Question 10.10: Laminar Boundary Layer on a Flat Plate A uniform free stream...

SOLUTION We are to calculate the boundary layer velocity profile (u as a function of x and y) as the laminar boundary layer grows along the flat plate.

Assumptions 1 The flow is steady, incompressible, and two-dimensional in the xy-plane. 2 The Reynolds number is high enough that the boundary layer approximation is reasonable. 3 The boundary layer remains laminar over the range of interest.

Analysis We follow the step-by-step procedure outlined in Fig. 10–93.

Step 1 The outer flow is obtained by ignoring the boundary layer altogether, since it is assumed to be very, very thin. Recall that any streamline in an irrotational flow can be thought of as a wall since there is no flow through a streamline. In this case, the x-axis can be thought of as a streamline of uniform free-stream flow, one of our building block flows in Section 10–5; this streamline can also be thought of as an infinitesimally thin plate (Fig. 10–96). Thus,

Outer flow: U(x) = V = constant (1)

For convenience, we use U instead of U(x) from here on, since it is a constant.

Step 2 We assume a very thin boundary layer along the wall (Fig. 10–97). The key here is that the boundary layer is so thin that it has negligible effect on the outer flow calculated in step 1.

Step 3 We must now solve the boundary layer equations. We see from Eq. 1 that dU/dx = 0; in other words, no pressure gradient term remains in the x-momentum boundary layer equation. This is why the boundary layer on a flat plate is often called a zero pressure gradient boundary layer. The continuity and x-momentum equations for the boundary layer (Eqs. 10–71) become

Boundary layer equations:

\begin{aligned}&\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \\&u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=U \frac{d U}{d x}+v \frac{\partial^{2} u}{\partial y^{2}}\end{aligned} (Eq. 10-71)

\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \quad u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=v \frac{\partial^{2} u}{\partial y^{2}} (2)

There are four required boundary conditions,

u = 0 at y = 0 u = U as y → ∞

𝜐 = 0 at y = 0 u = U for all y at x = 0 (3)

The last of the boundary conditions in Eq. 3 is the starting profile; we assume that the plate has not yet influenced the flow at the starting location of the plate (x = 0).

These equations and boundary conditions seem simple enough, but unfortunately no convenient analytical solution is available. However, a series solution of Eqs. 2 was obtained in 1908 by P. R. Heinrich Blasius (1883–1970). As a side note, Blasius was a Ph.D. student of Prandtl. In those days, of course, computers were not yet available, and all the calculations were performed by hand. Today we can solve these equations on a computer in a few seconds. The key to the solution is the assumption of similarity. In simple terms, similarity can be assumed here because there is no characteristic length scale in the geometry of the problem. Physically, since the plate is infinitely long in the x-direction, we always see the same flow pattern no matter how much we zoom in or zoom out (Fig. 10–98).

Blasius introduced a similarity variable 𝜂 that combines independent variables x and y into one nondimensional independent variable,

\eta=y \sqrt{\frac{U}{v x}} (4)

and he solved for a nondimensionalized form of the x-component of velocity,

f^{\prime}=\frac{u}{U}=\text { function of } \eta (5)

When we substitute Eqs. 4 and 5 into Eqs. 2, subjected to the boundary conditions of Eq. 3, we get an ordinary differential equation for nondimensional speed f′(𝜂 ) = u/U as a function of similarity variable 𝜂 . We use the popular Runge–Kutta numerical technique to obtain the results shown in Table 10–3 and in Fig. 10–99. Details of the numerical technique are beyond the scope of this text (see Heinsohn and Cimbala, 2003). There is also a small y-component of velocity 𝜐 away from the wall, but 𝜐 ≪ u, and is not discussed here. The beauty of the similarity solution is that this one unique velocity profile shape applies to any x-location when plotted in similarity variables, as in Fig. 10–99. The agreement of the calculated profile shape in Fig. 10–99 to experimentally obtained data (circles in Fig. 10–99) and to the visualized profile shape of Fig. 10–78 is remarkable. The Blasius solution is a stunning success.

