Reynolds number & Mach number for flow regime

[MaxKang]

"An adverse pressure gradient strongly favors transition to turbulent flow. In contrast, strong favorable pressure gradients (where p decreases in the downstream direction) tend to preserve initially laminar flow."

"High values of M∞ and low values of Re tend to encourage laminar flow"

Looking at those two statements from the "fundamentals of aerodynamics", I think the fluid velocity alone does not determine whether the fluid will be laminar or turbulent.(high V gives high mach number but also high Reynolds number). Can someone explain why/how the mach number comes into play?
I would also like to know why an adverse pressure gradient favors transition to turbulent flow.

Thank you!

 

[boneh3ad]

There is a lot more than just velocity that determines if the flow is laminar or not.

Regarding pressure gradients, the effect is not always the same depending on the instability mechanism. On a subsonic flat plate (or any two-dimensional flow), the primary instability mechanism is called a Tollmein-Schlichting (T-S) wave. T-S waves are generally more stable when the boundary-layer profile is "fuller", i.e. there is a larger velocity gradient at the wall and high-momentum fluid penetrates lower into the boundary layer. This effect can be achieved a number of ways, but one of them is through a favorable pressure gradient. The two other common means of achieving this are through light wall suction (e.g. through a porous wall) or by cooling the wall (in the case of air) or by heating the wall (in the case of water).

On the other hand, if you take an object like, say, a swept wing on an aircraft, T-S waves are not the dominant instability mechanism. In this case, you have what are called crossflow waves. Crossflow waves are actually destabilized by a favorable pressure gradient. Strategies for controlling crossflow are very limited, as you obviously don't want to enforce a strong adverse pressure gradient and risk separation (stall). Wall suction still works, as do a few other limited techniques, but for the most part, crossflow remains a very difficult problem if the goal is to laminarize a wing.

Mach number comes into play because of its correlation with compressibility of the flow. Flows that are highly compressible tend to be more stable and less likely to transition to turbulence than at lower Mach number. Without doing a little digging, I am not sure I have a better answer than that off the top of my head other than all of the numerical and experimental studies in the field show this to be true. I am sure there has been a more theoretical exploration but I don't have that answer off the top of my head. Additionally, once a flow moves into the supersonic regime, the fundamental instability mechanisms change. T-S waves and crossflow waves still exist, but there are a new class of instability waves that develop and become dominant at higher Mach numbers. That's a pretty rich area of study.

 

[aerohead_18]

There is a lot more than just velocity that determines if the flow is laminar or not.
On the other hand, if you take an object like, say, a swept wing on an aircraft, T-S waves are not the dominant instability mechanism. In this case, you have what are called crossflow waves. Crossflow waves are actually destabilized by a favorable pressure gradient. Strategies for controlling crossflow are very limited, as you obviously don't want to enforce a strong adverse pressure gradient and risk separation (stall). Wall suction still works, as do a few other limited techniques, but for the most part, crossflow remains a very difficult problem if the goal is to laminarize a wing.

What exactly are crossflow waves? I've never run across this term before and a quick google didn't turn up anything. Does it have to do with spanwise flow?

 

[boneh3ad]

It does. Consider a swept wing. The inviscid streamlines (just outside the boundary layer where viscosity is not important) are curved due to the shape and sweep of the wing. This means that there must be some sort of centripetal force corresponding to that curvature, which comes in the form of a pressure gradient in the spanwise direction. At the boundary-layer edge, these two (pressure gradient and the velocity along the curves) are in balance. However, as you approach the wall, the velocity is reduced, but if you recall from a boundary layer class, the pressure gradient is zero throughout the height of a boundary layer, so that centripetal force remains the same. The force imbalance induces a spanwise flow that is unstable to waves that move (or stand still) in roughly the spanwise direction.

In a flight-like environment, the stationary spanwise wave typically dominates and generates a series of co-rotating vortices aligned approximately with the inviscid streamlines. These draw high-momentum fluid from the free stream down into the boundary layer and push low-momentum fluid upward on the other side of them (since they are vortices, after all), which rapidly distorts the flow, leads to secondary instabilities of this newly-distorted flow, and ultimately leads to transition to turbulence.

A favorable pressure gradient destabilizes these flows because it accelerates the inviscid flow, leading to a larger spanwise pressure gradient to balance that, and therefore a larger spandwise induced flow.

It's an important but surprisingly obscure phenomenon given that it dominates laminar-turbulent transition on swept wings and on a number of very common supersonic shapes, like spinning cones (munitions, certain reentry vehicles, etc) and cones at angle of attack (nose cone of a plane). Usually, though, students don't ever really see it until graduate school. If you head over to Google Scholar, you can find a wealth of papers on the topic under "crossflow vortices" or "crossflow instability".

 

[aerohead_18]

Wouldn't the effect of those crossflow vortices be similar to that of VGs? Like do they delay boundary layer separation by inducing turbulence in the boundary layer? And do they produce induced drag like wingtip vortices?

 

[boneh3ad]

Well, anything that causes transition to turbulence will delay separation. VGs typically produce much stronger vortices, though. There has been some recent research on using so-called micro vortex generators to interact with unstable waves such as these, but that's still purely in the realm of academic research as far as I know.

Also, to the best of my knowledge, the direct effect of crossflow vortices on drag is essentially negligible. I am sure there is some small effect as compared to an undisturbed flow, particularly when the mean-flow distortion becomes large and the vortices become asymmetric, but even then, I'd imagine the periodic nature of the disturbance would tend to cause the local effect on viscous drag to cancel out over the span. Either way, when testing an swept wing in a wind tunnel or in flight, there is no way to avoid the development of crossflow waves, so any minor effect they have would already be included in those measurements. The only tractable way you could try to tackle it would be using simulations where you can essentially "turn off" the small initial disturbances required to initiate the instability, but those are notoriously poor at predicting viscous drag in the first place, particularly in situations where laminar-turbulent transition is involved. You could probably quantify this effect if you ran a pair of full direct numerical simulations (DNS) on a wing: one with a perfectly uniform free stream and one with small fluctuations that are more representative of reality, but this is well outside the capabilities of modern supercomputing.

 

[aerohead_18]

That makes sense, thanks for the explanations. I'll have to read more into this. Sorry to hijack the thread with the crossflow thing @MaxKang