Isı Transferi Problemi için Sabit Modelli Çoklu Ağ Şemalarının Değerlendirilmesi

Öz

Isı transferi problemleri ve çözümleri mühendislik ve teknolojinin hemen hemen tüm alanlarında kritik öneme sahipken, gerçek problemlerinin çoğu doğası gereği üç boyutludur. Bunları 2 boyutlu modellere basitleştirmek, makul modellerle pratik avantajlar sunar. Isı transferi probleminin temel sınıfı olan 2 boyutlu termal difüzyon problemi çalışma için seçilmiştir. Çoklu ağ yöntemleri, çözüm doğruluğunu korurken maliyet düşürme açısından da öne çıkan yöntemler olarak bilinmektedir. Sabit model şemaları kullanan çoklu ağ yöntemlerinin etkinliği incelenmiştir. Sayısal deney oluşturmak için temel sonlu hacim yöntemini, ağlar arası işlemleri ve yinelemeli çözücüleri uygulayan özgün bir kod oluşturulmuştur. Analitik Laplace çözümüne sahip bir referans durum seçilmiş ve sonuçlarla uygun şekilde doğrulanmıştır. Sabit modellere sahip çeşitli çoklu şebeke şemaları parametreler, tarama başına yinelemeler ve maksimum seyreltme düzeyinde araştırılmıştır. Sonuçlar maliyet ve performansın odak noktalarına göre derlenmiştir. Doğrudan yinelemeli yöntemlerin çoklu ağ şemalarıyla karşılaştırılması, çoklu ağ şemalarının etkinliğini kanıtlanmıştır. Maliyet ve performans sonuçlarının değerlendirilmesiyle, herhangi bir çoklu ağ şemasının, her ağ çözünürlüğünde minimum sayıda yinelemeyi korurken, mümkün olan en düşük seviyeye gelmesi gerektiği sonucuna varılmıştır.

Anahtar Kelimeler

• Sonlu Hacim Yöntemi
• Termal Difüzyon
• Yakınsama Oranı
• Çoklu Ağ Şemaları
• Hesaplama Maliyeti

Assessment of Multigrid Schemes with Fixed Patterns for a Two-Dimensional Heat Transfer Problem

Öz

Heat transfer problems and their solutions are of critical importance in almost all areas of engineering and technology, while many real-world problems are inherently three-dimensional. Simplifying them to 2D models offers practical advantages with reasonable models. With being the fundamental class of problem of heat transfer, the 2D thermal diffusion problem was selected for the study. Multigrid methods were referred to as standing out in terms of cost reduction while keeping solution accuracy. The effectiveness of multigrid methods employing fixed pattern schemes was subject to investigation. To set up numerical experimentation, an authentic code generation effort was given that implements a basic finite volume method, intergrid operations and iterative solvers. A reference case with an analytical Laplace solution was selected and properly validated by the results. A variety of multigrid schemes with fixed patterns were explored around parameters, iterations per sweep and maximum coarsening level. Results were compiled on the focal points of cost and performance. A comparison of the direct iterative methods with multigrid schemes proved the effectiveness of multigrid schemes. With the assessment of the cost and performance outcomes, it is concluded that any multigrid scheme should visit the maximum coarsest level possible while keeping a minimum number of iterations on each grid resolution.

Anahtar Kelimeler

• Finite Volume Method
• Multigrid Schemes
• Thermal Diffusion
• Computational Cost
• Convergence Rate

Kaynakça

1. Alpman, E. (2012). Blast Wave Simulations Using Euler Equations and Adaptive Grids, Journal of Thermal Sciences and Technology, 32 (2), 1-9.
2. Annaratone, D. (2010). Engineering Heat Transfer, Springer.
3. Arnone, A., Sestini, A. (1991). Multigrid Heat Transfer Calculations Using Different Iterative Schemes, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 19 (1), 1-11.
4. Axelsson, O. (1996) Iterative Solution Methods, Cambridge University Press.
5. Aydar, E., Ekmekçi, İ. (2012). Thermal Efficiency Estimation of the Panel Type Radiators with CFD Analysis, Journal of Thermal Sciences and Technology, 32 (2), 63-71.
6. Aykan, F.S., Dursunkaya, Z. (2008). İki Boyutlu Yüzeylerde Isıl Aşınma Sayısal Analizi, Isı Bilimi ve Tekniği Dergisi, 28 (1), 43-49.
7. Bali, T. (2006). Numerical Analysis of Laminar and Turbulent Swirl Flows, Journal of Thermal Sciences and Technology, 26 (1), 1-8.
8. Brandt, A. (1973). Multi-level Adaptive Technique (MLAT) for Fast Numerical Solution to Boundary Value Problems, Lecture Notes on Physics, 18, Springer.

