Python implementation of a numerical Blasius boundary-layer solver, using the shooting method to recover the canonical Blasius similarity solution and compute physically meaningful boundary-layer quantities.
This repository contains a full Python solution of the Blasius boundary-layer problem, implemented from first principles using numerical methods.
The Blasius ordinary differential equation (ODE) is solved using a shooting method combined with Brent’s root-finding algorithm, enforcing the far-field boundary condition that the dimensionless streamwise velocity converges to unity. The solver recovers the classical Blasius curvature value:
The code was developed for an academic fluid mechanics assignment and is written to prioritise clarity, physical interpretation, and numerical validation rather than minimalism.
The Blasius equation is:
with boundary conditions:
The third-order Blasius ODE is rewritten as a system of first-order equations and integrated numerically (e.g., using odeint or an equivalent integrator).
The unknown wall curvature is treated as the shooting parameter:
A residual is defined from the far-field mismatch, e.g.:
Brent’s method is then used to robustly solve the nonlinear root-finding problem:
The computed curvature value (B_found) is obtained numerically and is not imposed.
A separate spreadsheet value (B_given = 0.57) is used only where explicitly required in Part (b) of the assignment and does not influence the numerical solution.
The similarity variable is related to the physical wall-normal coordinate using the standard Blasius scaling (as used in the assignment).
Air at 100 °C is assumed, with kinematic viscosity:
The solution is evaluated at:
- Streamwise location:
x = 0.3 m - Free-stream speed:
U∞ = 4 m/s
The similarity domain is truncated at:
which is sufficient for the velocity profile to reach its asymptotic value. A grid of 1000 points is used, corresponding to a physical wall-normal resolution of approximately 10.5 μm, enabling accurate interpolation and numerical integration.
The shooting method converges to:
after 11 residual evaluations.
A consistency check confirms:
verifying correct enforcement of the far-field boundary condition.
The wall-normal location where:
is identified using interpolation on the computed velocity profile, and the corresponding physical velocities u_x and u_y are calculated.
The boundary-layer thickness delta_99 is computed numerically and compared against the empirical estimate:
with a percentage difference of approximately −1.8%.
The displacement thickness delta* is obtained via numerical integration of the velocity deficit (e.g., trapezoidal rule):
The code generates plots of:
- Dimensionless velocity profile:
f'(ζ) - Blasius stream function:
f(ζ) - A zoomed appendix plot highlighting the location corresponding to the given spreadsheet value of
B
All figures are generated programmatically using Matplotlib and saved at high resolution for inclusion in reports.
Figure captions used in the original academic submission are included in the repository for reference. Users are expected to run the script locally to regenerate the plots.
B_foundis computed numerically and represents the physically meaningful Blasius curvature.B_givenis not used in the shooting method and is included only to satisfy the requirements of a specific assignment sub-part.- Viscosity handling is explicitly documented to avoid ambiguity when converting between similarity and physical coordinates.
This repository is maintained as an academic reference and portfolio artefact.
It demonstrates numerical ODE solving, root finding, boundary-layer theory, and careful physical interpretation rather than serving as a general-purpose CFD tool.