Incompressible Flow over a Flat Plate: The Blasius Solution

Consider the incompressible, two-dimensional flow over a flat plate at 0° angle of attack, such as sketched in Figure 17.7. For such a flow, p = constant, p = constant, and dpe/dx = 0 (because the inviscid flow over a flat plate at a = 0 yields a constant pressure over the surface). Moreover, recall that the energy equation is not needed to calculate the velocity field for an incompressible flow. Hence, the boundary-layer equations, Equations (17.28) to (17.31), reduce to

3 и dv

— + — = 0

дх 3 у

ди ди 32u

Ud^ + V^ = Vd?

d-?=o

dy

where v is the kinematic viscosity, defined as v = р/р.

Equation (18.12) is of particular note. The function /(p) defined in Equation (18.11) has the property that its derivative /’ gives the a component of velocity as

/'(>?) =

Substitute Equations (18.8) to (18.10), (18.12), and (18.13) into the momentum equa­tion, Equation (18.2). Writing each term explicitly so that you can see what is hap­pening, we have

I V VoО r / 91? „Д.. I Voo „ Voo n

—— f + VvxVoo — f Voo J——- / =vV0o——- /

V x ox І V vx vx

[18.15]

Equation (18.15) is important; it is called Blasius’ equation, after H. Blasius, who obtained it in his Ph. D. dissertation in 1908. Blasius was a student of Prandtl, and his flat-plate solution using Equation (18.15) was the first practical application of Prandtl’s boundary-layer hypothesis since its announcement in 1904. Examine Equa­tion (18.15) closely. Amazingly enough it is an ordinary differential equation. Look what has happened! Starting with the partial differential equations for a flat-plate boundary layer given by Equations (18.1) to (18.3), and transforming both the inde­pendent and dependent variables through Equations (18.4) and (18.11), we obtain an ordinary differential equation for f(tj). In the same breath, we can say that Equa­tion (18.15) is also an equation for the velocity и because и = V*, fit]). Because Equation (18.15) is a single ordinary differential equation, it is simpler to solve than the original boundary-layer equations. However, it is still a nonlinear equation and must be solved numerically, subject to the transformed boundary conditions,

Atrj = 0: / = 0, /’ = 0

Atrj^-oo: f = 1

[Note that at the wall where rj — 0, /’ = 0 because и = 0, and therefore / = 0 from Equation (18.13) evaluated at the wall.]

Equation (18.15) is a third-order, nonlinear, ordinary differential equation; it can be solved numerically by means of standard techniques, such as the Runge-Kutta method (such as that described in Reference 52). The integration begins at the wall and is carried out in small increments Ay in the direction of increasing у away from the wall. However, since Equation (18.15) is third order, three boundary conditions must be known at rj = 0; from the above, only two are specified. A third boundary condition, namely, some value for /"(0), must be assumed-, Equation (18.15) is then integrated across the boundary layer to a large value of rj. The value of /’ at large eta is then examined. Does it match the boundary condition at the edge of the boundary layer, namely, is /’ = 1 satisfied at the edge of the boundary layer? If not, assume a different value of /"(0) and integrate again. Repeat this process until convergence

is obtained. This numerical approach is called the ‘‘shooting technique”; it is a clas­sical approach, and its basic philosophy and details are discussed at great length in Section 16.4. Its application to Equation (18.15) is more straightforward than the dis­cussion in Section 16.4, because here we are dealing with an incompressible flow and only one equation, namely, the momentum equation as embodied in Equation (18.15).

