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Viscosity and Boundary Layers

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How to investigate viscous effects

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Outline of the Boundary Layers• Introduction• Effects by Viscosity

– Drag– Pressure Distribution– Flow Separation

• Basic Boundary Layer Theory– Basic Theory and Definitions– Laminar Boundary layers– Transition– Turbulent boundary Layers

• Summary of Results

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Introduction• What’s Boundary layers ?

– A thin layer Appearing on the surface of bodies in viscous flow because the fluid seems to "stick" to the surface.

– A thin layer of fluid with lower velocity than the outer flow develops.

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• Description of Boundary Layer– No slip condition: the flow at the surface has no relative motion.

– The velocity in the boundary layer slowly increases until it reaches the outer flow velocity, Ue.

– The boundary layer thickness, δ , is defined as the distance required for the flow to nearly reach Ue.

– Take an arbitrary number (say 99%) to define what we mean by "nearly“.

Description of Boundary Layer

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航空宇航学院Boundary layer on airfoil.

• Boundary layer along surface of airfoil. – Generally starting out as a laminar flow, the boundary layer

thickens, undergoes transition to turbulent flow, and then continues to develop along the surface of the body, possibly separating from the surface under certain conditions.

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航空宇航学院laminar flow

• Laminar flow– Fluid moves in smooth layers or lamina.

– There is relatively little mixing and consequently the velocity gradients are small and shear stresses are low.

– The thickness of the laminar boundary layer increases with distance from the start of the boundary layer and decreases with Reynolds number.

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航空宇航学院Turbulent boundary layer

• Turbulent boundary layer– Flow is characterized by unsteady mixing due to eddies at

many scales. The result is higher shear stress at the wall, a "fuller" velocity profile,and a greater boundary layer thickness.

– The wall shear stress is higher because the velocity gradient near the wall is greater.

– The lower velocity fluid is also transported outward with the result that the distance to the edge of the layer is larger.

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Effects by viscosity

• Drag

• Pressure Distribution

• Flow Separation

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航空宇航学院Viscous Drag

• Viscous Drag– Skin Friction

Skin friction drag caused by shear stresses at the surface contribute a majority of the drag of most airplanes

We define the skin friction coefficient, Cf, by

The shear stress is then related to the viscosity by:

Cf is related to the drag coefficient by CD (skin friction) = Cf*Swetted/Sref.

where Swetted is the area "wetted" by the air and Sref is the reference area used to define the drag coefficient.

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This expression applies to a flat plate.

When the body has thickness, the local velocities on the surface may be higher than the freestream velocity and the skin friction is increased.

We usually write: CD = k * Cf * Swetted / Sref where k is a "form factor" that depends on the shape of the body.

The skin friction coefficient varies with Reynolds number, Mach number, and the character of the boundary layer.

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The plot below shows how Reynolds number and the location of the transition from laminar flow to turbulent flow, affects the skin friction coefficient.

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From the basic boundary layer theory combined with experimental fits, the following results are obtained:

For laminar boundary layers on flat plates:

For fully-turbulent flat plate boundary layers:

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– Pressure DragThe presence of the boundary layer creates a pressure or form drag on bodies

In an adverse pressure gradient, the skin friction drag is reduced, but pressure drag increases.

This increase in pressure drag compensates for some of the reduction in skin friction.

Pressure Drag

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• Effect of Boundary Layers on Pressures– The presence of the boundary layer changes the effective

shape of the body, leading to changes in the pressure distribution and to the overall lift and drag.

Effect of Boundary Layers on Pressures

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– The effective shape can be used to approximate the effect of the boundary layer using inviscid analysis methods combined with the boundary layer equations.

– Outside the boundary layer, the flow behaves much like an inviscid

(and usually irrotational) fluid.

– This leads to changes in the lift, drag, and moment compared with the inviscid solution.

– This change in pressure distribution leads to a non-zero pressure drag in addition to the skin friction drag

– The sum of the skin friction and pressure drag is often termed “profile drag”.

