1. Boundary Layer Analyses
Module 3.4: Wind Engineering
L. Sankar
School of Aerospace Engineering

2. 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.

3. 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.

4. 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.

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

6. Thwaites’ method IV
After q is found, we can also find skin friction coefficient
from the following empirical curve fits:

7. 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^ *rex^(-0.54));
if ret>retmax
trans = 1; itrans = i;

8. Reationship between l 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;

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

10. 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.

11. 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;

12. 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.

13. 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.

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

15. 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);
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).^(-0.777);
end
function cf = cfturb(rtheta,H);
cf = 0.246*(10.^(-0.678*H))*rtheta.^(-0.268);

16. Drag Prediction Squire-Young Formula