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Isı Aktarımı ile Birleştirilmiş MHD Denklemlerinin Chebyshev Spektral Kollokasyon Yöntemi ile Yaklaşımı

Bu çalı¸smada, enerji denklemi ile birle¸stirilmi¸s tam manyetohidrodinamik (MHD) denklemlerinin yakla¸sık çözümleri için Chebyshev spektral kollokasyon yöntemi (CSCM) önerilmektedir. ˙Iki boyutlu, zamana ba˘glı laminer ve sıkı¸stırılamaz MHD akı¸sı, ısı transferi Boussinesq yakla¸sımı kullanılarak termal etki ile birle¸stirilerek ele alınmaktadır. MHD akı¸sı, dikey olarak uygulanan manyetik alan etkisi altında olan kare kesitli bir kanalda gerçekle¸smekte, ayrıca akan sıvının elektriksel iletken olması nedeniyle manyetik indüksiyon göz önünde bulundurulmaktadır. Akım fonksiyonu, girdap (vortisite), sıcaklık, manyetik akım fonksiyonu ve akım yo˘gunlu˘gu türünden verilen temel denklemlerin uzaysal ayrıkla¸stırılması CSCM kullanılarak, zaman integrasyonu ise ko¸sulsuz kararlı geri farklar yöntemi kullanılarak çözüm elde edilmi¸stir. De˘gi¸sen Reynolds (Re), manyetik Reynolds (Rem), Rayleigh (Ra) ve Hartmann (Ha) sayıları de˘gerlerinin MHD akı¸sı ve ısı transferi üzerindeki etkileri ara¸stırılmaktadır.

Chebyshev Spectral Collocation Method Approximation to Thermally Coupled MHD Equations

In this study, a Chebyshev spectral collocation method (CSCM) approximationis proposed for solving the full magnetohydrodynamics (MHD) equations coupledwith energy equation. The MHD flow is two-dimensional, unsteady, laminar and incompressible,and the heat transfer is considered using the Boussinesq approximation forthermal coupling. The flow takes place in a square cavity which is subjected to a verticallyapplied external magnetic field, and the presence of the induced magnetic field is alsotaken into account due to the electrical conductivity of the fluid. The governing equationsgiven in terms of stream function, vorticity, temperature, magnetic stream function, andcurrent density, are solved iteratively using CSCM for the spatial discretisation, and anunconditionally stable backward difference scheme for the time integration. The inducedmagnetic field is obtained by means of its relation to the magnetic stream function. Thebehaviours of the flow and the heat transfer are investigated for varying values of Reynolds(Re), magnetic Reynolds (Rem), Rayleigh (Ra) and Hartmann (Ha) numbers.

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