I am new at CFD and I am confused by the difference between compressible and incompressible flow. As I see it, an incompressible flow can have variable density, which seems strange to me. Can some explain this difference to me?

If the flow is compressible, the density is a non-constant function of the pressure, the temperature, phase, composition, etc. When a fluid particle of some mass dm interacts with neighboring fluid particles via pressure forces, heat exchange, chemical reaction, etc., it undergoes compression or dilatation. That is, its specific volume (and thus density, which is the inverse of the sp vol) changes according to the changing pressure, tempreature, etc., that it encounters. Note that dm is the product of the density and the volume of the particle, and that the mass dm is a constant in Newtonian physics, barring nuclear reactions.

If the bulk compressibility of the fluid (the ratio of the change in specific volume to the change in pressure that causes it; this is a material property of the fluid) is small by comparison with the pressure variations encountered in the flow under consideration, then these pressure variations will cause only small changes in the density. Similarly, if the volume coefficient of thermal expansion is small relative to the temperature variations encountered by a particle, then the temperature variations will cause only small changes in the density. An incompressible flow is an idealized case, wherein the variations of pressure, temperature, etc., encountered by a particle cause zero change in the density of the particle. There is really no such thing as a truly incompressible fluid. In an incompressible flow, there is no change (with respect to time) in the specific volume (and thus density) of each particle of fluid. Thus, the density is constant along any given particle pathline (a pathline being the curve traced out over time by a given particle). However, in general, the density of different particles may be different from each other, due to the particles being at differing temperatures, of different composition, or at differing pressures (less likely), etc. Thus, the flow can be incompressible (the density is a constant along a pathline) and yet display variations in density (the density varies along curves that cross pathlines). One encounters this in stratified flows, for example. As a particular case, if all the fluid particles in an incompressible flow have the same density, then the density is constant everywhere in the flow domain and at all times, and the flow is said to be homochoric, I think. This is the case typically described in introductory textbooks of fluid mechanics.

That's an excellent description of the flow physics. I normally consider a case to be compressible when I have a Mach number greater than 0.3 or so. Ideally, no flow is incompressible since you would never have a constant atmospheric pressure with the constant atmospheric temperature. But, we tend to assume it to be incompressible for smaller velocities (less than 80-100 m/s) to make our flow physics simpler.

Thank you both for your comments. I am still confused by the concept of Favre (or density weighted) averaging and its application in variable density flows. Also, if my code allows density changes on a path line, but uses Favre averaging, is it an incompressible code?

First of all, to add to what I posted before, Anshul makes an excellent point about the role of the Mach number in indicating the degree of compressibility. The flow equations of an inviscid, non-conducting fluid can be formally expanded about the incompressible version in powers of the Mach number. This is a regular perturbation expansion, with the incompressible limit being achieved as the Mach number becomes zero. The bulk compressibility that I mentioned previously comes into play in determining the speed of sound, which is the reference speed in the Mach number.

Next, to answer your question about Favre averaging, the answer is no, your Favre-averaged code is not an incompressible code. The Favre-averaging effectively switches to a different representation of the flow variables than the standard Reynolds-averaged one. The switch is done in such a way that the resulting averaged equations formally (symbolically) depart less from the steady laminar compressible flow equations than do the standard Reynolds-averaged compressible flow equations. The Favre-averaged equations still involve the Reynolds-averaged density, and this is not in general a constant, as it would be for incompressible flow. It varies even along a pathline. Also, the turbulent density fluctuations are in general nonzero too, and they are accounted for in the Reynolds stress tensor, etc. Thus, the solution to the Favre-averaged equations can display compressible behavior and is not restricted to incompressible flow. See the turbulence modeling book by Wilcox, for example.

As I understand it, incompressibility is when the density is not a function of pressure. In a combusting flow it is common to have large changes in the flow field due to heat release (a drop in density and a correspoding increase in velocity). However, the large d_rho/dt term does not mean the flow is compressible unless density is a function of pressure. Is this correct? In other words, the density is varying due to combustion, but pressure does not play a part in it. The flow is therefore compressible in the sense that a drho_dt exists, but incompressible in that density is not a function of pressure. Can anyone confirm this for me?

A few authors may use the term "incompressible" to refer to the situation in which the density is not changed by changes in the pressure. This reflects the etymology of the term.

