Mikro boyuttaki ters basamak geometrisi içindeki akışın KTA yöntemiyle analizi

Özet

Karakteristik Tabanlı Ayırma Algoritması, sürekli rejimde yer alan sıkıştırılabilir ve sıkıştırılamaz viskoz akış problemlerinin Sonlu Elemanlar Yöntemi çözümünde geçerlidir. Bu çalışma kapsamında, Karakteristik Tabanlı Ayırma çözücüsü, kayma rejiminde yer alan mikro akış problemlerinin çözümünde de kullanılabilecek şekilde geliştirilmiştir. Mikro boyuttaki ters basamak geometrisi içindeki azot akışı, geliştirilen çözücü kullanılarak analiz edilmiştir. Bu geometri içindeki akış, ters basınç gradyeni ve akım ayrılması içermesi sebebiyle özellikle seçilmiştir. Gerçekleştirilen analizlerde, aynı giriş çıkış basınç oranı ve birbirinden farklı giriş parametrelerine sahip üç ayrı akış hesaplamalı olarak incelenmiştir. İncelenen bu üç akışa ait parametreler, söz konusu akış kayma rejiminde yer alacak şekilde belirlenmiştir. Kayma rejiminde yer alan akışlarda katı duvar yüzeylerinde meydana gelen kayma-hızı ve sıcaklık-sıçraması, Beskok ve Karniadakis’e ait ikinci mertebeden sınır şartlarının uygulanmasıyla hesaplanmıştır. Yapılan analizlerde, hesaplamalar için gereksinim duyulan bilgisayar zamanı ve bellek miktarını düşürmek için, pP2P1 tipi elemanlar P1P1 tipi elemanların yerine kullanılmıştır. Elde edilen sonuçlar, gerçekleştirilen uyarlamaları doğrulamak için literatürde yer alan diğer hesaplamalı sonuçlarla karşılaştırılmıştır. Gerçekleştirilen bu analizle, akışa ait parametrelerin kayma-hızı ve sıcaklık-sıçraması dağılımı ile ilişkisi araştırılmıştır. Bu amaçla, incelenen her üç akışa ait yerel Mach sayısı konturları, kanal boyunca Knudsen sayısı değişimi grafikleri ve tekrar duvara yapışma mesafeleri bu çalışmada sunulmuştur. Elde edilen sonuçlar ve karşılaştırmalar, geliştirilen çözücünün kayma rejiminde yer alan mikro akış problemlerinin çözümü ve analizinde kullanılabileceğini göstermektedir.

Abstract

Knudsen number (Kn) is a measure of rarefaction and it is defined as the ratio of mean free path (λ ) to the characteristic length of the flow (L). In continuum regime where Kn is smaller than 10-3, fluid flow problems can be solved using continuum models such as Navier-Stokes equations (N-S). The interval where Kn is in the range of 10-3 to 10-1 is called the slip regime. In this regime, N-S solvers can be used for simulation of fluid flow if slip-velocity and temperature- jump boundary conditions are employed on solid wall instead of usual no-slip and temperaturewall boundary conditions which are valid in the continuum regime. In recent years, extremely small sized devices have been manufactured due to the development in production technologies. These devices are combinations of electrical and mechanical devices. Their sizes are in the range of 1 mm to 1 micron and they are called Micro-Electro-Mechanical Systems (MEMS). Some applications of the MEMS are related directly or indirectly to fluid flow. Fluid flow through or around the MEMS differs from the larger devices. The increase in the surface to volume ratio due to the decreasing characteristic length affects transport of mass, momentum and energy through the surfaces. Furthermore, flow deviates from the thermodynamic equilibrium and slip-flow, temperature- jump, thermal creep, rarefaction, viscous dissipation, compressibility, intermolecular forces and other unconventional effects become important. It is reported that most of the MEMS devices work in slip regime at standard conditions. Thus, using appropriate models for numerical simulation of fluid flow through or around the MEMS would help to increase its productivity and to reach a better comprehension of their functions. In 1995, Zienkiewicz et al. introduced a unified algorithm designed to replace the Taylor-Galerkin (or Lax-Wendroff) methods that have been used in the solution of compressible flow problems in Finite Element Method (FEM) context. Then, they have published several papers concerning the basis and applications of this new algorithm. Finally they introduced an algorithm named as Characteristic- Based-Split (CBS) algorithm. Thus, fractional step process of Chorin is extended to solve the fluid dynamics equations of both compressible and incompressible forms. In this study, the CBS algorithm is modified to perform micro flow analysis. The second order slipvelocity and temperature-jump boundary conditions of Beskok and Karniadakis are implemented on solid wall to the micro flow analyses presented in this study. The second order formulation proposed by Beskok and Karniadakis includes a thermal creep term similar to the first order formulation found in literature. In this study, semi-implicit form of the CBS algorithm is used and in order to reduce the size of implicit part of the N-S FEM solver, pP2P1 type elements are used instead of P1P1.. Solution procedure and Galerkin weak form the auxiliary momentum, continuity, end of step momentum and energy equations of proposed CBS algorithm are given with details in this paper. Micro backward facing step is one of the MEMS devices and fluid flow through this device is quite complex since there are adverse pressure gradient and separation in flow field. In this study, micro backward facing step is selected as test geometry for micro flow analysis in slip regime. The solver is used to analyze nitrogen gas flow through this geometry for three different cases having different inlet Mach (M) and Reynolds numbers (Re). Inlet to outlet pressure ratio is equal to 2.32 for these three cases. To verify the performed implementations, obtained results are compared with available numerical results found in literature in terms of accuracy. Relationships between the flow conditions such as inlet M, Re, reattachment lengths and slip-velocity, temperature-jump distributions are numerically investigated. Local M contours and Kn variations are presented within this study for all cases. It is observed that increasing Kn at the channel inlet results in decreasing reattachment length values. Decreasing M values at the channel inlet results in increasing normalized slip-velocity values from entry to exit of the channel. On the other hand, temperature-jump values are smaller for the cases with lower inlet M. The performed analyses and comparisons show that the use of the proposed CBS algorithm with pP2P1 type elements is promising for fluid flow problems in slip regime.