non dimensionalisation eddy viscosity, mixing length and turbulent shear stress

Lmath · January 30, 2019, 10:20

The velocity profile is defined as law of the wall is defined as

, 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

and turbulent sheer stress $\tau_{turb} / \rho = -\overline{u^{'}v^{'}}$ as a function of

?

LuckyTran · January 30, 2019, 10:23

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*).

Lmath · January 30, 2019, 10:29

Quote:

Originally Posted by LuckyTran

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

. Then I want to relate each of these properties to $\frac{du^+}{dy^+}$ after which these will be evaluated in the limit

. Eventually I want to prove that in the overlap region the turbulent properties are independent of viscosity.

FMDenaro · January 30, 2019, 11:44

Quote:

Originally Posted by Lmath

The velocity profile is defined as law of the wall is defined as

, 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

and turbulent sheer stress $\tau_{turb} / \rho = -\overline{u^{'}v^{'}}$ as a function of

?

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

sbaffini · January 30, 2019, 13:22

Typically, $\nu_t = \nu f(y^+)$ , for some function

. For example, at high

, it typically holds that $\nu_t/\nu = 1/ky^+$ . In contrast, for small

, it typically holds that $\nu_t/\nu = C{y^+}^3$ .

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

January 30, 2019, 10:20	non dimensionalisation eddy viscosity, mixing length and turbulent shear stress	#1
Lmath New Member Join Date: Jan 2019 Posts: 7 Rep Power: 0	The velocity profile is defined as law of the wall is defined as , 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 and turbulent sheer stress $\tau_{turb} / \rho = -\overline{u^{'}v^{'}}$ as a function of ?

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January 30, 2019, 10:23		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,753 Rep Power: 66	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*).

January 30, 2019, 13:22		#5
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Typically, $\nu_t = \nu f(y^+)$ , for some function . For example, at high , it typically holds that $\nu_t/\nu = 1/ky^+$ . In contrast, for small , it typically holds that $\nu_t/\nu = C{y^+}^3$ . Of course, we're talking about equilibrium boundary layers here.