The Blasius exact solution for the boundary layer on a flat plate is a fundamental concept in fluid dynamics. It provides a similar answer to the boundary layer equations for laminar flow over a semi-infinite flat plate. This answer, which was found by Paul Richard Heinrich Blasius in 1908, gives us a way to speculate about the velocity profile and thickness of the boundary layer in these kinds of flows. The Blasius solution utilizes a similarity variable η to transform the partial differential equations of the boundary layer into a single ordinary differential equation, which can be solved numerically. The aim of conducting a CFD simulation and comparing it with the Blasius solution is to validate the numerical model’s accuracy in predicting boundary layer behavior, particularly the velocity profile.

Figure 1: Flat Plate Boundary Layer CFD Simulation- Blasius Solution Validation

Exact Blasius Solution for a single point

Given the similarity variable, the velocity magnitude in a point can be found by giving x and y coordination of the point, plus viscosity of the fluid:

So, considering  x=0.96, y=0.00099 ,V∞=10 ,ν=1e-6 → η=0.00099*√(10/(1e-6×0.96))= 3.20

Next, based on the tabulated Blasius solution table, three variables including f, f’, f” are calculated:

Figure 2: Blasius exact solution table

Table →f=1.56963 , f’=0.87461 , f”=0.13909

f’=u/V→ u = f’×V∞→u=0.87461*10=8.7461 m/s

It can be concluded, the exact Blasius solution predicts 8.7461 m/s for the point located in =0.96 , y=0.00099.

Post-processing

A comparative analysis between the CFD simulation and the Blasius exact solution was performed at x = 0.96m and y = 0.00099m from the plate surface. The Blasius exact solution predicted a velocity of 8.7461 m/s, while the CFD simulation yielded 6.983142 m/s, representing a relative difference of approximately 20.2%. This discrepancy can be attributed to several factors, including numerical discretization errors, mesh resolution particularly in the boundary layer region.

Error = (8.7461-6.983142)/(8.7461)=0.20×100=20 %

The velocity profile plot demonstrates the characteristic boundary layer development along the flat plate, with the velocity gradually increasing from zero at the wall (satisfying the no-slip condition) to the free stream velocity of 10 m/s. The vector plot clearly illustrates the boundary layer growth along the plate, showing the expected pattern of velocity gradients near the wall. Despite the quantitative difference at the specific point of comparison, the qualitative behavior of the flow field appears to capture the essential physics of the boundary layer development, including the velocity gradient near the wall and the asymptotic approach to the free stream velocity, which is consistent with classical boundary layer theory.

Figure 3: Velocity profile near the flat-plate boundary layer