An implicit spectral method for the numerical solution of unsteady flows with an application to rotating disk flow and heat transfer

Bu makalede zamana göre geçişken akışkanların çözümü için bir kapalı sayısal integrasyon yöntemi önerilmiştir. Nümerik metod diske dik dogrultuda Chebyshev dügümleme ve zamanda ise ileri adım tekniklerine dayanmaktadır. Klasik sonlu fark yöntemlerine haiz başlangıç ve sınır koşullarındaki süreksizliklerinden kaynaklanan nümerik salınımlardan uzak olmanın yanında, planlanan metod oldukça dayanıklı, şartsız kararlı, yüksek dogruluga sahip ve uygulaması çok kolay olması gibi avantajlardan yararlanmaktadır. Sonlu fark metodlarının aksine, şimdiki metod farklı dönüşümlere gerek kalmaksızın akış alanını kolayca çözümler. Burada geliştirilen nümerik algoritma klasik zamana baglı ve aniden harekete geçirilen, lakin uzun bir süreç sonunda çok iyi bilinen dönen diskten kaynaklanan von Karman dönen akış problemine uygulanmıştır. Enerji denklemi de bu metodla çözülerek radyal ve tegetsel yüzey sürtünme katsayıları, tork ve diskten ısı transfer oranı gibi çok mühim fiziksel parametreler nümerik olarak elde edilmiş ve bunların zamandan bagımsız degerlerine yakınsadıgı gösterilmiştir.

Zamana bağlı akışkanların sayısal çözümlemeleri için bir kapalı spektral yöntem ve dönen disk akışı ısı transferi problemine uygulaması

An implicit numerical integration scheme is proposed in the present paper for the solution of the transient flows. The numerical method is based on the spectral Chebyshev collocation technique in the direction normal to the disk and forward marching in time. Besides being free of the numerical oscillations caused by the discontinuities between the initial and boundary conditions inherent to the classical finite difference techniques, the devised technique benefits from the advantages of being robust, unconditionally stable, highly accurate and straightforward to implement. Unlike to the finite difference methods, it is also compact in the sense that it resolves the flow field without necessitating further transformations. The numerical algorithm developed here is applied to the classical time-dependent von Karman swirling flow due to a porous rotating disk impulsively set into motion which progresses into the well-known steady state after a long time. The energy equation is also treated by the method and the physical parameters of paramount interest as such the radial and tangential skin-friction coefficients, the torque and the rate of heat transfer from the disk surface are numerically calculated that are shown to approach their steady state counterparts.

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