CAPÍTULO 4 IMPLEMENTACIÓN Y PRUEBAS
4.3 R ESULTADOS DE LA APLICACIÓN
4.3.1 Impacto del agente PloRA en el Proyecto de Prensa
As a final illustration of the superposition method, we will consider a case of considerable utility. When the solutions for uniformflow and the doublet are superposed, the result, similar to the past case, defines a streamline pattern inside and around the outside surface of a body. In this case the body is closed and the exteriorflow pattern is that of ideal flow over a cylinder. The expressions for Y and f are
Y Yuniform flow Ydoublet
v∞y lsin q
r v∞rsinq l sinq r
v∞r l r
sinq f funiform flow fdoublet
v∞xlcosq
r v∞r cosq lcosq r
v∞rl r
cosq
It is useful, at this point, to examine the above expressions in more detail. First, for the stream function
Y v∞r l r
sinq
v∞r 1 l=v∞ r2
sinq
where, as we recall,l is the doublet strength. If we choose l such that l
v∞ a2 where a is the radius of our cylinder, we obtain
Y r; q v∞r sinq 1 a2 r2
which is the expression used earlier, designated as equation (10-11).
▶ 10.9
CLOSURE
In this chapter, we have examined potentialflow. A short summary of the properties of the stream function and the velocity potential is given below.
Stream function
1. A stream function Y(x, y) exists for each and every two-dimensional, steady, incompressibleflow, whether viscous or inviscid.
2. Lines for which Y(x, y) constant are streamlines.
3. In Cartesian coordinates,
vx@Y
@y vy @Y
@x (10-23a)
and in general,
vs@Y
@n (10-23b)
where n is 90° counterclockwise from s.
4. The stream function identically satisfies the continuity equation.
5. For an irrotational, steady incompressible flow,
Ñ2Y 0 (10-24)
Velocity potential
1. The velocity potential exists if and only if the flow is irrotational. No other restrictions are required.
2. Ñf v.
3. For irrotational, incompressible flow, Ñ2f 0.
4. For steady, incompressible two-dimensional flows, lines of constant velocity potential are perpendicular to the streamlines.
▶
PROBLEMS
10.1 In polar coordinates, show that Ñ v 1
r
@ rvq
@r
@vr
@q
ez
10.2 Determine the fluid rotation at a point in polar coordi-nates, using the method illustrated in Figure 10.1.
10.3 Find the stream function for aflow with a uniform free-stream velocityv∞. The free-stream velocity intersects the x axis at an anglea.
10.4 In polar coordinates, the continuity equation for steady incompressibleflow becomes
1 r
@
@r rvr 1 r
@vq
@q 0 Derive equations (10-10), using this relation.
10.5 The velocity potential for a given two-dimensionalflow field is
f 5 3
x3 5xy2
Problems ◀ 139
Show that the continuity equation is satisfied and determine the corresponding stream function.
10.6 Make an analytical model of a tornado using an irrotational vortex (with velocity inversely proportional to dis-tance 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 thefluid acceleration
10.7 For theflow 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?
10.8 In Problem 10.7, explain how one could obtain@vq/@q at the stagnation point, using only r and@vr/@r.
10.9 At what point on the surface of the circular cylinder in a potentialflow does the pressure equal the free-stream pressure?
10.10 For the velocity potentials given below,find the stream function and sketch the streamlines
a. f v∞L x
10.11 The stream function for an incompressible, two-dimensionalflow field is
y 2r3sin 3q
For thisflow field, plot several streamlines for 0 q p/3.
10.12 For the case of a source at the origin with a uniform free-stream plot the free-streamliney 0.
10.13 In Problem 10.12, how far upstream does theflow from the source reach?
10.14 Determine the pressure gradient at the stagnation point of Problem 10.10(a).
10.15 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 potentialflow, and that the hut is in the shape of a half-cylinder.
Opening
10.16 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?
10.17 Sketch the streamlines and potential lines of theflow due to a line source of at (a, 0) plus an equivalent sink at ( a, 0).
10.18 The stream function for an incompressible, two-dimensionalflow field is
y 3x2y y
For thisflow field, sketch several streamlines.
10.19 A line vortex of strength K at (x, y) (0, a) is combined with opposite strength vortex at (0, a). Plot the streamline pattern andfind the velocity that each vortex induces on the other vortex.
10.20 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 point
b. The body height as it crosses the y axis c. The body height at large x
d. The maximum surface velocity and its position (x, y) 10.21 When a doublet is added to a uniform stream so that the source part of the doublet faces the stream, a cylinder flow results. Plot the streamlines when the doublet is reversed so that the sink faces the stream.
10.22 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-streamfluid is sea-level air and the free-stream wind speed is 25 m/s.
10.23 For the stream function given by y 6x2 6y2
determine whether thisflow is rotational or irrotational.
10.24 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.
10.25 The stream function for steady, incompressibleflow is given byY y2 xy x2. Determine the velocity components for this flow, and find out whether the flow is rotational or irrotational.
10.26 The stream function for steady, incompressibleflow is given by
Y x; y 2x2 2y2 xy Determine the velocity potential for thisflow.