Kalita, Professor, Department of Mathematics, Indian Institute of Technology Guwahati for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for a degree. I take this opportunity to express my gratitude to all the faculty members of the Department of Mathematics who have provided direct or indirect help at various stages during my research work.

## General background

The geometric theories of incompressible viscous flows are also based on a divergence-free vector field that is very much part of the NS equations for incompressible viscous flows. Below we provide an overview of the motivation, objectives and a brief summary of the current work.

## Motivation

In particular, the physical problem of the formation and evolution of vortices in fluids, the N–S equations provide a framework for investigating different facets of the flow. Most of our observations are based on the numerical solution of the N–S equations in its two- or three-dimensional form and application of the recent developments of topological fluid dynamics.

## Objectives

Although most studies have agreed on the existence of Taylor-G¨ortler-like vortices (TGL) at moderate Reynolds numbers, the investigation of their topological aspects appears to be far from complete. In the process, we seek to bridge the gap between two opposing factions of thought by pinpointing what might be wrong with the assumptions of existing theories underlying the infinity conclusion.

## The work

The topological characteristics of corner vortices in a real 3D flow and its 2D idealization in a cavity were also compared. The second part of the work consists of the development of some theories on Moffatt vortices in incompressible viscous flows in bounded domains.

## Organization of the thesis

We provide our own set of evidence to establish the finiteness of a sequence of Moffatt eddies using the continuum hypothesis [12, 146] and the Kolmogorov length scale, which provide a non-zero scale for the smallest possible eddy structure in incompressible viscous flows. In Chapter 6, we establish the finiteness of Moffatt vortices, which in the existing literature has always been synonymous with the existence of an infinite sequence of angular vortices.

## Introduction

The aim of the present study is to rigorously investigate the prospect of quantifying the angular vortices in the famous lid-driven square cavity problem as Moffatt: whether it exists only for Stokes flow or also for flows with moderate and high Reynolds numbers . As such, the present study is more of an attempt to establish the Moffatt similarity of the vortex sequence in the pre-asymptotic regime.

## Problem description

Momentum in the x-direction dissipates transversely to the interior of the cavity due to viscosity. Thus, a sequence of vortices, starting from the main cavity-sized vortex to smaller ones, begins to develop at high Reynolds numbers.

## Basic formulations and numerical procedures

*The Numerical scheme**Solution of algebraic systems**Clustered grid generation**Approximation of the vorticity boundary conditions*

The importance of the parameter λ can be measured from figure 2.2(d) where one sees the. The boundary conditions for velocity on the top wall of the cavity are given by u= 1, v = 0.

## Grid independence and benchmarking

### Grid independence analysis

Furthermore, the overlap of the graphs for grids of size 161×161 and 321×321 clearly indicates the grid independence of our results. However, as noted by Kalita [71], grid independence studies based solely on the calculated values of the flow variables in certain parts of the physical domain sometimes do not reflect the true accuracy of the solution.

### Pressure gradient observation

Note that the same trend is observed on all three grids used in the calculation. For the vortices in the left corner, BL1 indicates the secondary vortex that is the first one to appear in the row.

### Quantification of Moffatt vortices

In figure 2.12 we present the three corner vortices captured by our calculation for Stokes flow in which one can clearly see the symmetry of the respective vortices at the left and right corners around the vertical center line. Note that in Moffatt's original paper [106], an asymptotic limit of the ratio of the size and intensity of the angular vortices was established only for Stokes flow.

## Self-similarity of Moffatt vortices

A close look at Tables 2.2 and 2.3 shows that for Re= 3200 neither the intensity nor the size of the vortices maintains a fixed ratio. The similarity in rotation of these vortices can be deduced from the fact that the vorticity curves show the same pattern over the range where it retains the same sign, albeit as a mirror image of the previous curve in the.

Conclusion

## Introduction

To the best of our knowledge, no study of cavity flow in a 2D versus 3D context is available in the literature. We also present an overview of 2D vis a vis 3D flows in the context of a lid-driven cavity.

## Critical points in the flow field

We now present the topology of corner vortices from our 2D calculations for Re = 1000 in figure 3.3(a) and the flow in the symmetry plane of the 3D lid-driven cavity in figure 3.3(b). A list of the critical points found by our calculation for Re= 1000, both for the pure 2D flow calculation and the flow in the plane of symmetry in the 3D calculation, is presented in table 3.1.

## Topological Implications

### Limit cycles in the lid-driven cavity flow

We present the traces of the limit cycles (the dark solid closed curves in Figure 3.8) in the symmetry plane in the lower right corners for the steady state flow in the 3D lid-driven cavity for Re = 1000. In steady 3D flows, the limit cycles in the normal plane with the vortex centers at the foci are stable.

### Critical points and the Poincar´e-Bendixson formula

As such, the Poincar'e-Bendixson formula will always hold in the 2D lid-controlled cavity flow. It is interesting to note that the critical points resulting from the 3D calculation (see Table 3.1) also obey the formula.

### Topological evolution of corner vortices

To the best of our knowledge, in light of [119], no previous research on lid-driven cavity flow (both 2D and 3D) has been conducted. A close-up of this vortex and the secondary vortex in the lower right corner of our 2D steady-state calculation for Re= 1000 is shown in Figure 3.11.

## Conclusion

Bendixson's formula to validate the calculated flow in the two-dimensional cavity, i.e., the possible number of critical points resulting from our calculation follows the formula. In the next chapter, we will discuss in detail the topological aspects of vortical structures in 3D cap-driven rectangular cavities.

## Introduction

Furthermore, no detailed topological studies for higher Reynolds numbers (Re) are available in the existing literature. The main objective of this workshop was to confirm the number of TGL vortex pairs in the transverse direction.

