Computational Fluid Dynamics: Dissipation Function Density Based Solver

[mertcan]

((((As a correction dissipation function in picture should have square of divergence of U))))

Hi, first of all I am aware that we have to discretize the non linear navier stokes equations to reach the almost exact solution, and pressure based or density based algorithms are deployed for that reason. But for the energy equation of navier stokes, dissipation function as in picture/attachment takes place and even if we apply finite difference or discretize the velocities we obtain square of those velocities and there is still a non linearity. But I know that pressure based or density based solvers should have linear algebraic matrix equations ( for instance pressure based algorithm uses segregated solution method which cares the linear algebraic system). MY QUESTİON is: How is the dissipation function treated especially for density based solver which include compressible flow ?? Dissipation function has non-linearity even if finite difference/discretization is applied so how do we handle this situation??

 

[NFuller]

MY QUESTİON is: How is the dissipation function treated especially for density based solver which include compressible flow ?? Dissipation function has non-linearity even if finite difference/discretization is applied so how do we handle this situation??

I don't think density based solvers are used to solve the Navier-Stokes equation because of the non-linearitites as you pointed out. For compressible flow, a common explicit method used is the MacCormack method and a common implicit method is the Beam-Warming scheme.

 

[mertcan]

I don't think density based solvers are used to solve the Navier-Stokes equation because of the non-linearitites as you pointed out. For compressible flow, a common explicit method used is the MacCormack method and a common implicit method is the Beam-Warming scheme.

thanks for return, but if we have steady state compressible viscous flow what kind of method is required? By the way, MacCormack or Beam Warming depends on time we can not use them I think

 

[NFuller]

thanks for return, but if we have steady state compressible viscous flow what kind of method is required? By the way, MacCormack or Beam Warming depends on time we can not use them I think

Those methods will still work, they will produce a solution which eventually converges to the steady state value. If the problem is at steady state then that must mean that you expect a laminar flow. What is the Reynolds number for the system? Can you give more details on the problem you are attempting to solve? There may be simplifications to the governing equation depending on the parameters of the system.

 

[mertcan]

Those methods will still work, they will produce a solution which eventually converges to the steady state value. If the problem is at steady state then that must mean that you expect a laminar flow. What is the Reynolds number for the system? Can you give more details on the problem you are attempting to solve? There may be simplifications to the governing equation depending on the parameters of the system.

but as far as I see MacCormack and Beam Warming has time steps or try to find solution time by time which mean unsteady state situation, How those still work (without asymptotic time limit because of computational expense)??
(By the way I know that for LOW mach number compressible flow may be considered as incompressible flow and SIMPLE SIMPLER PISO SIMPLEC algorithm may be used for steady state compressible(just for low mach number) but what method is used for STEADY STATE HIGH mach number compressible flow FOR HIGH AND LOW REYNOLD NUMBER??)

Also as far as I have searched in this link page 25 https://brage.bibsys.no/xmlui/bitstream/handle/11250/234395/441754_FULLTEXT01.pdf?sequence=1 there is compressible simple method which has segregated iteration but still some non linearities exist for instance non linear terms originated from dissipation function in equation 13's source term must be exist in page 19 . How do we explain this situation?

 

[mertcan]

Pardon, I would like to ask why I can not receive enough responses to my question. I consider there are many qualitative brains in this forum, why don't you help me about my question? Am I not explicit enough?? Thanks to only NFuller, I got some kind of clue.

By the way, If there is someone who cares my question he/she can deeply look at my post 5 and 1, thus I believe I am going to be understood well...

 

[mertcan]

Let me simplify my question a little bit : let's say that I have INCOMPRESSIBLE VISCOUS fluid and my reynolds number is LOW which means NO turbulence exists and NO time dependent situation takes places. I am aware that SIMPLE algorithm found by Patankar can be used and it has segregated iterative solution which includes linear matrix equations in a specific sequence. BUT due to viscous, in the source term of energy equation we have DISSIPATION term which is non-linear. Therefore my question is HOW DO WE USE SIMPLE algorithm for the equations/cases that have non linear source term like dissipation function? OR what should we do IN THAT SITUATION??

 

[bigfooted]

If you have the book of Patankar (numerical heat transfer and fluid flow), you can check the sections on source term linearization, which is already mentioned in the chapter that treats heat conduction. A (more) common approach nowadays is to solve it as an unsteady problem using a pseudo-timestepping method, usually together with a multigrid method. There is no accuracy in time, the timestepping is just used to converge quickly to the steady state solution using large timesteps.

Density based solvers are usually used for compressible flow only because of the slow convergence at low mach number (even with preconditioning or artificial compressibility methods).

 

[mertcan]

@bigfooted thank you for your return. I would like to ask some questions again if you don't mind, initially I searched the book chung, anderson... I did my best but not get satisfied, so my questions are :

1) In SUPERSONIC COMPRESSIBLE VISCOUS LAMINAR FLOW what kind of ALGORITHMS/METHODS SHOULD BE EMPLOYED??

2) In SUPERSONIC COMPRESSIBLE VISCOUS TURBULENCE FLOW what kind of ALGORITHMS/METHODS SHOULD BE EMPLOYED??

3) I think we can not use SIMPLE LIKE algorithms for previous questions because SIMPLE like algorithms have fixed point iterations and should reach unique point and all iterations have linear matrix equations as shown in my picture/attachement. For instance if source term of equation 3 in picture had non linear known velocity (like u^4 or u^5), then even if we know velocity from equation 2 or have the linearised form of those known velocities( like in Patankar ) we would still have power of velocities (u^4 u^5) and we would not have unique solution since all equations are not in linear matrix form. And real solution of fluid flow can not have different 2 solution at same time . Am I right??

4) Also in post 2 Nfuller does not agree that density based solvers are used. HE suggest Beam Warming or MacCormack like schemes. What do you say about that?

 

[bigfooted]

There are many methods to solve the compressible Navier Stokes equations. They all have advantages and disadvantages. The methods were created because they were either simple, accurate or could deal with special situations like show wave capturing, transonic flow,... If you are interested in shock-waves, look into (approximate) Riemann solvers. The book of Anderson, computational fluid dynamics also gives a nice introduction with some detailed explanation with which you can program your own solver.
The same holds for turbulence models: dozens of models exist and you have to have to consider the application when choosing the turbulence model. For aerodynamics problems the Spalart-Allmaras model is very popular, but it is never used for internal flow problems (k-epsilon/k-omega models are used instead).

SIMPLE methods cannot be used for compressible flow applications because they are used as a closure for the pressure in the incompressible form of the Navier Stokes equations. The continuity equation is used in the SIMPLE method.

Maybe @NFuller was thinking about something else? The Beam and Warming method as well as the MacCormack scheme can be used in a density based approach. MacCormack is probably one of the first you will program in a CFD course.

I suggest you pick a recent book on cfd and study it thoroughly if you want to master the topic. The book of Anderson is really good and a bit more compressible-oriented.