Presentation on theme: "Boundary Layer Analyses Module 3.4: Wind Engineering L. Sankar School of Aerospace Engineering."— Presentation transcript:
Boundary Layer Analyses Module 3.4: Wind Engineering L. Sankar School of Aerospace Engineering
Outline Thwaites Method for Computing Laminar Boundary Layers Michel’s Transition Criterion Head’s method for Turbulent Flow Squire-Young Formula for Drag Prediction See for background material.http://www.ae.gatech.edu/~lsankar/AE2020
Thwaites’ method I This 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 and B = 6 as the best empirical fit.
Thwaites’ Method II The above equation may be analytically integrated yielding For blunt bodies such as airfoils, the edge velocity u e is zero at x=0, the stagnation point. For sharp nosed geometries such as a flat plate, the momentum thickness is zero at the leading edge. Thus, the term in the square bracket always vanishes. The integral may be evaluated, at least numerically, when u e is known.
Thwaites’ method III After is found, the following relations are used to compute the shape factor H.
Thwaites’ method IV After is found, we can also find skin friction coefficient from the following empirical curve fits:
MATLAB 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 ==0 lambda = theta(i).^2*dueds(i)*Re; % test for laminar separation if lambda < lsep = 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 surface if 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 transition rex = Re*s(i)*ue(i); ret = Re*theta(i)*ue(i); retmax = 1.174*(rex^ *rex^(-0.54)); if ret>retmax trans = 1; itrans = i; end;
Reationship between and H function H = fH(lambda); if lambda < 0 if lambda==-0.14 lambda=-0.139; end; H = /(lambda+0.14); elseif lambda >= 0 H = *lambda *lambda.^2; end;
Skin Friction function L = fL(lambda); if lambda < 0 if lambda== lambda=-0.106; end; L = *lambda +(0.018*lambda)./(lambda+0.107); elseif lambda >= 0 L = *lambda - 1.8*lambda.^2; end; H(i) = fH(lambda); L = fL(lambda); cf(i) = 2*L./(Re*theta(i)); We invoke (or call this function) at each i-location as follows:
Transition prediction A number of methods are available for predicting transition. Examples: –Eppler’s method –Michel’s method Wind turbine designers and laminar airfoil designers tend to use Eppler’s method Aircraft designers tend to use Michel’s method.
Michel’s Method for Transition Prediction % test for transition rex = Re*s(i)*ue(i); ret = Re*theta(i)*ue(i); retmax = 1.174*(rex^ *rex^(-0.54)); if ret>retmax trans = 1; itrans = i; end;
Turbulent Flow A 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.
Head’s Method Von Karman Momentum Integral Equation: A new shape parameter H 1 : Evolution of H 1 along the boundary layer: These two ODEs are solved by marching from transition location to trailing edge.
Coding Closure Relations in Head’s Method function y=H1ofH(H); if H <1.1 y = 16; else if H <= 1.6 y = *(H-1.1).^(-1.287); else y = *(H ).^(-3.064); end; function H=HofH1(H1); if H1 <= 3.32 H = 3; elseif H1 < 5.3 H = *(H1- 3.3).^(-0.326); else H = *(H1-3.3).^( ); end function cf = cfturb(rtheta,H); cf = 0.246*(10.^(-0.678*H))*rtheta.^(-0.268);