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Solution: Start at the inside right corner, point A, and go around the complete path: Fig. The problem then becomes nonlinear, and is more difficult to solve than a problem purely in terms of the velocity potential. If you do indeed solve part a , can your final function F serve as b a velocity potential, or c a stream function? These functions define plane stagnation flow, Fig. If the flow possesses a velocity potential as defined by Eq.
If this is so, why do we back away from the full Navier-Stokes equation in solving potential flows? Interpret your result especially vis-a-vis the velocity potential. The water pressure far up-stream along the body centerline is kPa. Solution: a The nose radius is the distance a in Fig.
Does this manifold simulate a line source? If so, what is the equivalent source strength m? Solution: With that many small holes, equally distributed and presumably with equal flow rates, the manifold does indeed simulate a line source of strength Fig. Evaluate the circulation from Eq. Interpret your result. Solution: Break the path up into 1, 2, 3, 4 as shown.
A graph not requested is as follows: Fig. Using the r-momentum equation D. Find the location and magnitude of the lowest pressure. Assuming sea-level standard conditions at large r, a find the minimum pressure; b find the pressure at the match-point; and c show that both minimum and match-point pressures are independent of R. That is, these pressures depend only upon Vmax , wherever it occurs.
The jet volume flow is 0. If the jet is approximated as a line source, a locate the stagnation point S. Then, as in Fig. What should the pressure be at this point?
Solution: The surface velocity and surface contour are given by Eq. Is the pressure in c equal to the sum of those in a and b? What can you comment on this result? Is it always true that the pressure of a flow is equal to the sum of the pressures of its basic components? This is a special case in which the linear superposition of pressure works.
The two velocity fields are orthogonal to each other, and the inertia terms due to these basic flow fields do not interact with each other. The linear superposition of pressures does not work in general, even though it works for the velocity.
What is the flow pattern viewed from afar? The pattern viewed from afar is at right and represents a single source of strength 4m. Find the position of any stagnation point when the strengths at these two points are, respectively, a 1 and 4, b —1 and —4, and c 1 and —4. What is the pattern viewed from afar? Solution: The pattern viewed close-up is shown at right see Fig. Determine the resultant velocity induced by these two at point C.
Sum vertical and horizontal velocities from the sketch at right: Fig. Give a physical explanation of the flow pattern. Solution: The sum of stagnation flow plus a line source at the origin is The plot is below, using MATLAB, and represents stagnation flow toward a bump of height a. Solution: The velocities caused by each term—stream, vortex, and sink—are shown below. They have to be added together vectorially to give the final result: Fig.
Sketch the stream and potential lines in the upper half plane. If so, sketch the pressure coefficient along the wall, where po is the pressure at 0, 0. Find the minimum pressure point and indicate where flow separation might occur in the boundary layer. Beyond this point, pressure increases adverse gradient and separation is possible. Estimate a how far downstream and b how far normal to the paper the effects of the intake are felt in the ambient 8-m-deep waters.
The distance downstream from the sink is a and the distance normal to the paper is pa see Fig. Measured pressures at points A and B are kPa and 90 kPa, respectively, with uncertainties of 3 kPa each. Estimate the stream velocity and its uncertainty. Sketch the streamline and potential-line patterns.
Solution: This is a double-image flow and creates two walls, as shown. Estimate b the appropriate source strength m and c the pressure at the nose of the body.
Solution: We know, from Fig. If this tornado forms in sea-level standard air, at what radius will the local pressure be equivalent to 29 inHg? Solution: The vertical distance above the origin is a known multiple of m and a: Fig.
Determine the relationship between the maximum bump height h and the source strength m. How does the bump height decrease with distance sufficiently far from the center?
Find the resultant velocity induced by this pair at point A on the wall. Then we can sketch the source pair and the sink pair separately, as follows: You can see by symmetry that the new velocity normal to the wall is zero. Add up all the velocities to the right along the wall: 8.
Sketch the resulting streamlines and note any stagnation points. What would the pattern look like from afar? Viewed from afar, the pattern would look like a single source of strength 3 m.
Solution: a The straightforward, but unsatisfying, way to find length is to simply use Eq. What is the length? Estimate a the velocity at point A and b the location of point B where a particle approaching the stagnation point achieves its maximum deceleration. The numerical value of the maximum deceleration is P8.
Sketch your ideas of the body contours that would arise if the sources were a very weak; and b very strong. If you have solved Prob. Otherwise you have to work that out here. Find the body thickness for which cavitation will occur at point A. Trying out the answers to Prob. What are the height, width, and shoulder velocity of this oval? Solution: With reference to Fig. Similarly, Ans. The boat sails in seawater in standard atmosphere at 14 knots, parallel to the keel.
At a section 2 m below the surface, estimate the lowest pressure on the surface of the keel. Solution: Assume standard sea level pressure of , Pa. From Table A. Compute the resulting pressure and surface velocity at a the stagnation points, and b the upper and lower shoulders. What will be the lift per meter of cylinder width? Solution: Recall that Prob. From Eq. There are 10 bolts per meter of width on each side, and the inside pressure is 50 kPa gage.
Using potential theory for the outside pressure, compute the tension force in each bolt if the Fig. The intent is to use this cylinder as a stream velocimeter. The internal pressure is pi.
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