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

Doğrudan Benzetim Monte Carlo (DSMC) seyreltik ve mikro boyutlu gaz akışlarını çözümlemek için kullanılan molekül esaslı bir yöntemdir. DSMC hesaplama süresi olarak pahalı bir yöntem olduğundan genelde Euler ve Navier-Stokes gibi geleneksel sürekli ortam denklemlerinin kullanılmasının uygun olmadığı durumlarda kullanılmaktadır. Ortamın geleneksel sürekli ortam denklemlerine uygun olup olmadığına ise boyutsuz Knudsen sayısına  bakılarak karar verilmektedir. Knudsen sayısı yükseldikçe önce sınırlarda hız kayması ve sıcaklık atlaması oluşur. Takiben ısı akı vektörünün sıcaklık gradyeni, gerilme tensörünün ise gaz akış hız gradyeni ile olan doğrusal ilişkisi sona erer. Bu şartlar oluştuğunda hesaplamalar için deneylere uygun sonuçlar veren DSMC yöntemine ihtiyaç duyulmaktadır. DSMC yönteminde DSMC molekülleri her bir zaman adımında sahip oldukları hız nedeni ile konum değiştirirler. Bu konum değişikliği nedeni ile içinde yer aldıkları hücrelerde değişebilmektedirler. DSMC molekül çarpışmaları ve makro değer hesaplamaları hücre temelinde modellendiğinden her zaman adımı sonunda DSMC moleküllerinin yeniden konuşlandıkları hücrelere ait bilgilerin güncellenmesine ihtiyaç vardır. Geleneksel DSMC yönteminde moleküller hücreden hücreye takip edilerek son gittikleri hücre bilgisine ulaşılır. Eğer gaz akış bölgesi dikdörtgen bir geometriye dönüştürülebilir ise DSMC molekülünün gittiği son hücre yapısal ağ bilgisinden yararlanılarak basit bir aritmetik işlem ile bulunabilmektedir. Bu çalışmada DSMC hesaplama verimliliğini arttırmak için mikro ölçekli yakınsak-ıraksak bir soğuk gaz lülesi için koordinat dönüşümü ile kare şeklinde bir hesaplama bölgesi oluşturulmuş ve hücre bilgisi hesaplama süresi kısaltılmıştır. İlaveten her zaman adımı sonunda moleküllerin yeniden hücre numarasına göre sıraya dizilmesi işlemine olan ihtiyaca yeni bir yaklaşımla son verilmiştir. Anahtar Kelimeler: DSMC, mikro gaz akışları, mikro-lüle, genelleştirilmiş koordinatlar.In the last 25 years a number of Micro Electro Mechanical System (MEMS) were developed. These MEMS devices not only include the mechanical systems but also the fluids. Knowledge about fluid flows in this scale is not as mature as the mechanical properties of the MEMS. As their dimensions are between 1 mm and 1 micron, gas flows related with the MEMS devices have high Knudsen numbers similar to high atmosphere flights. If is higher than 0.1, instead of the classical continuum based Euler or Navier-Stokes (N-S) equations, higher order continuum based equations like Burnett equations or molecular models like DSMC should be used. This is due to the departure from local thermodynamic equilibrium with increasing number. First velocity slip and temperature jump are formed on the boundaries. Next, low order constitutive equations are lost their validity because relations both between shear stress and velocity gradient and heat conduction and temperature gradient are not linear any more. Additionally the ratio of flow surface area to flow volume is dramatically increased in micro gas flow conditions. So surface forces dominate the volume forces. Consequently, compressibility and viscous heating (dissipation) effects become more important in micro gas flows in addition to rarefaction effects. Even in low Mach numbers, large density and temperature gradients prevail. It is found that a micro scale gas flow can behave differently from the large-scale one, which is generally studied with hydrodynamic models. Our application of DSMC starts with the division of the computational domain into smaller cells. Linear dimensions of these cells are of the same order as the mean-free-path ( ) of the gas. A group of physical gas molecules are represented by one representative molecule that called DSMC molecule in this study. Every DSMC molecule carries position, velocity, cell number and if applicable internal energy information on it. In DSMC method molecule movements and collisions are separated from each other. As a first step, molecules move according to their velocities and initial conditions. Their velocities, positions and cell numbers are updated. In the collision step, stochastic approach is used and molecule velocities are updated according to the collision model chosen. Next step is the calculation of the macroscopic gas flow properties for each cell from the microscopic molecule information. For steady flows, time averaging is used for the calculation of macroscopic properties. DSMC method is computationally expensive. To shorten the computation time new approaches are needed.. Generally molecule movements are traced cell-by-cell in DSMC solvers both in structural and unstructured meshes. At each time step DSMC molecules move to a new position. Then each DSMC molecule is checked whether they left the cell or not. If it is determined that DSMC molecules left the cell then which cell they stopped is calculated. To do this either all the neighbor cells should be searched or which edge molecule left the origin cell should be determined. And then new cell is found. This procedure requires many mathematical calculations and time. If non-rectangular physical flow geometry can be converted a rectangular computational domain, then it is possible to calculate the DSMC molecule cell information in a very short time with a very simple mathematical operation. Additionally current DSMC solvers use complex and time consuming indexing mechanism to realize molecule collisions. All the molecules put in order 1-Dimensional arrays at the end of each time step. Molecule collision partners are selected from this array, which are required to be in the same cell. In this study using a new data structure which consist of a cell number and a molecule number in that cell. Each molecule completing its movement is renumbered according to its new cell information and what number in this cell. This new data structure is copied onto old data structure at the end of each time step. Consequently complex indexing mechanism is found to be obsolete now. A cold gas micro-nozzle test problem is chosen from the literature. In this problem working medium is Argon. A test study is performed with Argon gas flow through a convergent-divergent micro-nozzle to determine the efficiency of the new method. Both macro properties of Argon gas flow through the micro-nozzle and solution times of each method is reported. Cold gas flow through a micro-nozzle is chosen because they are thought important for micro-propulsion systems. Keywords: DSMC, micro gas flows, micro-nozzle, generalized coordinates