However, most writers in fluid mechanics use the term "incompressible" to refer to the density being constant along a pathline. Notice that the same incongruence in usage occurs in thermodynamics, where we can speak of a gas in a piston-cylinder combination undergoing compression or expansion at constant pressure. The change in density and volume is accompanied by temperature changes. Perhaps a better term would be constant-density flow, or nondilatational flow, but at this point we are stuck with historic usage. Thus, the combusting flow you mention should be referred to as a compressible flow.

A careful definition of incompressible flow can be found on pp. 17-26 of the excellent book "Principles of Ideal-Fluid Aerodynamics" by Krishnamurty Karamcheti.

Thank you. If the combusting flow I refer to is referred to as a compressible flow, then should the techniques used be those used for compressible flows? Since rho is not a function of pressure should it not be incompressible?

Thank you for the book reference. Does this book also give a careful definition of compressible flow?

The Karamcheti book, which is all about incompressible inviscid flow, begins with an introductory chapter which talks about fluid and flow properties. Therein, the author carefully distinguishes incompressible flow from compressible flow. He uses dimensional analysis to come up with five dimensionless parameters that are relevant to deciding whether a general flow can be considered to be incompressible or not, and whether it can be considered inviscid or not.

As I explained in my previous post, your combusting flow can be termed compressible (i.e., dilatational), even though the density of that fluid does not vary due to pressure. Because there are significant density variations (happens to be due to chemical heat release), the flow is termed compressible. If the reason that the density is not influenced by pressure variations is that there are no significant pressure variations in the flow, then the flow would be termed a compressible constant-pressure flow. If instead, the reason is the nature of the equation of state of the fluid, then I do not know what the appropriate description would be, although the flow still falls under the compressible category. Perhaps a compressible liquid flow?

Both of these descriptions sound odd, because we are used to thinking of the term "compressible" as having to do solely with the bulk compressibility of the fluid, i.e., the ability of a change in pressure to produce a change in density. However, from the mathematical point of view, the simplifications to the equations of motion gained by the assumption of constant density are much more significant than those gained by merely assuming that the density is not a function of pressure. Therefore, fluid dynamicists have traditionally used an adjective to label the constant density case. Unfortunately, that adjective happens to be "incompressible", rather than the better choices "constant-density" or "nondilatational". This leads to confusion of the sort that you are experiencing, but it is now too late to change this historical usage.

Thank you. In my experience, this "confusion of the sort that you (I) are experiencing" is quiet common.

You say above that: "If the reason that the density is not influenced by pressure variations is that there are no significant pressure variations in the flow, then the flow would be termed a compressible constant-pressure flow." This will be where delta_p = 0.0 (at least in a thermochemical sense). Therefore, temperature and density variations are allowable and the flow is considered compressible, but at constant pressure.

Thanks.

In its most general form "incompressibility" consists of decomposing pressure into two components: a thermodynamic component, pt, which is constant in space (but may vary in time) and a fluctuation, p. The assumption of incompressibility is introduced by using this constant background pressure pt as one of the variables of state.

Assuming our fluid is governed by a perfect gas law, if pt is held constant then an increase in temperature will reduce density and vice-versa. There is no assumption about constant density only constant background thermodynamic pressure.

d_rho can be defined as the product of the rho, d_p, compressibility factor. compressibility factor can be related to bulk modulus. now if d_rho can be greater or lesser depends upon the d_p, compressibility factor. suppose that for liquids compressibility factor is very less, hence d_rho caused by larger pressure difference also very less resulting in treating the flow to be incompressible. now for the gases the compressibility factor is large and hence small pressure changes with high velocities or large pressure ratios with lesser velocities can cause significant changes in the density with in the flow field making the flow to treat as the compressible flow.

now here relavance of Mach number comes into the physcis of the flow. any flow consist of kinetic energy (velocity head) and internal energy. the ratio of kinetic energy to internal energy is a function of square of Mach number if we treat gas as a perfect gas.see if the Mach number is less means the internal energy is less compared to kinetic energy which means that if u brought the flow to rest adiabatically, the difference between static temperature and stagnation temperature are less and thus allowing to decouple energy equation from momentuam equation. if the Mach number is greater means reverse is appicable.

pleaase remember that d_rho has to be caused by pressure difference. if the flow is not caused by pressure difference such as in the case of Natural convection flow can not be treated as the compressible one.in case of natural convection, pressure is constant (atmaspheric pressure) through out the flow field. so d_rho is Zero. so it is perfect incompressible flow.but still energy equation is coupled to momentuam equation due to density gradients.

regarding combustion i too donot know whether it has to treated as incompressible or compressible as flow here is not induced by pressure difference. but for analysis we treat combustion as mass addition which is source term in governing equation. for this type of flows governing equations has to be solved in conservation form which is usual practice in compressible flow. if in combustion you are aiming at capturing at detonation waves u should use numerical techniques which are used for compressible flows.

