Newman, S.J.

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      Aerodynamics & Flight Mechanics Research Group            Joukowski Aerofoil Modelling in MATLAB  S. J. Newman  Technical Report AFM-11/13  January 2011      1  UNIVERSITY OF SOUTHAMPTON   SCHOOL OF ENGINEERING SCIENCES   AERODYNAMICS AND FLIGHT MECHANICS RESEARCH GROUP    Joukowski Aerofoil Modelling in MATLAB      by     S. J. Newman     AFM Report No. AFM 11/13      January 2011       © School of Engineering Sciences, Aerodynamics and Flight Mechanics Research Group     2    COPYRIGHT NOTICE (c) SES University of Southampton All rights reserved. SES authorises you to view and download this document for your personal, non-commercial use. 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This document, is protected by worldwide copyright laws and treaty provisions.      3   Preliminary Discussion The 2D Potential Flow is built up using 3 basic components, namely:   Free Stream   Doublet   Vortex  The assumption is made whereby, the flow velocity is given by:    qui v       (1.)The Complex Potential is: From which we have: And:   Wi       (2.)  dWui vdZ  (3.) 2221 PdWqq uvdZqCU    (4.)      4        5    Free Stream If the incident freestream flow is angled by α to the X axis of velocity U, the Complex Potential (W) is given by:   Doublet A doublet placed at the origin is:  However, the doublet will create the flow of a freestream passing a circular cylinder. If the cylinder radius is rc, then the doublet strength specification can be modified to give:   i WU z e     (5.)  2DoubletDoubletWzdWdz z (6.)   222iiUaWezdW Uaedz z (7.)      6   Vortex A vortex placed at the origin is:   Total Hence, assembling the three components gives the overall result:     log VortexVortexWk iik dWdz zz   (8.)   222logiiVortexii VortexUaWU z e e k izik dW UaUe edz z z  z (9.)     7   Joukowski Transformation The Joukowski transformation takes a circle and transforms it into an aerofoil type of shape using the following complex expression:   Where a is the radius of the reference circle centred at the origin.  The circle which transforms to the aerofoil shape is defined as shown in Figure 1: δ βarcroffXY Figure 1 Using the above definitions, the circle is given by:   2 UaZU  (10.)      8   The reference angle, β, is given by: An example of the transformation geometry is shown in Figure 2:  Figure 2 ‐ Ref Circle (Green), Base Circle (Blue), Aerofoil (Red)    (c o s s i n)icirc c off re r i        (11.)  sinsinoffcrr  (12.)     9   The inverse transformation from the Z to the ζ plane is given by:  Note the use of both solutions of the inverse transformation quadratic equation. 22224() 024() 02ZZ aif real ZZZ aif real Z (13.)     10   Kutta Condition The fully‐developed flow over an aerofoil requires the flow to leave the trailing edge tangentially. This is known as the Kutta condition and is achieved by adjusting the vortex strength. The requirement is a result of the viscous nature of the fluid. Potential flow does not allow for viscosity so this is an analytical tweak to enable a necessary result of viscosity to be included in a potential flow model. In order for the Kutta condition to be met the vortex strength must be of the correct magnitude, given by: Whence the final flow Complex Potential is defined (in the ζ plane) by:      2s i n Ka U      (14.) 2222s i n l o g2s i niiiiUaWU e e a U iia U dW UaUe edz     (15.)     11   Examples Using the above transformation, the following figures show contours of stream function, speed and pressure coefficient: Kutta Condition Off Contours of Stream Function a = 1 rc = 1.0986 roff = 0.1  = 10 Kutta OffXY-3 -2 -1 0 1 2 3-3-2-10123      12  XYContours of Speed a = 1 rc = 1.0986 roff = 0.1  = 10 U = 1 Kutta Off  -3 -2 -1 0 1 2 3-3-2-101230.20.40.60.811.21.41.61.822.2      13  XYContours of Pressure Coefficient a = 1 rc = 1.0986 roff = 0.1  = 10 U = 1 Kutta Off  -3 -2 -1 0 1 2 3-3-2-10123-4-3.5-3-2.5-2-1.5-1-0.500.5 Kutta Condition On      14  Contours of Stream Function a = 1 rc = 1.0986 roff = 0.1  = 10 Kutta OnXY-3 -2 -1 0 1 2 3-3-2-10123      15  XYContours of Speed a = 1 rc = 1.0986 roff = 0.1  = 10 U = 1 Kutta On  -3 -2 -1 0 1 2 