Boundary Layer Equations

In 1904, Ludwig Prandtl, the well known German scientist, introduced the concept of boundary layer and derived the equations for boundary layer flow by correct reduction of Navier-Stokes equations.

He hypothesized that for fluids having relatively small viscosity, the effect of internal friction in the fluid is significant only in a narrow region surrounding solid boundaries or bodies over which the fluid flows.

Thus, close to the body is the boundary layer where shear stresses exert an increasingly larger effect on the fluid as one moves from free stream towards the solid boundary.

However, outside the boundary layer where the effect of the shear stresses on the flow is small compared to values inside the boundary layer (since the velocity gradient is negligible),---------

the fluid particles experience no vorticity and therefore,

the flow is similar to a potential flow.

Hence, the surface at the boundary layer interface is a rather fictitious one, that divides rotational and irrotational flow. Fig 28.1 shows Prandtl's model regarding boundary layer flow.

Hence with the exception of the immediate vicinity of the surface, the flow is frictionless (inviscid) and the velocity is U (the potential velocity).

In the region, very near to the surface (in the thin layer), there is friction in the flow which signifies that the fluid is retarded until it adheres to the surface (no-slip condition).

The transition of the mainstream velocity from zero at the surface (with respect to the surface) to full magnitude takes place across the boundary layer.

About the boundary layer

Boundary layer thickness is which is a function of the coordinate direction x .
The thickness is considered to be very small compared to the characteristic length L of the domain.
In the normal direction, within this thin layer, the gradient is very large compared to the gradient in the flow direction .

Now we take up the Navier-Stokes equations for : steady, two dimensional, laminar, incompressible flows.
Considering the Navier-Stokes equations together with the equation of continuity, the following dimensional form is obtained.

(28.1)

(28.2)

(28.3)

Fig 28.1 Boundary layer and Free Stream for Flow Over a flat plate

u - velocity component along x direction.
v - velocity component along y direction
p - static pressure
ρ - density.
μ - dynamic viscosity of the fluid

The equations are now non-dimensionalised.
The length and the velocity scales are chosen as L and respectively.
The non-dimensional variables are:



where $U_{\infty}$ is the dimensional free stream velocity and the pressure is non-dimensionalised by twice the dynamic pressure  .