RANS equation for compressible transitional state flow

Antony Biswas · August 10, 2018, 02:44

Hi,

As my undergraduate thesis is completely based on critical Reynolds Number region, can anyone give me proper mathematical analogy about Reynolds Average Navier Stoke equation to not able to solve in critical Reynolds Number region for compressible fluid?

LuckyTran · August 10, 2018, 11:54

RANS is just the mean momentum equation. Where RANS truly fails is where the concept of statistical averaging fails, e.g. when the mean is not unique. I can't think of any examples. The only one that comes to mind is if the flow becomes unbounded in finite time, which is currently the Millenium problem that we know does not have a proof. I don't have any examples of bounded variables with non-unique means. Put another way, when can we say that the mean momentum does not exist?

There are two caveats to RANS that are not really the fault of RANS.

When you write down the RANS equations, Reynolds stresses appear. A proper description/prediction of the Reynolds stresses completely satisfies the problem. Question is whether you can accurately model these reynolds stresses or not.

But usually when we say RANS we actually mean RANS with the Boussinesq eddy viscosity hypothesis. The reason for doing so is to rewrite the Reynolds stresses in terms of the velocity gradient. Now the issue is shifted a bit. Assuming the Boussinesq hypothesis works now all we need is a good prediction of the eddy viscosity. Question now is whether you can make this prediction or not. Note the validity of the Boussinesq hypothesis doesn't mean it represents the correct physics or not, it just simply means that for any given Reynolds stress, you can write down an eddy viscosity that gives you that Reynolds stresses using some magical equation that we don't even know yet.

But if your turbulence model does not give you the right eddy viscosity or the right Reynolds stress, you get the wrong solution. This isn't a RANS problem but a turbulence modeling problem. The real question is whether your turbulence model is good enough or not?

August 10, 2018, 02:44	RANS equation for compressible transitional state flow	#1
Antony Biswas New Member Antony Bidhan Biswas Join Date: Jul 2018 Posts: 2 Rep Power: 0	Hi, As my undergraduate thesis is completely based on critical Reynolds Number region, can anyone give me proper mathematical analogy about Reynolds Average Navier Stoke equation to not able to solve in critical Reynolds Number region for compressible fluid?

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August 10, 2018, 11:54		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,674 Rep Power: 66	RANS is just the mean momentum equation. Where RANS truly fails is where the concept of statistical averaging fails, e.g. when the mean is not unique. I can't think of any examples. The only one that comes to mind is if the flow becomes unbounded in finite time, which is currently the Millenium problem that we know does not have a proof. I don't have any examples of bounded variables with non-unique means. Put another way, when can we say that the mean momentum does not exist? There are two caveats to RANS that are not really the fault of RANS. When you write down the RANS equations, Reynolds stresses appear. A proper description/prediction of the Reynolds stresses completely satisfies the problem. Question is whether you can accurately model these reynolds stresses or not. But usually when we say RANS we actually mean RANS with the Boussinesq eddy viscosity hypothesis. The reason for doing so is to rewrite the Reynolds stresses in terms of the velocity gradient. Now the issue is shifted a bit. Assuming the Boussinesq hypothesis works now all we need is a good prediction of the eddy viscosity. Question now is whether you can make this prediction or not. Note the validity of the Boussinesq hypothesis doesn't mean it represents the correct physics or not, it just simply means that for any given Reynolds stress, you can write down an eddy viscosity that gives you that Reynolds stresses using some magical equation that we don't even know yet. But if your turbulence model does not give you the right eddy viscosity or the right Reynolds stress, you get the wrong solution. This isn't a RANS problem but a turbulence modeling problem. The real question is whether your turbulence model is good enough or not?