10.3. Boundary Layer on a Flat Plate: Blasius Solution

The simplest-possible boundary layer forms on a semi-infinite flat plate with a constant free-stream flow speed, U_e = U = constant. In this case, the boundary-layer equations simplify to:

∂u∂x+∂v∂y=0andu∂u∂x+v∂u∂y=ν∂2u∂y2,

(7.2, 10.18)

where (10.11) requires dp/dx = 0 because dU_e/dx = 0. Here, the independent variables are x and y, and the dependent field quantities are u and v. The flow is incompressible but rotational, so a guaranteed solution of (7.2) may be sought in terms of a stream function, ψ, with the two velocity components determined via derivatives of ψ (see (7.3)). Here, the flow is steady and there is no imposed length scale, so a similarity solution for ψ can be proposed based on (9.32):

ψ=Uδ(x)f(η)whereη=y/δ(x),

(10.19)

where x is the time-like independent variable, f is an unknown dimensionless function, and δ(x) is a boundary-layer thickness that is to be determined as part of the solution (it is not a Dirac delta-function). Here the coefficient Uδ in (10.19) has replaced UAξ^–n in (9.32) based on dimensional considerations; the stream function must have dimensions of length²/time. A more general form of (10.19) that uses UAx^–n as the coefficient of f(η) produces the same results when combined with (10.18).

The solution to (7.2) and (10.18) should be valid for x > 0, so the boundary conditions are:

u=v=0ony=0,

(10.20)

u(x,y)→Uasy/δ→∞,and

(10.21)

δ→0asx→0.

(10.22)

Here, we note that the boundary-layer approximation will not be valid near x = 0 (the leading edge of the plate) where the high Reynolds number approximation, Re_x = Ux/ν ≫ 1, used to reach (10.18) is not valid. Ideally, the exact equations of motion would be solved from x = 0 up some location, x₀, where Ux₀/ν ≫ 1. Then, the stream-wise velocity profile at this location would be used in the inlet boundary condition (10.14), and (10.18) could be solved for x > x₀ to determine the boundary-layer flow. However, for this similarity solution, we are effectively assuming that the distance x₀ is small compared to x and can be ignored. Thus, the boundary condition (10.22), which replaces (10.14), is really an assumption that must be shown to produce self-consistent results when Re_x ≫ 1.

The prior discussion touches on the question of a boundary layer’s downstream dependence on, or memory of, its initial state. If the external stream U_e(x) admits a similarity solution, is the initial condition forgotten? And, if so, how soon? Serrin (1967) and Peletier (1972) showed that for U_edU_e/dx > 0 (favorable pressure gradients) when considering similarity solutions, the initial condition is forgotten and that the larger the free-stream acceleration the sooner similarity is achieved. However, a decelerating flow will accentuate details of the boundary layer's initial state and similarity will never be found even if it is mathematically possible. This is consistent with the experimental findings of Gallo et al. (1970). Interestingly, a flat plate for which U_e(x) = U = const. is the borderline case; similarity is eventually achieved. Thus, a solution in the form (10.19) is pursued here.

The first solution steps involve performing derivatives of ψ to find u and v:

u=∂ψ∂y=Uδdfdη1δ=Uf′,v=−∂ψ∂x=−U(dδdxf+δdfdη(−yδ2)dδdx)=Uδ′(−f+ηf′),

(10.23, 10.24)

where a prime denotes differentiation of a function with respect to its argument. When substituted into (10.18), these relations for u and v produce:

Uf′Uf″(−yδ2)δ′+Uδ′(−f+ηf′)Uδf″=νUδ2f‴,or−[U2δδ′]ff″=[νUδ2]f‴,

(10.25)

since two terms on the left side of the first equality are equal and opposite. For a similarity solution, the coefficients in [,]-braces in (10.25) must be proportional:

CU2δδ′=νUδ2,orCδdδdx=νU,whichimplies:Cδ22=νxU+D,

where C and D are constants. Here (10.22) requires D = 0, and C can be chosen equal to 2 to simplify the resulting expression for δ:

δ(x)=[νx/U]1/2.

(10.26)

As described above, this result will be imperfect as x → 0 since it is based on equations that are only valid when Re_x ≫ 1. However, it is self-consistent since it produces a boundary layer that thins with decreasing distance so that u → U at any finite y as x → 0. When (10.26) is substituted into (10.25), the final equation for f is found:

d3fdη3+12fd2fdη2=0,orf‴+12ff″=0.

