Boundary Layer Solution Obtained with the Integral Method

The problem to be solved for boundary layer flow past a flat plate is

This problem is a boundary value problem, since the conditions on f are at two different positions. The problem is also complicated by the fact that one of the boundary conditions is at infinity.

In the Method of Weighted Residuals, there are five steps to the solution:

• 1. Write down a trial function or trial solution;
• 2. Make the trial solution satisfy the boundary conditions;
• 3. Substitute the trial solution into the differential equation to form the residaul;
• 4. Make the weighted residual zero;
• 5. Increase the complexity of the trial solution, and repeat, to assess the accuracy.

Firstly, write the function f ' as a quadratic function of h.

Secondly, make this function satisfy these conditions.

The conditions are chosen so that the velocity is zero at the wall, the velocity goes to 1.0 far from the wall at the edge of the boundary layer (here the value 4 is used), and the derivative of the velocity (or shear rate) goes to zero at the edge of the boundary layer. Imposing those conditions gives

and the trial solution is

In this formulation, there are no constants left to be satisfied, and the weighted residual is not used. The solution is illustrated in the figure, and we see that the solution is quite good compared with the numerical solution. It was certainly easy to derive.

The constant for the shear stress at the wall is

which is within 6% of the exact solution.

One problem with this method, though, is that we don't know how accurate it is if we have no solution for comparison. In the full Method of Weighted Residuals we would continue to higher polynomials and look for convergence as we gave the trial solution more degrees of freedom. We don't do that here.

We do, however, apply the Method of Weighted Residuals when we do not use the condition that the slope of the velocity be zero at the edge of the boundary layer. This time we will use the integral method to form the weighted residual. The trial function is taken as

We make the trial function satisfy the boundary conditions

While the polynomial is the same as before, this time we do not use the condition f"(4) = 0. When the boundary conditions are applied we get

Thus, the trial function is

and its derivatives are

The residual is

This completes the first three steps in the Method of Weighted Residuals.

The next step is to set the weighted residual to zero. There are many ways to do this, but we use the one that was used historically in fluid mechanics: we set the integral of the residual to zero. This is like saying we make the differential equation be satisfied on the average. The weighted residual is

which is

The integrals give

which leads to

When terms are combined, we get

The parameter d is obtained from the solution to

There are two solutions, one negative and one positive. We choose the negative one since the value of f"(4) is 0.25 + 24 d, and we want f" = 0 there, or close to it. Thus, d = - 0.021732. The solution is then

This gives for the shear stress constant

which is within 8% of the exact solution. The next step is to enhance the trial function and repeat the procedure, but we don't do that here. The accuracy is determined here by comparison with the numerical solution in the figure. This solution is very close to the first one derived without the residual. The value of f "(4) = - 0.01, which is close to zero, as desired.

It is also possible to employ the integral method directly on the equations of motion, before one gets to the ordinary differential equation, and that is the preferred method. However, the results are the same as obtained above.

Take Home Message: The integral method, one of the Methods of Weighted Residuals, gives quite reasonable solutions for the effort.