Using Symbolic Computing to Study Mass Transport of Interfacial Waves in Lagrangian Description

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Project title

Code/計畫編號

MOST105-2221-E022-008

Translated Name/計畫中文名

以符號計算解析拉格朗日描述下之內波質量傳輸

Project Coordinator/計畫主持人

Chia-Cheng Tsai

Funding Organization/主管機關

National Science and Technology Council

Co-Investigator(s)/共同執行人

許弘莒

Department/Unit

Department of Marine Environmental Engineering,NKUST

Website

https://www.grb.gov.tw/search/planDetail?id=11890456

Year

2016

Start date/計畫起

01-08-2016

Expected Completion/計畫迄

31-07-2017

Bugetid/研究經費

903千元

ResearchField/研究領域

土木水利工程

近年來，由於南海內波的觀測證據，與其海水混合及質量傳輸特性，內波的課題，已經成為一項熱門的研究領域。內波的成因，包含冰山融化在海水表層與水溫躍層等現象。我們知道，當非線性波浪行進時，會在波行方向產生質量傳輸現象，稱為史托克斯漂移。在本季畫中，我們將考慮非線性內波的質量傳輸現象，並同時考慮自由液面的效應。內波的質量傳輸，通常也會伴隨著沉積物、營養鹽、浮游生物、污染物質等的傳輸，了解其物理機制，有很高的研究價值，也是本計畫之主要目的。在本計畫中，我們將推導非線性內波問題之理論解。使用微擾法配合拉格朗日內波介面條件，我們將推導三階拉格朗日的內波解，觀察解答的型態，我們亦將撰寫自動化符號計算程式，以獲得五階拉格朗日的內波解，我們的解答將包含拉格朗日漂移、角頻率、波揚(降)等，我們也將探討這些物理量在內波介面的不連續現象。同時，我們也將驗證拉格朗日解的收斂性優於相對應的尤拉解。最後，我們也將發展四元樹自調適直接數值模式，並比較理論與數值模式的粒子軌跡與質量傳輸，並探討數值質量傳輸受不規則底床之影響。總的來說，我們將發展自動化符號計算程式，用以解析內波的質量傳輸。執行計畫的第一年，我們將直接解拉格朗日控制方程式；第二年，我們將發展尤拉－拉格朗日轉換；最後一年，我們將發展四元樹自調適直接數值模式，並與前述解答交叉比對。 Internal waves are ubiquitous in the interior of the oceans and have been the focus of much attention in recent years due to their role in oceanic mixing and mass transport. Internal waves can be observed in a variety of ways, one is by having a layer of fresh water over a layer of salt water such as the situation that the fresh snow water melts into the ocean. Another situation to the formation of internal waves occurs at the thermocline. It is well known that non-linear progressive surface waves generate a net mass transport in the propagation direction, which is known as the Stokes' drift. In this project, we consider the similar problem for the interfacial water waves with a free surface on a two-layer-fluid system. This transport of water occurs along with suspended mass such as sediment, nutrients, larvae, as well as contaminants. The details of such effects are crucial to the understanding of a wide range of physical situations. In this project, we provide a theoretical study on the interfacial water waves. Lagrangian matching conditions are introduced for the physical fields separated by the interface. And a perturbation analysis is carried out to the third order to find the particle trajectory in the Lagrangian description. Observing the derived solution, symbolic computations are introduced for obtaining the fifth order solution. The Lagrangian drifts, wave frequency, and set-up are also given as part of the solutions. Discontinuities across the interface will be discussed for all of these physical quantities. Better convergence for the Lagrangian solution than the Eulerian one is numerically demonstrated for the free surface dominated wave motion. In addition, the particle trajectories and mass transports will be compared with the results of the direct numerical simulation by Navier-Stokes equation, which will be accomplished by a quadtree-adaptive code with the volume of fluid method for both of the free and interfacial surfaces. Furthermore, the numerical model will be extended to the situation when irregular bathymetry is considered. Overall, we will implement symbolic computations for obtaining analytical solution of internal waves in Lagrangian description. In the first year, solution will be obtained by directly solving the Lagrangian governing equation. And in the second year, a symbolic Euler-Lagrange transform will be implemented for obtaining the same solution. Finally, the obtained solutions will be compared with the results of computational fluid dynamics in the third year.