Analysis of micro gas flows with DSMC (Direct Simulation Monte Carlo) method

Abstract

In the last 25 years a number of Micro Electro Me- chanical System (MEMS) were developed. TheseMEMS devices not only include the mechanical sys- tems but also the fluids. Knowledge about fluid flowsin this scale is not as mature as the mechanicalproperties of the MEMS. As their dimensions arebetween 1 mm and 1 micron, gas flows related withthe MEMS devices have high Knudsen numbers) (Knsimilar to high atmosphere flights. IfKnishigher than 0.1, instead of the classical continuumbased Euler or Navier-Stokes (N-S) equations, high- er order continuum based equations like Burnettequations or molecular models like DSMC should beused. This is due to the departure from local ther- modynamic equilibrium with increasingKnnum- ber. First velocity slip and temperature jump areformed on the boundaries. Next, low order constitu- tive equations are lost their validity because rela- tions both between shear stress and velocity gradientand heat conduction and temperature gradient arenot linear any more. Additionally the ratio of flowsurface area to flow volume is dramatically in- creased in micro gas flow conditions. So surfaceforces dominate the volume forces. Consequently,compressibility and viscous heating (dissipation)effects become more important in micro gas flows inaddition to rarefaction effects. Even in low Machnumbers, large density and temperature gradientsprevail. It is found that a micro scale gas flow canbehave differently from the large-scale one, which isgenerally studied with hydrodynamic models. Our application of DSMC starts with the division ofthe computational domain into smaller cells. Lineardimensions of these cells are of the same order asthe mean-free-path (  ) of the gas. A group of phys- ical gas molecules are represented by one repre- sentative molecule that called DSMC molecule inthis study. Every DSMC molecule carries position,velocity, cell number and if applicable internal en- ergy information on it. In DSMC method moleculemovements and collisions are separated from eachother. As a first step, molecules move according totheir velocities and initial conditions. Their veloci- ties, positions and cell numbers are updated. In thecollision step, stochastic approach is used and mo-lecule velocities are updated according to the colli- sion model chosen. Next step is the calculation of themacroscopic gas flow properties for each cell fromthe microscopic molecule information. For steadyflows, time averaging is used for the calculation ofmacroscopic properties. DSMC method is computationally expensive. Toshorten the computation time new approaches areneeded.. Generally molecule movements are tracedcell-by-cell in DSMC solvers both in structural andunstructured meshes. At each time step DSMC mole- cules move to a new position. Then each DSMCmolecule is checked whether they left the cell or not.If it is determined that DSMC molecules left the cellthen which cell they stopped is calculated. To do thiseither all the neighbor cells should be searched orwhich edge molecule left the origin cell should bedetermined. And then new cell is found. This proce- dure requires many mathematical calculations andtime. If non-rectangular physical flow geometry canbe converted a rectangular computational domain,then it is possible to calculate the DSMC moleculecell information in a very short time with a verysimple mathematical operation. Additionally currentDSMC solvers use complex and time consuming in- dexing mechanism to realize molecule collisions. Allthe molecules put in order 1-Dimensional arrays atthe end of each time step. Molecule collision part- ners are selected from this array, which are requiredto be in the same cell. In this study using a new datastructure which consist of a cell number and a mole- cule number in that cell. Each molecule completingits movement is renumbered according to its newcell information and what number in this cell. Thisnew data structure is copied onto old data structureat the end of each time step. Consequently complexindexing mechanism is found to be obsolete now. Acold gas micro-nozzle test problem is chosen fromthe literature. In this problem working medium isArgon. A test study is performed with Argon gasflow through a convergent-divergent micro-nozzle todetermine the efficiency of the new method. Bothmacro properties of Argon gas flow through the mi- cro-nozzle and solution times of each method is re- ported. Cold gas flow through a micro-nozzle is cho- sen because they are thought important for micro- propulsion systems.

Özet

Doğrudan Benzetim Monte Carlo (DSMC) seyreltik ve mikro boyutlu gaz akışlarını çözümlemekiçin kullanılan molekül esaslı bir yöntemdir. DSMC hesaplama süresi olarak pahalı bir yöntem ol- duğundan genelde Euler ve Navier-Stokes gibi geleneksel sürekli ortam denklemlerinin kullanılma- sının uygun olmadığı durumlarda kullanılmaktadır. Ortamın geleneksel sürekli ortam denklemleri- ne uygun olup olmadığına ise boyutsuz Knudsen sayısına  Knbakılarak karar verilmektedir.Knudsen sayısı yükseldikçe önce sınırlarda hız kayması ve sıcaklık atlaması oluşur. Takiben ısı akıvektörünün sıcaklık gradyeni, gerilme tensörünün ise gaz akış hız gradyeni ile olan doğrusal ilişkisisona erer. Bu şartlar oluştuğunda hesaplamalar için deneylere uygun sonuçlar veren DSMC yön- temine ihtiyaç duyulmaktadır. DSMC yönteminde DSMC molekülleri her bir zaman adımında sahipoldukları hız nedeni ile konum değiştirirler. Bu konum değişikliği nedeni ile içinde yer aldıklarıhücrelerde değişebilmektedirler. DSMC molekül çarpışmaları ve makro değer hesaplamaları hücretemelinde modellendiğinden her zaman adımı sonunda DSMC moleküllerinin yeniden konuşlandık- ları hücrelere ait bilgilerin güncellenmesine ihtiyaç vardır. Geleneksel DSMC yönteminde molekül- ler hücreden hücreye takip edilerek son gittikleri hücre bilgisine ulaşılır. Eğer gaz akış bölgesi dik- dörtgen bir geometriye dönüştürülebilir ise DSMC molekülünün gittiği son hücre yapısal ağ bilgi- sinden yararlanılarak basit bir aritmetik işlem ile bulunabilmektedir. Bu çalışmada DSMC hesap- lama verimliliğini arttırmak için mikro ölçekli yakınsak-ıraksak bir soğuk gaz lülesi için koordinatdönüşümü ile kare şeklinde bir hesaplama bölgesi oluşturulmuş ve hücre bilgisi hesaplama süresikısaltılmıştır. İlaveten her zaman adımı sonunda moleküllerin yeniden hücre numarasına göre sıra- ya dizilmesi işlemine olan ihtiyaca yeni bir yaklaşımla son verilmiştir.