1. PHAROS UNIVERSITY ME 253 FLUID MECHANICS II
Boundary Layer (Two Lectures)

2. Flow Past Flat Plate Dimensionless numbers involved
for external flow: Re>100 dominated by inertia, Re<1 – by viscosity

3. Boundary Layer
Flows over bodies. Examples include the flows over airfoils, ship hulls, etc.
Boundary layer flow over a flat plate with no external pressure variation. U
Dye streak U U U
laminar
turbulent
transition

4. Boundary layer characteristics
for large Reynolds number flow can be divided
into boundary region where viscous effect are
important and outside region where liquid can
be treated as inviscid

5. The Boundary-Layer Concept

6. The Boundary-Layer Concept

7. Boundary Layer Thicknesses

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

9. Boundary Layer Thickness
Boundary layer thickness is defined as the height above the surface where the velocity reaches 99% of the free stream velocity.

10. Boundary Layer(BL)
Three Thicknesses of a Boundary Layer d

11. Displacement Thickness
Volume flux:
Ideal flux:

12. Velocity Distribution
Ideal Fluid
Flow
Velocity
Defect
Solid
Boundary *
Velocity Defect
Equivalent Flow Rate

13. Eqn. for Displacement Thickness
By equating the flow rate for velocity defect to flow rate for ideal fluid
If density is constant, this simplifies to
* would always be smaller than 

14. Displacement Thickness Laminar B.L.

15. Eqn. for Momentum Thickness
By equating the momentum flux rate for velocity defect to that for ideal fluid
If density is constant, this simplifies to
 would always be smaller than * and 

16. Momentum Thickness
The rate of mass flow across an element of the boundary layer is (r u dy) and the mass has a momentum (r u2 dy ) The same mass outside the boundary layer has the momentum (r u ue dy)
Q is a measure of the reduction in momentum transport in the B. Layer

17. Empirical Equations of Laminar B. Layer Parameters
Boundary Layer Thickness
Momentum Thickness
Displacement Thickness
Skin Friction Coefficient

18. Skin Friction Coefficient

19. Boundary layer characteristics
Boundary layer thickness
Boundary layer displacement thickness:
Boundary layer momentum thickness (defined in terms of momentum flux):

20. Drag on a Flat Plate
Drag on a flat plate is related to the momentum deficit within the boundary layer
Drag and shear stress can be calculated by assuming velocity profile in the boundary layer

21. Boundary Layer Definition
Boundary layer thickness (d): Where the local velocity reaches to 99% of the free-stream velocity u(y=d)=0.99U
Displacement thickness (δ*): Certain amount of the mass has been displaced (ejected) by the presence of the boundary layer ( mass conservation). Displace the uniform flow away from the solid surface by an amount δ*
Amount of fluid
being displaced
outward
equals d*
U-u

22. Laminar Flat-Plate Boundary Layer: Exact Solution
Governing Equations

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

24. Laminar Flat-Plate Boundary Layer: Exact Solution
Results of Numerical Analysis

25. MOMENTUM INTEGRAL EQN BOUNDARY LAYER EQUATIONS
BOUNDARY CONDITIONS y=0, u=v=0 & y = δ , u= U
Integrating the momentum equation w.r.t y in the interval [0,δ]

26. MOMENTUM INTEGRAL EQN

27. DISPLACEMENT THICKNESS
MOMENTUM INTEGRAL EQN
DISPLACEMENT THICKNESS
MOMENTUM THICKNESS

28. MOMENTUM INTEGRAL EQN

29. MOMENTUM INTEGRAL EQN
VON KARMAN MOMENTUM INTEGRAL EQUATION

30. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLAT PLATE
BERNOULLI’S EQUATION
NO IMPOSED PRESSURE
VON KARMAN EQUATION
REVISED KARMAN EQUATION FOR NO EXTERNAL PRESSURE

31. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
Dimensionless velocity u/U can be expressed at any location x as a function of the dimensionless distance from the wall y/δ.
Boundary (at wall) conditions y=0, u=0 ; d2u/dy2 = 0
At outer edge of boundary layer (y=δ) u=U, du/dy=0
Applying boundary conditions a=0,c=0, b=3/2, d= -1/2
Velocity profile

32. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
INSERT INTO THIS EQUATION

33. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE

34. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
At the solid surface, Newton’s Law of Viscosity gives:

35. KARMAN POHLHAUSEN METHOD FOR FLOW OVER FLATE PLATE
BLASIUS SOLUTION
KARMAN POHLHAUSEN SOLUTION

36. Boundary Layer Parameters
BOUNDARY LAYER THICKNESS INCREASES AS THE SQUARE ROOT OF THE DISTANCE x FROM THE LEADING EDGE AND INVERSELY AS SQUARE ROOT OF FREE STREAM VELOCITY.
WALL SHEAR STRESS IS INVERSELY PROPORTIONAL TO THE SQUARE ROOT OF x AND DIRECTLY PROPORTIONAL TO 3/2 POWER OF U
LOCAL & AVERAGE SKIN FRICTION VARY INVERSELY AS SQUARE ROOT OF BOTH x & U