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– As the angle of attack changes, the boundary layer shape changes, with thicker boundary layers developing toward the aft part of the airfoil at higher angles of attack (because of the more severe adverse pressure gradients).

– The effective shape of the airfoil thus changes with angle of attack.

– If we look at the mean line of the effective shapes, it is clear that viscous effects cause an effective decambering of the airfoil shape.

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– This leads to changes in the lift curve slope (up to a 10% reduction in Cl at Reynolds numbers in the millions) and an aerodynamic center that is usually farther forward than is predicted by inviscidtheory.

– The effect is of increasing importance as Reynolds number is reduced.

Angle of attack

Viscous result

Separation causes large changes in effective airfoil shape here

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• Separation– What is Separation

When the flow near the surface reverses its direction and flows upstream, there must be a place, generally a bit farther upstream, where streamlines meet and then leave the surface.

– Why ?

It’s caused by the presence of an adverse pressure gradient.

– EffectsWhen this occurs, the assumptions that the u component of velocity is larger than the v component and that certain derivatives in the x direction may be ignored, no longer are valid.

Thus, coupling an inviscid analysis with a simple boundary layer calculation does not work.

One must resort to experiment or Navier-Stokes solutions.

Separation

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• Effects of the separation– The changes in the flow pattern, and associated

forces and moments are large.Drag usually increases substantiallyAirfoil lift usually dropsThe effect is generally Reynolds number dependent.

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• Detail explanation about separation– The presence of an adverse pressure gradient (increasing

pressure) causes a deceleration of the fluid. Just as when one coasts uphill, the fluid that starts up the (pressure) hill with little speed, starts rolling backward after a while.

– This picture explains why flow does not separate as readily at higher Reynolds numbers. In that case, the velocity profile is "fuller" with the high external velocities extending down closerto the surface.

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– Turbulent boundary layers also have greater velocity near the surface and are therefore better able to handle adverse pressuregradients.

– The laminar boundary layer is more likely to separateWhen this occurs, the laminar boundary layer leaves the surface and usually undergoes transition to turbulent flow away from the surface.

This process takes place over a certain distance that is inversely related to the Reynolds number, but if it happens quickly enough, the flow may reattach as a turbulent boundary layer and continue along the surface.

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• The laminar separation on low Reynolds number airfoils– The separation phenomenon has significant effects on airfoil

pressure distributions at low Reynolds numbers.

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• Comments on computing for separation• To compute when separation will occur, we can

solve the N-S equations or apply one of several separation criteria to solutions of the boundary layer equations.

• Laminar Separation Criteria

• Turbulent Separation Criteria

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Basic boundary layer theory• Boundary Layer Thickness

– Boundary layer thickness, δ• This is defined as the y- location where u/ue reaches 99%,

that is the u- velocity becomes 99% of the edge velocity.

– Displacement thickness, δ *• This is a measure of the outward displacement of the

streamlines from the solid surface as a result of the reduced u- velocity within the boundary layer. This quantity is defined as :

δ* = 1−ρu

ρeue

⎡

⎣ ⎢ ⎤

⎦ ⎥ 0

∞

∫ dy

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– Momentum Thickness, θ– This is a measure of the momentum loss within the

boundary layer as a result of the reduced velocities within the boundary layer.

– Shape factor, H is defined as the ratio of δ * to θ

* /H δ θ=

θ =ρu

ρeue0

∞

∫ 1−uue

⎡

⎣ ⎢ ⎤

⎦ ⎥ dy

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Boundary Layer Analyses

• Thwaites Method for Computing Laminar Boundary Layers

• Michel’s Transition Criterion

• Head’s method for Turbulent Flow

• Squire-Young Formula for Drag Prediction

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Thwaites’ method• 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:

ue

νddx

θ 2( )= A − Bθ 2

νdue

dx

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

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Thwaites’ method• The above equation may be analytically

integrated yielding

dxuuxu

xuxdxuu

x

xe

ee

ex

xe

e∫∫==

=⎥⎦

⎤⎢⎣

⎡ ==+=

0

566

62

0

56

2 45.0)(

)0()0(45.0 νθνθ

• 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 ue is known.