## Problem description and numerical methodology

Computer modeling of the flow in the rectangular cavity requires numerically solving the above-mentioned 3D N-S equations. To accurately capture the vortex structures, extreme clustering was applied near the solid boundaries.

## Preliminaries

*The concept of limiting streamlines/skin friction lines . 63**Mathematical criteria of separation and attachment**Topological rules**Vortex identification: λ 2 criterion*

If the roots C1, C2, and C3 are real and of the same sign, the critical point is a node. Neglecting these effects, the gradient symmetric part of the N-S equations can be obtained in the following form.

## Results and discussions

### Flow topology for Re = 1000

A graph of distinct critical points on the five solid walls of the cavity is provided in figure 4.7. A snapshot of the swirling motion of the fluid particles around the vortex core is presented in figure 4.11.

### Topology of Taylor-G¨ortler-like vortices

We note that U-shaped vortex structures are present along the broad direction of the cavity. Our study was also able to identify mushroom-shaped structures along the cavity space.

Conclusion

## Introduction

### Some essential topology

An intuitive notion of a topological mapping of the plane onto itself is delineated by Andronov et al. Any topological mapping of the plane itself is either a deformation of the above type (without tearing and folding) or a mirror reflection of the plane followed by such a deformation.”.

### Geometric theories of incompressible viscous flows

Therefore, if all ∂-singular points ofu in ∂M are non-degenerate, then the number of ∂-singular points of u is finite. A point p ∈ Ω of v (flow field) is called a boundary saddle Ω (half-saddle) if there are only three orbits connecting the pin Ω.

### Some essential theories of limit cycle

118] The set of all ω-limit points of a path Γ is called the ω-limit set of Γ and it is denoted by ω(Γ). The set of all α-boundary points of a path Γ is called the α-boundary set of Γ and it is denoted by α(Γ).

## Topological equivalence class of Moffatt vortices

### Equivalence class in terms of orientation

The members of the sequence of Moffatt eddies are of two types: (a) eddies with positive orientation (counterclockwise rotation), (b) eddies with negative orientation (clockwise rotation). Any two members in the odd sequence are (topologically) equivalent for both classes of Moffatt vortices.

### Equivalence class in terms of the nature of critical points 97

Below we give some of our newly developed theories of Moffatt vortices that would be used in the next chapter to prove our hypothesis about the finiteness of Moffatt vortices in the light of geometric theories of incompressible viscous flows.

### Centers of Moffatt vortices: topological fixed points and

Therefore, in the present case the center of the vortex Ci is the only possible one. Let Ci be the center of the i-th vortex Vi in the sequence of Moffatt vortices.

## Conclusion

However, the visualization of such vortices revealed the existence of only a few of them in the corner. Based on the newly developed theories by us in the previous chapter and some recent developments in geometric theory of incompressible viscous flows [36, 97], and using some elementary mathematical analysis [123], we prove that the sequence of Moffatt vortices in fluid flow around solid corners is finite 1.

## The backdrop

The mathematical origin of the infiniteness of Moffatt

### The concerns over infiniteness

At a certain distance from the source of the given force, no effect of it will be felt. Thus, the process of vorticity formation will end long before it reaches the corner.

### The unanswered questions

Note that the smallest length scale for incompressible viscous flows occurs in the turbulent regime. As such, the size of a vortex considered in the present study cannot fall below the Kolmogorov length scale.

## The notion of infiniteness: diagnosing the assumptions

So even if the vortices reach a molecular level, this minimum wavelength cannot be zero. As such, the largest number of vortices that have a one-to-one correspondence with the oscillating waves must be finite.

## Proof of finiteness of Moffatt vortices

Case-II: When β = 0, then the diameter of the extreme smallest vortex below the Kolmogorov length scale (see section 6.2.3.1), η (> 0) must decrease to have infinite number of vortices in the flow domain. ), feasible length scale to measure fluid vortices. We have already defined the largest neighborhood of the center of a vortex as a circular cell.

## Conclusion

Such a scenario can be seen from figure 6.2, where we present the vortex structure around the vortex core line from our own simulation of. In the next chapter, we use the same ideas to prove the finiteness of vortices for the more generalized case, namely in a bounded domain.

Introduction

## Finiteness of vortices

Therefore, the finiteness of the system of vortices is actually independent of the geometry of the flow domain. The instantaneous vortex will be attached to the fixed boundary, whereas the former will be pushed into the downstream of the flow domain.

## Conclusion

The second proof suggests that even if the flow domain is not simply connected (contains holes), the finiteness of the number of vortices is still guaranteed. The second proof of finiteness of vortices ensures that vortices correspond to the same separation point.

## Concluding remarks

Separation, reattachment and vortex structures in the flow were analyzed by post-processing the calculated flow through the identification of the critical points. The topological structures near these critical points have been explained in detail.

## Scope for future works

Our observations are consistent with recent developments in geometric theories of incompressible viscous flows. Geometrical theories of incompressible viscous flows as a discipline of topological fluid dynamics mainly revolve around the existence of critical points in the flow field.

## The problem and the governing equations

Due to the presence of the pressure term in the above equations (A.1-A.3), its direct solution is difficult to obtain. Due to the movement of the upper cover, the liquid in contact with it is set in motion and evolves inside the cavity.

## Numerical procedures

The scheme used

Associated algebraic systems and its solution

Strategy for boundary conditions

## Numerical results

Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Details of the start-up evolution of the Taylor-G¨ortler-like vortices inside a square lid driven cavity for 1000 ≤ Re ≤ 3200.