Derivation of the law of the wall

TurbJet · January 20, 2023, 14:56

In section 7.3.3 of Pope's book on p.304, he derived the law of the wall based on the mixing-length hypothesis. Start with the total shear stress which is Eq.(7.142) in his book

$\frac{\tau}{\tau_w} = \frac{\partial u^+}{\partial y^+} + \left(l_m^+\frac{\partial u^+}{\partial y^+}\right)^2$

where $(\cdot)^+$ denotes quantities in wall units,

is the mixing length, $\tau$ and $\tau_w$ are the total and wall shear stress, respectively.

This is a quadratic equation for $\frac{\partial u^+}{\partial y^+}$ , which should have solution

$\frac{\partial u^+}{\partial y^+} = \frac{-1\pm\sqrt{1+4(\tau/\tau_w)(l_m^+)^2}}{2(l_m^+)^2}$

However, Pope gives a very different answer

$\frac{\partial u^+}{\partial y^+} = \frac{2\tau/\tau_w}{1+\sqrt{1+4(\tau/\tau_w)(l_m^+)^2}}$

This is very strange, I can't see where this is coming from. Does anybody have any idea?

sbaffini · January 30, 2023, 09:45

Quote:

Originally Posted by TurbJet

In section 7.3.3 of Pope's book on p.304, he derived the law of the wall based on the mixing-length hypothesis. Start with the total shear stress which is Eq.(7.142) in his book

$\frac{\tau}{\tau_w} = \frac{\partial u^+}{\partial y^+} + \left(l_m^+\frac{\partial u^+}{\partial y^+}\right)^2$

where $(\cdot)^+$ denotes quantities in wall units,

is the mixing length, $\tau$ and $\tau_w$ are the total and wall shear stress, respectively.

This is a quadratic equation for $\frac{\partial u^+}{\partial y^+}$ , which should have solution

$\frac{\partial u^+}{\partial y^+} = \frac{-1\pm\sqrt{1+4(\tau/\tau_w)(l_m^+)^2}}{2(l_m^+)^2}$

However, Pope gives a very different answer

$\frac{\partial u^+}{\partial y^+} = \frac{2\tau/\tau_w}{1+\sqrt{1+4(\tau/\tau_w)(l_m^+)^2}}$

This is very strange, I can't see where this is coming from. Does anybody have any idea?

You may want to give a look here, under "Quadratic formula and its derivation".

The Pope's formula, which is itself taken from the van Driest paper, remains valid also when the mixing length goes to 0, which is not the case for the standard formula. However, when both valid, they are equivalent, as you can see from the link.

There may also be a reason linked to the stability of the formula, with the one in the book and paper being known also for its stability, but the validity for the mixing length equal 0 is the most prominent one in my opinion.

January 20, 2023, 14:56	Derivation of the law of the wall	#1
TurbJet Senior Member Join Date: Oct 2017 Location: United States Posts: 233 Blog Entries: 1 Rep Power: 10	In section 7.3.3 of Pope's book on p.304, he derived the law of the wall based on the mixing-length hypothesis. Start with the total shear stress which is Eq.(7.142) in his book $\frac{\tau}{\tau_w} = \frac{\partial u^+}{\partial y^+} + \left(l_m^+\frac{\partial u^+}{\partial y^+}\right)^2$ where $(\cdot)^+$ denotes quantities in wall units, is the mixing length, $\tau$ and $\tau_w$ are the total and wall shear stress, respectively. This is a quadratic equation for $\frac{\partial u^+}{\partial y^+}$ , which should have solution $\frac{\partial u^+}{\partial y^+} = \frac{-1\pm\sqrt{1+4(\tau/\tau_w)(l_m^+)^2}}{2(l_m^+)^2}$ However, Pope gives a very different answer $\frac{\partial u^+}{\partial y^+} = \frac{2\tau/\tau_w}{1+\sqrt{1+4(\tau/\tau_w)(l_m^+)^2}}$ This is very strange, I can't see where this is coming from. Does anybody have any idea?

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