Abstract




 
   

IJE TRANSACTIONS C: Aspects Vol. 30, No. 9 (September 2017) 1408-1416   

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  BUOYANCY TERM EVOLUTION IN THE MULTI RELAXATION TIME MODEL OF LATTICE BOLTZMANN METHOD WITH VARIABLE THERMAL CONDUCTIVITY USING A MODIFIED SET OF BOUNDARY CONDITIONS
 
M. Varmazyar, A. Mohammadi and M. Bazargan
 
( Received: March 12, 2017 – Accepted in Revised Form: July 07, 2017 )
 
 

Abstract    During the last few years, a number of numerical boundary condition schemes have been used to study various aspects of the no-slip wall condition using the lattice Boltzmann method. In this paper, a modified boundary condition method is employed to simulate the no-slip wall condition in the presence of the body force term near the wall. These conditions are based on the idea of the bounce-back of the non-equilibrium distribution. The error associated with the modified model is smaller than those of other boundary condition models available in the literature. Additionally, various schemes to simulate body forces have been studied. Based on the numerical results, the model demonstrating minimum error has been reported. Finally, it has been shown that the present model is capable of simulating the effect of high nonlinearity in the heat transfer equation in the presence of a variable thermal conductivity. This has been accomplished by employing a multi relaxation time scheme to model a Rayleigh-Benard natural convection current in a 2-D domain with high Rayleigh numbers. Previous studies reported that the onset of oscillation occurs at Ra≈30,000 and Pr=6.0. By the modified boundary condition method which is used in this study, the oscillation is removed until at least Ra≈ 45,000 and Pr=6.0. The results show that applying scheme 3 for the current boundary condition yields the least amount of error compared to the semi-empirical correlation. The Rayleigh-Benard convection problem has been revisited in the presence of a variable thermal conductivity and the simulation results remain stable for flows with a large variation of thermal conductivity ( = 0.7) and Rayleigh numbers up to 1,000,000 and Pr=0.7.

 

Keywords    Lattice Boltzmann Method, Boundary Condition, Multi Relaxation Time, Variable Thermal Conductivity, Rayleigh-Benard Convection

 

چکیده    در طول سال های اخیر، شرایط مرزی مختلفی در روش شبکه بولتزمن جهت تنظیم عدم لغزش بر روی دیواره معرفی شده است. این مقاله به معرفی یک روش جدید جهت حذف اثرات نیروی نزدیک دیواره پرداخته است. نتایج نشان می دهد خطای مدل پیشنهادی پایین تر از مدل های پیشنهادی موجود در منابع مرتبط می باشد. روش های مختلف اعمال نیروی حجمی نیز مورد ارزیابی قرار گرفته و بر اساس آن مدل دارای کمترین خطا معرفی گردیده است. در مسایل مهندسی و تحت گرادیان های شدید دما، نمی توان از تغییرات ضریب پخش حرارتی صرف نظر کرد و بر این اساس نشان داده شد که مدل انتخابیِ اعمال نیرو، قابلیت مدلسازی شرایط غیرخطی شدید تحت اثر تغییرات ضریب پخش حرارتی را نیز داراست. جهت افزایش پایداری و حصول دقت بالاتر از روش زمان آرامش چندگانه نیز استفاده شده است. در انتها، جهت اعتبارسنجی و ارزیابی مدل های پیشنهادی، مساله معروف جابجایی آزاد رایلی بنارد دوبعدی تحت شرایط ضریب پخش ثابت و متغیر در رایلی های بالا مورد مطالعه قرار گرفته است. مطالعات گذشته نشان می دهد که برای Pr=6.0، عدد نوسلت در Ra≈30,000 نوسانی می شود. با کمک شرط مرزی پیشنهاد شده در این مطالعه، نوسانات تا حدود Ra≈ 45,000 برای Pr=6.0 حذف گردید. نتایج نشان می دهد که شرط مرزی حاضر در کنار اسکیم اعمال نیروی نوع سوم می تواند کمترین خطا را در میان روش های موجود در قیاس با مدل نیمه تجربی مطالعات گذشته داشته باشد. جریان رایلی بنارد با ضریب هدایت حرارتی متغیر مورد مطالعه قرار گرفت. نتایج نشان می دهد که مدل پیشنهادی برای Pr=0.7 تا Ra=1000000 برای γ=0.7 پایدار است.

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