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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26 Aug 2019


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