Similarity of ideal gas flow at different scales

Vol. 46 No. 6SCIENCE IN CHINA(Series E)December 2003Similarity of ideal gas flow at different scalesWANG Moran(王沫然)& LI Zhixin(李志信)Department of Engineering Mechanics, Tsinghua University, Beijing 100084, ChinaCorrespondenceshouldbeaddressedtoWangMoran(email:moralwang99@mails.tsinghua.edu.cn)Received March 7. 2003Abstract The similarity of ideal gas flow at different scales is investigated analytically and numeri-cally. With the compressible and rarefied effects considered, two dimensionless parameters, Machnumber and Knudsen number, are proposed as the similarity criterions, because the Reynoldnumber can be expressed by the mach number and the Knudsen number of ideal gases. A DSMCmethod is used to simulate flows at different scales with the same Ma and Kn, including subsonicchannel flows and the supersonic flows over a hot plate. Comparisons between the results of different scales show that the normalized fields of macroscopic quantities are the same. This confirmsthe similarity. Especially, the results indicate that the micro flow are similar to the rarefied flow ofideal gas, which suggests that many transformations are available from the existing rarefied flowresults to the micro flowKeywords: similarity, ideal gas flow, direct simulation Monte CarloDOI:10.1360/02ye0072When the molecular mean free path becomes comparable with an important characteristiclength of the flow system, the continuum assumption breaks down and the particle nature of fluidmust be explicitly taken into account!. Such flows, generally named rarefied gas flows, havebeen investigated widely and deeply due to their important applications in aeronautics and astronautics. The rarefied gas flow can be described by the Boltzmann equation from a molecular pointof view. However, it is rather difficult to obtain analytical and numerical solutions of the boltz-mann equation. The direct simulation Monte Carlo(DSMC) method proposed by Bird offers avaluable tool for predicting the high Kn flow behavior. The dSMC method has been proved andwidely applied to the analysis of rarefied gas flow problems in the past several decades 3-9In recent years, MEMS/NEMS have been developed and the micro flow has become a hottopic!lo. In micro/nano gas flow, the Knudsen number can be quite high even though the gas is ofNomenclature a, sound speed; P, pressure: Cm, most probable molecular speed; Re, Reynolds number; d, molecular dameter; T, overall temperature; H, height of the computed domain; Tg, gas temperature on wall surfaces; k, Boltamann constant;Kn, Knudsen number; Tin translational temperature; Kn cal, local Knudsen number; Trot, rotational temperature; L, characteristic中国煤化工Mach numbermolecular mean velocity; n, number density; v,moCNMHG;H, dynamic viscosity: 7, mean free path: v, kinetic viscosity; P, density; Erot, rotational energyDer or nternal degree of freedom:or,totalcross-section;At, time step: e, translational energy; Ar, size of a cell; co, freestream; tr, translational mole: w, walot. rotational mode.662SCIENCE IN CHINA (Series E)high density due to the small characteristic length of the micro system. The classical N-s equations are not able to describe the flow accurately, so the dsmc method has been tried and usedto predict micro gas flow and heat transfer since the middle of the 1990s 2-18Since the rarefied gas flows on an original size scale have been investigated maturely and thunderstanding of micro/nano flow mechanism may have a significant bearing on the developmentand design of MEMS/NEMS, the current paper is an attempt to discuss the similarity of ideal gasflow at different scales. Firstly, the similarity is theoretically analyzed. Then, the DSMC method isdeveloped to simulate gas flows and heat transfer at different scales to verify the similarity numericallySimilarity analyIf the traditional incompressible flows in two different-Size systems are considered similarthe dimensionless parameter, Reynolds number, must be equal. For a compressible flow, a moredimensionless parameter, Mach number, plays an important role as well, while in the rarefied ormicro gas flow, the Knudsen number must be also considered since it affects the velocity and tem-perature boundary conditions directly. However, there is a relationship between the three parameters, and one can be expressed in terms of the other twoThe Reynolds number is the ratio of inertial forces to viscous onewhere v is the gas kinetic viscosityThe Mach number is the ratio of flow velocity to the speed of soundwhere a is the sound speed in gasThe Mach number is a dynamic measure of fluid compressibility and may be considered asthe ratio of inertial forces to elastic onesAccording to the molecular theory of gas, for the ideal gas made of smooth rigid elasticherical molecules with a repelling force from each other, the kinematic viscosity is related to themean free path/19)v=P=0.499Xvm≈vmwhere u is the gas dynamic viscosity, i the molecular mean free path and vm the mean molecularspeed, somewhat higher than the sound speed aarhi中国煤化工CNMHGwhere y is the gas specific heat rateNo 6SIMILARITY OF IDEAL GAS FLOW AT DIFFERENT SCALESCombining eqs. (1)-(4), the required relation can be obtainedKn2 ReEq(5)shows that as long as two of the three dimensionless numbers are specific, the third isalso determinate. Then, arbitrary two of the three numbers can be proposed as the similarity criteions In the present paper the Mach number and Knudsen number are selected as the similaritycriterions for ideal gas flow in different-scale systems. That is to say, when the pressure is not toohigh (P< 1000 atm), the temperature is not too high or too low (30 K