Flow Over Immersed Bodies_B

Momentum Integral Boundary Layer Equation for a Flat Plate

• Karman Solution (alternate method)  ↠  approx. closed form solution.
• Boundary layer theory  ↠  determination of drag (due to shear force)  ↠  using differential equation for laminar boundary layer flow (no exact solution)  ↠  alternate approx. method.

Ques: How to solve Boundary layer?

There are two solutions:

1. By Blasius Solution   ↠   limited to laminar boundary layer and for a flat plate only (without a pressure gradient).
2. By Momentum Integral Equation   ↠   used to obtain approx. information on boundary layer growth for the general case (laminar or turbulent boundary layers with or without pressure gradient).

The flow entering the CV at the leading edge of the plate is uniform while the velocity of the flow leaving the CV varies from the upstream velocities at the edge of the boundary layer to zero velocity on plate.

• Drag on a flat plate is related to the momentum deficit within the boundary layer.
• For a Laminar layer flow  ↠  CDF = CF which is a function of Reynold's number (Surface roughness is not important).
• For Turbulent flow, surface roughness affects the shear stress (friction drag coefficient).

Transition from Laminar To Turbulent Flow

Complex process of transition from laminar to turbulent flow is due to the instability of the flow field. Parameter that governs the transition to turbulent flow is Reynold's number based on the distance from leading edge of plate.

• Reynold's number at Transition location = Function (Surface roughness, Surface curvature, Disturbance in the flow outside the boundary layer).
• If flat plate with sharp leading edge transition take place at x from the leading edge  ↠  Rexcr = 2e5 (mostly used) to 3e6.
• Transition occur over a region (not at single point)  ↠  Rex = Rexcr.

Turbulent Boundary Layer on a Flat Plate

• Drag coefficient diagram (boundary layer flow)   ↠   Moody diagram.
• Pressure & Viscous forces balanced (pipe flow), fluid inertia remains constant  ↠  for Fully developed Horizontal pipe flow.
• Boundary Layer flow (Horizontal pipe)  ↠  Inertia & Viscous effects balanced, pressure pipe flow.

Boundary Layer with Pressure Gradient

In a situation where pressure increases downstream the fluid particles can move up against it by virtue of their K.E.

• Flow Reversal  ↠  inside the B.L. the velocity in a layer could reduce so much that k.e. of the fluid particles is no longer adequate to move the particles against the pressure gradient.
• Flow separation  ↠  the fluid layer higher up still have energy to move forward a rolling of fluid stream occurs.

• For laminar  ↠  separation at 90-degrees and drag is much high.
• For turbulent  ↠  separation after 90-degrees (at 100-135 degrees).
• Because of B.L. separation, the average pressure on the rear half of cylinder is less than that on the front half.

Note:

• Wind tunnel test

References:

• Material from Class Lectures + Book named Fundamentals of Fluid Mechanics by Munson, Young & Okiishi's (8th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube (Engineering Sights).