Question

Problem 3 We wish to represent flow through the sketched constriction in an otherwise straight, parallel wall, two-dimensional channel of height Y. Flow through the channel is inviscid, incompressible and uniform far upstream with velocity V One way to study this problem is to introduce a concentrated vortex (vortex point) at a height Y above the lower wall. n) Write a formula for the streami function ψ of the flow: the formula must be written in Cartesian is constant at the coordinate system with point A (see the figure) as the orig Check that lower wall. b) Find the strength Γ needed for a given constriction height l. c) Find the velocities at points A and B. Which one is higher? d) Is the flow irrotational everywhere? Why?

Answer 1

Here it is given to find the stream function

First, we need to know the stream functions of uniform stream and vortex.

The stream functions are given as

urine = Vooy, vvor ter- Inr

where $\small r = \sqrt{(x - a)^2 + (y-b)^2}$

where a and b are the a and y coordinates of line vortex. In this case,

a=0, b = Y

Hence the stream function for the flow can be written as

$\small \psi = \psi_{line} + \psi_{vortex}$

$\small \psi = V_{\infty}y - \frac{\Gamma}{2\pi}ln\sqrt{x^2 + (y - Y)^2}$

At B, the vertical velocity is 0 i.e., $\small v= 0 \implies -\frac{\partial \psi}{\partial x} = 0$

Hence the above equation becomes,

$\small 0 = V_{\infty}H - \frac{\Gamma}{2\pi}ln\sqrt{(H-Y)^2}$

$\small \Gamma = \frac{2V_{\infty}H\pi}{ln\sqrt{(H-Y)^2}}$

We know

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Hence $\small u_x = V_{\infty} - \frac{\Gamma}{2\pi}\frac{y-Y}{x^2 + (y - Y)^2}$

$\small u_x_B = V_{\infty} - \frac{\Gamma}{2\pi}\frac{H-Y}{ (H - Y)^2}$

$\small u_x_A = V_{\infty} + \frac{\Gamma}{2\pi}\frac{1}{Y}$

For irrotationality, $\small \nabla \times \vec{v} = 0$ i.e., $\small -\frac{\partial^2\psi}{\partial x^2} - \frac{\partial^2\psi}{\partial y^2} = 0$