Chapter 7, Problem 1P

The slope of the free surface of a steady wave in one-dimensional flow in a shallow liquid layer is described by the equation

$\frac{\partial h}{\partial x}=-\frac{u}{g}\frac{\partial u}{\partial x}$

Use a length scale, L, and a velocity scale, V₀, to nondimensionalize this equation. Obtain the dimensionless groups that characterize this flow.

To determine

The dimensionless groups that characterizes the given flow.

Explanation of Solution

Given:

The slope of the free surface of a steady wave in one dimensional flow in a shallow liquid is as follows:

  $\frac{\partial h}{\partial x}=-\frac{u}{g}\frac{\partial u}{\partial x}$        (I)

Calculation:

From Equation (I),

The length dimensional quantity which are having same unit of measurement are h and x.

The velocity dimensional quantity is u.

Consider the reference dimensional quantities as follows:

The length is $L$.

The velocity is $V_0$.

Dividing the dimensional quantity by its reference dimensional quantity yields non-dimensional quantity. The non-dimensional quantities are denoted with asterisk.

Divide the given dimensional quantities with their reference as follows:

  $\begin{array}{l}\frac{h}{L}=h^{*}\\ h=h^{*}L\end{array}$

  $\begin{array}{l}\frac{x}{L}=x^{*}\\ x=x^{*}L\end{array}$

  $\begin{array}{l}\frac{u}{V_0}=u^{*}\\ u=u^{*}{V}_0\end{array}$

Substitute $h=h^{*}L$, $x=x^{*}L$, and $u=u^{*}{V}_0$ in Equation (I).

  $\begin{array}{l}\frac{\partial \left(h^{*}L\right)}{\partial \left(x^{*}L\right)}=-\frac{u^{*}{V}_0}{g}\frac{\partial \left(u^{*}{V}_0\right)}{\partial \left(x^{*}L\right)}\\ \frac{\partial h^{*}}{\partial x^{*}}=-\frac{V_0{}^2}{gL}{u}^{*}\left(\frac{\partial u^{*}}{\partial x^{*}}\right)\end{array}$

Here,

The dimensionless group is $\frac{V_0{}^2}{gL}$ which is equal to the square of the Froude’s number.

Thus, the dimensionless groups that characterizes the given flow is $\underset{\_}{\frac{V_0{}^2}{gL}}$.