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**Flow Over Immersed Bodies**

Fluid Mechanics II Chapter 9 Flow Over Immersed Bodies

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**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

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**09_01 Flow Classification 2-Dimensional: Axi-symmetric:**

3-Dimenstional: 09_01 09_01

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**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

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**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

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**09_03 Surface Forces Pressure: Normal to surface**

Shear Stress: Tangent to surface 09_03

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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

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**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

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**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.

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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.

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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

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**Laminar Boundary Layer on Flat Plate**

Blasius Solution Momentum Integral Method

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**Experimental Skin Friction Drag Data**

Curve fit formula for turbulent boundary layer (Re > 500,000):

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**of Flat Plate with Roughness**

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

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**Drag Coefficient of Flat Plate**

09_01tbl Drag Coefficient of Flat Plate Empirical Formulas

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**Boundary Layer Flow Separation**

When flow separation occurs, there is also pressure drag.

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**Pressure (Form) Drag due to Flow Separation**

100% Pressure Drag Total Profile Drag = Skin Friction Drag + Form Drag

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**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

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**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.

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**09_15 Typical Experimental Data**

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

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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.

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

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09_16

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**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

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Test Data of 2D Objects 09_19

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**Axi-Symmetric Objects**

Test Data of Axi-Symmetric Objects 09_20

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**09_21 Test Data of 3D Objects Recommended films:**

Fluid Dynamics of Drag Part I-IV

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**Airfoil Characteristics**

09_22 Airfoil Characteristics

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P_09_60

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09_25 09_25

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