Boundary Layer Analyses - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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Boundary Layer Analyses

Outline

• Thwaites Method for Computing Laminar

Boundary Layers

• Michel’s Transition Criterion
• Head’s method for Turbulent Flow
• Squire-Young Formula for Drag Prediction

Thwaites’ Method II

The above equation may be analytically integrated

yielding

u dx

u x u

u x

u dx x

u

x

x

e e e

e

x

x

e e

∫ ∫ = =

0

5 6 6

6 2

0

5 6

2 0.^45

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 θ

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 θ is found, the following relations are used to compute

the shape factor H.

For 0 ≤ λ ≤ 0.

H = 2. 61 − 3. 75 λ + 5.24 λ

2

For − 0.1 ≤ λ ≤ 0

H = 2. 472 +

0.107 + λ

where ,

λ =

θ

2

ν

du e

dx

MATLAB Code from PABLO

%--------Laminar boundary layer

lsep = 0; trans=0; endofsurf=0; theta(1) = sqrt(0.075/(Redueds(1))); i = 1; while lsep ==0 & trans ==0 & endofsurf == lambda = theta(i).^2dueds(i)Re; % test for laminar separation if lambda < -0. lsep = 1; itrans = i; break; end; H(i) = fH(lambda); L = fL(lambda); cf(i) = 2L./(Retheta(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-coeffdx/2); f1 = f1^5; f2 = ppval(spues,xm); f2 = f2^5; f3 = ppval(spues,xm+coeffdx/2); f3 = f3^5; dth2ue6 = Kdx/18(5f1+8f2+5f3); theta(i) = sqrt((theta(i-1).^2ue(i-1).^6 + dth2ue6)./ue(i).^6); % test for transition rex = Res(i)ue(i); ret = Retheta(i)ue(i); retmax = 1.174(rex^0.46+22400*rex^(-0.54)); if ret>retmax trans = 1; itrans = i; end; end; Docsity.com

Reationship between λ and H

function H = fH(lambda);

if lambda < 0

if lambda==-0. lambda=-0.139; end;

H = 2.088 + 0.0731./(lambda+0.14);

elseif lambda >= 0

H = 2.61 - 3.75lambda + 5.24lambda.^2;

end;

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 = Res(i)ue(i); ret = Retheta(i)ue(i); retmax = 1.174(rex^0.46+22400rex^(-0.54)); if ret>retmax trans = 1; itrans = i; end;

[ ]

1. 46 0. 54 Re 1. 174 Re 22400 Re

Transition occurswhen

Re

Re

− ≥ +

=

=

x x

e

e x

u

u x

θ

θ ν

θ

ν

Head’s Method

( )

d

dx U

H

dU

dx

θ θ c (^) f

• 2 + = 2

Von Karman Momentum Integral Equation:

A new shape parameter H 1 : θ

δ δ

1

H ≡

Evolution of H 1 along the boundary layer:

( ) ( )

1 1 0 0306^13

0 6169

U

d

dx

U Hθ = H −

− .

.

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

Empirical Closure Relations

1. 064 1

2. 287 1

3. 3 1. 5501 0. 6778

4. 3 0. 8234 1. 1

If H 1.

−

−

= + −

= + −

≤

H H

else

H H

1. 678 0.^268

0. 246 10 Re

− −

H

f

C

Ludwig-Tillman relationship:

Turbulent separation occurs when H1 = 3.

Drag Prediction

Squire-Young Formula

2

5

, ,

, , ,

2

∞

  

   



= +

HTrailingEdge upper

TrailingEdge ETrailingEdge d upper

d d upper d lower

V

U

c

C

C C C