Criterion for Boundary Layer Separation

As we have seen, the boundary layer equations (8.110)-(8.113) generally lead to the conclusion that the tangential velocity in a thin boundary layer, $u$ , is large compared with the normal velocity, $v$ . Mathematically speaking, this result holds everywhere except in the immediate vicinity of singular points. But, if $v\ll u$ then it follows that the fluid moves predominately parallel to the surface of the obstacle, and can only move away from this surface to a very limited extent. This restriction effectively precludes separation of the flow from the surface. Hence, we conclude that separation can only occur at a point at which the solution of the boundary layer equations is singular.

As we approach a separation point, we expect the flow to deviate from the boundary layer towards the interior of the fluid. In other words, we expect the normal velocity to become comparable with the tangential velocity. However, we have seen that the ratio $v/u$ is of order ${\rm Re}^{-1/2}$ . [See Equation (8.18).] Hence, an increase of $v$ to such a degree that $v\sim u$ implies an increase by a factor ${\rm Re}^{1/2}$ . For sufficiently large Reynolds numbers, we may suppose that $v$ effectively increases by an infinite factor. Indeed, if we employ the dimensionless form of the boundary layer equations, (8.23)-(8.27), the situation just described is formally equivalent to an infinite value of the dimensionless normal velocity, $V_y$ , at the separation point.

Let the separation point lie at $x=x_0$ , and let $x<x_0$ correspond to the region of the boundary layer upstream of this point. According to the previous discussion,

$$v(x_0,y) = \infty$$ (8.121)

at all $y$ (except, of course, $y=0$ , where the boundary conditions at the surface of the obstacle require that $v=0$ ). It follows that the deriviative $\partial v/\partial y$ is also infinite at $x=x_0$ . Hence, the equation of continuity, $\partial u/\partial x+\partial v/\partial y=0$ , implies that $(\partial u/\partial x)_{x=x_0}=\infty$ , or $\partial x/\partial u=0$ , if $x$ is regarded as a function of $u$ and $y$ . Let $u(x_0,y)=u_0(y)$ . Close to the point of separation, $x_0-x$ and $u-u_0$ are small. Thus, we can expand $x_0-x$ in powers of $u-u_0$ (at fixed $y$ ). Because $(\partial x/\partial u)_{u=u_0}=0$ , the first term in this expansion vanishes identically, and we are left with

$$x_0-x = f(y)\,(u-u_0)^2 + {\cal O}\left[(u-u_0)^3\right],$$ (8.122)

or

$$u(x,y)\simeq u_0(y) +\alpha(y)\sqrt{x_0-x},$$ (8.123)

where $\alpha=1/\sqrt{f}$ is some function of $y$ . From the equation of continuity,

$$\frac{\partial v}{\partial y} = -\frac{\partial u}{\partial x} \simeq \frac{\alpha(y)}{2\sqrt{x_0-x}}.$$ (8.124)

Upon integration, the previous expression yields

$$v(x,y) \simeq \frac{\beta(y)}{\sqrt{x_0-x}},$$ (8.125)

where

$$\beta(y) = \frac{1}{2}\int^y \alpha(y')\,dy'.$$ (8.126)

The equation of tangential motion in the boundary layer, (8.111), is written

$$u\,\frac{\partial u}{\partial x} + v\,\frac{\partial u}{\partial y} = U\,\frac{dU}{dx} +\nu\,\frac{\partial^{\,2} u}{\partial y^{\,2}}.$$ (8.127)

As is clear from Equation (8.123), the derivative $\partial^{\,2} u/\partial y^{\,2}$ does not become infinite as $x\rightarrow x_0$ . The same is true of the function $U\,dU/dx$ , which is determined from the flow outside the boundary layer. However, both terms on the left-hand side of the previous expression become infinite as $x\rightarrow x_0$ . Hence, in the immediate vicinity of the separation point,

$$u\,\frac{\partial u}{\partial x} + v\,\frac{\partial u}{\partial y} \simeq 0.$$ (8.128)

Because $\partial u/\partial x = -\partial v/\partial y$ , we can rewrite this equation in the form

$$-u\,\frac{\partial v}{\partial y} +v\,\frac{\partial u}{\partial y} =- u^{\,2}\frac{\partial}{\partial y}\!\left(\frac{v}{u}\right)\simeq 0.$$ (8.129)

