Boundary Layer Analyses

AE 3903/4903 Airfoil Design

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:

ueddx

2 A B 2

duedx

Thwaites recommends A = 0.45 and B = 6 as the best empirical fit.

Thwaites’ Method IIThe above equation may be analytically integrated yielding

dxuuxu

xuxdxu

u

x

x

eee

ex

x

ee

0

566

62

0

56

2 45.0

)(

)0()0(

45.0

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 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 IIIAfter is found, the following relations are used to compute the shape factor H.

For 0 0.1

H 2.61 3.75 5.24 2

For 0.1 0

H 2.472 0.01470.107

where ,

2

du

e

dx

Thwaites’ method IV

2

62.0

21

09.0

e

wf

ew

uC

u

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 < -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; end;end;

Reationship between and Hfunction 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;

end;

Skin Frictionfunction L = fL(lambda);

if lambda < 0

if lambda==-0.107 lambda=-0.106;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;

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^0.46+22400*rex^(-0.54)); if ret>retmax trans = 1; itrans = i; end;

54.046.0 Re22400Re174.1Re

whenoccurs Transition

Re

Re

xx

e

ex

u

xu

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

d

dx UH

dU

dx

cf 2

2

Von Karman Momentum Integral Equation:

A new shape parameter H1:

*

1

H

Evolution of H1 along the boundary layer:

10 0306 31 1

0 6169

U

d

dxU H H

.

.

These two ODEs are solved by marching from transition location to trailing edge.

Empirical Closure Relations

064.31

287.11

6778.05501.13.3

1.18234.03.3

1.6 H If

HH

else

HH

268.0678.0 Re10246.0 H

fCLudwig-Tillman relationship:

Turbulent separation occurs when H1 = 3.3

Coding Closure Relations inHead’s Method

function y=H1ofH(H);

if H <1.1y = 16;

else if H <= 1.6 y = 3.3 + 0.8234*(H-1.1).^(-1.287); else y = 3.3 + 1.5501*(H-0.6778).^(-3.064); end;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 PredictionSquire-Young Formula

2

5

,,

,,

,

2

uppergeTrailingEdH

geTrailingEdEgeTrailingEdupperd

lowerdupperdd

V

U

cC

CCC