Blasius boundary layer

In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow.

Using scaling arguments, Ludwig Prandtl[1] argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate).

For steady incompressible flow with constant viscosity and density, these read: Here the coordinate system is chosen with

The term similarity refers to the property that the velocity profiles at different positions in the flow are the same apart from scaling factors.

Similarity scaling factors reduce the set of partial differential equations to a relatively easily solved set of non-linear ordinary differential equations.

Paul Richard Heinrich Blasius,[2] one of Prandtl's students, developed the similarity model corresponding to the flow for the case where the pressure gradient,

, along a thin flat-plate is negligible compared to any pressure gradient in the boundary layer region.

The self-similar solution exists because the equations and the boundary conditions are invariant under the transformation

This is a third-order non-linear ordinary differential equation which can be solved numerically, e.g. with the shooting method.

in the drag force formula is to account both sides of the plate.

The Blasius solution is not unique from a mathematical perspective,[7]: 131  as Ludwig Prandtl himself noted it in his transposition theorem and analyzed by series of researchers such as Keith Stewartson, Paul A.

[8] To this solution, any one of the infinite discrete set of eigenfunctions can be added, each of which satisfies the linearly perturbed equation with homogeneous conditions and exponential decay at infinity.

derivative of the first order Blasius solution, which represents the uncertainty in the effective location of the origin.

This boundary layer approximation predicts a non-zero vertical velocity far away from the wall, which needs to be accounted in next order outer inviscid layer and the corresponding inner boundary layer solution, which in turn will predict a new vertical velocity and so on.

The vertical velocity at infinity for the first order boundary layer problem from the Blasius equation is

The solution for outer inviscid and inner boundary layer are[7]: 134

Again as in the first order boundary problem, any one of the infinite set of eigensolution can be added to this solution.

[7]: 139  The solution for third-order correction does not have an exact expression, but the inner boundary layer expansion is of the form,

is the first eigensolution of the first order boundary layer solution (which is

is nonunique and the problem is left with an undetermined constant.

Suction is one of the common methods to postpone the boundary layer separation.

[9] Consider a uniform suction velocity at the wall

This parabolic partial differential equation can be marched starting from

Since the convection due to suction and the diffusion due to the solid wall are acting in the opposite direction, the profile will reach steady solution at large distance, unlike the Blasius profile where boundary layer grows indefinitely.

, both the boundary layer thickness and the solution are independent of

Stewartson[14] studied matching of full solution to the asymptotic suction profile.

Here Blasius boundary layer with a specified specific enthalpy

The equation for conservation of mass, momentum and energy become

Unlike the incompressible boundary layer, similarity solution exists only if the transformation

Since the boundary layer equations are Parabolic partial differential equation, the natural coordinates for the problem is parabolic coordinates.

A schematic diagram of the Blasius flow profile. The streamwise velocity component is shown, as a function of the similarity variable .
Developing Blasius boundary layer (not to scale). The velocity profile is shown in red at selected positions along the plate. The blue lines represent, in top to bottom order, the 99% free stream velocity line ( ), the displacement thickness ( ) and ( ). See Boundary layer thickness for a more detailed explanation.
A schematic depiction of the scaled velocities and -pressure gradient for Blasius boundary layer flow versus the scaled normal height above the wall, . The red line is the Blasius scaled velocity , the green line is the Blasius and the blue line is the Blasius .
Compressible Blasius boundary layer