^+?
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Old   January 30, 2019, 09:23
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You can non-dimensionalize anything anyway you want. The question is what you are going to do with these quantities later? Then you want to choose a non-dimensionalization that makes sense.

For example, the friction velocity isn't just a random non-dimensionalization of shear stress, but it is used because that's the characteristic velocity in the perturbatino when you do asymptotic analysis (i.e. u+ = y+ which happens only you scale u and y using u*).
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Old   January 30, 2019, 09:29
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Originally Posted by LuckyTran View Post
You can non-dimensionalize anything anyway you want. The question is what you are going to do with these quantities later?
I want to explicitly define these in terms of y^+. Then I want to relate each of these properties to \frac{du^+}{dy^+} after which these will be evaluated in the limit y^+>>1. Eventually I want to prove that in the overlap region the turbulent properties are independent of viscosity.
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Old   January 30, 2019, 10:44
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Quote:
Originally Posted by Lmath View Post
The velocity profile is defined as law of the wall is defined as u^+ = f(y^+), where u^+ = \frac{\bar{u}}{v^*}; y^+ = \frac{yv^*}{v} and v^*=\sqrt{\frac{\tau_w}{\rho}}.

How would one then non dimensionalize the eddy viscosity v_t and turbulent sheer stress \tau_{turb} / \rho = -\overline{u^{'}v^{'}} as a function of y^+?



Actually, the eddy viscosity function is naturally non-dimensional when is computed from a non-dimensional expression. Just use the correct velocity and pressure reference quantities to work directly in terms of u+ and Re_tau.

You can find in Sec.5 of this paper how to select the variables
https://www.researchgate.net/publica...ection_methods
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Old   January 30, 2019, 12:22
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Typically, \nu_t = \nu f(y^+), for some function f(y^+). For example, at high y^+, it typically holds that \nu_t/\nu = 1/ky^+. In contrast, for small y^+, it typically holds that \nu_t/\nu = C{y^+}^3.

Of course, we're talking about equilibrium boundary layers here.
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