# significance of energy equation and its conservation

I am interested to know the significance of the energy conservation laws when modelling fluids (or other materials). Am I correct in saying that if energy is conserved then stability is achieved. In a way, I need to know why it is important to show that energy is conserved when modelling different phenomena (which can be described by using mass and momentum equation, without the need for the energy equation) If you know any reading materials please let me know.

## 2 Answers

I am interested to know the significance of the energy conservation laws when modelling fluids (or other materials).

Energy conservation is usually explicitly modeled with a differential equation when modeling temperature or internal energy matters; the main examples I can think of are compressible flow applications and flows with thermic chemical reactions (like combustion).

Am I correct in saying that if energy is conserved then stability is achieved.

I assume you mean numerical stability? This property typically refers to numerical methods; if a numerical method is convergent, then it is stable. So, assuming you are using convergent numerical methods for your problem, and you assess convergence via calculating the energy, then you could reasonably deduce that your method is stable for your problem if energy is conserved.

In a way, I need to know why it is important to show that energy is conserved when modelling different phenomena (which can be described by using mass and momentum equation, without the need for the energy equation).

If changes in fluid temperature or internal energy are negligible, then the only changes in energy occur due to changes in pressure, density, velocity, or height; so you're computing the energy via Bernoulli's principle. Computing energy in this fashion would help validate numerical results, although there are better methods available (the method of manufactured solutions, for instance).

• Thank you Geoff. That helps. I am particularly looking for some literature that emphasises why in addition to solving mass, momentum (or other linear properties), energy conservation is of importance, and where perhaps new formulations has been introduced for energy conservation. – Hooman Nov 10 '15 at 11:29
• Geoff, you mentioned assessing convergence by calculating energy. Could you please explain that a bit more or provide a relatively simple reference I can read? – Hooman Nov 10 '15 at 23:12

Sometimes the energy equation drives the flow. Problems like natural convection are driven by temperature differences. The energy equation is frequently converted into a temperature advection-diffusion equation which can be coupled to the momentum equation through the Boussinesq approximation. If there is a heat source or variation in the temperature in the boundary conditions, then solving the energy equation too is the only way to incorporate that information into the overall problem. In compressible flow problems where shocks can create jumps in temperature, it is essential to include the energy equation to have a hope of getting the physics right. In these problems, all of the flow variables end up coupled through the conservation equations and the equation of state for the fluid being modeled.

• Thanks Bill, I guess my question is directed more towards cases other than the above (where energy equation closes the governing equations). Do you know of any literature for cases where the energy equation is not necessary to close the equations but used explicitly/implicitly to ensure energy conservation and thus numerical stability? I know that the kinetic energy is sometime solved to check stability, are there any other energy based methods for various cases? Also on the cases you mentioned, does the fact that energy equation is solved imply stable solutions? (I hope I'm making sense) – Hooman Nov 10 '15 at 17:14
• I don't know of such cases. I certainly have never done it. I also don't think this is a good test of stability, maybe accuracy of some sort, but not stability. What's your definition of stability here? – Bill Barth Nov 10 '15 at 18:07
• Thanks. sorry if I'm not not clear yet, they're based on what I've grasped so far. This may not be the same thing: when for instance with an incompressible isothermal flow, the kinetic energy equation (derived from the momentum equation) is solved. Although I'm not sure if its purpose is to demonstrate energy conservation and if it has any connection to stability. (By stability I mean numerical stability when convergence is obtained), I read somewhere that energy methods are used instead of using von-neumann analysis for nonlinear equations. I wonder if that's the same idea? thanks again. – Hooman Nov 10 '15 at 20:41
• No, I was clear on what you were saying, but I've never done or seen such such for checking stability. Any discretization issues surrounding approximating the continuity and momentum equations are likely to be similar to those one would have solving this KE equation and would likely infect it as well. Some people might compute (not solve) $\frac{1}{2}\rho\|v\|^2$ in an isothermal problem to make sure it stays constant, but that seems more like a check on numerical error not stability. – Bill Barth Nov 10 '15 at 21:30
• Hi Bill, I found something in the computational methods for fluid dynamics book that might be relevant to my question on page 160-161 link It would be good to get a second opinion on that in relation to my question. in particular 4th paragraph on page 161. – Hooman Nov 11 '15 at 17:54