# order of magnitude analysis

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 July 26, 2000, 14:53 order of magnitude analysis #1 atit koonsrisuk Guest   Posts: n/a 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 #2 Kalyan Guest   Posts: n/a You take the compressible momentum equation and non-dimensionalize 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 non-dimensional 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 #3 atit Guest   Posts: n/a 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 non-dimensional 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 non-dimension form of equations. Thank you very much sir. Best regards. Atit Koonsrisuk