(10.27)

The boundary conditions for (10.27) are:

df/dη=f=0atη=0,anddf/dη→1asη→∞,

(10.28, 10.29)

which replace (10.20) and (10.21), respectively.

A series solution of (10.27), subject to (10.28) and (10.29), was given by Blasius; today it is much easier to numerically determine f(η) (see Exercise 10.2), and Table 10.1 provides numerical results for f, f′= df/dη, and f″= d²f/dη² vs. η for 0 < η < 7.0. The resulting profile of u/U = f′(η) is shown in Figure 10.5. For η > 7.0, the table may continued via: f = η – 1.7208, df/dη = 1, and d²f/dη² = 0. The solution makes the profiles at various downstream distances collapse into a single curve of u/U vs. y[U/νx]^1/2, and is in excellent agreement with experimental data for laminar flow at high Reynolds numbers. The profile has a point of inflection (i.e., zero curvature) at the wall, where ∂²u/∂y² = 0. This is a result of the absence of a pressure gradient in the flow (see Section 10.7).

The Blasius boundary-layer profile has a variety of noteworthy properties. First of all, an asymptotic analysis of the solution to (10.27) shows that (df/dη – 1) ∼ (1/η)exp(–η²/4) as η → ∞ so u approaches U very smoothly with increasing wall-normal distance. Second, the wall-normal velocity is:

Table 10.1

Blasius Boundary-Layer Profile Functions

η	f	df/dη	d2f/dη2
0.0	0.0	0.0	0.3321
0.2	0.0068	0.0664	0.3320
0.4	0.0268	0.1328	0.3314
0.6	0.0611	0.1989	0.3299
0.8	0.1074	0.2646	0.3272
1.0	0.1667	0.3297	0.3228
1.2	0.2390	0.3937	0.3164
1.4	0.3252	0.4559	0.3074
1.6	0.4225	0.5163	0.2962
1.8	0.5310	0.5743	0.2826
2.0	0.6502	0.6297	0.2667
2.2	0.7823	0.6809	0.2481
2.4	0.9240	0.7282	0.2278
2.6	1.0741	0.7716	0.2063
2.8	1.2319	0.8109	0.1840
3.0	1.3969	0.8459	0.1614
3.2	1.5697	0.8756	0.1392
3.4	1.7478	0.9010	0.1181
3.6	1.9302	0.9226	0.0984
3.8	2.1164	0.9407	0.0804
4.0	2.3058	0.9555	0.0643
4.2	2.4983	0.9666	0.0508
4.4	2.6924	0.9758	0.0391
4.6	2.8883	0.9826	0.0296
4.8	3.0853	0.9878	0.0219
5.0	3.2833	0.9915	0.0160
5.2	3.4819	0.9942	0.0114
5.4	3.6809	0.9961	0.0080
5.6	3.8803	0.9975	0.0055
5.8	4.0799	0.9984	0.0037
Table Continued

η	f	df/dη	d2f/dη2
6.0	4.2796	0.9990	0.0024
6.2	4.4795	0.9994	0.0016
6.4	4.6794	0.9996	0.0010
6.6	4.8793	0.9998	0.0006
6.8	5.0793	0.9999	0.0004
7.0	5.2792	0.9999	0.0002

v=−∂ψ∂x=12νUx(−f+ηdfdη),orvU=12Rex1/2(−f+ηdfdη)∼0.86Rex1/2asη→∞,

a plot of which is shown in Figure 10.6. The wall-normal velocity increases from zero at the wall to a maximum value at the edge of the boundary layer, a pattern that is in agreement with the streamline shapes sketched in Figure 10.4.

The various thicknesses for the Blasius boundary layer are as follows. From Table 10.1, the distance where u = 0.99U is η = 4.92, so:

δ99=4.92νx/Uorδ99/x=4.92/Rex1/2.

(10.30a)

For air at ordinary temperatures flowing at U = 1 m/s, the Reynolds number at a distance of 1m from the leading edge of a flat plate is Re_x = 6 × 10⁴, and (10.30a) gives δ₉₉ = 2cm, showing that the boundary layer is indeed thin. The displacement and momentum thicknesses, (10.16) and (10.17), of the Blasius boundary layer are:

Figure 10.5 The Blasius similarity solution for the horizontal velocity distribution in a laminar boundary layer on a flat plate with zero-pressure gradient, U_e = U = constant. The finite slope at η = 0 implies a non-zero wall shear stress τ_w. The boundary layer's velocity profile smoothly asymptotes to U as η → ∞. The momentum θ and displacement δ∗ thicknesses are indicated by arrows on the horizontal axis.