Because $u$ does not, in general, vanish at $x=x_0$ , we conclude that

$$\frac{\partial}{\partial y}\!\left(\frac{v}{u}\right)\simeq 0.$$ (8.130)

In other words, $v/u$ is a function of $x$ only. From Equations (8.123) and (8.125),

$$\frac{v}{u} = \frac{\beta(y)}{u_0(y) \sqrt{x_0-x}}+ {\cal O}(1).$$ (8.131)

Hence, if this ratio is a function of $x$ alone then $\beta(y) = (1/2)\,A\,u_0(y)$ , where $A$ is a constant: that is,

$$v(x,y)\simeq \frac{A\,u_0(y)}{2\sqrt{x_0-x}}.$$ (8.132)

Finally, because Equation (8.126) yields $\alpha=2\,d\beta/dy=A\,du_0/dy$ , we obtain

$$u(x,y) \simeq u_0(y) + A\,\frac{du_0}{dy}\sqrt{x_0-x}.$$ (8.133)

The previous two expressions specify $u$ and $v$ as functions of $x$ and $y$ near the point of separation. Beyond the point of separation, that is for $x>x_0$ , the expressions are physically meaningless, because the square roots become imaginary. This implies that the solutions of the boundary layer equations cannot sensibly be continued beyond the separation point.

The standard boundary conditions at the surface of the obstacle require that $u=v=0$ at $y=0$ . It, therefore, follows from Equations (8.132) and (8.133) that

$$u_0(0) = 0,$$ (8.134)

$$\left.\frac{du_0}{dy}\right\vert _{y=0} =0.$$ (8.135)

Thus, we obtain the important prediction that both the tangential velocity, $u$ , and its first derivative, $\partial u/\partial y$ , are zero at the separation point (i.e., $x=x_0$ and $y=0$ ). This result was originally obtained by Prandtl, although the argument we have used to derive it is due to L.D. Landau (1908-1968) (Landau and Lifshitz 1987).

If the constant $A$ in expressions (8.132) and (8.133) happens to be zero then the point $x=x_0$ and $y=0$ , at which the derivative $\partial u/\partial y$ vanishes, has no particular properties, and is not a point of separation. However, there is no reason, in general, why $A$ should take the special value zero. Thus, in practice, a point on the surface of an obstacle at which $\partial u/\partial y=0$ is always a point of separation.

Incidentally, if there were no separation at the point $x=x_0$ (i.e., if $A=0$ ) then we would have $\partial u/\partial y<0$ for $x>x_0$ . In other words, $u$ would become negative as we move away from the surface, $y$ being still small. That is, the fluid beyond the point $x=x_0$ would move tangentially, in the region of the boundary layer immediately adjacent to the surface, in the direction opposite to that of the external flow: that is, there would be ``back-flow'' in this region. In practice, the flow separates from the surface at $x=x_0$ , and the back-flow migrates into the wake.

The dimensionless boundary layer equations, (8.23)-(8.27), are independent of the Reynolds number of the external flow (assuming that this number is much greater than unity). Thus, it follows that the point on the surface of the obstacle at which $\partial u/\partial y=0$ is also independent of the Reynolds number. In other words, the location of the separation point is independent of the Reynolds number (as long as this number is large, and the flow in the boundary layer is non-turbulent).

At $y=0$ , the equation of tangential motion in the boundary layer, (8.111), is written

$$\nu\left.\frac{\partial^{\,2} u}{\partial y^{\,2}}\right\vert _{y=0} = -\frac{1}{U}\,\frac{dU}{dx} = \frac{1}{\rho}\,\frac{dP}{dx},$$ (8.136)

where $P(x)$ is the pressure just outside the layer, and use has been made of Equation (8.6). Because $u$ is positive, and increases away from the surface (upstream of the separation point), it follows that $(\partial^{\,2} u/\partial y^{\,2})_{y=0}>0$ at the separation point itself, where $(\partial u/\partial y)_{y=0}=0$ . Hence, according to the previous equation,

$$\left(\frac{dU}{dx}\right)_{x=x_0} <0,$$ (8.137)

$$\left(\frac{dP}{dx}\right)_{x=x_0} > 0.$$ (8.138)

In other words, we predict that the external tangential flow is always decelerating at the separation point, whereas the pressure gradient is always adverse (i.e., such as to decelerate the tangential flow), in agreement with experimental observations.