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Thwaites’ method• After θ is found, the following relations are used to

compute the shape factor H.

For 0 ≤ λ ≤ 0.1

H = 2.61 − 3.75λ + 5.24λ 2

For − 0.1 ≤ λ ≤ 0

H = 2.472 + 0.01470.107 + λ

where,

λ =θ 2

νdue

dx

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Thwaites’ method

After θ is found, we can also find skin friction coefficient from the following empirical curve fits:

( )

2

62.0

21

09.0

e

wf

ew

uC

u

ρτ

λθ

µτ

=

+=

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航空宇航学院Thwaites’ method: 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 separationif 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 surfaceif 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 transitionrex = 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;

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Relationship between λ and Hfunction H = fH(lambda);

if lambda < 0

if lambda==-0.14lambda=-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;

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Skin Frictionfunction L = fL(lambda);

if lambda < 0

if lambda==-0.107lambda=-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:

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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.

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Michel’s Method for Transition Prediction

[ ]54.046.0 Re22400Re174.1Re

whenoccurs Transition

Re

Re

−+≥

=

=

xx

e

ex

u

xu

θ

θ νθ

ν

% test for transitionrex = 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;

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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.

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Head’s Method

( )ddx U

HdUdx

cfθ θ+ + =2

2

Von Karman Momentum Integral Equation:

A new shape parameter H1:θ

δδ *

1−

≡H

Evolution of H1 along the boundary layer:

( ) ( )10 0306 31 1

0 6169

Uddx

U H Hθ = −−

..

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

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Empirical Closure Relations

( )

( ) 064.31

287.11

6778.05501.13.3

1.18234.03.3

1.6 H If

−

−

−+=

−+=

≤

HH

elseHH

( ) 268.0678.0 Re10246.0 −−= θH

fC

Ludwig-Tillman relationship:

Turbulent separation occurs when H1 = 3.3

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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);

elsey = 3.3 + 1.5501*(H-0.6778).^(-3.064);

end;end;

function H=HofH1(H1);

if H1 <= 3.32H = 3;

elseif H1 < 5.3H = 0.6778 + 1.1536*(H1-

3.3).^(-0.326);elseH = 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);

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Drag PredictionSquire-Young Formula

25

,,

,,

,

2

+

∞⎟⎟⎠

⎞⎜⎜⎝

⎛=

+=uppergeTrailingEdH

geTrailingEdEgeTrailingEdupperd

lowerdupperdd

VU

cC

CCC

θ

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Some useful expressions for flat-plate boundary layers

• Laminar flows

• Turbulent flows

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3-D Boundary Layers on Wings

• Spanwise pressure gradients

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• Spanwise pressure gradient effects on the boundary layer

The gradients cause the boundary layer to flow outward, piling up tired, slow air near the tips and contributing to premature tip stall.

The streamwise growth of the boundary layer tends to cause early stall near the tips.

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The spanwise flow on the wing also tends to createstreamwise vorticity in the boundary layer

This cross-flow instability is very damaging to laminar boundary layers and quickly causes transition to turbulent flow.

Wings with sweep angles in excess of 30 to 40° require some sort of boundary layer control (e.g. suction) to maintain laminar flow.

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Coupled inviscid / viscous iterative methods

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Effect of boundary-layer displacement on the pressure distribution and lift of a modern airfoil

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Introduction to “Airfoil”• It is an airfoil analysis program that is an adaptation of

the original program "mcarfa" .

• It can be used to predict the aerodynamic characteristics of airfoils in subsonic, viscous flows.

• The computed aerodynamic characteristics include pressure distributions, lift, drag, pitch moment, transition position, and incipient separation on the airfoils.

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Method

• The program combines the potential-flow solution with boundary-layer theory in an iterative manner.

