Conjugate depth

In fluid dynamics, the conjugate depths refer to the depth (y1) upstream and the depth (y2) downstream of the hydraulic jump whose momentum fluxes are equal for a given discharge (volume flux) q. The depth upstream of a hydraulic jump is always supercritical. It is important to note that the conjugate depth is different from the alternate depths for flow which are used in energy conservation calculations.

Mathematical derivation

My diagram.

Beginning with an equal momentum flux M and discharge q upstream and downstream of the hydraulic jump:

M=\frac{y_1^2}{2}+\frac{q^2}{g y_1}=\frac{y_2^2}{2}+\frac{q^2}{g y_2}.

Rearranging terms gives:

\frac{q^2}{g}\left(\frac{1}{y_1}-\frac{1}{y_2}\right)=\frac{1}{2}\left(y_z^2-y_1^2\right).

Multiply to get a common denominator on the left-hand side and factor the right-hand side:

\frac{q^2}{g}\left(\frac{y_2-y_1}{y_1 y_2}\right)=\frac{1}{2}(y_2-y_1)(y_2+y_1).

The (y2y1) term cancels out:

\frac{q^2}{g}\left(\frac{1}{y_1 y_2}\right)=\frac{1}{2}(y_2+y_1)\qquad\text{where }q_1^2=y_1^2 v_1^2=y_2^2 v_2^2.

Divide by y12

\frac{v_1^2}{g}\left(\frac{1}{y_1 y_2}\right)=\frac{1}{2 y_1^2}(y_2+y_1)\qquad\text{recall }F r_1^2=\frac{v_1^2}{g y_1}.

Thereafter multiply by y2 and expand the right hand side:

F r_2^2=\frac{y_2^2}{2 y_1^2}+\frac{y_2}{2 y_1}.

Substitute x for the constant y2/y1:

F r_1^2=\frac{x^2}{2}+\frac{x}{2} \Rightarrow 0=\frac{x^2}{2}+\frac{x}{2}-F r_1^2.

Solving the quadratic equation and multiplying it by \tfrac{\sqrt{4}}{2} gives:

x=\frac{-\tfrac{1}{2}\pm\sqrt{(1/2)^2-4(1/2)(F r_1^2)}}{2(1/2)}=-\frac{1}{2}\sqrt{(1/4)-8(F r_1^2)}.

Substitute the constant y2/y1 back in for x to get the conjugate depth equation

\frac{y_2}{y_1}=\frac{1}{2}\left(\sqrt{1+8 F r_1^2} - 1\right).
This article is issued from Wikipedia - version of the Monday, February 24, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.