Combined Flow: Uniform and Vortex Flow

Just as we had used a combined uniform and source flow in a previous post, we will now combine the uniform flow at zero angle of attack with vortex flow. The main point of this post is to compute the circulation over a couple of closed loops and see how they change depending on where they are in the flow-field. The Cartesian velocity components can be seen in the equations below.

(1) $\begin{equation*} \begin{aligned} V_x &= V_\infty\cos\left(\alpha\right) + \frac{\Gamma dy}{2\pi r^2} \\[5pt] V_y &= V_\infty\sin\left(\alpha\right) + \frac{-\Gamma dx}{2\pi r^2} \end{aligned} \end{equation*}$

The code can be seen below.

% Vortex knowns
Vinf  = 1;									% Velocity
alpha = 0;									% Angle of attack [deg]
gamma = 30;								 % Vortex strength
X0    = 0;									% Vortex X origin
Y0    = 0;									% Vortex Y origin

% Create the grid
numX = 50;									      % Number of X points
numY = 50;									      % Number of Y points
X    = linspace(-10,10,numX)';    % Create X points array
Y    = linspace(-10,10,numY)';    % Create Y points array
[XX,YY] = meshgrid(X,Y);					% Create the meshgrid

% Solve for velocities
Vx = zeros(numX,numY);						 % Initialize X velocity
Vy = zeros(numX,numY);						 % Initialize Y velocity
r  = zeros(numX,numY);						 % Radius
for i = 1:1:numX							     % Loop over X-points
	for j = 1:1:numY						     % Loop over Y-points
		x       = XX(i,j);					   % X-value of current point
        y       = YY(i,j);				 % Y-value of current point
        dx      = x - X0;					% X distance
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        r       = sqrt(dx^2 + dy^2);		% Distance
        Vx(i,j) = Vinf*cosd(alpha) + (gamma*dy)/(2*pi*r^2);
        Vy(i,j) = Vinf*sind(alpha) + (-gamma*dx)/(2*pi*r^2);
	end
end

Two plots are shown in Fig. 1 and Fig. 2. Figure 1 is for the combined uniform and vortex flow when the vortex strength is $\Gamma = 10$ , whereas Fig. 2 is for the combined uniform and vortex flow when the vortex strength is $\Gamma = 30$ . In both plots, the red arrows are the velocity vectors, the black lines are streamlines, and the blue and magenta curves are used to compute the line integral in the circulation equation.

Figure 1: Combined uniform and vortex flow, where the vortex strength is $\Gamma = 10$ . Blue and magenta contours are used for circulation line integral, with respective values in the legend.

Figure 2: Combined uniform and vortex flow, where the vortex strength is $\Gamma = 30$ . Blue and magenta contours are used for circulation line integral, with respective values in the legend.

We can see from this resulting circulation calculations for both plots that when the vortex is enclosed in the loop, the circulation/vortex strength is accurately predicted, but when the loop does not enclose the vortex, the resulting circulation is zero. This will be important for our lift calculation for the airfoils we will be doing in the vortex panel method (VPM). It also helps shed some light on why the source panel method (SPM), which does not contain any vortex flows in its calculation, does not result in any circulation, and thus no lift.