Boundary Layer Analyses

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Boundary Layer Analyses

• AE 3903/4903 Airfoil Design
• L. Sankar
• School of Aerospace Engineering

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Outline

• Thwaites Method for Computing Laminar Boundary
  Layers
• Michels Transition Criterion
• Heads method for Turbulent Flow
• Squire-Young Formula for Drag Prediction
• See http//www.ae.gatech.edu/lsankar/AE2020 for
  background material.

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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.
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Thwaites Method II
The above equation may be analytically integrated
yielding
For blunt bodies such as airfoils, the edge
velocity ue is zero at x0, 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.
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Thwaites method III
After q is found, the following relations are
used to compute the shape factor H.
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Thwaites method IV
After q is found, we can also find skin friction
coefficient from the following empirical curve
fits
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MATLAB Code from PABLO
--------Laminar boundary layer lsep 0
trans0 endofsurf0 theta(1)
sqrt(0.075/(Redueds(1))) i 1 while lsep 0
trans 0 endofsurf 0 lambda
theta(i).2dueds(i)Re test for laminar
separation if lambda lt -0.09 lsep 1
itrans i break end H(i)
fH(lambda) L fL(lambda) cf(i)
2L./(Retheta(i)) if igt1, cf(i)
cf(i)./ue(i) end i i1 test for end of
surface if igt 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
f15 f2 ppval(spues,xm) f2 f25 f3
ppval(spues,xmcoeffdx/2) f3 f35 dth2ue6
Kdx/18(5f18f25f3) 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(rex0.4622400rex(-0.54)) if
retgtretmax trans 1 itrans i
end end
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Reationship between l and H
function H fH(lambda) if lambda lt 0 if
lambda-0.14 lambda-0.139 end H
2.088 0.0731./(lambda0.14) elseif lambda gt
0 H 2.61 - 3.75lambda 5.24lambda.2 en
d
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Skin Friction
function L fL(lambda) if lambda lt 0 if
lambda-0.107 lambda-0.106 end L
0.22 1.402lambda (0.018lambda)./(lambda0.107
) elseif lambda gt 0 L 0.22 1.57lambda
- 1.8lambda.2 end
We invoke (or call this function) at each
i-location as follows
H(i) fH(lambda) L fL(lambda) cf(i)
2L./(Retheta(i))
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Transition prediction

• A number of methods are available for predicting
  transition.
• Examples
• Epplers method
• Michels method
• Wind turbine designers and laminar airfoil
  designers tend to use Epplers method
• Aircraft designers tend to use Michels method.

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Michels Method for Transition Prediction
test for transition rex Res(i)ue(i) ret
Retheta(i)ue(i) retmax 1.174(rex0.46224
00rex(-0.54)) if retgtretmax trans 1
itrans i end
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Turbulent Flow

• A number of CFD methods, and integral boundary
  layer methods exist.
• The most popular of these is Heads method.
• This method is used in a number of computer
  codes, including PABLO.

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Heads 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.
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Empirical Closure Relations
Ludwig-Tillman relationship
Turbulent separation occurs when H1 3.3
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Coding Closure Relations inHeads Method
function yH1ofH(H) if H lt1.1 y 16 else
if H lt 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 HHofH1(H1) if H1 lt 3.32 H
3 elseif H1 lt 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.(-0.678H))rtheta.(-0.268)
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Drag PredictionSquire-Young Formula
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