[ugurtan666]

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!