#### Date of Submission

2-28-1993

#### Date of Award

2-28-1994

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Physics and Applied Mathematics Unit (PAMU-Kolkata)

#### Supervisor

Mazumdar, Himadri Pai (PAMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

The smooth flow of a fluid has sprung many surprises. A flow which at an instant of time is quite regular and orderly could produce on the slightest of disturbance a complex bewildering varieties of flows, broadly termed as turbulence. Direct numerical simulation of the Navier-Stokes equations have shown that it is quite possible that these turbulent flows are solutions of the Navier-Stokes equations. In fact it is by now well recognized that many non-linear systems produce chaos quite similar to turbulence. However the large number of scales and their complex interactions involved make turbulence difficult to understand. Direct numerical simulation of the 3D Navier- Stokes equations has been achieved for only low Reynolds number (Kim et. al. (1987)) and the prospects in the near future of a full scale simulation for a flow problem of practical interest seems quite bleak (Reynolds(1989)). Moreover the initial promise that chaos theory (Ruelle(1989)) could explain turbulence has faded away. The strong churning action and bursts which are found in turbulence are generally missing in the models of chaos.Most of the efforts in turbulence have been made towards obtaining better quantitative models. However the purpose of modeling is not only to predict the technologically relevant quantities but also to gain insight into the problem. It is difficult to define the word insight here. To quote Jimenez et. al. (1981),Apart from a certain amount of esthetic appeal that makes you feel you have understood a certain situation in a certain case, insight has something to do with minimum complexity and with the facility of including new features ...The study of turbulence as a history of flow events and patterns has yielded important insights. The importance of organized structures in the study of turbulence was realized in 1950's and later on, the flow visualization experiments by S.J.Kline and his co- workers (Runstadler et.al.(1963), Kline et.al.(1967), Kim et.al.(1971)) established the importance, of these structures for turbulent boundary layers. Brown and Roshko(1974) established the significance of organized structures in mixing layers. Recently Aubry et.al. (1988) have made an attempt to study the evolution of certain spatial modes obtained from Galerkin projection of the Navier-Stokes equation onto certain coherent structure functions obtained by what is known as the method of Proper Orthogonal Decomposition (Lumley(1970)). This study has established a bridge between Navier- Stokes equations and the dynamical systems theory. The motivation behind the present thesis work is the same as that of Aubry et.al.(1988), but our approach is quite different. Unlike Aubry et.al.(1988) we do not use the Fourier decomposition, instead we use the decomposition in terms of short waves, i.e., functions with rapidly vanishing tails.The study of the solutions of the non-linear equations in terms of short waves or wavelets has certain advantages over their study in terms of long waves that are ob- tained by Fourier decomposition. Some of the coherent phenomena in one dimension like solitons and shock waves form such short waves with rapidly vanishing tails, as we shall point out in Chapter 3 of this thesis.

#### Control Number

ISILib-TH248

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Venkatesan, S K. Dr., "Discrete Singularity Method and its Application to Incompressible Flows." (1994). *Doctoral Theses*. 184.

https://digitalcommons.isical.ac.in/doctoral-theses/184

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842961