1. Flow Over Immersed Bodies
Fluid Mechanics II
Chapter 9
Flow Over Immersed Bodies

2. Content Classification of External Viscous Flow
Fluid Dynamic Forces: Lift and Drag
Reynolds Number Effect
Boundary Layer: Laminar and Turbulent
Flow Separation
Experimental Drag Data
Airfoil and Wing Characteristics

3. 09_01 Flow Classification 2-Dimensional: Axi-symmetric:
3-Dimenstional:
09_01
09_01

4. Reynolds Number Effect
The Reynolds number, Re = U l/n , is the ratio between the inertial force and the viscous force.
Low Re: Mostly viscous flow
Moderate Re: Partial viscous flow around body
High Re: Viscous Boundary Layer near surface
09_04
09_04

5. 09_05 Flow Past Cylinder Low Re: Mostly viscous flow
Moderate Re: Partial viscous flow around body with separation and re-circulation flow in wake
High Re: Viscous Boundary Layer near surface till separation and wake
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6. 09_03 Surface Forces Pressure: Normal to surface
Shear Stress: Tangent to surface
09_03

7. Lift and Drag
The sum of forces due to pressure distribution and skin friction (shear stress) is the resultant force on a 2-D object.
This net force can be represented by its two components:
Lift: Component normal to the flow
Drag: Component in the flow direction
09_02
09_02

8. E_09_01 Example 9.1 (p. 329) Flow parallel to flat plate
Skin Friction Drag only:
D = lbf, L = 0
Flow normal to flat plate:
Pressure Drag only
D = 55.6 lbf, L = 0
Flow at an angle with plate:
Both Lift and Drag are present.
Drag consists of both pressure drag and skin friction drag.
E_09_01

9. Boundary Layer Flow Along a Smooth Flat Plate
09_06
Boundary Layer Flow Along a Smooth Flat Plate
Experimental observation:
At local Reynolds number (Rex = U x/n) around 5x105, transition from Laminar to Turbulent Boundary Layer Flow occurs. This Rex of 5x105 is known as the critical or transitional Reynolds number.

10. 09_09
Velocity Profiles
The gradient (du/dy) of the turbulent velocity profile at the wall (y=0) is higher than that of the laminar velocity profile.
Hence skin friction drag of turbulent boundary layer is higher than that of laminar one.

11. 09_08
Boundary layer thickness d (x): The location normal to surface at which the velocity reaches 0.99 of the velocity U in the inviscid free-stream. It increases in the x-direction along the plate.
Displacement thickness d*(x): The distance normal to the surface that the streamline passing d(x) is displaced from its original distance (h) at the leading edge of the plate. Hence,
d*(x) = d(x) – h
09_08

12. Laminar Boundary Layer on Flat Plate
Blasius Solution
Momentum Integral Method

13. Experimental Skin Friction Drag Data
Curve fit formula for turbulent boundary layer (Re > 500,000):

14. of Flat Plate with Roughness
Drag Coefficient
of Flat Plate with Roughness
Curve fitting of
Experimental Data
09_10
09_10

15. Drag Coefficient of Flat Plate
09_01tbl
Drag Coefficient of Flat Plate
Empirical Formulas

16. Boundary Layer Flow Separation
When flow separation occurs,
there is also pressure drag.

17. Pressure (Form) Drag due to Flow Separation
100% Pressure Drag
Total Profile Drag
= Skin Friction Drag
+ Form Drag

18. Development of velocity profile in the boundary layer on curved surface:
Flow separation occurs when the gradient of the velocity profile at the wall is zero, forming a re-circulating wake downstream.
09_12
09_12
Separation

19. Motor-controlled mechanism adjusts the model’s angle of attack.
Wind Tunnel Tests
Force transducer behind model senses lift, drag and pitching moment directly.
Motor-controlled mechanism adjusts the model’s angle of attack.

20. 09_15 Typical Experimental Data
Notice the sudden drop at the transition Re of 5x105 (Point E)
09_15

21. 09_15
For Re > 5x105, the boundary is turbulent, which has a fuller velocity profile.
Flow separation is delayed, resulting in a smaller wake, and hence the pressure drag.

22. Adding surface roughness on circular and spherical shapes triggers turbulence at lower Re, and hence helps to reduce the drag coefficient
09_18

23. 09_16

24. Benefit of Streamlining
Pressure drag is greatly reduced by preventing flow separation using a gradually tapering tail. Though skin friction increases with larger area, the total drag is much less. Hence streamlined bodies are made of smooth surfaces to reduce skin friction.
These objects have approximately the same drag:
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25. Test Data of 2D Objects
09_19

26. Axi-Symmetric Objects
Test Data of
Axi-Symmetric Objects
09_20

27. 09_21 Test Data of 3D Objects Recommended films:
Fluid Dynamics of Drag Part I-IV

28. Airfoil Characteristics
09_22
Airfoil Characteristics

29. P_09_60

30. 09_23

31. 09_25
09_25