Blasius’ analysis focus on the laminar boundary layer forming on a flat plate. The main aspect
of his work is the transformation of the partial differential equations,
for a flat plate boundary layer with zero pressure gradient,
into a single ordinary differential equation using a coordinate transformation.
We assume that the properties of the fluid are known and so the Reynolds number.
Blasius used the stream function to describe the two-dimensional flow.
This function automatically satisfies the continuity equation by definition. The
coordinate transformation is performed introducing a non-dimensional coordinate for the Y direction
normal to the wall defined as eta and based on the boundary layer thickness correlation with X.
We can then define a scaled form of the stream function based on the dependent variable function
F of ETA. With this in mind, we can see that the stream function is proportional to the function F,
the velocity components are proportional to the first derivative of F and the 2nd and 3rd
derivative of F are proportional to the 1st and the 2nd derivative of the velocity. Note that
the prime superscript represents the derivative of F with respect to ETA. Substituting these
relations into the momentum equation, we get the final form of the Blasius boundary layer equation
for a flat plate. This equation can be solved once set the appropriate boundary conditions,
that, for our case, are: F equal to 0 at the wall because the wall forms a streamline:
the first derivative at the wall equal to zero, to impose the no-slip condition; and then the first
derivative of F equal to 1 as eta goes to infinity to have the velocity equal the freestream velocity
outside the boundary layer. Blasius solved this equation using an analytical
series solution technique. Nowadays, thanks to the availability of computers, we can obtain a
numerical solution of this equation and calculate it with a high degree of accuracy. In order to
do so, we need to cast the Blasius equation as a set of coupled 1st order ODEs.
Welcome to the bonus content of this lecture! Here I'm going to explain briefly how
we can solve numerically the Blasius equation. A possible method is to use the shooting algorithm
to find what value of H would satisfy the boundary condition at Infinity. The first step is to guess
a value of H at the wall. Then we solve the ODEs marching along the non-dimensional coordinate
until the 1st derivative of F stops changing. Next we check if our first derivative satisfies
the boundary condition. If it doesn’t we can adjust the H value either decreasing or increasing
it if the first derivative is higher or lower than one respectively. The algorithm is repeated
until the boundary condition is satisfied and we reached our solution.
Reported here is a tabular form of the values of F and its derivatives on the left, as you can see
the solution is a value of 2nd derivative of F at the wall equal to 0.332.
On the right is a plot of F prime versus the non-dimensional coordinate.
This is directly a representation of the velocity profile inside the boundary layer.
Once the solution is obtained we can calculate the velocity components using these relations. Hence
we can calculate the boundary layer thickness locating the point where the velocity is 99%
of the freestream velocity. That occurs at approximately ETA equal to 5. Using the
solution we can also estimate the displacement thickness and momentum thickness. The two terms
can be rewritten in these forms. Last, we can estimate the wall shear stress based on the
velocity gradient at the wall and calculate then the friction and the drag coefficient.
Note that the expression is only accounting for one side of the plate. It's interesting to compare
the exact solution with the approximate solution from the integral analysis. The relations are
pretty close to each other, indeed the integral analysis is within 10% of the exact solution,
but unlike the Blasius solution these values were obtained without complicated math.
The Blasius solution provides a self-similar solution, meaning that the solution is the same
if the independent and dependent variables of the governing equations are appropriately scaled. This
can be seen in the comparison of experimental results with the Blasius solution. Different
Reynolds number provide the same profile when the variables on the two axes are scaled properly.