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Andrei Chernousov July 4, 1999 10:18

1D governing equations
Hi, anybody!

My question (may be of academic significance) concerns the formulation of conservation laws which must we use to more adequately describe unsteady flows in channels (wall-bounded flows!) as a 1D phenomena.

Comparing the computed and the experimental pressure curves in ducts with finite amplitude air waves I revealed that there is a small but quite pronounced overprediction of the compression vave velocity on its vay from the open end of the duct. The amplitude of the wave is affected by friction and ever heat transfer, but the quasy-stationary source terms allowed to predict the amplitude well. I use conservative numerical scheme which solves test problems well too, so I suspect that the usual 1D formulation is not so adequate for wall-bounded flows, because the information about the actual distributions of parameters is lost in implied averaging. The resulting model does not account properly for correlations between nonuniform flow parameters. I think some correlation coefficients/functions must be introduced in the very 1D formulation to fix this (remaining) inadequacy.

The question is: did anybody heard of such a corrections to the well known 1D governing equations?

John C. Chien July 4, 1999 12:26

Re: 1D governing equations
(1). I think the reason why there are 1-D, 2-D, and 3-D flow problems is because they are all different. (2). So, 1-D channel flow is different from 2-D channel flow, and 2-D channel flow is also different from 3-D channel flow. (3). I am sure that the result of flow in 3-D curved channel is not the same as that in a 2-D channel. (4). It is not easy to link 1-D results to 2-D results by a simple parameter. But, in a narrow range of Reynolds number and Mach number, you can try to find the correlation if the geometry is similar. (5). I am sure that there is a good reason why turbomachinery are designed by using 1-D code, 2-D code and 3-D code. (1-D code will have a lot of empirical data in it)

Andrei Chernousov July 4, 1999 14:38

Re: 1D governing equations
The geometry I am writing about is a simplest one: a cylindrical circular tube. The air flow within this tube is unsteady, but the characteristic ratio (wave length) / (diameter) is in range of 100...150. Under these conditions the one-dimensional (hydraulical) formulation seems to be _satisfactory_ but the waves appear to move about 3% slower than obtained from simulation.

I intend to correct this error by considering certain realistic hypotheses on the parameter profiles instead of using averaged 1D equations. It is doubtful that this systematic error haven't been discovered yet, and I want to know is there any experience on this point.

Andrzej Matuszkiewicz July 5, 1999 23:53

Re: 1D governing equations
Are you saying the systematic error of 3% ? Congratulations! There are few theories that can claim such an excellent agreement between numerical prediction and experimental data. However, if you are looking for perfection you may want to check the following points before you jump into 2D or 3D interpretation:

1. Are you sure that it is not experimental date that is subject to a systematic error. All pressure transducers have some small bias errors. Do not believe what manufacturers tell you about their precision. Usually it is much less than they claim. 2. Probably wave speeds have been measured by digital signal processing. Thess techniques have their own bias errors. 3. Do you use a prefect gas equation of state? Wave speeds are sensitive to the form of equation of state. Consider using a real gas equation of state. 4. You are right. The closure equations for one dimensional flows depend on the velocity profile. So, you may obtain different wall friction than that from a correlation you are using, if you perform averaging over a more realistic velocity profile. I derived such equations for a two-phase counter-current gas-liquid flow. In this case the frictional terms have different analytical forms than that for a single-phase flow. However, do not expect that this term will effect the wave velocity unless it is in a differential form. Waves in a hyperbolic system are tied to space derivatives. These kind of models were used in one-dimensional two-phase flows in channels to explain wave propagation.


Andrei Chernousov July 6, 1999 02:26

Re: 1D governing equations
Thank you!

I had already considered all these points and can say that:

1. Pressure transducers used in experiments were tested on shock wave front and very noticeable vibrations were registered, but their period was about 0.1 msec, much less than about 7 msec - the wave length in time, with the shape and the amplitude of experimental pressure curves reproduced well by simulation.

2. The author of the signal processing program assured me that any serious temporal bias is impossible. This is affirmed also by the absence of such bias for the front of the initial rarefaction wave registered by two distant transducers.

3. The perfect gas equation of state with R=287.1 J/(kg*K) and cp/cv=1.40, to my opinion, is quite accurate for air in this case where the pressure varies from 60000 to 130000 Pa, nearly ambient temmperature - possible errors cannot affect such an overestimation of the sound speed.

The remaining cause (it seems to me now) of the remaining (temporal systematic error of 3%!) bias is the need for

4. the corrected hyperbolic system. The correction due to considering velocity and, say, stagnation temperature profiles can sriously (or maybe cannot - that is the question!) change characteristic velocities of the hyperbolic system. And I realize that the source terms cannot do it, but namely stationary coefficients of friction gave excellent results in shape and amplitude.

Well, am I right? Are corrected governing equations for single-phase flows used anywhere? (Anyway, I'll explore this approach to perfection).


Appendix: non-corrected equations for cylindrical duct are (ro)t + (ro*u)x = 0 (ro*u)t + (ro*u*u+p)x = friction (ro*E)t + (ro*u*E+p*u)x = heat_exchange p/ro = R*T

M. Akram July 13, 1999 15:06

Re: 1D governing equations
May I draw your attention to the numerical scheme you are using. Many had speed and amplitude errors. I have not worked with flow through ducts and can not tell you direct refrences. You may look in Jr Comp Phys for phase errors.

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