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

38. 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)

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

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

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

42. Example
Assume a laminar boundary layer has a velocity profile as u(y)=U(y/d) for 0yd and u=U for y>d, as shown. Determine the shear stress and the boundary layer growth as a function of the distance x measured from the leading edge of the flat plate.
u=U  y
u(y)=U(y/d) x

43. Note: In general, the velocity distribution is not a straight line
Note: In general, the velocity distribution is not a straight line. A laminar flat-plate boundary layer assumes a Blasius profile. The boundary layer thickness d and the wall shear stress tw behave as:

44. Laminar Boundary Layer Development
Boundary layer growth: d  x
Initial growth is fast
Growth rate dd/dx  1/x, decreasing downstream.
Wall shear stress: tw  1/x
As the boundary layer grows, the wall shear stress decreases as the velocity gradient at the wall becomes less steep.

45. Momentum Integral Relation for Flat-plate BLy P
Free stream U
Stream line d h x x CV
U=const P=const
Steady & incompressible

46. Momentum Integral Relation for Flat-plate BL
Outlet
Inlet
Continuity

47. Meantime
For flat plate boundary layer

50. Shape factor

51. Skin-friction coefficient
Drag coefficient

52. Boundary Layer Equation
Inviscid

53. 3 Boundary Layer Equation

55. Boundary Layer Equation
2-D,steady,incompressible,neglect body force

56. For BL,
External flow
(Inviscid Flow)

57. Euler Equation
Laminar flow
Turbulent flow

58. Blasius 1908

60. DISPLACEMENT AND MOMENTUN THICKNESS
Typical distribution of d, d* and q
U =10m/s, n=17x10-6 m2/s d d* q

61. Influence of Adverse Pressure Gradient
Adverse Pressure Gradient dp/dx>0 can cause flow separation

62. Pressure Gradients in Boundary-Layer Flow

63. Boundary Layer and separation
Flow decelerates
Flow accelerates
Constant flow
Flow reversal
free shear layer
highly unstable
Separation point

64. Flow Separation Separation Boundary layer q Wake Inviscid curve
Stagnation point
1.0
Turbulent
Laminar
-1.0
-2.0
-3.0 q

65. Drag Coefficient: CD Stokes’ Flow, Re<1 Supercritical flow
turbulent B.L.
Relatively constant CD

66. Drag
Drag Coefficient
with or

67. Influence of Adverse Pressure Gradient
Adverse Pressure Gradient dp/dx>0 can cause flow separation

68. WHY DOES BOUNDARY LAYER SEPARATE?
Adverse pressure gradient interacting with velocity profile through B.L.
High speed flow near upper edge of B.L. has enough speed to keep moving through adverse pressure gradient
Lower speed fluid (which has been retarded by friction) is exposed to same adverse pressure gradient is stopped and direction of flow can be reversed
This reversal of flow direction causes flow to separate
Turbulent B.L. more resistance to flow separation than laminar B.L. because of fuller velocity profile
To help prevent flow separation we desire a turbulent B.L.

69. EXAMPLE OF FLOW SEPARATION
Velocity profiles in a boundary layer subjected to a pressure rise
(a) start of pressure rise
(b) after a small pressure rise
(c) after separation
Flow separation from a surface
(a) smooth body
(b) salient edge

70. BOUNDARY LAYER SEPARATION
Separation takes place due to excessive momentum loss near the wall in a boundary layer trying to move downstream against increasing pressure, i.e., which is called adverse pressure gradient.
MOMENTUM EQUATION
AT WALL v=u=0

71. BOUNDARY LAYER SEPARATION
In adverse gradient, the second derivative of velocity is positive at the wall, yet it must be negative at the outer layer (y=δ) to merge smoothly with the mainstream flow U(x).
It follows that the second derivative must pass through zero somewhere in between, at the point of inflexion, and any boundary layer profile in an adverse must exhibit a characteristic S –shape.

72. BOUNDARY LAYER SEPARATION
In FAVOURABLE GRADIENT, profile is rounded, no point of inflexion, no separation.
In ZERO PRESSURE GRADIENT, point of inflexion is at the wall itself. No separation.
In ADVERSE GRADIENT, point of inflexion (PI) occurs in the boundary layer, its distance from the wall increasing with the strength of the adverse gradient
CRITICAL CONDITION is reached where the wall shear is exactly zero (∂u/∂y =0). This is defined as separation point
SHEAR STRESS AT WALL IS ZERO

73. BOUNDARY LAYER SEPARATION

74. BOUNDARY LAYER SEPARATION
The mathematical explanation of flow-separation :
The point of separation may be defined as the limit between forward and reverse flow in the layer very close to the wall, i.e., at the point of separation
This means that the shear stress at the wall, But at this point, the adverse pressure continues to exist and at the downstream of this point the flow acts in a reverse direction resulting in a back flow.

75. EXAMPLE: SLATS AND FLAPS