Step 4 We next calculate several quantities of interest in this boundary layer. First, based on a numerical solution with finer resolution than that shown in Table 10–3, we find that u/U = 0.990 at 𝜂 ≅ 4.91. This 99 percent boundary layer thickness is sketched in Fig. 10–99. Using Eq. 4 and the definition of 𝛿, we conclude that y = 𝛿 when

\eta=4.91=\sqrt{\frac{U}{v x}} \delta \rightarrow \frac{\delta}{x}=\frac{4.91}{\sqrt{ R e _{x}}} (6)

This result agrees qualitatively with Eq. 10–67, obtained from a simple order ofmagnitude analysis. The constant 4.91 in Eq. 6 is rounded to 5.0 by many authors, but we prefer to express the result to three significant digits for consistency with other quantities obtained from the Blasius profile.

\frac{\delta}{L} \sim \frac{1}{\sqrt{\operatorname{Re}_{L}}} (Eq. 10-67)

Another quantity of interest is the shear stress at the wall \tau_{w},

\left.\tau_{w}=\mu \frac{\partial u}{\partial y}\right)_{y=0} (7)

Sketched in Fig. 10–99 is the slope of the nondimensional velocity profile at the wall (y = 0 and 𝜂 = 0). From our similarity results (Table 10–3), the nondimensional slope at the wall is

\left.\frac{d(u / U)}{d \eta}\right)_{\eta=0}=f^{\prime \prime}(0)=0.332 (8)

After substitution of Eq. 8 into Eq. 7 and some algebra (transformation of similarity variables back to physical variables), we obtain

Shear stress in physical variables: \tau_{w}=0.332 \frac{\rho U^{2}}{\sqrt{\operatorname{Re}_{x}}} (9)

Thus, we see that the wall shear stress decays with x like x^{-1 / 2}, as sketched in Fig. 10–100. At x = 0, Eq. 9 predicts that \tau_{w} is infinite, which is unphysical. The boundary layer approximation is -not appropriate at the leading edge (x = 0), because the boundary layer thickness is not small compared to x. Furthermore, any real flat plate has finite thickness, and there is a stagnation point at the front of the plate, with the outer flow accelerating quickly to U(x) = V. We may ignore the region very close to x = 0 without loss of accuracy in the rest of the flow.

Equation 9 is nondimensionalized by defining a skin friction coefficient (also called a local friction coefficient),

Local friction coefficient, laminar flat plate: C_{f, x}=\frac{\tau_{w}}{\frac{1}{2} \rho U^{2}}=\frac{0.664}{\sqrt{\operatorname{Re}_{x}}} (10)

Notice that Eq. 10 for C_{f, x} has the same form as Eq. 6 for 𝛿/x, but with a different constant—both decay like the inverse of the square root of Reynolds number. In Chap. 11, we integrate Eq. 10 to obtain the total friction drag on a flat plate of length L.

Step 5 We need to verify that the boundary layer is thin. Consider the practical example of flow over the hood of your car (Fig. 10–101) while you are driving downtown at 20 mi/h on a hot day. The kinematic viscosity of the air is ν = 1.8 × 10^{-4} ft ^{2} / s. We approximate the hood as a flat plate of length 3.5 ft moving horizontally at a speed of V = 20 mi/h. First, we approximate the Reynolds number at the end of the hood using Eq. 10–60,

Reynolds number along a flat plate: \operatorname{Re}_{x}=\frac{\rho V x}{\mu}=\frac{V x}{v} (Eq. 10-60)

Since \operatorname{Re}_{x} is very close to the ballpark critical Reynolds number, \operatorname{Re}_{x, c r}=5 \times 10^{5}, the assumption of laminar flow may or may not be appropriate. Nevertheless, we use Eq. 6 to estimate the thickness of the boundary layer, assuming that the flow remains laminar,

\delta=\frac{4.91 x}{\sqrt{\operatorname{Re}_{x}}}=\frac{4.91(3.5 ft )}{\sqrt{5.7 \times 10^{5}}}\left(\frac{12 in }{ ft }\right)=0.27 \text { in } (11)

By the end of the hood the boundary layer is only about a quarter of an inch thick, and our assumption of a very thin boundary layer is verified.

Discussion The Blasius boundary layer solution is valid only for flow over a flat plate perfectly aligned with the flow. However, it is often used as a quick approximation for the boundary layer developing along solid walls that are not necessarily flat nor exactly parallel to the flow, as in the car hood. As illustrated in step 5, it is not difficult in practical engineering problems to achieve Reynolds numbers greater than the critical value for transition to turbulence. You must be careful not to apply the laminar boundary layer solution presented here when the boundary layer becomes turbulent.