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# Flow Over Immersed Bodies

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

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

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

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

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 09_05 09_05

09_03 Surface Forces Pressure: Normal to surface
Shear Stress: Tangent to surface 09_03

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

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

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.

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.

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

Laminar Boundary Layer on Flat Plate
Blasius Solution Momentum Integral Method

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

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

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

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

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

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

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.

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

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.

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

09_16

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: 09_14

Test Data of 2D Objects 09_19

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

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

Airfoil Characteristics
09_22 Airfoil Characteristics

P_09_60

09_23

09_25 09_25

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