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July 26, 2000, 14:53 
order of magnitude analysis

#1 
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I read from one paper that from order of magnitude analysis, the order of pressure gradient term will be one over Mach number squared, which is made the big trouble for low speed compressible flow. I would like to know how to do order of magnitude analysis and how to derive until I get the order of pressure gradient term. Could you please suggest me? Thanks.


July 26, 2000, 15:56 
Re: order of magnitude analysis

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You take the compressible momentum equation and nondimensionalize it using reference quantities.
Lo = reference length Uo = reference velocity to = reference time scale = Lo/Uo To = reference temperature ro = reference density Po = reference pressure = ro*R*To (equation of state) R : universal gas constant. Using these quantities obtain the momentum equation for the nondimensional momenta. Retain the unsteady term and the convection term on the left and the pressure gradient term and the viscous terms on the right. In front of the pressure gradient, you have Po/(ro * Uo * Uo) Using the facts that "a*a = (gamma)*R*To" and "Po = ro*Uo*Uo" (where gamma = ratio of specific heats), you end up with [ (a*a) / (Uo*Uo*gamma) ] in front of the pressure gradient term. This is nothing but [1/(Gamma * M * M)]. You would also end up with 1/Re in front of the viscous term where Re = Uo*Lo/mu (the Reynolds number, mu = reference viscosity) 

July 27, 2000, 05:17 
Re: order of magnitude analysis

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Dear Mr.Kalyan, Thank you very much for your suggestion. So the idea that we get from the new equation is that if the flow has very low Mach number, the term with one over Mach number squared will approach infinity. Actually I do not familiar with the nondimensional form. I know only that we work with this form because the quatities will not depend on the amount of it. Anyway from this analysis, we know that the term with one over Mach number squared will approach infinity when the Mach number is very small. Can I interpret that, in the dimensional form, the pressure gradient will approach infinity when the Mach number is very small? Please explain to me about the concept of nondimension form of equations. Thank you very much sir.
Best regards. Atit Koonsrisuk 

July 27, 2000, 11:59 
Re: order of magnitude analysis

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When the Mach number tends to zero, the pressure gradient does NOT tend to infinity. Pressure gradient is actually finite. Just because the (1/M*M) in front of the pressure gradient is going to inifinity (as M approaches zero), the pressure gradient need not tend to infinity. In a way, for the product of (1/M*M) and the pressure gradient to be finite, the pressure gradient should become smaller and smaller as M is approaching zero.
This can be understood in the following way. The effect of pressure gradient on the velocity field (fluid flow acceleration, d{ro*u}/dt) becomes more significant as the Mach number reduces (since M appears in the denominator of the pressure gradient term). What this means physically is that the pressure field is more rapid compared to the velocity (or the vorticity field). This is no surprise since, at low M, it is expected that the speed of sound is much higher than the flow velocity (by definition). In fluid flow, the pressure field has to adjust to changes in the velocity field and viceversa. But when M is very small, the pressure field adjusts very quickly to the changes in the velocity field. The adjustment of the velocity field to changes in the pressure field is not as fast. In the limit of zero M (which means that u is finite and the speed of sound is infinite), the pressure waves travel at infinite speed. So if the velocity field tends to create any pressure gradients, the speed of sound being infinite makes the pressure uniform instantaneously. So, there are no thermodynamic pressure gradients at low speeds. Even if they are finite, they are very small. In the so called the zero Mach number formulation for low speed flows, the pressure is separated out into two compoenents : a thermodynamic component that is constant in space and a auxiliary pressure that couples the momentum equations with the mass conservation equation (in addition to accounting for any minor pressure gradients that exist in the flow). This formulation needs the flow to be incompressible but not constant density. I think it has been discussed in the reacting flow book by Boris and Oran. If you can not find it in that book, I can send you the titles of some papers where you can find more informtation on this formulation. 

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