2Outline Thwaites Method for Computing Laminar Boundary Layers Michel’s Transition CriterionHead’s method for Turbulent FlowSquire-Young Formula for Drag PredictionSee for background material.
3Thwaites’ method IThis is an empirical method based on the observation that most laminar boundary layers obey the following relationship.Ref: Thawites, B., Incompressible Aerodynamics, Clarendon Press, Oxford, 1960:Thwaites recommends A = 0.45 andB = 6 as the best empirical fit.
4Thwaites’ Method II The above equation may be analytically integrated yieldingFor blunt bodies such as airfoils, the edge velocity ue is zero at x=0, the stagnation point. For sharp nosed geometries such as a flat plate, the momentum thickness q is zero at the leading edge. Thus, the term in the square bracket always vanishes.The integral may be evaluated, at least numerically, when ue is known.
5Thwaites’ method IIIAfter q is found, the following relations are used to computethe shape factor H.
6Thwaites’ method IVAfter q is found, we can also find skin friction coefficientfrom the following empirical curve fits:
7MATLAB Code from PABLO %--------Laminar boundary layer lsep = 0; trans=0; endofsurf=0;theta(1) = sqrt(0.075/(Re*dueds(1)));i = 1;while lsep ==0 & trans ==0 & endofsurf ==0lambda = theta(i).^2*dueds(i)*Re;% test for laminar separationif lambda < -0.09lsep = 1; itrans = i;break;end;H(i) = fH(lambda); L = fL(lambda); cf(i) = 2*L./(Re*theta(i));if i>1, cf(i) = cf(i)./ue(i); end;i = i+1;% test for end of surfaceif i> n endofsurf = 1; itrans = n; break; end;K = 0.45/Re; xm = (s(i)+s(i-1))/2; dx = (s(i)-s(i-1)); coeff = sqrt(3/5);f1 = ppval(spues,xm-coeff*dx/2); f1 = f1^5; f2 = ppval(spues,xm); f2 = f2^5;f3 = ppval(spues,xm+coeff*dx/2); f3 = f3^5; dth2ue6 = K*dx/18*(5*f1+8*f2+5*f3);theta(i) = sqrt((theta(i-1).^2*ue(i-1).^6 + dth2ue6)./ue(i).^6);% test for transitionrex = Re*s(i)*ue(i); ret = Re*theta(i)*ue(i); retmax = 1.174*(rex^ *rex^(-0.54));if ret>retmaxtrans = 1; itrans = i;
8Reationship between l and H function H = fH(lambda);if lambda < 0if lambda==-0.14lambda=-0.139;end;H = /(lambda+0.14);elseif lambda >= 0H = *lambda *lambda.^2;
9Skin Friction function L = fL(lambda); if lambda < 0 end;L = *lambda +(0.018*lambda)./(lambda+0.107);elseif lambda >= 0L = *lambda - 1.8*lambda.^2;We invoke (or call this function) at each i-location as follows:H(i) = fH(lambda); L = fL(lambda); cf(i) = 2*L./(Re*theta(i));
10Transition prediction A number of methods are available for predicting transition.Examples:Eppler’s methodMichel’s methodWind turbine designers and laminar airfoil designers tend to use Eppler’s methodAircraft designers tend to use Michel’s method.
11Michel’s Method for Transition Prediction % test for transitionrex = Re*s(i)*ue(i); ret = Re*theta(i)*ue(i);retmax = 1.174*(rex^ *rex^(-0.54));if ret>retmaxtrans = 1; itrans = i;end;
12Turbulent FlowA number of CFD methods, and integral boundary layer methods exist.The most popular of these is Head’s method.This method is used in a number of computer codes, including PABLO.
13Head’s Method Von Karman Momentum Integral Equation: A new shape parameter H1:Evolution of H1 along the boundary layer:These two ODEs are solved by marching from transition location to trailing edge.