How to check a numerical simulation is energy conservative?

[hoomanya]

I am modelling a 1D fluid wave propagation problem and I needed to know how I can check that my results are energy conservative. please reply, urgent. thank you.

 

[phyzguy]

Well, in your 1D simulation, each cell has a fluid density, and a fluid velocity, right. So if the density of cell i is given by \rho_i, and the velocity is given by u_i, then if you take the total \frac{1}{2} \sum_i{ \rho_i u_i^2}, then this should be conserved. Of course, your simulation might have other things, like gravitation, in which case you would need to add in the gravitational potential energy as well.

 

[Chestermiller]

You are using a finite difference scheme, and you want to know if the finite difference scheme automatically conserves energy, correct?

 

[hoomanya]

if the density of cell i is given by \rho_i, and the velocity is given by u_i, then if you take the total \frac{1}{2} \sum_i{ \rho_i u_i^2}, then this should be conserved. Of course, your simulation might have other things, like gravitation, in which case you would need to add in the gravitational potential energy as well.

thank you. do I take the total over the cells, so say I have 10 cells, I add the kinetic energy term over for the 10 cells and show that they equal zero? what about time?

 

[hoomanya]

You are using a finite difference scheme, and you want to know if the finite difference scheme automatically conserves energy, correct?

thanks, I'm using finite volume. my problem is transient.

 

[Chestermiller]

thanks, I'm using finite volume. my problem is transient.

Does your finite difference scheme automatically conserve energy (i.e., is this a mathematical characteristic of your finite difference equations)?

Chet

 

[hoomanya]

Does your finite difference scheme automatically conserve energy (i.e., is this a mathematical characteristic of your finite difference equations)?

Chet

I'm not sure. What do you mean by automatically? You mean by formulation? so if I add the discretised terms, the terms at the internal boundaries cancel out and I'm left with the terms at the two left and right boundaries.

 

[Chestermiller]

I'm not sure. What do you mean by automatically? You mean by formulation? so if I add the discretised terms, the terms at the internal boundaries cancel out and I'm left with the terms at the two left and right boundaries.

The latter.

 

[phyzguy]

thank you. do I take the total over the cells, so say I have 10 cells, I add the kinetic energy term over for the 10 cells and show that they equal zero? what about time?

Yes, you add the kinetic energy for all of the cells. It won't necessarily equal zero, but it should be the same value at each time step, assuming no energy is flowing in or out of the boundaries. As chestermiller says, you could examine the equations that propagate the fluid quantities from the current time step to the next time step. These should automatically conserve the conserved quantities (mass, momentum, energy), because they should just move these quantities from one cell to an adjacent cell.

 

[hoomanya]

Yes, you add the kinetic energy for all of the cells. It won't necessarily equal zero, but it should be the same value at each time step, assuming no energy is flowing in or out of the boundaries. As chestermiller says, you could examine the equations that propagate the fluid quantities from the current time step to the next time step. These should automatically conserve the conserved quantities (mass, momentum, energy), because they should just move these quantities from one cell to an adjacent cell.

Thanks a lot. Does the type of boundary condition used have any influence in this?

 

[hoomanya]

Hi, so I've added my density, velocity, K.E (against space) and sum of K.E at nodes (agains time) plot, before the wave reaches the boundary so that the results are independent of the type of BC choses. The sum of KE's tend to increase at the starts as velocity is zero at the start, I wonder if internal energy is present? Am I missing something here? Thanks!

 

[phyzguy]

Right. I should have said this earlier. It is the total energy that is conserved, so you need to include the internal energy. The internal energy will be decreasing as your initial fluid pulse expands. Do you know how to calculate the internal energy? If so, you should add this to the KE and plot the total energy.

 

[hoomanya]

Right. I should have said this earlier. It is the total energy that is conserved, so you need to include the internal energy. The internal energy will be decreasing as your initial fluid pulse expands. Do you know how to calculate the internal energy? If so, you should add this to the KE and plot the total energy.

Hi, thanks again. Not quite sure, I think by integrating du = -p dv which I am guessing will give u - u_0 = -p (v-v_0), but not quite sure what the specific volume will be. I am guessing that's where density comes in.

 

[Chestermiller]

Is this a continuum model? If so, the differential energy balance equation should already involve the internal energy. Please write the differential energy balance equation you are using.

Chet

 

[hoomanya]

Is this a continuum model? If so, the differential energy balance equation should already involve the internal energy. Please write the differential energy balance equation you are using.

Chet

I'm not sure what you mean by a continuum model. I am not using a differential energy balance equation.

 

[phyzguy]

Hi, thanks again. Not quite sure, I think by integrating du = -p dv which I am guessing will give u - u_0 = -p (v-v_0), but not quite sure what the specific volume will be. I am guessing that's where density comes in.

What are the fluid variables that are carried at each point in the code? You should have access to either the internal energy, or to a temperature and/or pressure from which you can calculate the internal energy. Your code must have equations for calculating the fluid variables at time t+δt in terms of the variables at time t. Do you have access to these equations?

 

[hoomanya]

What are the fluid variables that are carried at each point in the code? You should have access to either the internal energy, or to a temperature and/or pressure from which you can calculate the internal energy. Your code must have equations for calculating the fluid variables at time t+δt in terms of the variables at time t. Do you have access to these equations?

I am solving the isothermal Euler equations and relating pressure to density using the bulk modulus of the material. I have an initial density (and hence pressure) wave and initial velocity is zero.

 

[phyzguy]

I am solving the isothermal Euler equations and relating pressure to density using the bulk modulus of the material. I have an initial density (and hence pressure) wave and initial velocity is zero.

Well, I'm not 100% certain, but I think what you are saying does not make physical sense. You start with a fluid with an initial density and pressure. The fluid expands and drives a wave in both directions. The kinetic energy of the waves must come from the internal energy of the fluid. So the fluid cannot be isothermal. The temperature has to decrease as the wave expands. At least, this is how it seems to me.

 

[hoomanya]

Well, I'm not 100% certain, but I think what you are saying does not make physical sense. You start with a fluid with an initial density and pressure. The fluid expands and drives a wave in both directions. The kinetic energy of the waves must come from the internal energy of the fluid. So the fluid cannot be isothermal. The temperature has to decrease as the wave expands. At least, this is how it seems to me.

Hi, so I had a think about what you said, and in the problem essentially, sound waves propagate as a result of interplay between density and pressure. Essentially, I think the rare-fractions and the compressions have enough time to exchange heat and maintain at a constant temperature depending on the speed of sound? (dp/drho = speed of sound squared). The fluid I've used is water. Please correct me if I'm wrong.

 

[phyzguy]

So if why you are saying is correct, where does the kinetic energy of the waves come from?

 

[hoomanya]

So if why you are saying is correct, where does the kinetic energy of the waves come from?

I was thinking of the fundamental equation of thermodynamics and how Tds = du - pdv
and so changes in the internal energy could be due to pdv or Tds independently. Can the changes in internal energy therefore no be just due the changes in pdv?

 

[phyzguy]

I just don't know the answer. Maybe you are right, but I'm not sure.

 

[hoomanya]

I just don't know the answer. Maybe you are right, but I'm not sure.

thanks very much for your help this far.