[borhan_sd@yahoo.com]

Hi all,in my problem the air velocity is 2m/s in a very short pipe and there is about 4.5 mm HG pressure drop in the pipe,I want to determine the flow type,but I understood that the division of <2300 - between 2300 and 4000 and more than 4000 about Re. number is of incompressible flow,

Shall I assume thet the air is incompressible?
And then how should I determine the flow type considering compressiblity or incompressibity of the air in this problem?

[FMDenaro]

Originally Posted by borhan_sd@yahoo.com

Hi all,in my problem the air velocity is 2m/s in a very short pipe and there is about 4.5 mm HG pressure drop in the pipe,I want to determine the flow type,but I understood that the division of <2300 - between 2300 and 4000 and more than 4000 about Re. number is of incompressible flow,

Shall I assume thet the air is incompressible?
And then how should I determine the flow type considering compressiblity or incompressibity of the air in this problem?

in standard condition, you have the sound velocity of 340 m/s therefore your flow is in incompressible regime.

[Rami]

It seems you confuse two different properties:

1. Flow is incompressible if (as Denaro said) the Mach number is small (usually Ma<0.3 is still considered incompressible).
2. Flow is laminar, transitional, turbulent or inviscid according to the Reynolds number. Roughly speaking n pipe flow, fro Re<2300 the flowis laminar, from about 4000 - it is turbulent (transitional in between), and as Re approached infinity, the viscous terms approach 0 so the flow becomes inviscid.

[borhan_sd@yahoo.com]

Thanks for your replies ,I mean that the division you mentioned in your second part about types of regime (less than 2300 =laminar and between 2300 and 4000= transient and more than 4000=turbulent) is allocated to incompressible flows and the division of the types of flow and Re numbers is different for compressible flows,so for determining regime types we should firstly determine that our flow is incompressible or not.

My teacher told me that because of gas flow this flow is compressible and only liqiuds are incompressible.
Considrering your opinion it seems that her idea is not right,and compressibility depends on Ma. number not on gas or liquid nature,Is it right?

[sbaffini]

As a matter of fact, all the substances are compressible to some degree, even concrete.

In practice, in fluid dynamics, compressibility is related to the capability of pressure forces to induce density variations. This can be related to the Mach number (M) of the flow. A low M means that even a strongly compressible fluid like air is actually experiencing only negligible compression and can be consedere incompressible.

The fact that liquids, like water, are considered incompressible is coming from the fact that "In fresh water, sound travels at about 1497 m/s at 25 °C." (cit. Wikipedia). So an object traveling at 340 m/s in water would still cause negligible (to some degree) compressibility effects in water.

On the other side, turbulent-laminar flow states are determined, to a large extent, by the Re number only if M is negligible.

[Martin Hegedus]

Low Mach number flows can be assumed "incompressible" because changes in density have a negligible affect on momentum and conservation of mass.

In other words:
d(rho*V)=rho*dV + V*drho. If both V and drho are small then that term can be neglected and thus,

d(rho*V) = rho*dV.

However, the pressure and density changes are of the same order.

For example, stagnate flow isentropically from M=0.001 to 0.0 and the following pressure and density ratios result.
p2/p1 = 1.000000700
d2/d1 = 1.000000500

[Aeronautics El. K.]

Mohamed Gad-El-Hak in the book "Flow control; passive, active and reactive flow management" says (I quote):
"In a flow dominated by viscous effects - such as that inside of a microduct - density changes may be significant even in the limit of zero Mach number".

I'm not arguing... I'm just stating that for the sake of completeness.

[mercy.gh]

Hello,
In addition to the interesting conversation;
Measuring the Mach number also demands attention to the temperature of your environment, since the speed of sound would change correspondingly.
I am trying to simulate a combustion chamber with a high velocity fuel -oxygen injection.Really close to the injectors(before combustion happens) flow is definitely compressible (velocity of injection is 500 and sound almost 380) but in a small distance, when combustion happens Temperature will rise highly and the velocity of sound will increase to 800 which makes the Mach number much less!

Should I consider this compressible or in-compressible flow?