Hello,

I was wondering if someone could tell me which equations are solved in a compressible flow. If for example I solve the standard incompressible momentum equations and using the SIMPLE algorithm to solve the pressure correction equation, what equations would I solve in a compressible flow? Would I have a transport equation for some other variable? How would I link the density to the pressure?

Thanks.

It may be wiser to get this sort of fundamental knowledge from a book. Compressible flow for a single material requires the determination of the fluid velocity and two independent thermodynamic variables. These are determined by solving equations for the conservation of mass, momentum and energy. There are a range of choices for the pair of variables of state (e.g. pressure, density, energy, entropy, enthalpy,...). The material will require an equation of state (e.g.perfect gas) to describe its thermodynamic behaviour.

Incompressible flow involves the additional assumption that there is a constant background thermodynamic pressure. It is also common to make the further assumption of isentropy and drop the conservation of energy equation.

Thank you. If I could construct an Enthalpy equation or a Temperature equation, how do I specify boundary conditions for the Enthalpy equation? If I have combustion in a premixed environment and assume adiabatic walls, how can I use a transport equation for the Enthalpy given that to the best of my knowledge, the Enthalpy in a premixed combustion system can be considered adiabatic? Are there different forms of Enthalpy that could resolve my misunderstanding?

See my post below "BC for compressible flow"

"Incompressible flow involves the additional assumption that there is a constant background thermodynamic pressure. It is also common to make the further assumption of isentropy and drop the conservation of energy equation."

The essential distinction between compressible and incompressible flows has to do with time scales. Blow into a whistle. There is a convective velocity of maybe a meter per second and sound at maybe 340 m/s. The Navier-Stokes equations support two widely disparate time scales; an acoustic and a convective. The question then becomes how does one seperate the evolution of the fast scale from the slow scale. By expanding each variable in a power series, one may project the equations onto a "slow manifold" where there is little influence of the fast time scale. One now ends up with a relation which in NOT time dependent. For isothermal, zero Mach number flow, the constraint is DIV(U)=0. Once you allow the temperature to vary, the constraint changes. If you solve the wrong constraint then you are not adhering to the "slow manifold."

Isentropic flow is a flow where the thermodynamic forces effectively vanish. This means these scalars (simple gases) are negligible;

1) (rate-of-deformation)rate-of-deformation)

2) (temperature gradient).(temperature gradient)

In gas mixtures, the diffusion driving force vector would also be negligible.

The question of how the compressible equations are solved cannot be answered within this venue. John Anderson has written many introductory books that might be much more useful than anything anyone here could write in a few paragraphs.

Is you interest combustion at low Mach numbers?. Do you want low Mach number variable density incompressible flow or truly compressible flow? The latter is considerably more difficult at low Mach numbers but required if non-linear acoustic phenomena are an important part of your study.

Assuming your primary interest is combustion at low Mach numbers without acoustics then sticking with incompressible flow will make life easier. You will need to understand how the fluids thermodynamic state is determined and this is almost always done by solving additional transport equations. This might involve an equation based on the conservation of energy or it might only involve assumptions based around the transport of fuel, air and turbulence. There are a range of assumptions that can be made and I would recommend reading a book on combustion to familiarise yourself. In the books section I would suggest Poinsot and Veynante as a good place to start.

To answer you questions. I would endorse Guillaume's suggestions if you really want compressible flow. To solve an energy transport equation and then assume adiabatic walls would be strange. One usually solves an energy transport equation in order to avoid having to make such assumptions. I suspect the resolution of your misunderstanding lies in understanding how the combustion modelling assumptions relate to the fluids thermodynamic state. I would expect the imposition of reasonable boundary conditions in your case to follow from this.

I am not sure if the inclusion of my text in you reply means agreement or disagreement because I do not wholly follow your answer.

You are quite correct to point out that the constant background pressure is only constant in space and not time. This is particularly relevant if the original poster is interested in piston engine flows.

My second point was meant to indicate why the energy equation has disappeared in many incompressible codes and to indicate this does not follow from the assumption of incompressibility.

