3D Ideal Flow For Non-Lifting Bodies

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Juliet Goodman
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1 3D Ideal Flow For Non-Lifting Bodies

2 3D Doublet Panel Method 1. Cover body with 3D onstant strength doublet panels of unknown strength (N panels) 2. Plae a ontrol point at the enter of eah panel (N ontrol points 3. Write down an expression for the normal veloity at eah.p. produed by the free stream veloity and N panels 4. Solve resulting N equations for N panel strengths 5. With panel strengths ompute tangent veloity at.p.s and thus surfae pressure, and rest of flow V

3 An Issue With Doublet Panels Consider a ut through the body surfae no flow V=V tang V<V tang V<<V tang Panels appear as a series of disrete vorties with strengths equal to the differene between adjaent panel strengths. At the ontrol points we see almost none of the tangential veloity generated by nearby vorties. This is wrong. It is as though the surfae were a ontinuous vortex sheet, and we alulated the tangential veloity at the.p. ignoring the disontinuous jump between the inside and outside. This jump, equal to half the loal sheet strength is alled the prinipal V=V tang value and must be added to the tangent veloity alulated at a ontrol point V=0 The prinipal value for a doublet panel an be estimated as This must be added to the tangential veloity alulated at any ontrol point

4 3D Doublet Panel Code Non-lifting bodies Handling vetors in Matlab Speifying a 3D body Speifying Panel Geometry Panel Influene solving for the panel strengths Getting the surfae pressure

5 %3D doublet panel method for ayli flow around 3D bodies. lear all; vinf=[1;0;0]; %free stream veloity %Speify body of revolution geometry th=-pi/2:pi/20:pi/2;xp=sin(th);yp=os(th);n=20; [r,r,nw,sw,se,ne,no,we,so,ea]=bodyofrevolution(xp,yp,n); % determine surfae area and normal vetors at ontrol points (assumes ounter lokwise around om a=0.5*v_ross(r(:,sw)-r(:,ne),r(:,se)-r(:,nw));n=a./v_mag(a); %determine influene oeffiient matrix oeff npanels=length(r(1,:));oef=zeros(npanels); for n=1:npanels mn=ffil(r(:,n),r(:,nw),r(:,sw))+ffil(r(:,n),r(:,sw),r(:,se))+ffil(r(:,n),r(:,se),r(:,ne))+f oef(n,:)=n(1,n)*mn(1,:)+n(2,n)*mn(2,:)+n(3,n)*mn(3,:); end %determine result vetor and solve matrix for filament strengths rm=(-n(1,:)*vinf(1)-n(2,:)*vinf(2)-n(3,:)*vinf(3))'; oef(end+1,:)=1;rm(end+1)=0; %prevents singular matrix - sum of panel strengths on losed body is z ga=oef\rm; %Determine veloity and pressure at ontrol points ga=repmat(ga',[3 1]); for n=1:npanels %Determine veloity at eah.p. without prinipal value mn=ffil(r(:,n),r(:,nw),r(:,sw))+ffil(r(:,n),r(:,sw),r(:,se))+ffil(r(:,n),r(:,se),r(:,ne))+f v(:,n)=vinf+sum(ga.*mn,2); end %Determine priniple value of veloity at eah.p., -grad(ga)/2 gg=v_ross((r(:,we)-r(:,no)).*(ga(:,we)+ga(:,no))+(r(:,so)-r(:,we)).*(ga(:,so)+ga(:,we))+(r(:, v=v-gg/2; %veloity vetor p=1-sum(v.^2)/(vinf'*vinf); %pressure %plotting of pressure distribution and veloity vetors

6 Vetors In Matlab Use arrays with 3 rows, one for eah omponent. E.g. V Easy to make funtions for vetor operations like dot and ross produts, and magnitude With these it is now easier to make more omplex funtions E.g. f fil (r,r s,r e ) %3D doublet panel method for flow around 3D bodies. lear all; vinf=[1;0;0]; %free stream veloity funtion =v_dot(a,b); =zeros(size(a)); (1,:)=a(1,:).*b(1,:)+a(2,:).*b(2,:)+a(3,:).*b(3,:); (2,:)=(1,:);(3,:)=(1,:); funtion =v_ross(a,b); =zeros(size(a)); (1,:)=(a(2,:).*b(3,:)-a(3,:).*b(2,:)); (2,:)=(a(3,:).*b(1,:)-a(1,:).*b(3,:)); (3,:)=(a(1,:).*b(2,:)-a(2,:).*b(1,:)); funtion q=ffil(r,rs,re); r1(1,:)=rs(1,:)-r(1,:); r1(2,:)=rs(2,:)-r(2,:); r1(3,:)=rs(3,:)-r(3,:); r2(1,:)=re(1,:)-r(1,:); r2(2,:)=re(2,:)-r(2,:); r2(3,:)=re(3,:)-r(3,:); r0(1,:)=r1(1,:)-r2(1,:);r0(2,:)=r1(2,:)-r2(2,:);r0(3,:)=r1(3,:)-r2(3,:); =v_ross(r1,r2); 2=v_dot(,); q=./2.*v_dot(r0,r1./v_mag(r1)-r2./v_mag(r2))/4/pi;