The solution of Equation (18.15) is plotted in Figure 18.2 in the form of f'(r) — m/Voc as a function of r. Note that this curve is the velocity profile and that it is a function of г] only. Think about this for a moment. Consider two different x stations along the plate, as shown in Figure 18.3. In general, и = u(x, у), and the velocity profiles in terms of и = и (у ) at given x stations will be different. Clearly, the variation of и normal to the wall will change as the flow progresses downstream. However, when plotted versus r), we see that the profile, и = и (if), is the same for all x stations, as illustrated in Figure 18.3. This result is an example of a self­similar solution—solutions where the boundary-layer profiles, when plotted versus a similarity variable r) are the same for all x stations. For such self-similar solutions, the governing boundary-layer equations reduce to one or more ordinary differential equations in terms of a transformed independent variable. Self-similar solutions occur only for certain special types of flows—the flow over a flat plate is one such example. In general, for the flow over an arbitrary body, the boundary-layer solutions are nonsimilar; the governing partial differential equations cannot be reduced to ordinary differential equations.

Numerical values of /, /’, and f" tabulated versus r can be found in Ref­erence 42. Of particular interest is the value of /" at the wall; /"(0) = 0.332. Consider the local skin friction coefficient defined as С/ = zw/ . From

/'(r?) = u/Voe

Figure 1 8.2 Incompressible velocity profile for a flat plate; solution of the Blasius equation.

du df

__ — у J— — у

~ — [3] 00 r, — Kcx

ay ay

which is a classic expression for the local skin friction coefficient for the incompress­ible laminar flow over a flat plate—a result that stems directly from boundary-layer theory. Its validity has been amply verified by experiment. Note that Cf oc Re“l/2 oc x-1/2; that is, cf decreases inversely proportional to the square root of distance from the leading edge. Examining the flat plate sketched in Figure 17.7, the total drag on the top surface of the entire plate is the integrated contribution of rw{x) from x = 0 to x = c. Letting Cf denote the skin friction drag coefficient, we obtain from Equation (1.16)

Cf = – f cf dx [18.21]

c Jo

Substituting Equation (18.20) into (18.21), we obtain

where Rec is the Reynolds number based on the total plate length c.

An examination of Figure 18.2 shows that f = 0.99 at approximately rj — 5.0. Hence, the boundary-layer thickness, which was defined earlier as that distance above the surface where и = 0.99ue, is V

Note that the boundary-layer thickness is inversely proportional to the square root of the Reynolds number (based on the local distance x). Also, 8 oc. rl/2; the laminar boundary layer over a flat plate grows parabolically with distance from the leading edge.

The displacement thickness 5*, defined by Equation (17.3), becomes for an in­compressible flow

8* = [ (l ——’j dy [18.24]

Jo ue)

In terms of the transformed variables f and rj given by Equations (18.4) and (18.12), the integral in Equation (18.24) can be written as

s* = f [1 – ҐШ dr] = УттЧт – /(hi)] [1 8.25]

у voo Jo V boo

where T]X is an arbitrary point above the boundary layer. The numerical solution for f(rj) obtained from Equation (18.15) shows that, amazingly enough, r/, – f(t]) =

1.72 for all values of r] above 5.0. Therefore, from Equation (18.25), we have

8* = 1.72.

Note that, as in the case of the boundary-layer thickness itself, 8* varies inversely with the square root of the Reynolds number, and <5* a x,/2. Also, comparing Equations (18.23) and (18.26), we see that 8* = 0.34<5; the displacement thickness is smaller than the boundary-layer thickness, confirming our earlier statement in Section 17.2.

The momentum thickness for an incompressible flow is, from Equation (17.10),

or in terms of our transformed variables,

Гш~ Ґh

9 = ПГ [18.27]

V Гоо Jo

Equation (18.27) can be integrated numerically from r) = 0 to any arbitrary point Г)і > 5.0. The result gives

Note that, as in the case of our previous thicknesses, в varies inversely with the square root of the Reynolds number and that 9 ос xl/2. Also, 9 = 0.39<5*, and 9 = 0.135. Another property of momentum thickness can be demonstrated by evaluating в at the trailing edge of the flat plate sketched in Figure 17.7. In this case, x = c, and from Equation (18.28), we obtain

0.664c

9X=C = —=

VRi;

Comparing Equations (18.22) and (18.29), we have

„ 2 0X=C

Equation (18.30) demonstrates that the integrated skin friction coefficient for the flat plate is directly proportional to the value of в evaluated at the trailing edge.

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