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Chapter 6: Problem 126

Consider the flow field formed by combining a uniform flow in the positive \(x\) direction and a source located at the origin. Obtain expressions for the stream function, velocity potential, and velocity field for the combined flow. If \(U=25 \mathrm{m} / \mathrm{s}\), determine the source strength if the stagnation point is located at \(x=-1 \mathrm{m} .\) Plot the streamlines and potential lines.

Short Answer

Expert verified
The stream function for the combined flow is \(唯 = Uy + \frac{m}{2蟺} 胃\), the velocity potential is \(桅 = Ux + \frac{m}{2蟺} ln r\), and the velocity field is \(V_r = \frac{m}{2蟺r} - Ucos胃\), \(V_胃 = \frac{1}{r} \frac{d桅}{d胃} = U sin胃\). The source strength is \(50蟺\) m虏/s. Streamlines and potential lines plotting will show the path of a fluid particle and regions of equal potential respectively.

Step by step solution

01

Defining Uniform flow and Source flow

Uniform flow is defined by \(唯_1 = Uy\), \(桅_1 = Ux\) and Source flow is defined by \(唯_2 = \frac{m}{2蟺} 胃\), \(桅_2 = \frac{m}{2蟺} ln r\), where \(m\) is source strength, \(r\) is radial distance from the source, and \(胃\) is the angle.
02

Combine the flow definitions

The combined flow is obtained by superposing the individual flows. So, the stream function would be \(唯 = 唯_1 + 唯_2 = Uy + \frac{m}{2蟺} 胃\) and the velocity potential would be \(桅 = 桅_1 + 桅_2 = Ux + \frac{m}{2蟺} ln r\).
03

Calculate velocity field

The velocity field is obtained by taking the gradient of the velocity potential. This gives \(V_r = \frac{d桅}{dr} = \frac{m}{2蟺r} - Ucos胃\), \(V_胃 = \frac{1}{r} \frac{d桅}{d胃} = U sin胃\).
04

Find the source strength

The stagnation point is where velocity is zero. Setting \(V_r = 0\) and using the given \(x = -1\) m which gives \(r = 1\) m and \(胃 = 蟺\), we find the source strength \(m\). This gives \(\frac{m}{2蟺} - 25cos蟺 = 0\). Solving for \(m\) gives \(m = 50蟺\) m虏/s.
05

Plot streamlines and potential lines

Plotting the streamlines requires setting the stream function \(唯\) to a constant for various values. Similarly, potential lines are plotted by setting the velocity potential function \(桅\) to a constant. These can be done using a software tool like MATLAB or Python. Note that the physical interpretation is streamline shows path of a fluid particle and potential line shows regions of equal potential.

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Most popular questions from this chapter

Consider the flow represented by the stream function \(\psi=A x^{2} y,\) where \(A\) is a dimensional constant equal to 2.5 \(\mathrm{m}^{-1} \cdot \mathrm{s}^{-1}\). The density is \(1200 \mathrm{kg} / \mathrm{m}^{3}\). Is the flow rotational? Can the pressure difference between points \((x, y)=(1,4)\) and (2,1) be evaluated? If so, calculate it, and if not, explain why.

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An open-circuit wind tunnel draws in air from the atmosphere through a well- contoured nozzle. In the test section, where the flow is straight and nearly uniform, a static pressure tap is drilled into the tunnel wall. A manometer connected to the tap shows that static pressure within the tunnel is \(45 \mathrm{mm}\) of water below atmospheric. Assume that the air is incompressible, and at \(25^{\circ} \mathrm{C}, 100 \mathrm{kPa}\) (abs). Calculate the air speed in the wind-tunnel test section.

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The stream function of a flow field is \(\psi=A x^{2} y-B y^{3},\) where \(A=1 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1}, B=\frac{1}{3} \mathrm{m}^{-1} \cdot \mathrm{s}^{-1},\) and the coordinates are measured in meters. Find an expression for the velocity potential.
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