روشي جديد براي اعمال شرط مرزي فشار بر مرزهاي منحني‌شكل در روش شبكه‌ي بولتزمن

A New Method For Pressure Boundary Conditions In The Lattice Boltzmann Method For Domains With Curved Boundaries

در اين مطالعه روشي جديد براي پياده‌سازي شرط مرزي فشار روي مرزهاي منحني در روش شبكه‌ي بولتزمن پيشنهاد شده است. در اين روش كه مبتني بر برون‌يابي و اصل برهمنهي1 است، تابع توزيع روي نقطه‌ي مرزي به دو قسمت تعادلي و غيرتعادلي تقسيم شده است: قسمت تعادلي به‌كمك چگالي مشخص مرزي و برون‌يابي سرعت از نقاط مجاور، و قسمت غيرتعادلي نيز با استفاده از شرط كمانش محاسبه مي‌شود. سپس مقادير مجهول تابع توزيع در نقاط مجاور مرز، به‌كمك ميان‌يابي محاسبه و طوري اصلاح مي‌شود كه فشار (چگالي) مورد نظر روي مرز اعمال شود. نتايج عددي به دست آمده نشان مي‌دهد كه روش مذكور از مرتبه‌ي دوم دقت است. مثال‌هاي حل شده براي حالت دوبعدي طرح شده است، اما از اين روش به‌راحتي مي‌توان براي مسايل سه‌بعدي نيز بهره‌مند شد.

The Lattice Boltzmann model is an alternative kinetic based method, capable of solving hydrodynamics for various systems. Major advantages of this model lend themselves to the fact that the solution for particle distribution functions is explicit, easy to implement, and natural to be parallelized. Because the method uses a regular Cartesian lattice in space, implementation of the Dirichlet pressure boundary condition has always been a challenge for curved entities in Lattice Boltzmann simulations. The difficulty comes from the fact that the unknown boundary velocity cannot be determined on curved surfaces using common assumptions in the popular Zou-He scheme. The lack of a certain solution for flows having non-straight, constant pressure boundaries encourages the need to develop exact boundary conditions for such flows. In this paper, a method has been developed for imposing a pressure boundary condition for curved entities in the Lattice Boltzmann method. The proposed formulation is based on the so-called superposition interpolation scheme, where the unknown distribution functions of the boundary are divided into equilibrium and non-equilibrium parts. The equilibrium part is calculated based on the known value for the boundary density and the extrapolated value of the velocity. The non-equilibrium part is determined employing the bounce-back scheme. Finally, the unknown populations are corrected in such a way that the desired pressure is achieved on the surface. Fully developed flows in a 2-D channel and in a 2-D inclined channel with pressure boundary conditions at inlet and outlet are used to illustrate the accuracy of the scheme. The numerical results for the benchmark flows prove the second order accuracy of the proposed scheme. While the method is preliminarily established for 2D problems, its extension to 3D flows is quite straight forward and turns it into an efficient tool for simulating flows dealing with complex constant pressure geometries. One important industrial application of this new boundary condition is for flows through a generic packed bed involving adsorbent particles of constant surface pressure which is a subject for further investigation.