low speed aerodynamics notes 8:viscous flo · title: microsoft powerpoint - ae2020f07_viscous07...

Low Speed AerodynamicsLow Speed Aerodynamics

Notes 8:Viscous Flow

Copyright 2007 N. Komerath. Other rights may be specified with individual items. All rights reserved.

Shear Stress vs. Rate-of-Strain Relations

The direction is indicated by the second subscript. The units of stressare Force/Area, such as Newtons/m2, psi, or psf.

xy uy

vx

yz

vz

wy

These can be written in compact form Here u is used to represent any of the velocity components and

zx uz

wx

These can be written in compact form. Here u is used to represent any of the velocity components, andx is used to represent any of the spatial coordinates. i and j representing x,y,z in turn.

uj ui

ij xi

x j


Normal stresses due to viscosity

yy 2 vy

ux

vy

wz

xx 2 u

u

v

w

y x y z

xx x x y z

2 w u v w

zz 2 w

z u

xvy

wz

Stokes’ Hypothesis: 2

3

So the normal stresses due to viscosity add up to zero.


2D Navier-Stokes Equations For Incompressible FlowCons of mass:Cons. of mass:

Cons. of x-momentum:

Cons. of y-momentum:y


Exact Solution to the Steady 2D Navier-Stokes Equations For Incompressible Flow

Plane Parallel Steady Flow Between Two Infinite Plates: Couette Flow

y,v hx,uy,

From the problem, there cannot be any change in u along x: So d(u )/dx is zero. Also, v = 0. So u is u(y) (y)


Case 1: Flow between two plates, driven by plate motion

Boundary conditions: u=v=0 at y=0 u = U and v=0 at y=h.

UU

hy,v

u(y) y dp y h Uy

x,u

u(y) 2 dx

y h h

If dp/dx=0, u(y) Uy Linear velocity profileu(y)h

Shear stress Skin Friction Coefficient


Case 1: Flow between two stationary plates, driven by pressure gradient

x,u

u(y) y2

dpd

y h 2 dx

( Note: Sign of dp/dx must be opposite to that of u )


Hagen-Poiseuille Flow: Steady axisymmetric, fully-developed incompressible laminar flow through a constant-diameter pipe

u-momentum equation reduces to Integrating,

F fl t i fi it t i ( 0) C 0 Al 0 t th ll RFor flow to remain finite at axis (r=0), C=0. Also, u = 0 at the wall, r=R.

Parabolic Velocity Profile


Boundary Layer EquationsBoundary Layer EquationsBoundary Layer EquationsBoundary Layer Equations

Newton’s 2nd Law

Conservation Equations in integral form

Convert to differential form

Simplify by use of OM analysis


Boundary Layer ApproximationsBoundary Layer Approximationsy y ppy y ppAt very high Reynolds number (~100,000+) based on distance along the surface in the freestream direction,

1. The spatial rate at which properties change across a boundary layer is very large, compared to the rate at which things change along the flow direction In other words derivatives with respect to rate at which things change along the flow direction. In other words, derivatives with respect to y are much higher than derivatives with respect to x. d( )/dy >> d()/dx

2 Order of magnitude analysis shows that the v-momentum equation reduces to 2. Order of magnitude analysis shows that the v-momentum equation reduces to

In other words, static pressure is constant across a boundary layer.


Boundary Layer EquationsBoundary Layer Equations

a)The variation of pressure across the boundary layer is small. ) p y yThus, we can assume that pressure varies only with respect to x. The pressure field may be computed at the edge of the boundary layer from inviscid flow theories, without any knowledge of the boundary layer characteristics.

b) The v- component of velocity within the boundary layer is of the order of (U∞/c) where is the boundary layer thickness.

c) The boundary layer thickness is very small, and for laminar flows it varies inversely as the square root of Reynolds number U∞c/.

