BİR BOYUTLU TAŞINIM SÜREÇLERİNDE ÖLÇEKLEME ANALİZİ VE KENDİNE BENZEŞİM

Year 2018, Volume: 23 Issue: 1, 235 - 246, 24.04.2018

Ali Ercan

https://doi.org/10.17482/uumfd.330886

Cited By: 1

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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