• The interrelationship between the potential-flow solution and the boundary-layer effects is included .

• Providing significant improvements in prediction accuracy.

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Modifications to "mcarfa"

• Simplification of the input data procedure

• Ability to generate NACA airfoil geometry data

• Compacting output file

• Displaying airfoil shape and pressure distributions in a graphic manner

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Running Procedure (1)• Issue the command "airfoil" • Select the option and input parameters through the

interface– Airfoil Definition Option:

1 -- Generate NACA airfoil2 -- input airfoil data fileSelect 1 or 21 <--- selected by use

Note: if 2 is selected, an input file, which defines the geometry of an airfoil, should be set up before the program is invoked. The input file format is given in section 3.

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Running Procedure (2)

NACA Airfoil Selection:1 -- NACA 4-Digit Airfoi2 -- NACA Standard 5 Digit Airfoil1 <--- selected by user

Enter NACA 4-Digit Airfoil Name:Input Format: NACA XXXXNACA 4412 <--- defined by user

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Running Procedure (3)Enter name of output file ===>demo.out <--- defined by user

Enter Parameters:Reference chord length = ? (ft)1.0 <--- input parameterangle of attack = ? (in deg)4.0 <--- input parameterMach number = ? ( 0.05 < M < Mcr)0.1 <--- input parameterReynolds number = ? (in mllions)0.8 <--- input parameter

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Running Procedure (4)

• Obtain the computed results – Open output file "demo.out" using Text

Editor to get the aerodynamic characteristics of the airfoil defined by user.

– Issue the M-file "airfoil.m" under Matlabenvironment to display the airfoil shape and pressure distributions on the airfoil

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---- Output of Airfoil Program -----

TITLE -- NACA 4412

Mach number = 0.100 Reynolds Number = 0.800 millionAngle of Attack = 4.000 Ref. Chord = 1.000 feetCL = 0.8584 CD = 0.0101 CM(C/4) = -0.0925

Transition Point:Upper x/c = 0.31117Lower x/c = 0.95974

Separation (Percent of Surface):Upper = 2.272Lower = 0.000

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---- Pressure Distribution on Upper Surface ----x/c Zu Cp

0.00183 0.00980 -0.286800.00668 0.01780 -0.840130.01413 0.02571 -1.102470.02453 0.03365 -1.201440.03776 0.04147 -1.221300.05337 0.04900 -1.210700.07107 0.05615 -1.193840.09018 0.06281 -1.177000.11100 0.06907 -1.165770.13289 0.07489 -1.170680.15481 0.07992 -1.171410.17790 0.08437 -1.15723

0.20142 0.08822 -1.14189

……… …….. ………

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Airfoil Shape and Pressure Distributions

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Limitations

• Only NACA 4-digit Airfoil and NACA Standard 5 Digit Airfoil ordinates can be generated automatically.

• Mach number must be greater than 0.05 and less than criteria Mach number.

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Introduction to XFOIL

• XFOIL is a software which goal was to combine the speed and accuracy of high-order panel methods with the new fully-coupled viscous/inviscid interaction methods.

• It was developed by Mark Drela, MIT and Harold Youngren, Aerocraft, Inc.

• It consists of a collection of menu-driven routines which perform various useful functions .

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Introduction to XFOIL• Functions

– Viscous (or inviscid) analysis of an existing airfoil

– Airfoil design and redesign by interactive specification of a surface speed distribution via screen cursor or mouse.

– Airfoil redesign by interactive specification of new geometric parameters

– Blending of airfoils

– Drag polar calculation with fixed or varying Reynolds and/or Mach numbers.

– Writing and reading of airfoil geometry and polar save files

– Plotting of geometry, pressure distributions, and polar.

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Homework

• Compare the results from using panel method program PABLO for inviscid flows and viscous flows and using Airfoil panel method coupled with boundary-layer theory in an iterative manner.– Lift coefficient

– Lift coefficient slope

– Drag coefficient

– Pressure distribution