ME 4943 Course Project - Part I

Transcription

1 ME 4943 Course Project - Part I Numerical Implementation of the Source Panel Method by Jeffrey A. Kornuta Mechanical Engineering Louisiana State University October 9, 27 1

2 Contents 1 Verification of Algorithm N = N = Method Applied to Arbitrary Shape The Square Appendices 8 A MATLAB Code 8 A.1 Part I A.2 Part II List of Figures 1 Visualizations of (a) N = 3 and (b) N = 5 for N = 3, R = 1 m, and V = 2 m/s Visualizations of (a) v(θ) and (b) C p (θ) for N = 3, R = 1 m, and V = 2 m/s Visualizations of (a) v(θ) and (b) C p (θ) for N = 5, R = 1 m, and V = 2 m/s Visualization of square with N = 8 panels Visualizations of (a) v(θ) and (b) C p (θ) for N = 8, (2 2) m, and V = 2 m/s

3 1 Verification of Algorithm Two cases are considered: 1. N = 3, R = 1 m, and V = 2 m/s 2. N = 5, R = 1 m, and V = 2 m/s. 1 Geometry, N = 3 1 Geometry, N = !.2.4!.4.6!.6.8! (a) N = 3!1!1! (b) N = 5 Figure 1: Visualizations of (a) N = 3 and (b) N = 5 for N = 3, R = 1 m, and V = 2 m/s. The following governing equations are utilized by the source panel method to generate the velocity and pressure coefficient distribution at the surface of a shape: V cos β i + λ N i 2 + λ j 2π I ij =, where (1) j=1,j i I ij = (ln r ij ) ds j. (2) n i Solving for λ = {λ 1, λ 2,..., λ N } using Eq.1 and Eq.2, one can solve for the velocities at the surface of the circle using the following equation: V i = V sin β i + N j=1,j i 3 j λ j 2π s j s j (ln r ij ) ds j. (3)

4 Finally, the pressure coefficients can be found using the following: C p,i = 1 ( Vi V ) 2. (4) Refer to pages in Anderson s Fundamentals of Aerodynamics, Fourth edition for details regarding notation. 1.1 N = 3 4 Velocity v(θ) 1 Pressure coefficient C p (θ) Velocity (m/s) 1 C p θ (degrees) (a) v(θ) θ (degrees) (b) C p (θ) Figure 2: Visualizations of (a) v(θ) and (b) C p (θ) for N = 3, R = 1 m, and V = 2 m/s. Note that with only three panels, the geometry forms a triangle [Figure 1a]. Although when the analytical and numerical values of v(θ) and C p (θ) are plotted against each other, the marked points lie on the curve [Figure 2], lines connecting the numerically calculated points would reveal an extremely inaccurate representation of the analytical solutions. Thus, more panels would be needed to give a good approximation of the velocity and pressure coefficient distributions. 4

5 1.2 N = 5 4 Velocity v(θ) 1 Pressure coefficient C p (θ) Velocity (m/s) 1 C p θ (degrees) (a) v(θ) θ (degrees) (b) C p (θ) Figure 3: Visualizations of (a) v(θ) and (b) C p (θ) for N = 5, R = 1 m, and V = 2 m/s. Note that with 5 panels, the geometry closely represents a circle [Figure 1b]. When the analytical and numerical values of v(θ) and C p (θ) are plotted against each other, the marked points not only lie on the curve [Figure 3], but lines that would connect the numerically calculated points would reveal an accurate representation of the analytical solutions. Thus, this number of panels gives a good approximation of the velocity and pressure coefficient distributions. In fact, reducing the number of panels to as little as 2 would yield a fair representation. These results suggest that the algorithm used for this scenario is valid. 5

6 2 Method Applied to Arbitrary Shape 2.1 The Square 1.5 Geometry, N = x (m) y (m) Figure 4: Visualization of square with N = 8 panels. The algorithm previously defined [Appendix A.1] will now be applied to an arbitrary shape in this case, a (2 2) m square [Figure 4, Appendix A.2]. For this square, the number of panels N = 8, and from the plots generated, this number seems to indicate that the surface is well-resolved [Figure 5]. As one may notice, the overall characteristics of these curves coincide with the results produced by the circle [Figure 3]. The stagnation points are located in the same locations, and the average maxima and minima values remain similar. These similarities suggest that the results for the square are sound. The only noticeable peculiarities lie with the values obtained for v(θ) and C p (θ) around the corners of the square. At these corners, lim R R, where R is the radius of curvature at the edges; thus, values of v (and correspondingly C p ) yield singularities at these points. These points are clearly seen in Figure 5 where there exist sharp spikes. 6

