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10.2 In polar coordinates, the continuity equation for steady incompressible flow becomes...Derive equations (10-10), using this relation.... (10-10)
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10.3 Make an analytical model of a tornado using an irrotational vortex (with velocity inversely proportional to distance from the center) outside a central core (with velocity directly proportional to distance). Assume that the core diameter is 200 ft and the static pressure at the center of the core is 38 psf below ambient pressure. Find
a. The maximum wind velocity
b. The time it would take a tornado moving at 60 mph to lower the static pressure from –10 to –38 psfg
c. The variation in stagnation pressure across the tornado; Euler’s equation may be used to relate the pressure gradient in the core to the fluid acceleration
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10.4 In Problem 10.7, explain how one could obtain ∂vθ/∂θ at the stagnation point, using only r and ∂vr/∂r.Problem 10.7For the flow about a cylinder, find the velocity variation along the streamline leading to the stagnation point. What is the velocity derivative ∂vr/∂r at the stagnation point?
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10.5 For the velocity potentials given below, find the stream function and sketch the streamlines
a. ...
b. ...
c. ...
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10.6 The stream function for an incompressible, twodimensional flow field is...For this flow field, plot several streamlines for 0 ≤ θ ≤ π/3.
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10.7 Calculate the total lift force on the Arctic hut shown below as a function of the location of the opening. The lift force results from the difference between the inside pressure and the outside pressure. Assume potential flow, and that the hut is in the shape of a half-cylinder....
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10.8 Consider three equally spaced sources of strength m placed at (x, y) = (–a, 0), (0, 0), and (a, 0). Sketch the resulting streamline pattern. Are there any stagnation points?
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10.9 The stream function for an incompressible, twodimensional flow field is...For this flow field, sketch several streamlines.
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10.10 A source of strength 1.5 m2/s at the origin is combined with a uniform stream moving at 9 m/s in the x direction. For the half-body that results, find
a. The stagnation pointb. The body height as it crosses the y axisc. The body height at large xd. The maximum surface velocity and its position (x, y)
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10.11 A 2-m-diameter horizontal cylinder is formed by bolting two semicylindrical channels together on the inside. There are 12 bolts per meter of width holding the top and bottom together. The inside pressure is 60 kPa (gage). Using potential theory for the outside pressure, compute the tension force in each bolt if the free-stream fluid is sea-level air and the free-stream wind speed is 25 m/s.
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10.12 In Example 3 we began finding the equation for the stream function by integrating equation (3). Repeat this example, but instead begin by integrating equation (4) and show that no matter which equation you begin with, the results are identical.Example 3The steady, incompressible flow field for two-dimensional flow is given by the following velocity components: vx = 16y – x and vy = 16x + y. Determine the equation for the stream function and the velocity potential.First, let’s check to make sure continuity is satisfied:...So continuity is satisfied, which is a necessary condition for us to proceed.We defined the stream function as...and...Thus,...We can begin by integrating equation (4) or equation (5). Either will result in the same answer. (Problem 10.24 will let you verify this.) We will choose to begin by integrating equation (4) partially with respect to y:...where f1(x) is ... an arbitrary function of x.The result is that we now have two equations for vy, equations (5) and (7). We can now equate these and solve for f2(x):...Solving for f2(x),...So...The integration constant C is added to the above equation since f is a function of x only. The final equation for the stream function is...The integration constant C is added to the above equation since f is a function of x only. The final equation for the stream function is...The constant C is generally dropped from the equation because the value of a constant in this equation is of no significance. The final equation for the stream function is...One interesting point is that the difference in the value of one stream line in the flow to another is the volume flow rate per unit width between the two streamlines.Next we want to find the equation for the velocity potential. Since a condition for the velocity potential to exist is irrotational flow, we must first determine whether the flow in this example is irrotational.To do this, we must satisfy equation (10-1)....So the flow is irrotational, as required.We now want to determine the equation for the velocity potential. The velocity potential is defined by equation (10-15):...or......Differentiating with respect to y and equating to vx,...Thus,...and...so that the final equation for the velocity potential is...
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10.13 The stream function for steady, incompressible flow is given by...Determine the velocity potential for this flow.
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