Fluid Dynamics I

PROBLEM SET 5

Due November 16, 1998

1. (Reading, Milne-Thomson sec. 13.52 on ``stationary vortex filaments in the presence of a circular cylinder'' in 3rd edition.) Consider the following model of flow past a circular cylinder of radius a with two eddies downstream of the body. Consider two point vortices, of opposite strengths, the upper vortex having clockwise circulation -G(i.e. G > 0) located at the point c = be^iq, thus adding a term (iG/ 2p)ln(z-c) to the complex potential w, the other having circulation G at the point `c = be^-iq. Here b > a > 0.

Using the circle theorem write down the complex potential for the entire flow field, and determine by differentiation the complex velocity. Sketch the positions of the vortices and all vortex singularities within the cylinder, indicating their strengths.

2. Continuing problem 1, verify that x = ±a, y = 0 remain stagnation points of the flow. Show that the vortices will remain stationary behind the cylinder (i.e. not move with the flow) provided that

Show (by dividing both sides of the last equation by their conjugates and simplifying the result) that this relation implies b-a²/b = 2bsinq, that is, the distance between the exterior vortices is equal to the distance between a vortex and its image vortex.

3. Using the method of Blasius (see Milne-Thomson), show that the moment of a body in 2D potential flow, about the axis perpendicular to the plane, is given by

where Re denotes the real part and C is any simple closed curve about the body. Using this, verify by the residue method that the moment on a circular cylinder with a point vortex of circulation G at its center, in uniform flow, experiences zero moment.

4. Compute, using the Blasius formula, the force exerted by a simple source at the point c = be^iq, b > a upon a circular cylinder at the origin of radius a. The complex potential of a simple source is Q/2pln(z-c). (Use the circle theorem and residues). Verify that the cylinder is pulled toward the source.

5.Investigate the pressure distribution on a lifting flat plate of chord 4a with arbitrary circulation G, in uniform flow at angle of attack a in uniform flow of speed U. Give an expression for the pressure as a function of position, for both top and bottom surfaces. Describe the nature of the singularities at the leading and trailing edges. Apply the Kutta condition, and show in this case that singularity at the trailing edge disappears and

where the upper/lower sign refers to the upper/lower side of the plate.

6. Let the airfoil parameters k,b be independent of y in Prandtl's lifting-line theory. Also, assume the wing has the usual left-right symmetry of an airplane. Show, using Prandtl's theory, that for a given lift the induced drag is minimized for a wing having an elliptical planform. Show that in that case the coefficient of induced drag, C_Di = 2 x Drag/(rU²) and lift coefficient C_L = 2 x Lift/(rU² S) are related by

where S is the wing area and A is its aspect ratio 4b²/S, 2b= wingspan.