Learning Objective
After completion the reading of this chapter, you should be able to
  1. Define the boundary layer theory and its three regions (laminar, transition and turbulent) and the critical Reynolds number for internal and external flow.
  2. Derivation the fluid flow and heat transfer laws of physics (mass, momentum and energy) in full – details
  3. Definition the turbulence and the different between Navier – Stokes equations and the Reynolds equations.
  4. Derivation the boundary layer thickness, momentum thickness and energy thickness
  5. Knowledge of the hydrodynamics boundary layer governing equation.
  6. Derivation the Momentum Integral Equation for laminar and turbulent regions.
  7. Starting from the Momentum Integral Equation to find an expressions for the rate of growth of boundary layer thickness for laminar, transition and turbulent regions.
  8. Derivation an expression of drag coefficient in terms of Reynolds number for each boundary layer thickness
  9. Recognize the laminar and turbulent velocity profile
Introduction
The boundary layer theory is an interesting subject for the researchers among the world due to its wide range of applications in aerospace engineering like aerodynamics, flows over aircrafts like missiles, airplane, road vehicles and ships. One of the crucial applications of boundary layer is the determination of the drag coefficient of flat plate at zero incidences, flows over airfoil, ships, road vehicles and aircraft. The calculation of the drag is very important as it effects on the fuel consumptions and stability of the body. These days, the fuel resources are decreases and its prices goes up and the fuel consumption is highly influenced by the boundary layer over the road vehicles and aircrafts [1]. Also, one of the problems from the boundary layer is the separation and the stall phenomenon and for this reason there are many aerodynamics modifications to control or delay the separation as it leads to more fuel consumptions. Finally, there are applications in heat transfer between the fluid and the body as in the combustion chamber of spark ignition engines. The flow over a thin flat plate is the first case study of the boundary layer equations of Prandtl (4 February 1875 – 15 August 1953) solved later exactly by Blasius (9 August 1883 – 24 April 1970) in his PhD dissertation on 1908.
When a fluid flows past a solid surface, the velocity of the fluid at that solid surface must be the same as that of the solid surface. If the solid surface is stationary, then the fluid velocity at the surface is zero. So that there is a region close to the surface where the velocity increases from zero at the solid surface to the mean stream velocity (\(U_{\infty}\)). In this way, the boundary layer is a narrow region near the solid surface over which both velocity gradient and shear stress are large. It is also known as shear layer theory. The boundary layer theory can be divided into two main types which they are hydrodynamics and thermal layers. The present work illustrates the hydrodynamics boundary layer. The hydrodynamics boundary layer can be divided into two three region or zones, laminar, transition and turbulent as indicated in Figure 1. The well – known Reynolds number is used to distinguish between each layer. For this reason before discussing the hydrodynamics boundary layer it is required to write a section illustrates the concepts of turbulence, Reynolds number and the three laws of physics (mass, energy and momentum of fluid). Form the first look on the schematic diagram it can be noted that the laminar flow is parallel and the fluid flows in on layers gliding smoothly on the adjacent layers. The viscous forces are higher than the inertia forces which makes the laminar flow with small Reynolds number and thus there is no tendencies towards turbulence, eddies formations and instabilities. Beside that the velocity profile is parabolic. So as the flow moves further, there will be eddies formation with higher increasing in Reynolds number which it an indicator on the turbulence had been begun. The velocity profile in turbulent flow regime is logarithmic.