0.8 Multidimensional laminar flow  (Page 6/8)

Laminar flow along flat plate

The classical system for the study of laminar boundary layers is the flow of a fluid in uniform translation past a flat plate. The free stream velocity is constant and the pressure gradient is zero. The classical solution to this problem is the doctoral thesis of H. Blasius (1908). The equation of continuity is satisfied exactly by expressing the velocity as the curl of the stream function. Since the system does not have a characteristic length, a similarity transformation makes it possible to combine the two independent variables $(x,y)$ into a single independent variable, $\eta =y\sqrt{\frac{U_{\infty }}{\nu x}}$ . The equations reduce to a quasilinear third order ordinary differential equation for the dimensionless stream function. The solution is given as a series solution. Its derivation is tedious and will not be discussed here. The reader is referred to Schlichting (1960) for details.

An alternative approach is to keep the velocity components as the dependent variables and approximately satisfy the continuity equation by expressing the transverse velocity component in the equation of motion as follows.

This is the approach taken in BSL. They express the solution as a cubic polynomial in $\eta$ .

Analogy with wall suddenly set in motion

The previously mentioned solutions may be accurate solutions to the boundary layer equations but do not offer much physical insight. Here we will derive the solution to the boundary layer flow by using the flow due to a wall suddenly set in motion discussed in Chapter 8. Since the wall extends to infinity, there is no dependence on $x$ and the equations of motion, initial condition, and boundary condition are as follows.

The solution derived by a similarity transform in Chapter 7 is,

The coordinates can be transformed such that the plate is stationary and the fluid is initially in uniform translation past the plate. The initial condition, boundary conditions, and solution are then as follows.

This exact solution describes the diffusion of momentum from the uniformly translating fluid to the stationary plate. It describes growth of the boundary layer as a function of the similarity variable, $\eta =\frac{y}{\sqrt{4\nu t}}$ . There is no convection of momentum because there is no dependence on $x$ and the transverse component of velocity is zero.

Suppose now that the plate is not doubly infinite but only exists along the positive $x$ axis and the flow is in the positive $x$ direction.. Since there is now dependence on the $x$ coordinate, boundary conditions depend on $x$ and the convective terms in the equations of motion no longer vanish. Now consider the steady state flow past this semi-infinite plate. Assume that the flow is undisturbed until $x=0$ . The differential equations are the boundary layer equations with zero velocity gradients. The equations and boundary conditions are as follows.