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KULIAH XKULIAH XEXTERNAL EXTERNAL
INCOMPRESSIBLE INCOMPRESSIBLE VISCOUS FLOWVISCOUS FLOW
Nazaruddin SinagaNazaruddin Sinaga
Main TopicsMain Topics• The Boundary-Layer Concept• Boundary-Layer Thickness• Laminar Flat-Plate Boundary Layer: Exact Solution• Momentum Integral Equation• Use of the Momentum Equation for Flow with Zero
Pressure Gradient• Pressure Gradients in Boundary-Layer Flow• Drag• Lift
The Boundary-Layer Concept
The Boundary-Layer Concept
Boundary Layer Thickness
Boundary Layer Thickness
• Disturbance Thickness, where
Displacement Thickness, *
Momentum Thickness,
Boundary Layer LawsBoundary Layer Laws
1. The velocity is zero at the wall (u = 0 at y = 0)
2. The velocity is a maximum at the top of the layer (u = um at = )
3. The gradient of BL is zero at the top of the layer (du/dy = 0 at y = )
4. The gradient is constant at the wall (du/dy = C at y = 0)
5. Following from (4): (d2u/dy2 = 0 at y = 0)
Navier-Stokes EquationNavier-Stokes EquationCartesian CoordinatesCartesian Coordinates
Continuity
X-momentum
Y-momentum
Z-momentum
Laminar Flat-PlateLaminar Flat-PlateBoundary Layer: Exact SolutionBoundary Layer: Exact Solution
• Governing Equations
• For incompresible steady 2D cases:
Laminar Flat-PlateBoundary Layer: Exact Solution
• Boundary Conditions
Laminar Flat-PlateBoundary Layer: Exact Solution
• Equations are Coupled, Nonlinear, Partial Differential Equations
• Blassius Solution:– Transform to single, higher-order, nonlinear, ordinary
differential equation
Boundary Layer Procedure
• Before defining and * and are there analytical solutions to the BL equations?– Unfortunately, NO
• Blasius Similarity Solution boundary layer on a flat plate, constant edge velocity, zero external pressure gradient
Blasius Similarity Solution• Blasius introduced similarity variables
• This reduces the BLE to
• This ODE can be solved using Runge-Kutta technique
• Result is a BL profile which holds at every station along the flat plate
Blasius Similarity Solution
Blasius Similarity Solution
• Boundary layer thickness can be computed by assuming that corresponds to point where U/Ue = 0.990. At this point, = 4.91, therefore
• Wall shear stress w and friction coefficient Cf,x can be directly related to Blasius solution
Recall
Displacement Thickness• Displacement thickness * is the imaginary
increase in thickness of the wall (or body), as seen by the outer flow, and is due to the effect of a growing BL.
• Expression for * is based upon control volume analysis of conservation of mass
• Blasius profile for laminar BL can be integrated to give
(1/3 of )
Momentum Thickness• Momentum thickness is another
measure of boundary layer thickness.
• Defined as the loss of momentum flux per unit width divided by U2 due to the presence of the growing BL.
• Derived using CV analysis.
for Blasius solution, identical to Cf,x
Turbulent Boundary Layer
Illustration of unsteadiness of a turbulent BL
Black lines: instantaneousPink line: time-averaged
Comparison of laminar and turbulent BL profiles
Turbulent Boundary Layer
• All BL variables [U(y), , *, ] are determined empirically.
• One common empirical approximation for the time-averaged velocity profile is the one-seventh-power law
• Results of Numerical Analysis
Momentum Integral Equation
• Provides Approximate Alternative to Exact (Blassius) Solution
Momentum Integral Equation
Equation is used to estimate the boundary-layer thickness as a function of x:
1. Obtain a first approximation to the freestream velocity distribution, U(x). The pressure in the boundary layer is related to the freestream velocity, U(x), using the Bernoulli equation
2. Assume a reasonable velocity-profile shape inside the boundary layer
3. Derive an expression for w using the results obtained from item 2
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Simplify Momentum Integral Equation(Item 1)
The Momentum Integral Equation becomes
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Laminar Flow– Example: Assume a Polynomial Velocity Profile (Item 2)
• The wall shear stress w is then (Item 3)
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Laminar Flow Results(Polynomial Velocity Profile)
Compare to Exact (Blassius) results!
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow– Example: 1/7-Power Law Profile (Item 2)
Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow Results(1/7-Power Law Profile)
Pressure Gradients in Boundary-Layer Flow
DRAG AND LIFTDRAG AND LIFT• Fluid dynamic forces are
due to pressure and viscous forces acting on the body surface.
• Drag: component parallel to flow direction.
• Lift: component normal to flow direction.
Drag and Lift• Lift and drag forces can be found by
integrating pressure and wall-shear stress.
Drag and Lift
• In addition to geometry, lift FL and drag FD forces are a function of density and velocity V.
