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 http://www.ae.gatech.edu/~lsankar/AE2020 for background material.

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 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.

Thwaites’ method III After q is found, the following relations are used to compute the shape factor H.

Thwaites’ method IV After q 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 < -0.09 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^0.46+22400*rex^(-0.54)); if ret>retmax trans = 1; itrans = i;

Reationship between l and H function H = fH(lambda); if lambda < 0 if lambda==-0.14 lambda=-0.139; end; H = 2.088 + 0.0731./(lambda+0.14); elseif lambda >= 0 H = 2.61 - 3.75*lambda + 5.24*lambda.^2;

Skin Friction function L = fL(lambda); if lambda < 0 end; L = 0.22 + 1.402*lambda +(0.018*lambda)./(lambda+0.107); elseif lambda >= 0 L = 0.22 + 1.57*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));

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^0.46+22400*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 H1: Evolution of H1 along the boundary layer: These two ODEs are solved by marching from transition location to trailing edge.

Empirical Closure Relations Ludwig-Tillman relationship: Turbulent separation occurs when H1 = 3.3

Coding Closure Relations in Head’s Method function y=H1ofH(H); if H <1.1 y = 16; else if H <= 1.6 y = 3.3 + 0.8234*(H-1.1).^(-1.287); y = 3.3 + 1.5501*(H-0.6778).^(-3.064); end; function H=HofH1(H1); if H1 <= 3.32 H = 3; elseif H1 < 5.3 H = 0.6778 + 1.1536*(H1-3.3).^(-0.326); else H = 1.1 + 0.86*(H1-3.3).^(-0.777); end function cf = cfturb(rtheta,H); cf = 0.246*(10.^(-0.678*H))*rtheta.^(-0.268);

Drag Prediction Squire-Young Formula