© Pritchard

Introduction to Fluid Mechanics

Chapter 9

External Incompressible Viscous Flow

© Pritchard

Main TopicsThe Boundary-Layer ConceptBoundary-Layer ThicknessesLaminar Flat-Plate Boundary Layer: Exact

SolutionMomentum Integral EquationUse of the Momentum Equation for Flow with

Zero Pressure GradientPressure Gradients in Boundary-Layer FlowDragLift

 External incompressible viscous flow

 Boundary-layer thicknesses

•

A boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant or the velocity gradient is high.

 Boundary-layer thicknessesThe thickness of a layer of free-stream flow equal to the mass flow rate that has been displaced or reduced due to the existence of a bounding surface.

The thickness of a layer of free-stream flow that has the momentum equal to the momentum reduction due to the existence of a bounding surface (mass flow reduction is accompanied by momentum reduction).

  Momentum integral equation

  Momentum integral equation

  Momentum integral equation/Laminar flow

  Laminar flow solution

  Laminar flow solution

Turbulent flow (Semi-empirical approach)

Turbulent flow (Semi-empirical approach)

Flow over a flat plate parallel to the flow: Friction Drag

Friction Drag

  

  Pressure gradients in boundary-layer flow

Flow about immersed bodies/Friction drag + pressure drag 

 Pressure Drag

  Pressure drag (shear stress does not contribute much to the drag in this case). Because of the boundary separation, the average pressure in the wake is considerably less than that on the front, thus a large pressure drag is developed even though the viscous shear is small or normal to the fluid stream, which does not have a contribution to the drag. The pressure drag may be explained through the following facts: (1) The pressure over the frontal surface is increased due to the stagnation effect (the kinetic energy is converted into pressure head), and (2) Since the boundary layer is destroyed in the wake, large circulation and strong mixing may cause significant losses of mechanical energy and consequently a lower mechanical energy component, resulting in a lower pressure head over the rear surface.

 Pressure drag results  

• A larger b may increase the travel distance from the center to the lateral edges and require a larger pressure near the center.

 Pressure drag results  

• For the case of square prism, h is the same as that in Fig. 9.10.• For the case of Ring, sudden contraction and expansion would cause additional losses, resulting in a lower pressure over the rear surface.• The case of Hemisphere (open end facing flow) may cause flow reversal, resulting in a larger pressure on the open end surface.

 

Flow over a sphere and cylinder: Total drag = friction drag + pressure drag (both drags may be important)

    

• When Re is very small (either a very low velocity or a very small diameter, Fig. 9.11), there is no flow separation and the wake is laminar. The drag is predominantly friction drag that can be evaluated by the Stokes theory. However, at a large Re, pressure drag would be dominant. • A turbulent boundary layer may delay the boundary layer separation, resulting in a smaller wake region and subsequently a smaller pressure drag.• The roughness of the surface may promote an earlier transition from a laminar boundary layer to a turbulent boundary layer, resulting in a smaller pressure drag, such as the case of a golf ball with dimples.

 

Flow over a sphere    

 

 Streamlining    

To reduce the drag on a body, particularly to reduce the pressure drag by reducing the size of turbulent wake. However the skin friction drag may be increased because the surface area is increased. In practice, there is an optimum amount of fairing or tapering at which the total drag (pressure + friction) is minimized.

 

 Lift

    

 External incompressible viscous flow

http://www.av8n.com/how/htm/airfoils.html

 

 Lift

    

• The chord of an airfoil is the straight line joining the leading edge and the trailing edge.• The angle of attack α, is the angle between the airfoil chord and the free stream velocity vector. • The upper surface may be called suction surface and the lower surface may be called pressure surface. The lift on a body can also be related to the circulation around the profile. • The wing section shape is obtained by combining a mean line and a thickness distribution.• An aircraft wing with a curved mean line is said to be cambered.• A cambered wing may generate a lift at zero angle of attack.• As angle of attack increases, the lift coefficient increase smoothly until a maximum is reached. Further increases in angle of attack produce a sudden decrease in CL, and the airfoil is said to have stalled.• Airfoil stall results when flow separation occurs over a major portion of the upper surface of the airfoil (With the flow separation, the boundary layer is destroyed and replaced by large scale circulation. As a result, the flow path near the surface is being enlarged due to the elimination of the so called displacement layer associated with the boundary layer).