7 Speifying a 3D Body First we must hoose shape. E.g. Body of revolution: %Speify body of revolution geometry th=-pi/2:pi/20:pi/2;xp=sin(th);yp=os(th);n=20; [r,r,nw,sw,se,ne,no,we,so,ea]=bodyofrevolution(xp,yp,n) Then we generate the oordinates, in vetor form, of all the points on the body. These points (in a retangular array, beome the panel orner points r xp + yp n (number of irumferential points) funtion [r,r,nw,sw,se,ne,no,we,so,ea]=bodyofrevolution(xp,yp,n) %Define verties of panels x=repmat(xp,[n+1 1]); %grid of x points al=[0:n]/(n)*2*pi; %vetor of irumferential angles (about x axis) y=os(al)'*yp; z=sin(al)'*yp; r=zeros([3 size(x)]);r(1,:)=x(:);r(2,:)=y(:);r(3,:)=z(:); %position vetor of verties ri=reshape(1:prod(size(x)),size(x)); %index of verties

8 no Panel Geometry nw ne ea Control point loations have loations r Panel verties have loations r we n se r = r r sw so Defining the indies, also defines how the body loses r=zeros([3 size(x)]);r(1,:)=x(:);r(2,:)=y(:);r(3,:)=z(:); %position vetor of verties ri=reshape(1:prod(size(x)),size(x)); %index of verties %panel i,j has orners i,j i+1,j i+1,j+1 i,j+1 nw=ri(1:end-1,1:end-1);sw=ri(2:end,1:end-1); %indies of upper and lower left orners ne=ri(1:end-1,2:end);se=ri(2:end,2:end); %indies of upper and lower right orners % determine panel enters (ontrol points) and indies r=(r(:,nw)+r(:,sw)+r(:,ne)+r(:,se))/4;r=reshape(r,[3 size(nw)]); i=reshape(1:prod(size(nw)),size(nw)); % determine indies of panels bordering eah ontrol point no=i([end 1:end-1],:);so=i([2:end 1],:); %Panels above and below. we=i(:,[1 1:end-1]);ea=i(:,[2:end end]); %Panels to left and right.

9 Panel Geometry Defining the indies, also defines how the body loses Seam 2nd index inreasing w n e s 1st index inreasing

10 Panel Geometry r se -r nw nw ne Determine area vetor and outward normal of eah panel by ross produt r sw -r ne sw n se a n = = [r,r,nw,sw,se,ne,no,we,so,ea]=bodyofrevolution(xp,yp,n); % determine surfae area and normal vetors at ontrol points (assumes ounter lokwise around ompass by RH rule points out of surfae) a=0.5*v_ross(r(:,sw)-r(:,ne),r(:,se)-r(:,nw));n=a./v_mag(a);

11 Panel Influene Veloity due to panel m at ontrol point n: V( r ) = [ f fil ( r nw sw ) + f fil ( r sw se ) ne + f fil ( r se ne ) + f fil ( r ne nw ) ] Γ nw sw Panel m Γ se or V( r ) = C ( n, m) Γ Summed veloity at ontrol point n is thus: ( n, m) V( r ) V + C Γ = m (w/o tangential veloity due to prinipal value) r (m) r (n) Normal omponent is ( n, m) V( r ). n 0 = V n + C. n Γ = So, to get the Γ (m) s we need to solve the simultaneous equations: m Control point n Nx1 matrix of freestream omponents V n m ( n, m) = C.n Γ NxN oeffiient matrix Nx1 matrix of panel strengths

12 Panel Influene %determine influene oeffiient matrix oef npanels=length(r(1,:));oef=zeros(npanels); for n=1:npanels mn=ffil(r(:,n),r(:,nw),r(:,sw))+ffil(r(:,n),r(:,sw),r(:,se)) +ffil(r(:,n),r(:,se),r(:,ne))+ffil(r(:,n),r(:,ne),r(:,nw)); oef(n,:)=n(1,n)*mn(1,:)+n(2,n)*mn(2,:)+n(3,n)*mn(3,:); end %determine result vetor and solve matrix for filament strengths rm=(-n(1,:)*vinf(1)-n(2,:)*vinf(2)-n(3,:)*vinf(3))'; oef(end+1,:)=1;rm(end+1)=0; %prevents singular matrix ga=oef\rm;

13 Determining Surfae Pressure Total veloity at ontrol point r we no n so ea Γ V( r ) = V + C Tangential veloity due to prinipal value at r m = Using the gradient theorem and values of Γ at neighboring ontrol points we an show that ( rwe rno )( Γwe + Γno ) + ( rso rwe )( Γ + ( rea rso )( Γea + Γso ) + ( rno rea )( Γ ( r r ) ( r r ) no ( n, m) so Γ + we Tangential veloity due to prinipal value at r ea so no + Γwe ) n + Γea ) We an then use Bernoulli V( r ) C ( r ) = 1 to ompute the pressure p 2 V 2

14 Determining Surfae Pressure %Determine veloity and pressure at ontrol points for n=1:npanels %Get veloity at eah.p. w/o prinipal value mn=ffil(r(:,n),r(:,nw),r(:,sw))+ffil(r(:,n),r(:,sw),r(:,se)) +ffil(r(:,n),r(:,se),r(:,ne))+ffil(r(:,n),r(:,ne),r(:,nw)); v(:,n)=vinf+sum(ga.*mn,2); end %Determine priniple value, -grad(ga)/2 gg=v_ross((r(:,we)-r(:,no)).*(ga(:,we)+ga(:,no))+(r(:,so)-... v=v-gg/2; %veloity vetor p=1-sum(v.^2)/(vinf'*vinf); %pressure

15 Using the Code Plotting the pressure, streamlines Deforming the body shape Changing the shape More than one body

Physics of an Flow Over a Wing

Wings in Ideal Flow Physics of an Flow Over a Wing Werle, 1974, NACA 0012, 12.5 o, AR=4, Re=10000 Bippes, Clark Y, Rectangular Wing 9 o, AR=2.4, Re=100000 Head, 1982, Rectangular Wing, 24 o, Re=100000