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Chapter 6: Problem 127

Consider the flow field formed by combining a uniform flow in the positive \(x\) direction and a source located at the origin. Let \(U=30 \mathrm{m} / \mathrm{s}\) and \(q=150 \mathrm{m}^{2} / \mathrm{s} .\) Plot the ratio of the local velocity to the freestream velocity as a function of \(\theta\) along the stagnation streamline. Locate the points on the stagnation streamline where the velocity reaches its maximum value. Find the gage pressure there if the fluid density is \(1.2 \mathrm{kg} / \mathrm{m}^{3}\)

Short Answer

Expert verified
The points where the velocity reaches its maximum value on the stagnation streamline are \(\theta = 0\) and \(\theta = 2 \pi\). The gauge pressure at these points for a fluid density of \(1.2 \mathrm{kg} / \mathrm{m}^{3}\) is 0 Pa.

Step by step solution

01

Determining the relationship between the local and freestream velocities

The velocity V at any point in the flow field can be calculated by the combination of the uniform flow source strength. This is calculated as \(V = U + \frac{q}{2\pi r}\). Then the ratio of the local to the freestream velocity \(V/U\) is given by: \(1+ \frac{q}{2 \pi r U}\)
02

Plotting the ratio as a function of \(\theta\)

The magnitude of velocity \(V\) as a function of \(\theta\) along the stagnation streamline (where \(r = \frac{q}{2 \pi U}\)) is defined as: \(V/U = 1 + cos \theta\). This equation can be plotted to show the ratio of the local to the freestream velocity with respect to \(\theta\).
03

Identify the maximum points

The maximum value of the velocity ratio function \(1 + cos \theta\) is obtained when \(cos \theta = 1\), i.e., when \(\theta = 0\) and \(\theta = 2 \pi\).
04

Compute the gauge pressure

Using Bernoulli's equation \(p - p_0 = 0.5 * \rho * (V^2 - U^2)\) where \(p_0\) is the far-field pressure, we can calculate the gauge pressure at the points of maximum velocity (i.e., \(\theta = 0\) and \(\theta = 2 \pi\)). Substituting the given values we get, \(p - p_0 = 0.5 * 1.2 * (30^2 - 30^2) = 0 Pa\).

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Most popular questions from this chapter

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

To model the velocity distribution in the curved inlet section of a water channel, the radius of curvature of the streamlines is expressed as \(R=L R_{0} / 2 y .\) As an approximation, assume the water speed along each streamline is \(V=10 \mathrm{m} / \mathrm{s}\) Find an expression for and plot the pressure distribution from \(y=0\) to the tunnel wall at \(y=L / 2,\) if the centerline pressure (gage) is \(50 \mathrm{kPa}, L=75 \mathrm{mm},\) and \(R_{0}=0.2 \mathrm{m} .\) Find the value of \(V\) for which the wall static pressure becomes 35 kPa.

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