## Boundarylayer Theory

For fluids with small viscosity such as air and water with a high degree of accuracy, we can consider frictionless flow over entire fluid except for their regions around the contact areas. Here, because of high velocity gradients, we could not properly neglect frictions (Newton's viscosity law) so we consider these regions apart from the main flow, terming the boundary layers.

In 1904 Ludwing Prandti introduced the concept of boundary layer and showed how Navier-Stokes equations could be simplified. This concept literally revolutionized the science of fluid mechanics. According to Prandti's boundary layer concept viscous effects at high Reynolds number (Re = pjL = inertia Forces = v2l 2) are confined in thin layers adjacent to solid bound-

Viscous Forces p^v/pL2' J J

aries. Outside these solid boundary layers the flow may be considered inviscid (l = 0) and can, thus be described by Euler equation. Within the boundary layer the velocity component in the main flow direction (x) changes from vx = 0 (at the solid boundary) to vx = v^(the free stream velocity at the edge of the boundary layer).

We consider qualitatively the boundary layer flow over a flat plate. As it is shown in figure 2.9, a laminar region begins at the leading edge and grows in thickness. A transition region is reached where the flow changes from laminar to turbulent, with a consequent thickening of the boundary layer. The transition depends partly on the Reynold number( pv^x ) where x is the distance downstream from the leading edge. Transition occurs in the range Rex = 3 x 105 to Rex = 106. In the turbulent region we find, that as we get vo

Dm Millón zone

Laminar flow

Dm Millón zone

Laminar flow

Figure 2.9: Details of boundary layers

near the boundary the turbulence becomes suppressed to such a degree that viscous effects predominate, leading us to formulate the concept of a viscous sublayer shown darkened in the diagram.

There is actually a smooth variation from boundary layer region to the region of constant velocity. The velocity profile merges smoothly into the main-stream profile. There are several definitions of boundary-layer thickness that are quite useful. One can consider that the thickness is the distance 6 from the wall out to where the fluid velocity is 99 percent of the mainstream velocity. The experimental determination of the boundary layer thickness as defined above is very difficult because the velocity approaches asymptotically the free stream value The edge of the boundary layer is poorly defined. For this reason alternative thicknesses which can be measured more accurately are often used. The displacement thickness 6* defined as distance by which the boundary would have to be displaced if the entire flow were imagined to be frictionless and the same mass flow maintains at any section (figure 2.10). Thus, considering a unit width along the z across an infinite flat plate at zero angle of attack for incompressible fluid:

idy v

Hence, changing the lower limit on the second integral and solving for 6* we get:

0 0