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Free Powerpoint TemplatesPage 1
KULIAH XKULIAH XEXTERNAL EXTERNAL
INCOMPRESSIBLE INCOMPRESSIBLE VISCOUS FLOWVISCOUS FLOW
Nazaruddin SinagaNazaruddin Sinaga
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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
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The Boundary-Layer Concept
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The Boundary-Layer Concept
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Boundary Layer Thickness
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Boundary Layer Thickness
• Disturbance Thickness, where
Displacement Thickness, *
Momentum Thickness,
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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)
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Navier-Stokes EquationNavier-Stokes EquationCartesian CoordinatesCartesian Coordinates
Continuity
X-momentum
Y-momentum
Z-momentum
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Laminar Flat-PlateLaminar Flat-PlateBoundary Layer: Exact SolutionBoundary Layer: Exact Solution
• Governing Equations
• For incompresible steady 2D cases:
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Laminar Flat-PlateBoundary Layer: Exact Solution
• Boundary Conditions
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Laminar Flat-PlateBoundary Layer: Exact Solution
• Equations are Coupled, Nonlinear, Partial Differential Equations
• Blassius Solution:– Transform to single, higher-order, nonlinear, ordinary
differential equation
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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
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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
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Blasius Similarity Solution
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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
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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 )
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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
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Turbulent Boundary Layer
Illustration of unsteadiness of a turbulent BL
Black lines: instantaneousPink line: time-averaged
Comparison of laminar and turbulent BL profiles
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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
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• Results of Numerical Analysis
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Momentum Integral Equation
• Provides Approximate Alternative to Exact (Blassius) Solution
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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
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Simplify Momentum Integral Equation(Item 1)
The Momentum Integral Equation becomes
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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)
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Laminar Flow Results(Polynomial Velocity Profile)
Compare to Exact (Blassius) results!
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow– Example: 1/7-Power Law Profile (Item 2)
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Use of the Momentum Equation for Flow with Zero Pressure Gradient
• Turbulent Flow Results(1/7-Power Law Profile)
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Pressure Gradients in Boundary-Layer Flow
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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.
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Drag and Lift• Lift and drag forces can be found by
integrating pressure and wall-shear stress.
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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).
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Drag
• Drag Coefficient
with
or
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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
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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
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Drag• Flow over a Flat Plate Parallel to the Flow: Friction
Drag (Continued)
Laminar BL:
Turbulent BL:
… plus others for transitional flow
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Drag Coefficient
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Drag• Flow over a Flat Plate Perpendicular to the
Flow: Pressure Drag
Drag coefficients are usually obtained empirically
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Drag• Flow over a Flat Plate Perpendicular to the
Flow: Pressure Drag (Continued)
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Drag• Flow over a Sphere : Friction and Pressure Drag
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Drag• Flow over a Cylinder: Friction and Pressure
Drag
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Streamlining• Used to Reduce Wake and Pressure Drag
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Lift• Mostly applies to Airfoils
Note: Based on planform area Ap
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Lift
• Examples: NACA 23015; NACA 662-215
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Lift• Induced Drag
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Lift• Induced Drag (Continued)
Reduction in Effective Angle of Attack:
Finite Wing Drag Coefficient:
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Lift
• Induced Drag (Continued)
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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.
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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!
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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
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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
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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.
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Streamlining
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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).
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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.
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CD of Common Geometries
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CD of Common Geometries
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CD of Common Geometries
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Flat Plate Drag
• Drag on flat plate is solely due to friction created by laminar, transitional, and turbulent boundary layers.
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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
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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
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Cylinder and Sphere Drag
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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
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Effect of Surface Roughness
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Lift• Lift is the net force (due
to pressure and viscous forces) perpendicular to flow direction.
• Lift coefficient
• A=bc is the planform area
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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.
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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.
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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
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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.
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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.
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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.
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Lift Generated by Spinning
Superposition of Uniform stream + Doublet + Vortex
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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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Free Powerpoint TemplatesPage 81
The EndThe End
Terima kasihTerima kasih
81
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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
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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
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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
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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
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Blassius exact solution IVBoundary layer over a flat plate
86
Solution
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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
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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
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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
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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
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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,