• Dimensional analysis gives 2 dimensionless parameters: lift and drag coefficients.
• Area A can be frontal area (drag applications), planform area (wing aerodynamics), or wetted-surface area (ship hydrodynamics).
Drag
• Drag Coefficient
with
or
Drag• Pure Friction Drag: Flat Plate Parallel to
the Flow• Pure Pressure Drag: Flat Plate
Perpendicular to the Flow• Friction and Pressure Drag: Flow over a
Sphere and Cylinder• Streamlining
Drag• Flow over a Flat Plate Parallel to the Flow: Friction
Drag
Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available
Drag• Flow over a Flat Plate Parallel to the Flow: Friction
Drag (Continued)
Laminar BL:
Turbulent BL:
… plus others for transitional flow
Drag Coefficient
Drag• Flow over a Flat Plate Perpendicular to the
Flow: Pressure Drag
Drag coefficients are usually obtained empirically
Drag• Flow over a Flat Plate Perpendicular to the
Flow: Pressure Drag (Continued)
Drag• Flow over a Sphere : Friction and Pressure Drag
Drag• Flow over a Cylinder: Friction and Pressure
Drag
Streamlining• Used to Reduce Wake and Pressure Drag
Lift• Mostly applies to Airfoils
Note: Based on planform area Ap
Lift
• Examples: NACA 23015; NACA 662-215
Lift• Induced Drag
Lift• Induced Drag (Continued)
Reduction in Effective Angle of Attack:
Finite Wing Drag Coefficient:
Lift
• Induced Drag (Continued)
Fluid Dynamic Forces and Moments
Ships in waves present one of the most difficult 6DOF problems.
Airplane in level steady flight: drag = thrust and lift = weight.
Example: Automobile DragScion XB Porsche 911
CD = 1.0, A = 25 ft2, CDA = 25 ft2 CD = 0.28, A = 10 ft2, CDA = 2.8 ft2
• Drag force FD=1/2V2(CDA) will be ~ 10 times larger for Scion XB
• Source is large CD and large projected area
• Power consumption P = FDV =1/2V3(CDA) for both scales with V3!
Drag and Lift
• For applications such as tapered wings, CL and CD may be a function of span location. For these applications, a local CL,x and CD,x are introduced and the total lift and drag is determined by integration over the span L
Friction and Pressure Drag• Fluid dynamic forces are comprised
of pressure and friction effects.• Often useful to decompose,
– FD = FD,friction + FD,pressure
– CD = CD,friction + CD,pressure • This forms the basis of ship model
testing where it is assumed that– CD,pressure = f(Fr)– CD,friction = f(Re)
Friction drag
Pressure drag
Friction & pressure drag
Streamlining• Streamlining reduces drag
by reducing FD,pressure, at the cost of increasing wetted surface area and FD,friction.
• Goal is to eliminate flow separation and minimize total drag FD
• Also improves structural acoustics since separation and vortex shedding can excite structural modes.
Streamlining
Streamlining via Active Flow Control
• Pneumatic controls for blowing air from slots: reduces drag, improves fuel economy for heavy trucks (Dr. Robert Englar, Georgia Tech Research Institute).
CD of Common Geometries• For many geometries, total drag CD is
constant for Re > 104 • CD can be very dependent upon
orientation of body.• As a crude approximation,
superposition can be used to add CD from various components of a system to obtain overall drag. However, there is no mathematical reason (e.g., linear PDE's) for the success of doing this.
CD of Common Geometries
CD of Common Geometries
CD of Common Geometries
Flat Plate Drag
• Drag on flat plate is solely due to friction created by laminar, transitional, and turbulent boundary layers.
Flat Plate Drag• Local friction coefficient
– Laminar:
– Turbulent:
• Average friction coefficient
– Laminar:
– Turbulent:
For some cases, plate is long enough for turbulent flow, but not long enough to neglect laminar portion
Effect of Roughness• Similar to Moody Chart
for pipe flow• Laminar flow unaffected
by roughness• Turbulent flow
significantly affected: Cf can increase by 7x for a given Re
Cylinder and Sphere Drag
Cylinder and Sphere Drag• Flow is strong function of
Re.• Wake narrows for turbulent
flow since TBL (turbulent boundary layer) is more resistant to separation due to adverse pressure gradient.
• sep,lam ≈ 80º
• sep,turb ≈ 140º
Effect of Surface Roughness
Lift• Lift is the net force (due
to pressure and viscous forces) perpendicular to flow direction.
• Lift coefficient
• A=bc is the planform area
Computing Lift• Potential-flow approximation gives accurate
CL for angles of attack below stall: boundary layer can be neglected.
• Thin-foil theory: superposition of uniform stream and vortices on mean camber line.