d) We will assume the airfoil surface to be flat, and use a Cartesian coordinate system, neglecting surface curvature. This is because the boundary layer thickness is very small compared to the airfoil s rface radi s of c r at re


to the airfoil surface radius of curvature.

u- momentum equationu momentum equation

For high Reynolds number flows, our order of magnitude analysis says that the u- momentum equation may be approximated as

v- momentum equationv momentum equation


Boundary Layer FeaturesBoundary Layer FeaturesBoundary Layer FeaturesBoundary Layer Features

y

V l it fil

( )

Velocity profile

x

(x)Du/dy at the wall


Boundary Layer ParametersBoundary Layer Parameters

1. Boundary Layer Thickness : Defined as the y- location where u/ue reaches 0.99%, that is the u- velocity becomes 99% of the edge velocity.

2. Displacement Thickness * : 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

u dyuu

ee

01*

3. Momentum Thickness : This is a measure of the momentum loss within the boundary layer as a result of the reduced velocities within the boundary layervelocities within the boundary layer.

dyuu

uu

eee

01


Shape Factor HShape Factor HShape Factor HShape Factor H

4. Shape Factor H : This quantity is defined as the ratio */q .4. Shape Factor H : This quantity is defined as the ratio /q .

For laminar flows H varies between 2 and 3. It is 3.7 near separation point. Thus excessively large values of H (above 3) indicate that the boundary layer y g ( ) y yis about to separate. In turbulent flows, H varies between 1.5 and 2.


Wall Shear Stress and DragWall Shear Stress and Drag

Surface Shear Stress: Surface Shear Stress: The shear stress at the wall can be found from definition of shear stress

u

Skin friction Coefficient cf:

wallwwall y

Skin friction Coefficient cf:

2

21

ee

wf

uc

Skin Friction Drag, D : Shear stress may be numerically integrated over the entire solid surface to give the skin friction drag force along the x- axis:

2

dxD

Ski F i ti D C ffi i t Cd Th d f i ll di i li d b th

aceEntireSurf

wdxD

Skin Friction Drag Coefficient Cd: The drag force is usually non-dimensionalized by the freestream dynamic pressure times the chord of the airfoil c, giving the skin friction drag coefficient along the x- axis, Cd.

DCd21


cu 221

Thwaites' Integral method for Laminar Incompressible Boundary Layers

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).:C a e do ess, O o d, 960)

(1) dduBA

ddu ee 22

dxdx

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


The above equation may be analytically integrated yielding

2 0.45 5x d 2 ue

6x0

2

ue6 ue

5

0 dx 2

x0 e

ue6

For blunt bodies such as airfoils, the edge velocity ue is zero at x=0, the stagnation point. For h d i h fl l h hi k i h l di sharp nosed geometries such as a flat plate, the momentum thickness q is zero at the leading

edge. In these cases, the term in the square bracket vanishes.

The integral may be evaluated, at least numerically when ue is known.


After is found, the following relations are used to compute the shape factor H and the shear stress at the wall w.

1.00 2245753612 HFor 1.00 24.575.361.2 H

01.0 0147.04722H

For

For 01.0

107.0

472.2H

2where

dxdue

2 62.009.0

e

wallu

D it th i i i i th b f l Despite the empiricism in the above formulas, Thwaites' integral method is considered to be the bestof integral boundary layer methods.


Transition

Consider the 2-D unsteady, incompressible viscous flow past an airfoil at a sufficiently high Reynolds number

000300Re cU

We find that the boundary layer over both the upper and lower surfaces may be divided into three regions They are (1) laminar region (2)

000,300Re

three regions. They are (1) laminar region, (2) transitional region, and (3) turbulent region.

