explanation of $y$ dependence

The instantaneous velocity perturbations in a 2D (in the mean) turbulent boundary layer only vary in the wall normal $(y)$ direction because the flow is unconstrained in the other directions $(x, z)$ and therefore any velocity perturbation is unnaffected by a change of position in the streamwise or spanwise direction.

$u'=a_0 + a_1 y + a_2 y^2 + a_3 y^3 + ... $

$v'=b_0 + b_1 y + b_2 y^2 + b_3 y^3 + ... $

$w'=c_0 + c_1 y + c_2 y^2 + c_3 y^3 + ... $

At the wall, $y=0$ and the "no-slip" condition applies so $u'=v'=w'=0$ and $a_0=b_0=c_0=0$.

Continuity also requires that ${\partial u' \over \partial x}={\partial v' \over \partial y}={\partial w' \over \partial z}=0$

From this it can be seen that ${\partial v' \over \partial y}=b_1  + 2 b_2 y + 3 b_3 y^2 + ... = 0$ and hence $b_1=0$

Looking at leading order terms in the vicinity of the wall we can therefore say

$u'=a_1 y $

$v'=b_2 y^2 $

$w'=c_1 y $

Thus we can now show $y$ dependence of the Reynolds stress perturbations:

$u'v'= a_1 y. b_2 y^2  \sim y^3 $

$u'u' \sim y^2$

$v'v' \sim y^4$

$w'w' \sim y^2$

Looking at $T'$ and $Q'$

we can also look at the dominant terms in some other useful quantities

${\partial {u_i}' \over \partial x_j} \sim \left( {\partial u' \over \partial y}+{\partial w' \over \partial y} \right) \sim 1$ where the following quantities become small due to

i) lack of streamwise and spanwise dependence ${\partial u' \over \partial x} \rightarrow 0$, $ {\partial w' \over \partial x} \rightarrow 0$, ${\partial u' \over \partial z} \rightarrow 0$, $ {\partial w' \over \partial z} \rightarrow 0$

ii) continuity ${\partial u' \over \partial x} ={\partial v' \over \partial y} ={\partial w' \over \partial z} = 0$

so the term ends up being independent of the $y$ direction.

$T'={\partial {u_i}' \over \partial x_j}{\partial {u_i}' \over \partial x_j} \sim \left( {\partial u' \over \partial y}{\partial u' \over \partial y} + {\partial w' \over \partial y}{\partial w' \over \partial y} \right)  \sim 1 $

There is also a suggestion for the following quantity:

$Q'={\partial {u_i}' \over \partial x_j} {\partial {u_j}' \over \partial x_i} \sim {\partial v' \over \partial x} {\partial u' \over \partial y} + {\partial v' \over \partial z} {\partial w' \over \partial y} $

strictly speaking $Q'$ has no $y$ dependence due to the assumption that there is no dependence of fluctuations on the streamwise $(x)$ and spanwise directions $(z)$. However the streamwise and spanwise dependences are one order lower than all the equivalent terms in $T'$. In the case of LES where we are always working locally it is reasonable to expect that the $y$ dependence of $Q'$ is at least one order greater than the $y$ dependence of $T'$.

If we make the approximation that first order terms in $x$ and $z$ retain their $y$ dependence and second order terms lose their dependence we can propose approximate conditions such as:

$ {\partial v' \over \partial x} \sim y^2 $ and $ {\partial v' \over \partial x} {\partial v' \over \partial x} \sim 0 $

and hence

$Q \sim y^2$

Strictly speaking $T' \sim 1 $ and $Q' \sim 0 $ however the fact that $Q'$ has less dependence on the streamwise and spanwise directions than $T'$ suggests that it has more of a residual $y$ dependence than $T'.$

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Topic revision: r3 - 2010-07-22 - 16:45:53 - NeilAshton
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24 Apr 2019


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