Originally Posted by TurbJet (Post 720762)

One more: if transonic flow is nonlinear, and supersonic is linear, does it imply that transonic flow tends to be turbulent, and supersonic flow tends to be laminar?

Originally Posted by LuckyTran (Post 720766)

The general non-linear potential flow equation is valid for all regimes. Andersons' point is that when the flow is entirely subsonic (M<1)or entirely supersonic (M>1), the linearized perturbation equation can be used. If you have a transonic case (when M=0 somewhere) the linearized potential equation makes no sense. Just look at the equation! There's a (1-M^2) in it.

Hypersonic means high-subsonic where non-linearities in the equation of state take over and to use the linearized perturbation method no longer makes any sense. That part does not need any explanation. Hypersonic flows is more-or-less defined as flows where the linearized perturbation equation cannot be applied.



sub/super/trans/anything sonic has nothing to do with turbulence. How fast the flow is moving says nothing about any viscous effects. Especially in the context of potential flow, where the flow is assumed to be irrotational it's even more meaningless to talk about turbulence. You can have supersonic turbulent, subsonic tubulent, supersonic laminar, or subsonic laminar flow.


Also Anderson is referring to the perturbations in the potential equation when he refers to linear/non-linear. Your linear/non-linear refers to what.....?

Originally Posted by TurbJet (Post 720859)

Anyway, so I think, the nonlinearity of perturbation in transonic flow is due to this "in-between" Mach number, so the nonlinear part is comparable to the linear part, which makes the linearization invalid?

Originally Posted by LuckyTran (Post 720935)

Right. Linearity/non-linearity is a property so you need to be specific when you say what is linear. For example y = Ax^2 + B is linear in A and B but non-linear in x. So y can be linear or non-linear depending on what it is you are talking about....






It's not just a strange in-between Mach number. The perturbation parameter is 1-M. When M is 1, the first order perturbation is 0. You have to actually have to do the algebra to show that the attempt at linearization results in a different equation. Meaning, it's non-trivial to show that the linearization perturbation equation does not work, but the reason why it does not work should not be a surprise.

Originally Posted by TurbJet (Post 721030)

I am not familiar with aerodynamics. So, physically speaking, is there anything different between transonic and supersonic/subsonic flows ? (I mean other than those obvious things, like shock waves)

Originally Posted by FMDenaro (Post 721033)

The equation for the strady potential is elliptic for subsonic condition and hyperbolic for supersonic condition. They are both linear and cannot describe the non-linearity nature in the shock wave. In transonic condition you assume that elliptic, parabolic and hyperbolic features of propagation of waves cohexist.
Have a reading here
https://ntrs.nasa.gov/archive/nasa/c...0020078395.pdf

Originally Posted by TurbJet (Post 721074)

Um, that's very interesting. I read this part in Anderson's book about the potential equation will turn to hyperbolic for hypersonic.

But, it's really hard for me to wrap my mind around it. I mean, typically, hyperbolic equation has a variable in time, and so it's convective in nature. However, for this hyperbolic potential equation, both variables are only in space, no one is in time. So how does this equation behave? Is it still convective?

Finally, computing resources are very powerful today. Under many circumstances, we can directly solve NS (either by applying DNS/LES/RANS) for aerodynamic problems. So I am wondering is it still necessary/meaningful to study this potential theory/equation?

Originally Posted by TurbJet (Post 721074)

Finally, computing resources are very powerful today. Under many circumstances, we can directly solve NS (either by applying DNS/LES/RANS) for aerodynamic problems. So I am wondering is it still necessary/meaningful to study this potential theory/equation?

Originally Posted by FMDenaro (Post 721087)

No, the hyperbolic character appears also in steady flow, it is a mathematical characteristics of a PDE with a solution f(x1,x2, ..xn).


In general, I think that the study of potential flows in real applications is less relevant today but has still some meaning for educational purpose. However, is a very fast tool for preliminary results.

Originally Posted by LuckyTran (Post 721143)

I agree with this sentiment but the rocket folks still use it.... It's still good enough for rocket science. It will be around for probably another 25 years in this field alone. And these are the folks that have access to the largest supercomputing clusters in the world.

I wouldn't underestimate its educational purpose either. It is a critical ingredient in boundary layer theory.

Originally Posted by FMDenaro (Post 721207)

Steady model drives to steady solution, no moving in time.
BL theory requires the effects of the viscosity and is no longer in the framework of potential flows. Classically, the potential solution is used as external solution in the BL theory

Originally Posted by TurbJet (Post 721318)

So I am wondering what kind of numerical schemes can I use to solve this kind of steady hyperbolic equations? Can I do central in space?

Originally Posted by LuckyTran (Post 721320)

Are you talking about solving the potential equation or solving NS? For NS these flows are strongly upwind biased. 1st order upwind works really nicely! A supersonic flow doesn't care what happens downstream.