Transition Strip: rough strip taped toairfoil close to max thickness location, to triggerdisturbances that cause quick transition to turbulence

(1) (3)disturbances that cause quick transition to turbulence.(2)


The turbulent region has unsteady flow, although the fluctuations are small and occur about some " t d " fl l l "steady" mean flow levels. The velocity profile is fuller than in the laminar flow case. Skin friction is higher than in laminar flow.

yy

turbulent


xlaminar

Incompressible Boundary Layer on Flat Plate at Incompressible Boundary Layer on Flat Plate at zero angle of attack: Laminar vs Turbulentzero angle of attack: Laminar vs Turbulentzero angle of attack: Laminar vs. Turbulentzero angle of attack: Laminar vs. Turbulent

Laminar Turbulent

0.37x 5.0x

Re0.2 5.0xRe0.5

0 664x 0 664 0.664xRe0.5 c f

0.664Re0.5

Cf 0.074Rec

0.2Cf

1.328Rec

0.5


Incompressible Turbulent Skin Friction Correlation At Incompressible Turbulent Skin Friction Correlation At Large Reynolds NumbersLarge Reynolds NumbersLarge Reynolds NumbersLarge Reynolds Numbers

Shultz / Grunow equation, modified by Kulfan, Boeing Co..

45.2))(Re(295.0 xLogC fi 106 < Rex < 109f


Courtesy, Dr. R. Kulfan, BOEING Co

Comparison of Incompressible Local Skin Friction PredictionsComparison of Incompressible Local Skin Friction Predictions

0.0050

14 15 1 7

Cfix Cfi . * log(Re * ) .

n Fr

ictio

n ,C

fi

0.0040

Cfi Log x 0 295 2 45. * ( (Re )) .

ible

Loc

al S

kin

0.0030

Inco

mpr

essi

0.0020

0.0010

Theory: Modified Schultz / Grunow Theory: Karmen - Schoenerr

106 10 7 10 8 10 9Reynolds Number, Rex

0.0000


Page 8Courtesy, Dr. R. Kulfan, BOEING Co

Airfoil Drag vs. Angle of AttackNote: When viscous drag and/or flow separation are present, airfoils do have drag that changes with Note: When viscous drag and/or flow separation are present, airfoils do have drag that changes with angle of attack. In other words, CD0 of an aircraft is not completely independent of angle of attack, over a large range of angle of attack. This has nothing to do with tip vortices etc. So when we must use an averaged value for CD0 when we use lifting line theory and say that for a whole aircraft (note g D0 g y y (nomenclature)

ARCCC L

DD )(

2

0

c

eARDD )(0

cd The more usual and general way, is to represent the drag as

20 LDD KCCC 0 LDD KCCC

K will now be > 1 K will now be

eAR)(


Aircraft Drag Polar

cLmax

cL

Best L/D? Can you prove it?

cD

cD0


Relation between Boundary Layer Profile and Skin Friction Relation between Boundary Layer Profile and Skin Friction DragDragDragDrag

y A positive pressure gradient dp/dx would cause the boundary layer velocity profile to become boundary layer velocity profile to become “unhealthy” – develop an inflexion point.

Beyond that, 3 things can happen:

xlaminar

turbulentBeyond that, 3 things can happen:

1. Laminar separation2. Transition to turbulence.x

wwallu

3. Turbulent separation

wallwwall y

TurbulentTurbulent boundary layer hashigh velocities close to surface, large du/dy high skin friction large du/dy – high skin friction drag (bad). However, very stable profile _ delays separation (good!)


delays separation (good!)

Separation

Stagnationg


Separation Control: Surface Blowing

CavitiesCavities


Moving Wall


Boundary Layer Suctiony y


Vortex Interactions & Ground EffectTechnique: Think of the velocity induced at the axis of each vortex by every other vortex.A solod wall is treated as an “Image plane” with an imaginary mirror image of the vortex on the other side. on the other side.


7’ X 10’ section

UH 1H

Rotor Wake Evolution in Axial Flight

UH-1H model rotor

30’ X 31’ settling chamber

Copyright 2007 N. Komerath. Other rights may be specified with individual items. All rights reserved. Vortex Pairing Sequence

low speed aerodynamics notes 8:viscous flo · title: microsoft powerpoint - ae2020f07_viscous07...

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