AAE 3610: You Consider a Flow Over a Vortex Sheet at an Angle of Attack: Aerodynamics Assignment, WMU, US

Project Brief:

You consider a flow over a vortex sheet at an angle of attack (see Fig. 1).  The relevant formulas are given below.

• The velocity of uniform flow is where α is the angle of attack.
• The velocity of the ith point vortex is ; γi is the strength of the ith vortex and  is the radial distance between the ith vortex at  and a point  in space defined as
• As shown in Fig. 1, you place N vortices on the x-coordinate in the uniform flow. The total velocity is given by the superposition of the uniform flow and a sheet of vortices distributed on the x-axis, i.e., the coordinates of the leading edge.

• The distribution of the vortex sheet strength is given by an ad-hoc function, The amplitude Am and the exponent n will be determined based on the non-penetration boundary condition. 

• The given parameters are the freestream velocity U∞, the angle of attack α, the chord length c, and the coordinates of the leading edge (x0, y0), and the total number of vortices N. The adjustable parameters are the amplitude Am and the exponent n.

Tasks:

• Make a Matlab program to calculate the velocity (ux, uy) over the vortex sheet (a flat plate) using the above formulas.  Please reference the program, “Vortex_Sheet.m”.
• Calculate the normal velocity component uy on the vortex sheet and observe if it is zero.  It should be zero to meet the non-penetration boundary condition,  Calculate the absolute value or norm ||uy||.
• Determine the amplitude Am and the exponent n to meet the non-penetration boundary condition on the vortex sheet.  A manual optimization procedure is used to minimize the absolute value or norm ||uy||.
• Using the optimal values of Am and n, calculate and plot the distribution of
• Plot the velocity field over the vortex sheet.
• Calculate the total circulation Γ by integrating the distribution of γi.
• Calculate the sectional lift coefficient  Based on the KJ theorem
• Write a small technical paper with figures in Word to formulate the theory, discuss the results, and make conclusions.

Fig. 1. A vortex sheet in a uniform flow

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