Chapter 6, Problem 6.34P

To determine

To express:

The time-averaged x -momentum (6.21) equation by direct substitution of Eqs. (6.19) into the momentum equation (6.14).

Answer to Problem 6.34P

The following equation is derived:

$\rho [\frac{d\overline{u}}{dt}+\frac{\partial }{\partial x}(\overline{u^{\text{'}2}})+\frac{\partial }{\partial y}(\overline{u^{\text{'}}{v}^{\text{'}}})+\frac{\partial }{\partial z}(\overline{u^{\text{'}}{w}^{\text{'}}})]=-\frac{\overline{\partial p}}{\partial x}+\rho g_x+\mu {\left(\overline{V}\right)}^2\overline{u}$

Explanation of Solution

Given Information:

Convective acceleration:

$\frac{du}{dt}=\frac{\partial }{\partial x}(u^2)+\frac{\partial }{\partial y}(uv)+\frac{\partial }{\partial z}(uw)$

Concept Used:

Equation 6.21:

$\rho \frac{d\overline{u}}{dt}=-\frac{\partial \overline{p}}{\partial x}+\rho g_x+\frac{\partial }{\partial x}(\mu \frac{\partial \overline{u}}{\partial x}-\rho \overline{u^{\text{'}2}})+\frac{\partial }{\partial y}(\mu \frac{\partial \overline{u}}{\partial y}-\rho \overline{u^{\text{'}}{v}^{\text{'}}})+\frac{\partial }{\partial z}(\mu \frac{\partial \overline{u}}{\partial z}-\rho \overline{u^{\text{'}}{w}^{\text{'}}})$

Equation 6.19:

$\begin{array}{l}u=\overline{u}+u^{\text{'}}\\ v=\overline{v}+v^{\text{'}}\\ w=\overline{w}+w^{\text{'}}\\ p=\overline{p}+p^{\text{'}}\end{array}$

Equation 6.14:

Continuity: $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$

Momentum: $\rho \frac{dV}{dt}=-\overline{V}p+\rho g+\mu \overline{V^2}V$

x -momentum equation is given by:

$\rho \frac{du}{dt}=-\frac{\partial p}{\partial x}+\rho g_x+\mu \overline{V^2}u$

Therefore, $\rho {[\frac{\partial }{\partial x}(u^2)+\frac{\partial }{\partial y}(uv)+\frac{\partial }{\partial z}(uw)]}^{}=-\frac{\partial p}{\partial x}+\rho g_x+\mu \overline{V^2}u\dots \left(1\right)$

Calculation:

Substituting the values in equation 1,

$\rho {[\frac{\partial }{\partial x}\{{(\overline{u}+u^{\text{'}})}^2\}+\frac{\partial }{\partial y}\{(\overline{u}+u^{\text{'}})(\overline{v}+v^{\text{'}})\}+\frac{\partial }{\partial z}\{(\overline{u}+u^{\text{'}})(\overline{w}+w^{\text{'}})\}]}^{}=-\frac{\partial }{\partial x}(\overline{p}+p^{\text{'}})+\rho g_x+\mu \overline{V^2}(\overline{u}+u^{\text{'}})$

$\rho [\frac{\partial }{\partial x}\{({\overline{u}}^2+u^{\text{'}}{}^2+2\overline{u}{u}^{\text{'}})\}+\frac{\partial }{\partial y}\{(\overline{u}\overline{v}+u^{\text{'}}\overline{v}+\overline{u}{v}^{\text{'}}+u^{\text{'}}{v}^{\text{'}})\}+\frac{\partial }{\partial z}\{(\overline{u}\overline{w}+u^{\text{'}}\overline{w}+\overline{u}{w}^{\text{'}}+u^{\text{'}}{w}^{\text{'}})\}]=-\frac{\partial }{\partial x}(\overline{p}+p^{\text{'}})+\rho g_x+\mu \overline{V^2}(\overline{u}+u^{\text{'}})$

$\rho [\frac{\partial }{\partial x}({\overline{u}}^2)+\frac{\partial }{\partial y}(\overline{u}\overline{v})+\frac{\partial }{\partial z}(\overline{u}\overline{w})+\frac{\partial }{\partial x}(u^{\text{'}}{}^2+2\overline{u}{u}^{\text{'}})+\frac{\partial }{\partial y}(u^{\text{'}}\overline{v}+\overline{u}{v}^{\text{'}}+u^{\text{'}}{v}^{\text{'}})+\frac{\partial }{\partial z}(u^{\text{'}}\overline{w}+\overline{u}{w}^{\text{'}}+u^{\text{'}}{w}^{\text{'}})]=-\frac{\partial }{\partial x}(\overline{p}+p^{\text{'}})+\rho g_x+\mu \overline{V^2}(\overline{u}+u^{\text{'}})$

Time average of entire equation is to be taken. Also, we know:

$\begin{array}{l}\overline{u^{\text{'}}}=0\\ \overline{v^{\text{'}}}=0\\ \overline{w^{\text{'}}}=0\\ \overline{p^{\text{'}}}=0\end{array}$

After the time average is taken and on substitution of the above values:

$\rho [\frac{d\overline{u}}{dt}+\frac{\partial }{\partial x}(\overline{u^{\text{'}2}})+\frac{\partial }{\partial y}(\overline{u^{\text{'}}{v}^{\text{'}}})+\frac{\partial }{\partial z}(\overline{u^{\text{'}}{w}^{\text{'}}})]=-\frac{\overline{\partial p}}{\partial x}+\rho g_x+\mu {\overline{V}}^2\overline{u}$

Conclusion:

The equation is $\rho [\frac{d\overline{u}}{dt}+\frac{\partial }{\partial x}(\overline{u^{\text{'}2}})+\frac{\partial }{\partial y}(\overline{u^{\text{'}}{v}^{\text{'}}})+\frac{\partial }{\partial z}(\overline{u^{\text{'}}{w}^{\text{'}}})]=-\frac{\overline{\partial p}}{\partial x}+\rho g_x+\mu {\overline{V}}^2\overline{u}$.