My point was to clarify the words "pressure" and "isentropy." When you speak of the pressure, it is in the context of a pressure poisson equation. The pressure poisson equation is not the true incompressibility constraint but rather the result of a differentiation of the true constraint. The semi-discretized incompressible Navier-Stokes equations are an index-2 DAE system with a constraint manifold defined by DIV(U)=0. So, you have

dU/dt = F(U,p)

0 = G(U)

Take the derivative of G(U)

dG/dt = (dG/dU)*dU/dt + (dG/dp)*dp/dt

or, since dG/dt = dG/dp = 0,

0 = (dG/dU)*F

OK, DIV(U) = G(U) but the pressure poisson equation equals (dG/dU)*F. By solving the pressure poisson equation, you are enforcing the derivative of the constraint rather than the constraint. The constraint has a physical meaning, that is a flowfield where the fast scale has been supressed. The issue of the proper value of G(U) or (dG/dU)*F becomes more important when one goes beyond strictly incompressible flow and tries to suppress the fast scale while large temperature and density variations exist. By the way, my post never said the word pressure. When you speak of background pressure, what you are implicitly talking about is how one enforces the constraint 0 = (dG/dU)*F while trying to mimic solving 0 = G(U).

The constant background pressure is the thermodynamic pressure used with one other variable to determine the thermodynamic state of an incompressible fluid. The thermodynamic pressure is constant in space and hence no acoustic waves can exist.

The small "pressure" component which occurs in the momentum equation (or a Poisson equation if one takes the divergence of the momentum equation) for an incompressible fluid varies in space and is not the thermodynamic pressure. The role of this quantity is to balance the books and enforce continuity. It is not the quantity I referred to earlier in the thread as thermodynamic pressure. Of course, if it was then an incompressible fluid would be a compressible fluid.

Again, I do not wholly understand your posting (I have no idea what an index-2 DAE system is) and I also cannot see the definition of incompressibility.

Incompressibility is defined as DIV(U) = 0 for an isothermal flow of a simple gas at zero Mach number. It corresponds to projecting the compressible equations onto a "slow manifold" where all fast time scales have been supressed. My point in all of this is to, hopefully, make practitioners more aware of what is going on at a fundamental level. Since the semi-discretized equations are a bunch of ODEs plus a time independent equation, it is called a differential algebraic system. The constraint equation, DIV(U)=0, does not contain the algebraic variable (the pressure) and is hence an index-2 constraint. When you differentiate it to get the Poisson equation, it then becomes (dG/dU)*F(U,p) so that the constraint now has an explicit pressure dependence. It is now index-1. The idea that you are dancing around arises because enforcing a constraint is different from enforcing the derivative of the constraint. What happens when you solve the index-reduced constraint is that the solution will walk away from the original mainfold as time progresses. While this may be a conquerable problem in strictly incompressible flow, once you allow variable density, low-Mach number flows, things get ugly. You must know the revised form of the index-2 constraint if you are to have any hope of suppressing the fast time scale. Further, if you were to take that constraint and differentiate it, how do you keep the solution on both the index-1 and index-2 manifolds.

By moving to incompressible-like equations, one exchanges the fast acoustic scale of the compressible equations for an algebraic constraint such as DIV(U)=0. I think there are many cases where a good stiff integrator stepping over the acoustic scales (with an error controller) is a cheaper solution than projecting out the acoustic time scale. More poeple need to think about this. The value of the low-Mach number equations on tight grids is debatable.

By the way, acoustics do not exist because DIV(U) = 0 has been enforced. This has the effect of supressing the fast "acoustic" time scale.

I am not sure I am dancing and I do not understand your definition of incompressibility. What does "at zero Mach number" do to the governing equations if we discard the trivial case that the velocity is zero? Later in the paragraph you seem to state strict incompressiblility requires constant density? Would typical combustion codes with strong density gradients but no mechanism for acoustic waves be incompressible or something else?

