Hi all,

I am trying to figure out wether my CFD code (FLUENT) is telling me the truth - at least qualitatively. Since my question is not FLUENT-related, I decided to post it here in the main forum.

I am simulating 3-D pressure-driven, compressible, constant-cross section microchannel flow, with ideal gas assumption (continuity assumption still valid, Knudsen number is low enough). Since the pressure drops along the channel, so does density, and velocity increases in order to satisfy the continuity equation. As a consequence the temperature drops, and the speed of sound decreases. That way, provided that the pressure ratio (inlet/outlet) is suitable, the Mach number can exceed 1, i.e. I get supersonic flow, even without using a Laval nozzle.

Is that possible or is it complete junk?

Thank you very much in advance for your comments,

Ingo Meisel

Yes, it is possible.Check your book on thermodynamics and see how a heat pump (refrigerator) cycle works

If the flow enters subsonically, you cannot achieve supersonic flow without a laval nozzle.

Oh, I didn't read properly: you can reach Mach 1 without laval nozzle, but not above.

Are you sure that I can reach at most Mach 1? I mean, the pressure will continue to drop along the channel, even at the location where Mach 1 is reached. Therefore the density will decrease, too, enforcing a velocity increase (since mass flux is the same in any channel cross section) which *must* lead to supersonic flow. Or where's my error in reasoning?

Many thanks for your help,

Ingo Meisel

The pressure doesn't develop as you think, but I'm not able to explain it in detail now. Search for Fanno and Rayleigh flow.

Dirk

There are two kinds of problems, stationary and moving shock waves.

The stationary shock will require a nozzel. Otherwise, the downstream density cannot decrease indefinitely, or even fast enough.

The moving shock might be produced by restraining gas under high pressure by a diaphram, with the area beyond the diaphram at very low pressure. The diaphram bursts and the gas expands into the low pressure region. Obviously the density and pressure downstream beyond the shock does not change until the shock reaches it.

I would like to remind all who are contributing to this thread that this is a Micro Channel. I have contacted Dr. Ingo Meisel who kindly sent me the geometry and BC's. I repeated the analysis and all I can tell you guys is that the results will surprise a lot of us who are used to analyze missles and turbines. For obvious reasons I cannot disclose any of the results, but if you are really interested in micro flows (nano flows if you do not like the previous terminology) you better start by doing some sort of analysis. Enjoy the nano flows, I find it exciting and once again I thank Dr. Ingo Meisel for the opportunity he gave me.

If flow is subsonic at the inlet it has to stay bounded by sonic velocity for constant diameter channel (see any thermodynamics or gas dynamics textbook). However, when pressure decreases the density decreases as well. Thus, the molecular mean free path increases. Even if at the inlet Kn<<1, it may be that at the end of channel Kn>=1, so continuity assumption do not hold any more?

Angen

Good point, but before continuing this line of thoughts you are adviced to download a free tutorial on microchannel flows www.ecn.purdue.edu/microfluidics As I said in my previous posting, it is an exciting branch of Fluid Mechanics. Enjoy reading that tutorial, also you can google for microchannel flows

I did my PhD in the area of flows with Kn>=1 at the time when hte buzz word "nanoflows" has not been invented yet. Still, thanks for advice. I think some reading never hurts, so I will check it out if I have time.

Angen

I understand why you're suspicious. I would be, too.

Regardless if your assumptions hold for micro channel flow, the ideal gas continuity flow that you describe will not result in supersonic flow, if your inlet is subsonic. You will at most reach sonic flow, right at the channel exit. You could change the behavior by heating/cooling, but I am not sure that's what you're doing, at least you didn't mention it. What happens in the channel: starting from zero displacement thickness at the inlet, a boundary layer grows to reach its maximum displacement thickness at the exit. That's where your effective throat is located, and that's where you get Mach number 1, provided that the pressure gradient is strong enough. To go beyond Mach one, the flow would need to expand which can only be achieved by an increasing effective cross section (boundary layer miraculously vanishes) or by cooling. There is no mechanism to go beyond Mach 1, unless you have some numerical phenomenon (artificial/numerical dissipation? violation of continuity? bad boundary conditions? violation of the second law of thermodynamics?) that somehow kicks you over the edge. I would definitely run some tests like checking the conservation of mass, momentum, energy over the domain, by integrating the boundary fluxes. Don't trust your results at this point.

Dirk's comments are absolutely right. I suggest you take a good gas dynamics book and review Fanno and Rayleigh flow, because this is exactly what you're looking at. Again, how well your ideal gas assumption and continuity assumption are applicable to micro flow is a completely different issue. You're probably not really doing micro flow, here.

Hi all,

many thanks for your comments.

@Angen: I checked the Knudsen number, and for my assumed flow situation it is always around 10^(-4), even at the channel outlet. Therefore the continuity assumption is valid.

@Mani: No, I do not heat/cool the channel flow; that's why I didn't mention it. And secondly, you're right, this is not necessarily a micro flow situation. I just mentioned it to give a few details where my motivation came from; I have to admit that this background information is useless here.

I did a full 3-D simulation of such a channel flow with a CFD code, and it turned out that increasing the applied pressure drop beyond some threshold value (which depends on the channel geometry, the mean pressure level, the temperature level and the fluid properties) did not increase the resulting mass flux any further. Instead, the flow quantities started to oscillate slightly (fluctuation amplitudes in the percentage range) at the channel exit. And the exit Mach number, averaged across the channel exit, did not go beyond M=1.04 as expected by most of you.

But what I still do not understand is the following:

Fanno flow is a 1-D model derived from general/integral considerations about the irreversible (since friction is taken into account), compressible flow in a constant-cross section channel. Therefore this model cannot give any information about what's going within an arbitrary cross section of the channel, say the exit cross section! At the channel exit I do obtain a Mach number distribution with peak values well beyond M=1 in the bulk. In fact, this *must* occur since a MEAN value of M=1 implies that smaller as well as greater Mach number values must occur: smaller near the walls, and greater in the bulk.

Since Fanno flow is merely an *approximate* model, I am still not convinced that with a more realistic (CFD) model, M>1 cannot occur at least in portions of the channel cross section.

I'd be happy to get some enlightenment on this issue!

Thank you again for your replies,

yours sincerely,

Ingo Meisel

From a physical standpoint, assuming that the flow is properly modeled using the NS equations then there is no way to accelerate beyond M=1 in a duct without turning the streamlines in an expansion or adding energy. As the flow moves down a constant area duct it will accelerate due to boundary layer thickening. If the channel is long enough or the BL grows rapidly enough then the flow will hit M=1. But to get beyond that you will need to look for some mechanism to either add energy to the flow or cause the flow to expand (i.e. thin the BL).

Yes, Cooling is the key point, it naturally occurs because the flow is throttled, The Joul-Thomson(Kelvin) effect, and that effect depends on the inlet temperature, pressure and the final pressure. If these happen to be on a curve (Isenthalpic curve) having a maximum then this process takes place. In real industrial life, this is the process used in the liquefaction of gases and getting Oxygen, Nitrogen from Air. Now since the solution in this case is obtained by a numerical procedure, I guess we ought to establish the validity of the continuum hypothesis based on cell dimensions and not on the microchannel dimensions. In this case and contrary to CFD practice, one should decrease the number of cells in order to validate the results.

Now I'm interested too. Could you also send send me the geometry and BC or the casefile itself Dr. Meisel.

Thanks in advance