BİR BOYUTLU TAŞINIM SÜREÇLERİNDE ÖLÇEKLEME ANALİZİ VE KENDİNE BENZEŞİM

Abstract

Konveksiyon-difüzyon denklemi nehirlerdeki kirleticilerin yayılması, çözülmüş maddenin haliç ve sahil sularına dağılımı, gözenekli ortamda akış ve taşınım, ve atmosferdeki kirleticilerin taşınması gibi yer bilimlerindeki çeşitli akım ve taşınım süreçlerini modellemek için yaygın bir şekilde kullanılmaktadır. Bu çalışmada tek boyutlu konveksiyon-difüzyon denkleminin kendine benzeşim koşulları tek parametreli Lie grubu nokta ölçeklendirme dönüşümleri kullanılarak araştırılmıştır. Sayısal simülasyonlarla, tek boyutlu noktasal kaynaklı taşıma sürecinin ölçeklendirilmiş bir mekanla özdeşleşebileceği gösterilmiştir. Ölçeklendirme parametresi veya uzunluk boyutunun ölçekleme katsayısı değiştirilerek daha büyük veya daha küçük mekansal boyutlarda taşınım sürecinin gerçekleştiği simetrik problemler elde edebilir. Lie grubu ölçeklendirme yaklaşımı ile elde edilen ölçeklendirme ilişkileri, farklı mekan ve zaman ölçeklerindeki taşınım süreçlerini anlamamızı kolaylaştırabilir ve bir boyutlu taşınımın önemli olduğu süreçlerinde fiziksel modellerin oluşturulmasında ilave esneklik sağlayabilir.

Keywords

• Lie grup değişim yöntemi,ölçekleme analizi,kendine benzerlik,konveksiyon-difizyon denklemi

Scaling Analysis and Self-Similarity of One-Dimensional Transport Process

Abstract

Convection-diffusion equation has been widely used to model a variety of flow and transport processes in earth sciences, including spread of pollutants in rivers, dispersion of dissolved material in estuaries and coastal waters, flow and transport in porous media, and transport of pollutants in the atmosphere. In this study, the conditions under which one-dimensional convection-diffusion equation becomes self-similar are investigated by utilizing one-parameter Lie group of point scaling transformations. By the numerical simulations, it is shown that the one-dimensional point source transport process in an original domain can be self-similar with that of a scaled domain. In fact, by changing the scaling parameter or the scaling exponents of the length dimension, one can obtain several different down-scaled or up-scaled self-similar domains. The derived scaling relations obtained by the Lie group scaling approach may provide additional understanding of transport phenomena at different space and time scales and may provide additional flexibility in setting up physical models in which one dimensional transport is significant.

Keywords

• Lie group transformations,scaling,self-similarity,convection-diffusion equation

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