Thermodynamic Entropy vs Numerical Entropy
 

Hi everyone,

I have just started to work on the Riemann problems in Euler equations. I am confronting a mind blowing question about the eigenvectors. Here is my question.

Assume that an airfoil, let's say NACA0012, is to be solved at subsonic regime in 2D. Governing equations are Euler Equations. Actually the airfoil is adiabatic and frictionless. The farfield flow conditions are set as initial conditions which means that free stream pressure, temperature and Mach are set everywhere in the domain at t=0.

I have some assumptions and results which cause confusion in my mind.

Assumption 1

When the solution converges to steady state, I would expect the thermodynamic entropy must be the same everywhere in the domain. I would also expect the pressure and temperature, so does the thermodynamic entropy, in the farfield must be the same between the converged state and initial state.

If it is true, the thermodynamic entropy must be kept constant throughout the timeline because the entropy cannot be decreased, it can only be kept constant or increased due to second law of thermodynamics.

Assumption 2

In Euler equations, entropy increase can only be possible in two ways.

1) Contact discontinuity to cause an increase in total enthalpy

2) Shock wave behaviour from genuinely nonlinear acoustic eigenmodes i.e. decrease in total pressure.

Assumption 3

Shock wave behavior of a genuinely nonlinear eigenvector requires a conflict of characteristics in similarity solution as it happens in Burgers equation. In the absence of conflict, the rarefaction wave is generated which is totally isentropic.

Here comes my question,

If Assumption 1 is true, the thermodynamic entropy must be the same everywhere in the domain at each time step because in the farfield the pressure and temperature are expected to be unchanged between initial and converged steady state solution. It can only be possible in two ways.

1) Entropy is increased and then decreased but it violates the second law of thermodynamics.

2) Entropy is constant everywhere and every time. It is not a violation.

If Assumption 1 and 2 are true, there should not be any shockwave or contact discontinuity because once those waves occur, thermodynamic entropy increases and it cannot be decreased back due to the second law.

According to Assumption 3, there should not be any conflict in eigenmode u+a or u-a in the trailing edge of airfoil. However, the eigenmodes u-a and u+a might cause a conflict when the initial state in the vicinity of leading edge has freestream velocity. This is exactly where my confusion starts.

For that reason, I wanted to separate entropy as thermodynamic and numerical entropy. I know that the thermodynamic entropy should be constant everywhere and everytime in this problem but I am not sure if it must be true for the numerical entropy.

It is not hard to calculate entropy of air at given temperature and pressure and I have some questions about it.

1) Would I see a variation in entropy if I solved this problem numerically?

2) What exactly happens in this problem in physical manner? Let's assume a bullet gains a subsonic velocity in a couple of microseconds. It would be exactly imposition of farfield boundary condition in the vicinity of bullet.

3) If I have some mistake, I cannot find it. Could you help me out?

Thank you for your patience and time!

what literature have read about the numerical entropy?


As a more general comment for general unsteady Euler equations:


1) depending on the geometry and BCs also subsonic inlet can produce singularity due to the coalescence of characteristics of the same family.
2) if you have isoentropic and unsteady flows, the entropy is constant only along the flow trajectories. If the inlet conditions are homogeneous the flow is homoentropic, at least until a singularity is generated.
3) the goal is to have numerical entropy that fulfills the real thermodynamics conditions. In this sense, you need to assess that the production of numerical entropy is never negative.

Thank you for your valuable answer. Actually I am trying to understand the Riemann problem solution and types of eigenmodes in Euler equations.

I am following Toro's books Riemann Solvers and Numerical Methods for Fluid Dynamics. So far, I did not find anything about numerical entropy in that book but I just made some thoughts so that I can have some conclusions.

Just as in your third comment, the entropy that is calculated from numerical solution must follow the theoretical entropy. I thought that the entropy must be kept constant in a 2D subsonic problem of an airfoil with Euler equations. It can only be possible if all active, or nonzero amplitude, eigenmodes are isentropic which means zero amplitude contact discontinuity and rarefaction solution of acoustic eigenmodes everywhere in the domain for this specific problem.

My question is a bit different than that. Toro states that the type of nonlinear eigenmodes (rarefaction or shock) must be determined for exact Riemann solution. I do not understand its necessity because the type of eigenvectors are known and left and right states are known as well. Each jump must have a scalar multiple of corresponded eigenvector. I do not understand why I must check the type of nonlinear eigenmodes.

This confusion makes me think about this simple 2D subsonic Euler problem. Maybe I should have more thought on this problem but I cannot comprehend it in a subtle way.

Thank you very much for your time again!

I suggest have a look also to the Leveque textbook for FV in hyperbolic problems.
There is a lot of literature about numerical entropy.
Be aware, that speaking about exact Riemann is typical of 1d problem.
Are you talking about a multidimensional Riemann solver?

Yes I am talking about a multidimensional Riemann solver. Actually I could not see any note about it but I am gonna check it out again. I have Leveque textbook as well but I have still in some confusion.

A multidimensional Riemann solver has indeed more theoretical issues. I personally don't work using Euler equations but you can find a lot of literature.

But I think that your numerical entropy analysis makes sense only for correct shock capturing. In case of regular solutions that is less relevant.