3-3-2-101230.511.52      16  XYContours of Pressure Coefficient a = 1 rc = 1.0986 roff = 0.1  = 10 U = 1 Kutta On  -3 -2 -1 0 1 2 3-3-2-10123-5-4-3-2-10 Matlab File  % %   Joukowski Aerofoils % %   SJN 5/11/09 % clear all xmax=5; nx=150; a=1; deltadeg=10; roff=.1;   U=1; alfdeg=5;   kutta=1;   %-------------------------------------------------------------------------- delta=deltadeg*pi/180; offset=roff*(-cos(delta)+1i*sin(delta)); circrad=abs(offset-a); beta=asin(roff*sin(delta)/circrad); %--------------------------------------------------------------------------      17    alf=alfdeg*pi/180; kuttagamma=2*circrad*U*sin(alf+beta); if kutta~=0     vortex=kuttagamma; else     vortex=0; end %-------------------------------------------------------------------------- x=linspace(-xmax,xmax,nx); y=x; [X,Y]=meshgrid(x,y); Z=X+1i*Y; ZETA=.5*(Z-sqrt(Z.^2-4*a^2)).*(real(Z)<0)+...     .5*(Z+sqrt(Z.^2-4*a^2)).*(real(Z)>=0); %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> % CONTOURS OF STREAM FUNCTION W=U*exp(1i*-alf)*ZETA...     +U*exp(1i*alf)*circrad^2./(ZETA-offset)...     +vortex*1i*log(ZETA-offset); v1=-40:.1:40; contour(X,Y,imag(W),v1,'b'); hold on if kutta~=0     title(['Contours of Stream Function a = ',num2str(a),...         ' r_c = ',num2str(circrad),...         ' r_o_f_f = ',num2str(roff),...         ' \delta = ',num2str(deltadeg),'\circ',' Kutta On']); else     title(['Contours of Stream Function a = ',num2str(a),...         ' r_c = ',num2str(circrad),...         ' r_o_f_f = ',num2str(roff),...         ' \delta = ',num2str(deltadeg),'\circ',' Kutta Off']); end xlabel('X'); ylabel('Y'); %-------------------------------------------------------------------------- theta=linspace(0,2*pi,101); ctheta=cos(theta); stheta=sin(theta); ZETACIRC=circrad*(ctheta+1i*stheta)+offset; ZCIRC=ZETACIRC+a^2./ZETACIRC; fill(real(ZCIRC),imag(ZCIRC),'r','FaceAlpha',1); %axis(xmax*[-1 1 -1 1]); axis(3*[-1 1 -1 1]); %axis([1.5 2.5 -.5 .5]); %axis equal grid on %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> figure % CONTOURS OF SPEED   dWdZ=U*exp(1i*-alf)...     -U*exp(1i*alf)*circrad^2./(ZETA-offset).^2 ...     +vortex*1i./(ZETA-offset);     Q=abs(dWdZ); v2=0:.05:3; contour(X,Y,Q,v2); colorba   rhold on %--------------------------------------------------------------------------      18  theta=linspace(0,2*pi,101); ctheta=cos(theta); stheta=sin(theta); ZETACIRC=circrad*(ctheta+1i*stheta)+offset; ZCIRC=ZETACIRC+a^2./ZETACIRC; fill(real(ZCIRC),imag(ZCIRC),'r','FaceAlpha',1); %axis(xmax*[-1 1 -1 1]); axis(3*[-1 1 -1 1]); %axis([1.5 2.5 -.5 .5]); %axis equal grid on if kutta~=0     title(['Contours of Speed a = ',num2str(a),...         ' r_c = ',num2str(circrad),...         ' r_o_f_f = ',num2str(roff),...         ' \delta = ',num2str(deltadeg),'\circ',' U = ',num2str(U),' Kutta On']); else     title(['Contours of Speed a = ',num2str(a),...         ' r_c = ',num2str(circrad),...         ' r_o_f_f = ',num2str(roff),...         ' \delta = ',num2str(deltadeg),'\circ',' U = ',num2str(U),' Kutta Off );  ']end xlabel('X'); ylabel('Y');   %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> figure % CONTOURS OF PRESSURE COEFFICIENT   CP=1-Q.^2/U^2; v3=-10:.1:1; contour(X,Y,CP,v3); colorba   rhold on %-------------------------------------------------------------------------- theta=linspace(0,2*pi,101); ctheta=cos(theta); stheta=sin(theta); ZETACIRC=circrad*(ctheta+1i*stheta)+offset; ZCIRC=ZETACIRC+a^2./ZETACIRC; fill(real(ZCIRC),imag(ZCIRC),'r','FaceAlpha',1); %axis(xmax*[-1 1 -1 1]); axis(3*[-1 1 -1 1]); %axis([1.5 2.5 -.5 .5]); %axis equal grid on if kutta~=0     title(['Contours of Pressure Coefficient a = ',num2str(a),...         ' r_c = ',num2str(circrad),...         ' r_o_f_f = ',num2str(roff),...         ' \delta = ',num2str(deltadeg),'\circ',' U = ',num2str(U),' Kutta On']); else     title(['Contours of Pressure Coefficient a = ',num2str(a),...         ' r_c = ',num2str(circrad),...         ' r_o_f_f = ',num2str(roff),...         ' \delta = ',num2str(deltadeg),'\circ',' U = ',num2str(U),' Kutta Off']); end      19  xlabel('X'); ylabel('Y');      20    

Joukowski aerofoil modelling in MATLAB

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