7 6 Velocity v(θ) 1 Pressure coefficient C p (θ) Velocity (m/s) C p θ (degrees) (a) v(θ) θ (degrees) (b) C p (θ) Figure 5: Visualizations of (a) v(θ) and (b) C p (θ) for N = 8, (2 2) m, and V = 2 m/s. In reality, this type of behavior would not occur. Since singularities of this kind are rarely seen in nature, these results suggest that a deficiency in the algorithm exists. In brief, this deficiency is characterized by two things: the absence of infinitely small radii of curvature, and the presence of viscous effects in the fluid. In actuality, edges cannot physically have an infinitely small radius of curvature, so these singularities would not exist in this scenario since R >. Also, the boundary layer of the surrounding fluid would separate at these sharp edges if they were indeed that sharp. Viscosity would cause the boundary layer to peel off at the corners, forming a wake. This scenario is void of singularities and would prove to be a much more reliable model to represent this shape, as compared to the inviscid model described by the source panel method. 7

8 Appendices A MATLAB Code A.1 Part I % Jeff Kornuta % ME Project Part I-1 clear all; clf; clc; % ask user for required information N = input( Enter number of panels: ); fprintf( \n ); U = input( Enter freestream velocity [m/s]: ); fprintf( \n ); R = 1; % plot a plane-jane circle for t = 1:361 theta = (t-1)*pi/18; u(t) = cos(theta); v(t) = sin(theta); %hold off %plot(u,v, r ) %axis square % now plot polygon (endpoints) hold on for i = 1:(N+1) dtheta = 2*pi/N; beta(i) = pi-(i-1)*dtheta; Phi(i) = beta(i)-pi/2; theta(i) = beta(i)+dtheta/2; tempx(i) = R*cos(theta(i)); tempy(i) = R*sin(theta(i)); %plot(tempx,tempy) 8

9 %pause(.25) % resize X, Y, and Beta to have one less element X(i) = tempx(i); Y(i) = tempy(i); if (beta(i) < ) beta(i) = beta(i)+2*pi; Beta(i) = beta(i)*18/pi; % plot midpoints of panels if (i ~= N) x(i) = (X(i+1)+X(i))/2; y(i) = (Y(i+1)+Y(i))/2; else x(i) = (X(1)+X(N))/2; y(i) = (Y(1)+Y(N))/2; % find length of segments S = sqrt((x(1)-x(n))^2+(y(1)-y(n))^2); %plot(x(i),y(i), o ) %pause(.25) % that was fun, now let s find I(i,j) for j = 1:N if (i ~= j) A(i,j) = -(x(i)-x(j))*cos(phi(j)) - (y(i)-y(j))*sin(phi(j)); B(i,j) = (x(i)-x(j))^2 + (y(i)-y(j))^2; % don t divide by zero if (B(i,j) == ) B(i,j) = 1e-12; C(i,j) = sin(phi(i)-phi(j)); D(i,j) = (y(i)-y(j))*cos(phi(i)) - (x(i)-x(j))*sin(phi(i)); E(i,j) = (x(i)-x(j))*sin(phi(j)) - (y(i)-y(j))*cos(phi(j)); 9

10 % please, stop dividing by zero if (E(i,j) == ) E(i,j) = 1e-12; I(i,j) = C(i,j)/2*log((S^2+2*A(i,j)*S+B(i,j))/B(i,j)) +... (D(i,j)-A(i,j)*C(i,j))/E(i,j)*(atan((S+A(i,j))/E(i,j))-... atan(a(i,j)/e(i,j))); else I(i,j) = pi; % set up system of equations for j = 1:N M(i,j) = I(i,j)/(2*pi); b(i,1) = -U*cos(beta(i)); % get those lambdas & check em lambda = inv(m)*b Sum = sum(lambda) % find velocities & pressure coefficients Vs = ; for j = 1:N if (i ~= j) M2(i,j) = ((D(i,j)-A(i,j)*C(i,j))/(2*E(i,j)))... *log((s^2+2*a(i,j)*s+b(i,j))/b(i,j))-c(i,j)... *(atan((s+a(i,j))/e(i,j))-atan(a(i,j)/e(i,j))); Vs = Vs+lambda(j)/(2*pi)*M2(i,j); V(i) = -(U*sin(beta(i))+Vs); Cp(i) = 1-(V(i)/U)^2; 1

11 % plot velocities & pressure coefficients for i = 1:361 t = (i-1)*pi/18; Cpreal(i) = 1-4*(sin(t))^2; Vreal(i) = -2*U*sin(t); clear t; t=[:36]; hold off %plot(t,vreal) plot(t,cpreal) axis square hold on %plot(beta,v, + ) plot(beta,cp, + ) xlabel( \theta (degrees) ); %ylabel( Velocity (m/s) ); %title( Velocity v(\theta) ); ylabel( C_p ); title( Pressure coefficient C_p(\theta) ); 11

