What is the exact cause of flow separation in a viscous fluid?

Question

I recently got into a lengthy debate about the exact nature of boundary layer separation. In common parlance, we have a tendency to talk about certain geometries as being too "sharp" for a viscous flow to remain attached to them. The flow can't "turn the corner" so to speak, and so it separates from the body. While I think this way of thinking can properly predict in which situations a flow might separate, I think it gets the underlying Physics completely wrong. From my understanding, what's happeneing is the adverse streamwise pressure gradient precludes the boundary layer from progressing downstream past a certain point, and the upstream flow subsequently has nowhere to go but up and off of the body. This is a very different causal relationship from the first explanation, where the flow lacks a sufficient streamwise-normal pressure gradient to overcome the centrifugal forces of a curved streamline. But which is correct?

Considering that normal shockwaves can produce extreme adverse pressure gradients (even along a streamline that is not curved), I figured that shock-induced flow separation might be a way to settle this matter. Any thoughts?

Answer 1

From my understanding, what's happeneing is the adverse streamwise pressure gradient precludes the boundary layer from progressing downstream past a certain point, and the upstream flow subsequently has nowhere to go but up and off of the body.

This is correct, in a sense. The effect of an adverse pressure gradient is to decelerate the flow near the body surface. This can be seen, for example, by examining the boundary layer equation in two dimensions.

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=\nu\frac{\partial^2 u}{\partial y^2}-\frac{1}{\rho}\frac{\partial p}{\partial x}$$

If you consider steady flow and assume normal velocities to be small, then by inspection, we can see that an adverse pressure gradient causes $u$ to decrease in the streamwise ($x$) direction.

As you suspected, separation requires that the flow near the boundary stagnates. Moreover, separation occurs when the flow actually reverses. $$ \frac{\partial u}{\partial y}_{y=0}=0; \quad \text{Flow Stagnation / Impending Reversal} $$ Additionally, it requires that the pressure gradient be simultaneously adverse, so that the the flow does not accelerate again. $$ \frac{\partial p}{\partial x}>0 \quad \text{Adverse Pressure Gradient}$$

So, in short, you're correct. However...

This is a very different causal relationship from the first explanation, where the flow lacks a sufficient streamwise-normal pressure gradient to overcome the centrifugal forces of a curved streamline.

The two statements are essentially the same - there are any number of ways to physically describe what's going on- but I think you've got the causality mixed between the two. The curvature of a body, and thus its attending streamlines, jacks up the adversity of the pressure gradient along that body (assuming you're past the point of minimum pressure). So it's the adverse pressure gradient that ultimately leads to separation. In a perfect world, where viscosity didn't exist, the flow would speed up as it hits the forward part of a curved body. The pressure would drop as it reaches the widest point of the body, streamlines are "squeezed" together, and the flow reaches a maximum velocity. On the afterbody, the flow would decelerate and the pressure would increase until both reach their upstream values. It's a simple trade between kinetic energy (velocity) and potential energy (pressure). In a real viscous flow, some of that kinetic energy is dissipated in the heat-generating nuisance that is a boundary layer, so that when the transfer from kinetic back to potential energy occurs on the afterbody of a curved surface, there isn't enough kinetic energy, the flow stagnates and reverses, and you get flow separation.

I can't comment on shock-induced separation, as I work in hydrodynamics and don't worry about compressibility. I'm no authority in that area, either, so if somebody takes issue with my explanation, feel free to criticize.

Answer 2

In classic Prandtl's Boundary Layer Theory (BTL) in 1904 from Navier-Stokes (N-S)equations, the fluid particles are driven by pressure gradient $dp/dx$. If p falls along $x-$direction, $dp/dx<0$ and we call the pressure gradient is “favorable.” If otherwise, the pressure rises along the streamline, i.e., $dp/dx>0$ and we say the pressure gradient is “adverse” which in most cases is unfavorable. In the "unfavorable" case, the boundary layer becomes thicker and thicker in a decelerating flow region which grows rapidly and can develop a slow reverse flow at the wall where $du/dn_w=0$, $n_w $ is the normal at wall and the streamline intersects the wall at this point of separation.

There is the other formulation describing the fluid motions of equations states that the fluid particles follow the curvature of the boundary without separation if $\partial p/\partial n=U^2/R$ and separate tangentially if $\partial p/\partial n<U^2/R$, where $U$ is the tangential fluid velocity, and $R$ is the radius of the boundary.

This is closely related to the BIG mysterious mechanism of the separation which must be the compound of inertial and viscous effects.

But back to your question, "exact cause of flow separation in a viscous fluid", I assume that the viscosity is not the only cause.

Besides, I do not agree with the following statement

For Engineering understanding of flow separation, Faltinsen 1990 states "A consequence of separation is that pressure forces due to viscous effects are more important than shear forces. There is some confusion about what is precisely meant by separation in unsteady flow...".