Learning Objective
After completion the reading of this chapter, you should be able to
- Define the boundary layer theory and its three regions (laminar,
transition and turbulent) and the critical Reynolds number for
internal and external flow.
- Derivation the fluid flow and heat transfer laws of physics (mass,
momentum and energy) in full – details
- Definition the turbulence and the different between Navier – Stokes
equations and the Reynolds equations.
- Derivation the boundary layer thickness, momentum thickness and energy
thickness
- Knowledge of the hydrodynamics boundary layer governing equation.
- Derivation the Momentum Integral Equation for laminar and turbulent
regions.
- 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.
- Derivation an expression of drag coefficient in terms of Reynolds
number for each boundary layer thickness
- 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.