Strict incompressibility in gas mixtures means isothermal and, I suspect, a constant gas-constant ( implies constant mean molecular mass ). Once you permit these things to vary, you have to look at the equation of state and ask how perturbations in the respective variables are related. Ignoring mixtures, there are two low-Mach number possibilities depending on density, temperature, and pressure perturbations. One is acoustic and the other is non-acoustic. If you're really curious, these papers discuss the topic in detail.

http://scitation.aip.org/getabs/serv...cvips&gifs=Yes

http://scitation.aip.org/getabs/serv...cvips&gifs=Yes

Zero Mach number refers to an asymptotic expansion like

u = u0 + e*u1 + e^2*u2 + ... e^i*ui

where e = gamma*Ma^2. In the limit of e -> 0, u = u0, the incompressible velocity.

Combustion codes would be "low-Mach number" codes that attempt to project the compressible equations onto a "slow manifold" in the presence of composition and temperature variations. For them, the questions are what is the appropriate equation for the slow manifold (the one where all fast scales are suppressed) and do they want to solve the constraint or the derivative of the constraint.

They look at the expansion

u = u0 + e*u1

You appear to be confusing asymptotic expansions of the compressible equations and their truncation and/or limit at zero Mach number with the generally accepted meaning of incompressibility in CFD. The assumption of incompressibility is to decompose pressure into a large spatially constant part and a small spatially varying deviation. Omitting this small spatially varying component when using the pressure as a variable of state is the assumption of incompressibility. Holding the thermodynamic state constant everywhere is a bigger assumption and inappropriate for combustion simulations for example.

Low Mach number asymptotic expansions such as the ones you discuss are certainly useful for deriving alternative approximations and/or guiding implementations of compressible flow for use at low Mach number. The only point I would add is that there are more than two alternative expansions since there are multiple physically significant time and space scales with which to scale.

What on Earth is a slow manifold?

In the simplest terms it is a "set" ( locally Euclidean n-dimensional smooth surface) for which the solutions of the, in this case, low Mach number Navier-Stokes equations are, in some sense, close. For the slow manifold to exist you need to remove all acoustic waves in the small Mach number expansion. This last step cannot be performed however because of the spontaneous emission of acoustic waves that are required in such an expansion; i.e. Lighthill radiation.

The term "slow" comes from the removal of the (fast) acoustic waves from the low Mach number expansion and the term manifold follows from the fact that the set that is being constructed is a smooth surface.

There are clear mathematical ways to derive the incompressible limit of the compressible NSEs. If you want to resort to hand waving derivations, what can I say. While your ultimate conclusions may be right, my point here is to mention where all of these popular notions are based. Kreiss explained how one projects onto a slow manifold. You must impose a boundedness on the time derivatives of the integration vector. Decomposing the pressure is one of the things that gets done after one is well into the analysis and has written the pressure in terms of an expansion. Read that paper by Zank and Mattheus or the one by Bruce Bayley et al. There is no confusion here because it is the expansions that ultimately tell you how to treat pressure.

You are missing the essential point with the low Mach number equations. It is no longer compressible flow. There is a constraint manifold to which the solution must adhere just like with incompressible flow. There are issues with this manifold and how one solves it. Yes, one can further bust up time and space into dual variables if on likes. Some authors keep short and long spatial scales and some authors keep fast and slow times. But that does not change the way perturbations to the equation of state are treated. In gas mixtures, there are likely to be more than two cases but inevitiable one is looking for the nonacoustic versions.

You and I continue to talk at completely different levels. If people are going to solve equations, I would contend that they should understand, in some detail, their derivation, their mathematical character, and the appropriate numerical methods needed to solve them. That is not directed at you but rather to the community.

This sequence of posts from me has been about mentioning things that I think many incompressible types are unaware. Toward that end, there isn't much else I can usefully say that Zank and Mattheus haven't already said. Nor is the topic of DAEs worth continuing. Index-2 numerics is very complicated.