Figure 10.6 Surface-normal velocity component, v, in a laminar boundary layer on a flat plate with constant free-stream speed U. Here the scaling on vertical axis, (v/U)Rex, causes it to be expanded compared to that in Figure 10.5.

δ∗=1.72νx/U,andθ=0.664νx/U.

(10.30b,c)

These thicknesses are indicated along the abscissa of Figure 10.5.

The local wall shear stress, is τw=μ(du/dy)y=0=(μU/δ)(d2f/dη2)η=0, so it and the skin friction coefficient are:

τw=0.332ρU2/Rex,andCf≡τw12ρU2=0.664Rex.

(10.31, 10.32)

The wall shear stress therefore decreases as x^–1/2, a result of the thickening of the boundary layer and the associated decrease of the velocity gradient at the surface. Note that the wall shear stress at the plate's leading edge has an integrable singularity. This is a manifestation of the fact that boundary-layer theory breaks down near the leading edge where the assumptions Re_x ≫ 1, and ∂/∂x ≪ ∂/∂y are invalid. The drag force per unit width on one side of a plate of length L is:

FD=∫0Lτwdx=0.664ρU2LReL,

where Re_L ≡ UL/v is the Reynolds number based on the plate length. This equation shows that the drag force is proportional to the 3/2-power of the velocity. This is a higher power than that in low Reynolds number flows where drag is proportional to the first power of velocity. But, it is a lower power than that in high Reynolds number flow past a blunt body where drag is typically proportional to the square of velocity.

The overall drag coefficient for one side of the plate, defined in the usual manner, is:

CD=FD12ρU2L=1.33ReL.

(10.33)

It is clear from (10.32) and (10.33) that:

CD=1L∫0LCfdx.

which says that the overall drag coefficient is the spatial average of the local skin friction coefficient. However, carrying out an integration from x = 0 may be of questionable validity because the equations and solutions are valid only for Re_x ≫ 1. Nevertheless, (10.33) is found to be in good agreement with laminar flow experiments for Re_L > 10³.

Example 10.3

Using the information in Table 10.1 plot streamlines and δ₉₉ in the Blasius boundary layer for a 1.0m/s airflow over a 3.0-m-long surface.

Solution

Start with (10.19) and insert (10.26) to reach:

ψ=νUxf(yU/νx).

 

The goal is to use this formula and Table 10.1 to plot an x-y curve that represents ψ = constant. To get started denote the first two entries on the ith row of Table 10.1 by η_i and f_i, and look for an algebraic parameterization of the streamline's coordinates at discrete locations: x_i = x(η_i, f_i) and y_i = y(η_i, f_i). The first parameterization can be found directly from the above equation:

ψ=νUxifiorxi=ψ2/(fi2νU).

 

The second comes from (10.26):

ηi=yiU/νxioryi=ηiψ/(fiU).

 

Thus, once a value of ψ is selected, x_i-y_i coordinate pairs on this streamline can be obtained by evaluating the equations for x_i and y_i using the η_i and f_i entries in Table 10.1. For the conditions given, such a streamline plot is provided in Figure 10.7. Here, a few additional η and f values from high in the boundary (η > 7.0) were needed to plot streamlines starting from x = 0. And, the darker line is δ₉₉ from (10.30a). For x < 0, the streamlines are straight and horizontal.

This figure shows several important phenomena. First, even at this modest size and flow speed the boundary layer's thickness (centimeters) is much less than the corresponding development length (meters). Second, there is a kink in the streamlines at x = 0. This occurs because the boundary-layer equations are parabolic so the plate has no upstream influence. This kink would be absent if the full equations of fluid motion were used near the plate's leading edge. And third, streamlines that originate in the outer irrotational flow continually enter the boundary layer with increasing downstream distance.

Figure 10.7 Blasius boundary-layer streamlines and 99% thickness for a 1.0 m/s air flow over a 3.0-m-long surface. At x = 3 m, the Reynolds number based on downstream distance, Re_x, is 200,000. The lighter curves are streamlines from the solution of (10.27), and the stream function values are in m²/s. The heavier curve is the overall boundary-layer thickness, δ₉₉ from (10.30a); it reaches 3.3 cm at x = 3.0 m.