1. EXTERNAL INCOMPRESSIBLE VISCOUS FLOW Nazaruddin Sinaga
KULIAH X
EXTERNAL INCOMPRESSIBLE VISCOUS FLOW
Nazaruddin Sinaga

4. Main 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

5. The Boundary-Layer Concept

6. The Boundary-Layer Concept

7. Boundary Layer Thickness

8. Boundary Layer Thickness
Disturbance Thickness, d where
Displacement Thickness, d*
Momentum Thickness, q

14. Boundary Layer Laws The velocity is zero at the wall (u = 0 at y = 0)
The velocity is a maximum at the top of the layer (u = um at =  )
The gradient of BL is zero at the top of the layer (du/dy = 0 at y =  )
The gradient is constant at the wall (du/dy = C at y = 0)
Following from (4): (d2u/dy2 = 0 at y = 0)

15. Navier-Stokes Equation Cartesian Coordinates
Continuity
X-momentum
Y-momentum
Z-momentum

16. Laminar Flat-Plate Boundary Layer: Exact Solution
Governing Equations
For incompresible steady 2D cases:

17. Laminar Flat-Plate Boundary Layer: Exact Solution
Boundary Conditions

18. Laminar Flat-Plate Boundary Layer: Exact Solution
Equations are Coupled, Nonlinear, Partial Differential Equations
Blassius Solution:
Transform to single, higher-order, nonlinear, ordinary differential equation

19. 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

20. 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

21. Blasius Similarity Solution

22. Blasius Similarity Solution
Boundary layer thickness can be computed by assuming that  corresponds to point where U/Ue = At this point,  = 4.91, therefore
Wall shear stress w and friction coefficient Cf,x can be directly related to Blasius solution
Recall

23. 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 )

24. 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

25. Turbulent Boundary Layer
Black lines: instantaneous
Pink line: time-averaged
Illustration of unsteadiness of a turbulent BL
Comparison of laminar and turbulent BL profiles

26. 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

28. Results of Numerical Analysis

29. Momentum Integral Equation
Provides Approximate Alternative to Exact (Blassius) Solution

30. Momentum Integral Equation
Equation is used to estimate the boundary-layer thickness as a function of x:
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
Assume a reasonable velocity-profile shape inside the boundary layer
Derive an expression for tw using the results obtained from item 2

31. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Simplify Momentum Integral Equation (Item 1)
The Momentum Integral Equation becomes

32. 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 tw is then (Item 3)

33. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Laminar Flow Results (Polynomial Velocity Profile)
Compare to Exact (Blassius) results!

34. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Turbulent Flow
Example: 1/7-Power Law Profile (Item 2)

35. Use of the Momentum Equation for Flow with Zero Pressure Gradient
Turbulent Flow Results (1/7-Power Law Profile)

36. Pressure Gradients in Boundary-Layer Flow

37. DRAG 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.

38. Drag and Lift
Lift and drag forces can be found by integrating pressure and wall-shear stress.

39. 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).

40. Drag
Drag Coefficient
with or

41. 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

42. 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

43. Drag
Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued)
Laminar BL:
Turbulent BL:
… plus others for transitional flow

44. Drag Coefficient

45. Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag
Drag coefficients are usually obtained empirically

46. Drag
Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag (Continued)

47. Drag
Flow over a Sphere : Friction and Pressure Drag

48. Drag
Flow over a Cylinder: Friction and Pressure Drag

49. Streamlining
Used to Reduce Wake and Pressure Drag

50. Lift
Mostly applies to Airfoils
Note: Based on planform area Ap

51. Lift
Examples: NACA 23015; NACA

52. Lift
Induced Drag

53. Lift Induced Drag (Continued) Reduction in Effective Angle of Attack:
Finite Wing Drag Coefficient:

54. Lift
Induced Drag (Continued)

55. 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.

56. Example: Automobile Drag
Scion 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=1/2V2(CDA) will be ~ 10 times larger for Scion XB
Source is large CD and large projected area
Power consumption P = FDV =1/2V3(CDA) for both scales with V3!

57. 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

58. 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

59. 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.

60. Streamlining

61. 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).

62. 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.

63. CD of Common Geometries

64. CD of Common Geometries

65. CD of Common Geometries

66. Flat Plate Drag
Drag on flat plate is solely due to friction created by laminar, transitional, and turbulent boundary layers.

67. Flat Plate Drag
Local friction coefficient
Laminar:
Turbulent:
Average friction coefficient
For some cases, plate is long enough for turbulent flow, but not long enough to neglect laminar portion

68. 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

69. Cylinder and Sphere Drag

70. 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

71. Effect of Surface Roughness

72. Lift
Lift is the net force (due to pressure and viscous forces) perpendicular to flow direction.
Lift coefficient
A=bc is the planform area

73. 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:
Kutta condition required at trailing edge: fixes stagnation pt at TE.

74. 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.

75. Effect of Foil Shape
Thickness and camber influences pressure distribution (and load distribution) and location of flow separation.
Foil database compiled by Selig (UIUC)

76. 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.

77. 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.

78. 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.

79. Lift Generated by Spinning
Superposition of Uniform stream + Doublet + Vortex

80. 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.

81. The End
Terima kasih

82. Derivation of the boundary layer equations II

83. Blassius exact solution I Boundary layer over a flat plate
= Ue1
Variable transformation: ,
Stream function definition: ,
Wall boundary condition:

84. Blassius exact solution II Boundary layer over a flat plate
Ordinary differential equation
Boundary conditions
The analytical solution of the ordinary differential equation was obtained by Blasius using series expansions

85. Blassius exact solution III Boundary layer over a flat plate
Velocity along x1-direction:
Velocity along x2-direction:
Boundary layer thickness:
Displacement thickness:
Wall shear stress:

86. Blassius exact solution IV Boundary layer over a flat plate

87. Von Karman integral momentum equation I
Momentum conservation along x1-direction

88. Von Karman integral momentum equation II
Solution
Considerations: p(x1) = pe(x1)

89. Approximate solutions I Linear, quadratic, cubic and sinusoidal velocity profiles
Assumption of a self-similar velocity profile
U1*= f (x2*)
Specifications of the boundary conditions
3. Resolution of the Von Karman integral momentum equation

90. Approximate solutions II Example: quadratic velocity profiles
General form boundary conditions = velocity profile along x1-dir
Von Karman integral momentum equation solution
boundary layer thickness

91. Approximate solutions III Example: quadratic velocity profiles
Displacement thickness:
from the definition =>
Velocity along x2-direction
from continuity equation
Wall shear stress
from =>