The Blasius Boundary Layer

The incompressible boundary layer on a flat plate in the absence of a pressure gradient is usually referred to as the Blasius boundary layer (Ref. 1). The steady, laminar boundary layer developing downstream of the leading edge eventually becomes unstable to Tollmien-Schlichting waves and finally transitions to a fully turbulent boundary layer. Due to its fundamental importance, this type of flow has become the subject of numerous studies on boundary-layer flow, stability, transition, and turbulence. This application considers the first section of the plate where the boundary layer remains steady and laminar, and compares results from incompressible, two-dimensional, single-phase-flow simulations obtained in COMSOL Multiphysics to the Blasius similarity solution. The solutions converge ideally with respect to both mesh refinement and discretization order.

Model Definition

Consider a homogeneous free-stream flow with speed U0 parallel to an infinitely thin, flat plate located along the positive x-axis. The flow is assumed to be steady, symmetric with respect to y, and homogeneous in the z direction. Due to friction, the flow adjacent to the plate is retarded and a thin boundary layer, where the velocity gradually grows from zero to the free-stream value, develops downstream of the leading edge (see Figure 1).

Figure 1: The boundary layer on a flat plate. δ(x) is the boundary-layer thickness, such that u(x, δ(x)) = U0.

A reasonably accurate solution for the flow field can be found by considering the boundary-layer approximation to the steady, incompressible Navier-Stokes equations

(1)

(2)

Introducing a stream function,

and the similarity transformation,

Equation 1 and Equation 2 reduce to the ODE

(3)

COMSOL solves Equation 3 on the interval η ∈ [0, 10] with the boundary conditions

by rewriting the equation as a system of two equations,

and implementing the system within the Coefficient Form PDE interface.

Using the Laminar Flow interface for single-phase flow, the model solves the steady, incompressible Navier-Stokes equation in a domain (x, y) ∈ ([-1, 2.1], [0, 0.5]) m with the leading edge of the plate located at x = 0 m. The working fluid is air at a temperature of T = 20 °C and U0 = 0.75 m/s. The simulations uses discretizations with linear basis functions for velocity and pressure (P1+P1) on three different meshes.

Results and Discussion

Figure 2 shows the similarity solution u/U0 = f'(η). At η = 4.99, the deviation from the free-stream value is 1%. This value can be used to define the boundary-layer thickness,

Figure 3 shows a comparison between the Blasius similarity solution and the results from the two-dimensional simulations at xE = 2 m, corresponding to a Reynolds number of Rex = 1.0·105. Only the results from the P1+P1 simulation on the coarse mesh show a significant deviation from the similarity solution. To quantify differences in the results, define the following measure,

Here, η∞ = 10, for which the similarity solution has converged to its asymptotic value to within the numerical precision in the computations.

Figure 3: Comparison between the similarity solution and the two-dimensional simulations.

Table 1 displays deviations from the similarity solution together with the number of degrees of freedom (DOF) for the three simulations. The convergence is displayed in Figure 4 where the mesh size h is calculated as the maximum cell side in the mesh. The curve is close to straight line, which means that the model is in a mesh convergence regime; that is, the solution converges toward the correct solution when the mesh is refined.

Table 1: Deviation from the Blasius Solution.
1/h	10	20	40
ε	6.10·10-2	3.33·10-2	1.68·10-2
DOF	2016	7749	30375

Figure 4: Convergence rate as a function of inverse maximum cell side.

Notes About the COMSOL Implementation

The relative tolerance is set to 10−4 in all the solvers to ensure that the equation systems become well converged. All meshes have monotonically increasing element sizes away from the plate, with distributions employing geometric sequences. A nonlocal coupling is set up to enable evaluation of the similarity solution in the two-dimensional model.

Reference

1. H. Blasius, “Grenzschichten in Flüssigkeiten mit kleiner Reibung,” Z. Math. Phys., vol. 56, pp. 1–37, 1908 (Engl. transl. in NACA TM 1256).

COMSOL_Multiphysics/Fluid_Dynamics/blasius_boundary_layer

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

In the text field, type f.

Click

In the table, enter the following settings:


f
fprime

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
U0	0.75[m/s]	0.75 m/s	Inlet velocity
nu	1.506137e-5[m^2/s]	1.5061E-5 m²/s	Kinematic viscosity
xE	2[m]	2 m	Evaluation location
b0	nu/U0	2.0082E-5 m	B-L scale
N	1	1	Mesh refinement factor