9. Brandt, A., Livne, O.E. (2011). Multigrid Techniques: 1984 guide with applications to fluid dynamics, Society for Industrial and Applied Mathematics.
10. Briggs, W.L., Henson, V.E., McCormick, S.F. (2000). A Multigrid Tutorial, Society for Industrial and Applied Mathematics.
11. Dawood, A.S., Burns, P.J (1992). Steady Three-Dimensional Convectivie Heat Transfer in A Porous Box Via Multigrid, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, 22 (2), 167-198.
12. Doğan, A., Akkuş, S., Başkaya, Ş. (2012). Numerical Analysis of Natural Convection Heat Transfer from Annular Fins on a Horizontal Cylinder, Journal of Thermal Sciences and Technology, 32 (2), 31-41.
13. Galante, G., Rizzi, R.L. (2007). A Multigrid-Schwarz Method for the Solution of Hydrodynamics and Heat Transfer Problems in Unstructured Meshes, 19th International Symposium on Computer Architecture and High Performance Computing, 87-94.
14. Internet, (2023). The Julia Programming Language, Julia Micro-Benchmarks, https://julialang.org/benchmarks/
15. Karaaslan, S., Hepkaya, E., Yücel, N. (2013). CFD Simulation of Longitudinal Ventilation Systems in a Scaled Short Tunnel, Journal of Thermal Sciences and Technology, 33 (1), 63-77.
16. Kürekçi, N.A., Özcan, O. (2012). An Experimental and Numerical Study of Laminar Natural Convection in a Differentially-Heated Cubical Enclosure, Journal of Thermal Sciences and Technology, 32 (1), 1-8.
17. Lai, Y.G., Przekwas, A.J. (1996). A Multigrid Algorithm for A Multiblock Pressure-Based Flow and Heat Transfer Solver, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 30 (2), 239-254.
18. Lygidakis, G.N., Nikolos, I.K. (2014). Using a Parallel Spatial/Angular Agglomeration Multigrid Scheme to Accelerate the FVM Radiative Heat Transfer Computation—Part I: Methodology, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 66 (6), 471-497.
19. Lygidakis, G.N., Nikolos, I.K. (2014). Using a Parallel Spatial/Angular Agglomeration Multigrid Scheme to Accelerate the FVM Radiative Heat Transfer Computation—Part II: Numerical Results, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 66(6), 498-525.
20. Mançuhan, E., Küçükada, K., Alpman, E. (2011). Mathematical Modeling and Simulation of the Preheating Zone of a Tunnel Kiln, Journal of Thermal Sciences and Technology, 31 (2), 79-86.
21. Onur, N., Turgut, O., Arslan, K. (2011). Three-Dimensional Numerical Analysis of Forced Convection Flow and Heat Transfer in a Curved Square Duct, Journal of Thermal Sciences and Technology, 31 (2), 13-24.
22. Patil, P.V., Prasad, K. (2014). Numerical Solution for Two Dimensional Laplace Equation with Dirichlet Boundary Conditions, IOSR Journal of Mathematics, 6, 66-75.
23. Sert, Z., Timuralp, Ç., Tekkalmaz, M. (2019). Heat Transfer in Three-Dimensional Rectangular Cavities with Pins, Journal of Thermal Sciences and Technology, 39 (1), 39-49.
24. Şimşek, B., Uslu, S., Ak, M.A. (2020). Validation of Aerodynamic Heating Prediction Tool, Journal of Thermal Sciences and Technology, 40 (1), 53-63.
25. Tang, L., Joshi, Y.K. (1999). Application of Block-Implicit Multigrid Approach to Three-Dimensional Heat Transfer Problems Involving Discrete Heating, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, 35 (7), 717-734.
26. Trottenberg, U., Oosterlee, C.W., Schüller, A. (2001). Multigrid, Academic Press.
27. Uğurlubilek, N. (2012). Numerical Investigation of Heat Transfer and Flow in a Twisted-Shaped Square Duct, Journal of Thermal Sciences and Technology, 32 (2), 121-131.
28. Uzuner, M.K., Başol, A.M., Mischo, B., Jenny, P. (2023). Numerical Analysis and Diffuser Vane Shape Optimization of a Radial Compressor with the Open-Source Software SU2, Journal of Thermal Sciences and Technology, 43 (2), 233-242.
29. Versteeg, H., Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Pearson.
30. Vierendeels, J., Merci, B., Dick, E. (2004). A multigrid method for natural convective heat transfer with large temperature differences, Journal of Computational and Applied Mathematics, 168, 509-517.
31. Wang, Q., Joshi, Y. (2006). Algebraic Multigrid Preconditioned Krylov Subspace Methods for Fluid Flow and Heat Transfer on Unstructured Meshes, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 49 (3), 197-221.
32. Wesseling, P. (1992). An Introduction to Multigrid Methods, John Wiley & Sons.
33. Yan, J., Thiele, F. (1998). Performance and Accuracy of a Modified Full Multigrid Algorithm for Fluid Flow and Heat Transfer, Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 34 (3), 323-338.
34. Yetik, Ö., Mahir, N. (2020). Flow and Forced Heat Transfer from Tandem Square Cylinders Near a Wall, Journal of Thermal Sciences and Technology, 40 (1), 99-112.
35. Yıldızeli, A., Çadırcı, S. (2023). Numerical Investigation of Plate Cooling Using Multiple Impinging Jets in Different Alignments, Journal of Thermal Sciences and Technology, 43 (1), 1-10.