# Verify that the fluctuating component of velocity (2D assumption) satisfies the following equation: [begin{equation*}frac{partial v_{x}^{prime}}{partial x}+frac{partial v_{y}^{prime}}{partial

## Question:

Verify that the fluctuating component of velocity (2D assumption) satisfies the following equation:

$\begin{equation*}\frac{\partial v_{x}^{\prime}}{\partial x}+\frac{\partial v_{y}^{\prime}}{\partial y}=0 \tag{17.52}\end{equation*}$

The velocity components can be expanded as a Taylor series in $$y$$. A cubic approximation to the velocity is then as follows:
$v_{x}^{\prime}=a_{0}+a_{1} y+a_{2} y^{2}+a_{3} y^{3}$
and $v_{y}^{\prime}=b_{0}+b_{1} y+b_{2} y^{2}+b_{3} y^{3}$
Use the no-slip condition to show that $$a_{0}=0$$ and $$b_{0}=0$$.
Also, since $$v_{x}$$ near the wall does not change with $$x$$, show that $$b_{1}=0$$.
Hence suggest an expression for the expansion of turbulent shear stress near the wall. How does it compare with the Prandtl mixing-length theory? Does the linear relation of the turbulent stress with $$y$$ hold?

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