In the context of the Navier-Stokes equations, the slow manifold is related to the removal of events happening associated with acoustics. Kreiss looked the problem of when a PDE supports different widely seperated time scale phenomena.

http://scholar.google.com/scholar?hl...e=off&q=kreiss+different+time+scales

From an earlier post, if you blow into a whistle, two things come out: a convective flow of about 1 m/s and sound at about 340 m/s. The idea now is to manipulate the PDE and find the evolution of the convective part with the acoustic part removed. From an aeroacoustic point of view, you might want to know the PDE that describes the evolution of the acoustics without the convection. When Kreiss talks about a slow manifold of the NSE, he means the evolution of the flow when the acoustic contribution has been removed. To project onto the slow manifold is to remove the acoustic contribution within the PDE. The essential feature of the incompressible and "low Mach number" equations is to remove the fast acoutic time scale by projecting the compressible equations onto the slow manifold. One may then ask what that manifold looks like. It is a time independent condition and when the velocities are small and the temperature is constant, it is DIV(U) =0. You can also ask how one finds the slow manifold. Part of doing this requires that the higher local time derivatives of the vector that you are integrating don't become infinite as the Mach number goes to zero. By doing this, the semi-discretized PDE now is a system of ODEs subject to a time independent constraint. This fundamentally changes the character of the problem and makes numerical solution much more difficult. ODEs subject to constraints are called DAEs (differential algebraic equations). The character of the constraint can have a dramatic influence on how hard it is to solve the problem and numerical methods achieve different order of accuracy depending on the "index" of the constraint. The index tells you how sensitive the solution is to perturbations (read numerical error) in the RHS of the equations and the constraint. An ODE is an index-0 DAE. DIV(U) = 0 is an index-2 constraint. To remove a bit of perturbation sensitivity, most people differentiate the index-2 constraint and solve an index-1 constraint on pressure. The problem here is "drift-off" from the true constraint manifold.

http://www.amazon.com/exec/obidos/tg...glance&s=books

I think you are misinterpreting the problem here. The existence or not of the slow manifold has little or nothing to do with the derivation of the incompressible Navier-Stokes equations. There are two ways to view the incompressible Navier-Stokes equations:

(1) as the lead order term in an asymptotic expansion in powers of the Mach number M ( and also log(M) !);i.e. it is the zero Mach number solution,

or

(2) solutions to the full equations for which the velocity field is solenoidal (=> conservation of volumes)

In either case the pressure, which is the sum of a thermodynamic and dynamic contribution, in the equations of motion is solely dynamic in origin and its only purpose is ensure that div(u) = 0.

The slow manifold is the extension of (1) to higher order in M under the further restriction that there are no acoustic waves present. Whether such an entity exists for the Navier-Stokes equations is debatable.

At this point, I am bewildered by the responses I have been getting here.

The slow manifold is absolutely fundamental to an informed derivation of the incompressible ( or low-Mach number) equation. It is the means by which the acoustic component is removed. That IS the incompressible NSEs. Where do you think the solenoidal idea has its basis? It comes SPECIFICALLY from the requirement that first, second, ..., and higher local time derivatives of the integration variables remain bounded as Ma -> 0. This procedure is exactly what is meant by projecting the equations onto the slow manifold. Does the slow manifold exist?? Yes it exists except maybe in some bizarre circumstances. The trick is to find out what it is for the flow you care about.

You guys might read the papers by Zank + ... (1991) and Bayley + ... (1992). Until you do, this conversation is going absolutely nowhere. If you really think the existence of the slow manifold is debatable then read some of Kreiss's papers on the subject.

I give up!!

Actually I think you should read more carefully what I've written - you should also note that there are a number of demonstrations of the non-existence of the slow manifold in fluid mechanics (specifically in the removal of gravity waves in the stratified flows where one of the demonstration relies upon the concept of Lighthill radiation from acoustics).

Also the idea behind the solenoidal condition arises from electromagnetism (hence the name) and has been around a lot longer than the slow manifold concept. Specifically the incompressible Navier-Stokes equations can be derived by assuming the flow is solenoidal and then deducing the the form of the equations - this step works irrespective of the existence of the slow manifold (as does the small Mach number asymptotics). Have a look at "Mathematical topics in fluid mechanics" by Pierre-Louis Lions.

The existence of the slow manifold, like the inertial manifold, has implications upon the global character and properties of the Navier-Stokes equations (existence and smoothness for all times) which still have not been proven.