• Java-applet panel codes available online: http://www.aa.nps.navy.mil/~jones/online_tools/panel2/
• Kutta condition required at trailing edge: fixes stagnation pt at TE.
Effect of Angle of Attack• Thin-foil theory shows that
CL≈2 for < stall
• Therefore, lift increases linearly with
• Objective for most applications is to achieve maximum CL/CD ratio.
• CD determined from wind-tunnel or CFD (BLE or NSE).
• CL/CD increases (up to order 100) until stall.
Effect of Foil Shape• Thickness and camber
influences pressure distribution (and load distribution) and location of flow separation.
• Foil database compiled by Selig (UIUC)http://www.aae.uiuc.edu/m-selig/ads.html
Effect of Foil Shape
• Figures from NPS airfoil java applet.
• Color contours of pressure field
• Streamlines through velocity field
• Plot of surface pressure
• Camber and thickness shown to have large impact on flow field.
End Effects of Wing Tips• Tip vortex created by
leakage of flow from high-pressure side to low-pressure side of wing.
• Tip vortices from heavy aircraft persist far downstream and pose danger to light aircraft. Also sets takeoff and landing separation at busy airports.
End Effects of Wing Tips• Tip effects can be
reduced by attaching endplates or winglets.
• Trade-off between reducing induced drag and increasing friction drag.
• Wing-tip feathers on some birds serve the same function.
Lift Generated by Spinning
Superposition of Uniform stream + Doublet + Vortex
Lift Generated by Spinning
• CL strongly depends on rate of rotation.
• The effect of rate of rotation on CD is small.
• Baseball, golf, soccer, tennis players utilize spin.
• Lift generated by rotation is called The Magnus Effect.
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81
Derivation of the boundary layer equations II
82
02
1
1
1
*
*
*
*
x
U
x
U
22
122
12
12
1
11
1*
*
L**
**
*
**
x
UL
Rex
*p
x
UU
x
UU
*x
*p
2
0
Blassius exact solution I Boundary layer over a flat plate
83
Variable transformation: ,
Stream function definition:
,
21 x
U
12 x
U
11 xX 1
22 x
UxX
21
2,1 XGx
UXX
Wall boundary condition:
= Ue1
p
Blassius exact solution II Boundary layer over a flat plate
84
Ordinary differential equation
Boundary conditions
The analytical solution of the ordinary differential equation was obtained by Blasius using series expansions
0232
3
22
2
dX
Gd
dX
GdG
002 XG
0022
XdX
dG
1122
XdX
dG
Blassius exact solution IIIBoundary layer over a flat plate
85
Solution
Velocity along x1-direction:
Velocity along x2-direction:
Boundary layer thickness:
Displacement thickness:
Wall shear stress:
21 dX
dGU
G
dX
dGXU
x 222 2
11
1Re
1
1860402
x
URe
.
11Re
15
x
x
1Re
17208.1 1
xd x
1Re
1332.0 2
xw U
Blassius exact solution IVBoundary layer over a flat plate
86
Solution
Von Karman integral momentum equation I
pF AD 1
87
Momentum conservation along x1-direction
ddxdx
dpdx
dx
dppdp
ddxxpF CD
11
11
111
ddxdx
dppd
ddxxpF BC
11
111
2
1
2
1
11 dxF wAD
CDBCAB MMMF 1111
p
p d)dxx
pp(
11
88
Solution
Considerations: p(x1) = pe(x1)
Von Karman integral momentum equation II
20
21
12
01
11
1
dxUx
dxUx
Udx
dPew
01
11
dx
dp
dx
dUU ee
e
1ee UU
02
11
1
2 1 dxU
U
U
U
xUw
Approximate solutions I Linear, quadratic, cubic and sinusoidal velocity profiles
89
1. Assumption of a self-similar velocity profile U1
*= f (x2*)
2. Specifications of the boundary conditions
3. Resolution of the Von Karman integral momentum equation
1
0211
1
2 1 dxUUx
Uw
Approximate solutions II Example: quadratic velocity profiles
90
Velocity profile
General form + boundary conditions = velocity profile along x1-dir
Von Karman integral momentum equation solution
2*23
*221
*1 xCxCCU 0)0( *
2*1 xU
1)1( *2
*1 xU
0)1( *2*
2
*1
x
x
U
2*2
*2
*1 2 xxU
**
*******
*)()(
U
LxUdxxxxx
xUw
2
15
2212
1
21
0 222222
1
2
Re*
** 1
151x
*1
*
1Re
1477.5 x
x
boundary layer thickness
Approximate solutions III Example: quadratic velocity profiles
*
U
Lw
2
91
Displacement thickness:
from the definition =>
Velocity along x2-direction
from continuity equation
Wall shear stress
from =>
201
11 dxU
U
ed
1Re
1826.1 *
1*
xd x
1Re
1913.02
x
UU
1
2
3650x
w
U
Re,
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