Numerical solution and stability analysis of transient MHD duct flow

Öz

This paper simulates the 2D transient magnetohydrodynamic (MHD) flow in a rectangular duct in terms of the velocity of the fluid and the induced magnetic field by using the radial basis function (RBF) approximation.  The inhomogeneities in the Poisson’s type MHD equations are approximated using the polynomial functions (1+r) and the particular solution is found satisfying both the equations and the boundary conditions (no-slip and insulated walls).  The Euler scheme is used for advancing the solution to steady-state with a time increment and a relaxation parameter which are determined for achieving stable solution.  It is shown that, as Hartmann number increases, the fluid becomes stagnant at the center of the duct, the flow is flattened and boundary layers are developed on the Hartmann and side walls.  These are the well-known characteristics of the MHD duct flow.  The stability analysis is also carried in terms of the spectral radius of the coefficient matrix of the discretized coupled system.  Stable solutions are obtained with RBF by using quite large time increment and suitable relaxation parameters on the expense of explicit Euler time-integration scheme used.

Anahtar Kelimeler

• MHD duct flow,RBF,Euler time-integration,stability

Zamana bağımlı MHD kanal akışının nümerik çözümü ve kararlılık analizi

Öz

Bu çalışmada, dikdörtgen kesit içerisindeki iki boyutlu zamana bağlı olan MHD akışı, sıvının hızı ve indüklenen manyetik alan cinsinden radyal baz fonksiyon yaklaştırımı kullanılarak sunulmuştur.  Poisson tipinde olan MHD denklemlerindeki homojen olmayan kısımlar, polinom fonksiyonları (1+r) ile yaklaştırılmıştır ve hem denklemleri hem de kaymaz ve iletken olmayan sınır koşullarını sağlayan özel bir çözüm bulunmuştur.  Euler yöntemi, kararlı çözümü veren zaman aralığı ve yumuşama katsayıları ile kullanılmıştır. Hartmann sayısı artıkça sıvının kanal ortasında durgunlaştığı, akışın düzleştiği, Hartmann ve yan duvarlardaki sınır tabakalarının geliştiği gösterilmiştir.  Bunlar MHD kanal akışının en iyi bilinen özellikleridir.  Ayrıca, kararlılık analizi, ayrıklaştırılmış birbirine bağlı olan sistemdeki katsayı matrisinin spektral yarıçapı doğrultusunda yapılmıştır.  Açık Euler zaman integrasyonu yöntemi kullanılmasına rağmen RBF ile oldukça geniş zaman aralığı ve uygun yumuşama parametreleri kullanılarak kararlı çözümler elde edilmiştir.

Anahtar Kelimeler

• MHD kesit akışı,RBF,Euler zaman integrasyonu,kararlılık analizi

Kaynakça

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