12 A.2 Part II % Jeff Kornuta % ME Project Part I-2 clear all; clf; clc; % set required information N = 8; U = 2; S =.1; % plot a square hold off t = [-1:.1:1]; one = ones(1,21); negone = -1*ones(1,21); % plot(negone,t) % hold on % plot(t,one) % plot(one,t) % plot(t,negone) % axis([ ]) % axis square % find endpoints, midpoints, and angles for i = 1:11 X(i) = -1; Y(i) = (i-1)*s; if (i ~= 11) x(i) = -1; y(i) = Y(i)+S/2; beta(i) = pi; Phi(i) = pi/2; for i = 12:31 X(i) = -1+S*(i-11); Y(i) = 1; 12

13 x(i-1) = X(i)-S/2; y(i-1) = Y(i); beta(i-1) = pi/2; Phi(i-1) = ; for i = 32:51 X(i) = 1; Y(i) = 1-S*(i-31); x(i-1) = 1; y(i-1) = Y(i)+S/2; beta(i-1) = ; Phi(i-1) = 3*pi/2; for i = 52:71 X(i) = 1-S*(i-51); Y(i) = -1; x(i-1) = X(i)+S/2; y(i-1) = Y(i); beta(i-1) = 3*pi/2; Phi(i-1) = pi; for i = 72:8 X(i) = -1; Y(i) = -1+S*(i-71); x(i-1) = X(i); y(i-1) = Y(i)-S/2; beta(i-1) = pi; Phi(i-1) = pi/2; % get those last ones x(n) = -1; y(n) = -S/2; beta(n) = pi; Phi(N) = pi/2; % that was fun, now find theta values to keep things tidy for i = 1:2 theta(i) = pi-atan(abs(y(i))/abs(x(i))); for i = 21:4 13

14 theta(i) = atan(y(i)/x(i)); for i = 41:6 theta(i) = 2*pi-atan(abs(y(i))/abs(x(i))); for i = 61:8 theta(i) = pi+atan(abs(y(i))/abs(x(i))); % plot endpoints & midpoints % plot(x,y, r+,x,y, g. ) % axis square % axis([ ]) % title( Geometry, N = 8 ) % xlabel( y (m) ) % ylabel( x (m) ) %that was fun, now let s find I(i,j) for j = 1:N if (i ~= j) A(i,j) = -(x(i)-x(j))*cos(phi(j)) - (y(i)-y(j))*sin(phi(j)); B(i,j) = (x(i)-x(j))^2 + (y(i)-y(j))^2; %don t divide by zero if (B(i,j) == ) B(i,j) = 1e-12; C(i,j) = sin(phi(i)-phi(j)); D(i,j) = (y(i)-y(j))*cos(phi(i)) - (x(i)-x(j))*sin(phi(i)); E(i,j) = (x(i)-x(j))*sin(phi(j)) - (y(i)-y(j))*cos(phi(j)); %please, stop dividing by zero if (E(i,j) == ) E(i,j) = 1e-12; I(i,j) = C(i,j)/2*log((S^2+2*A(i,j)*S+B(i,j))/B(i,j)) +... (D(i,j)-A(i,j)*C(i,j))/E(i,j)*(atan((S+A(i,j))/E(i,j))-... atan(a(i,j)/e(i,j))); else I(i,j) = pi; 14

15 %set up system of equations for j = 1:N M(i,j) = I(i,j)/(2*pi); b(i,1) = -U*cos(beta(i)); %get those lambdas & check em lambda = inv(m)*b Sum = sum(lambda) %find velocities & pressure coefficients Vs = ; for j = 1:N if (i ~= j) M2(i,j) = ((D(i,j)-A(i,j)*C(i,j))/(2*E(i,j)))... *log((s^2+2*a(i,j)*s+b(i,j))/b(i,j))-c(i,j)... *(atan((s+a(i,j))/e(i,j))-atan(a(i,j)/e(i,j))); Vs = Vs+lambda(j)/(2*pi)*M2(i,j); V(i) = -(U*sin(beta(i))+Vs); Cp(i) = 1-(V(i)/U)^2; axis square subplot(1,2,1),plot(theta*18/pi,v, o ) xlabel( \theta (degrees) ); ylabel( Velocity (m/s) ); title( Velocity v(\theta) ); subplot(1,2,2),plot(theta*18/pi,cp, sq ) xlabel( \theta (degrees) ); ylabel( C_p ); title( Pressure coefficient C_p(\theta) ); 15