# Why transonic flow is highly nonlinear?

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 January 4, 2019, 00:28 Why transonic flow is highly nonlinear? #1 Senior Member   Join Date: Oct 2017 Location: United States Posts: 228 Blog Entries: 1 Rep Power: 8 Hi, According to J.D.Anderson, Fundamentals of aerodynamics (2017), it is the nonlinear velocity potential equation can be (and should be) applied to both transonic/hypersonic flows, whereas we can apply a simpler version -- linearized potential equation to supersonic flow. I can't wrap my mind around this. I can understand hypersonic flow having strong nonlinearity, but why supersonic doesn't? And wireder, why transonic is highly nonlinear? (and this makes me remember that I read somebody said that transonic flow is harder to solve than supersonic due to strong nonlinearity) I mean, based on the flow regime, supersonic "sits" between transonic and hypersonic. Intuitively, if transonic/hypersonic flow is highly nonlinear, supersonic should be as well. But the theory says the other way. So why? Can anyone explain this? Thanks. Last edited by TurbJet; January 4, 2019 at 16:22.

January 4, 2019, 01:26
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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=1 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.

Quote:
 Originally Posted by TurbJet 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?
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.....?

Last edited by LuckyTran; January 5, 2019 at 15:56.

 January 4, 2019, 04:19 #3 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,197 Rep Power: 66 The key is the linearization method. You introduce a small perturbation to linearize the equation but you need to assess that all disregarded terms are small compared to the basis function. In transonic flows you deduce that you cannot disregard the non linear terms. That has nothing to do with turbulence. The nature of the global unsteady viscous model is in the non linear NS equations system.

January 4, 2019, 16:49
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Quote:
 Originally Posted by LuckyTran 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.....?
Thanks for your reply. I was stuck at the nonlinearity in Navier-Stokes eqs, so for the nonlinearity I was thinking something like the nonlinearity in NS. 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?

As for the turbulence part, please forget it. I know that was stupid and I don't even know what was I thinking.

January 5, 2019, 15:56
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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....

Quote:
 Originally Posted by TurbJet 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?

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.

January 6, 2019, 18:26
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Quote:
 Originally Posted by LuckyTran 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.
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)

January 6, 2019, 18:35
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Quote:
 Originally Posted by TurbJet 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)

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.
https://ntrs.nasa.gov/archive/nasa/c...0020078395.pdf

January 7, 2019, 00:21
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Quote:
 Originally Posted by FMDenaro 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
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?

January 7, 2019, 03:45
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Quote:
 Originally Posted by TurbJet 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?

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.

January 7, 2019, 12:36
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 Originally Posted by TurbJet 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?
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.

January 7, 2019, 20:23
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 Originally Posted by FMDenaro 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.
So I am wondering how does this solution behave? Does it move? or just stay where it is?

January 7, 2019, 20:26
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 Originally Posted by LuckyTran 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.
For designing nozzles or the aerodynamic shape of rockets?

BTW, could you give some examples how this is used in boundary layer theory? I know linearization plays a crucial part in BLT, but I've never seen potential theory coming into play.

 January 8, 2019, 03:05 #13 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,197 Rep Power: 66 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

January 8, 2019, 15:11
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 Originally Posted by FMDenaro 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
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 for both x/y directions?

January 8, 2019, 15:19
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Quote:
 Originally Posted by TurbJet 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?

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.

January 8, 2019, 16:13
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 Originally Posted by LuckyTran 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.
I was talking about the potential equation for supersonic flow.