# How does a weak shock-wave boundary-layer interaction create wave drag (other than through direct shock losses)?

I'm trying to understand the physical cause of wave drag, beyond the simple statement "the presence of shockwaves increases the drag".

As far as I understand, in the case of a weak BLSWI (so without separation), the drag increase is due to:

• Direct shock losses (momentum deficiency of freestream flow through shockwave)
• A change in the state of the boundary layer due to the compression waves in side the BL

But how exactly does the second point work? If the BL becomes thicker after the SW, wouldn't that mean that the drag decreases because the velocity gradient and therefore the shear stress decrease?

I feel like I'm missing some tiny aspect...

Edit: I've found the following statement in this report, which maybe explains my problem better:

The increase in drag occurs directly because of the wave drag associated with the presence of shock waves. However, the drag also increases because the boundary layer thickness increases due to the sudden pressure rise on the surface due to the shock wave, which leads to increased profile drag. Lynch has estimated that at drag divergence the additional transonic drag is approximately evenly divided between the explicit shock drag and the shock induced additional profile drag

Why does the increase in BL thickness cause more drag? Shouldn't the velocity gradient at the wall reduce, leading in less shear stress? Or should I see it in a way that the already turbulent BL "takes up more energy in its vortical structures" from the flow, due to the increase in thickness?

• I've seen a nice explanation of wave drag (and lift) around here showing that wave drag is simply because at supersonic speeds there is no pressure recovery beyond the thickest point. It is, therefore, not actually caused by shock waves themselves, just correlated with their presence, because where there is supersonic flow, there are shock waves. – Jan Hudec Dec 31 '18 at 1:21
• So the effect you describe would be pressure drag, caused by a more negative pressure between the thickest point and the shock wave, when compared to the subsonic case? That would make sense. But still, the question remains, what's up with the thicker boundary layer? Shouldn't that even reduce the drag? – Daniel Dec 31 '18 at 12:07

As you know, wall shear stress $$\tau_w$$ (and thus friction drag) can be described as: $$\tau_w = \mu \frac{\partial u}{\partial y} \vert_{y=0}$$ where $$\mu$$ is the dynamic viscosity of the fluid and $$\frac{\partial u}{\partial y} \vert_{y=0}$$ is the fluid velocity gradient at the wall.

The shock-wave boundary-layer interaction (SBLI) can change these two factors through several mechanisms:

1. A change in boundary-layer thickness: As you noted correctly, an increase in boundary-layer (BL) thickness will decrease the velocity gradient, compared to an otherwise similar BL velocity profile. However, my conclusion after reading several publications is that the increase in BL thickness doesn't always occur. BL thickness seams to scale with shock strength and very weak shocks actually produce a thinner BL.

2. A decrease in fluid velocity: The shock-wave will obviously result in a decreased velocity outside of the boundary layer, which decreases the velocity gradient. This will also decrease wall shear stress, so it cannot be the dominant mechanism either, since it contradicts our experience of increased friction.

3. An increase in turbulence: The turbulence of the BL is significantly amplified by the shock-wave through different mechanisms. The image below visualises the distribution of reynolds shear stresses, which are a measure for turbulence. A high magnitude (positive or negative) of reynolds stresses means high turbulence: Reynolds shear stress distribution of a SBLI. Notice the strong negative values in the lower BL behind the SBLI. Negative values indicate slow fluid moving upwards and fast fluid moving downwards. Also notice the value is zero very close to the wall, because the wall prevents vertical fluctuations. Image Source (I added the white markings)
The increased turbulence causes an increased momentum exchange between the lower and slower part of the BL with the higher and faster part. This momentum exchange changes the velocity profile of the BL. There are higher velocities closer to the wall. This results in a higher velocity gradient at the wall. I believe this must be the main cause of increased wall shear stress. The stronger turbulence will probably decay downstream, so the velocity profile will slowly change back to the normal turbulent flat plate type.

4. Increase in pressure and temperature by the shock-wave: The rising temperature and rising pressure (not so much) increase the dynamic viscosity of the fluid and thus increase shear stress.

Most likely there are other factors that also play a role. I am no expert in SBLIs, I only read through a fair amount of literature to come to my conclusions.

Literature that I found especially helpful:

• Great, thank you for the detailed answer! So, to summarise, the increase in drag due to a weak shockwave would be because of an increased turbulence in the BL after the shockwave (therefore friction drag)? I guess that the increase in pressure drag due to the lack of pressure recovery after the thickest point (see comment on my qeustion) also plays a role although that is only correlated to the presence of shock waves, and not caused by. – Daniel Jan 7 '19 at 12:29
• Exactly. The increased turbulence in the BL should be the main cause for increased friction drag. However, total-pressure loss across the shockwave always plays a role for drag. How much either effect contributes to shock-induced drag will greatly depend on the flow in each specific situation. Intuitively I would expect the total-pressure loss to be the stronger effect in most cases. – Felix L. Jan 12 '19 at 21:44
• Alright, but the static pressure still increases across the shock wave, right? So intuitively I would think that this increased static pressure provides some "pressure thrust", if it acts after the thickest point. – Daniel Jan 13 '19 at 10:42
• Yes, static pressure increases across the shock wave. But in contrast to subsonic flow, supersonic flow accelerates when it expands.This is what happens after the thickest point. The static pressure decreases with flow acceleration. This is why you have an area of reduced pressure beyond the thickest point. Because of the total pressure loss, static pressure is even lower than how it would have been without the shock, if that makes sense. – Felix L. Jan 16 '19 at 16:34