Kalita for the award of the degree of Doctor of Philosophy and that this work has not been applied for a degree elsewhere. It is acknowledged that the work contained in the thesis entitled Higher Order Compact Schemes for Incompressible Viscous Flows on Non-Rectangular Geometries by Swapan Kumar Pandit, a student in the Department of Mathematics, Indian Institute of Technology, Guwahati, for the award of the degree of Doctor of Philosophy was completed under under our control and that this work has not been submitted elsewhere for a degree.

## Background

The leading term in the truncation error (TE) of this expansion determines the order of accuracy of the scheme. Furthermore, in practical situations, most flow phenomena require a refined mesh to accurately capture the flow physics.

## Motivation

Recently, another popular approach to find higher-order compactness has been to use the original differential equation to replace the leading terms of the truncation error (TE) of the standard central difference approximation. Spotz and Carey extended the idea in Mackinnon and Johnson to the eddy formulation of the steady-state stream function of the 2D Navier-Stokes (N-S) equations.

Objectives

## The Work

In this work, we firstly proposed a higher order scheme in the ψ(stream function)- ζ(vorticity) formulation of the Navier-Stokes equations to solve stable incompressible viscous flows. It is seen to efficiently capture steady-state solutions of the N-S equations with both Dirichlet and Neumann boundary conditions.

## Organization of the work

Over the past few decades, numerical schemes and simulation tools for incompressible flows, as a subset of the computational fluid dynamics (CFD) discipline, have made tremendous progress. Almost all of these efforts involving high-order compact schemes have used simple rectangular uniform space lattices for a number of reasons, such as the lack of curvilinear lattice techniques and the limited flexibility of HOC schemes.

## Mathematical Formulations and Discretization Procedure

### Governing Equations

The scheme can be applied to both steady-state convection-diffusion and reaction-diffusion equations, and can also be easily accommodated in N-S type solution equations with a small modification of the convection coefficients. With the introduction of the current function as given in equation (2.4), the conservation of mass is automatically satisfied.

### Transformation of the Governing Equations

By introducing the flow function into equation (2.5), a Poisson equation is found that relates the flow function and the vorticity. This ψ-ζ formulation ((2.6) and (2.7)) has major advantages over the primitive variable form: first, it automatically satisfies the continuity equation and second, it decouples the pressure calculation from the velocity calculation.

### Fourth-order accurate discretization scheme

It is seen that the expressions for the fourth-order derivatives (for example, see equation (2.15)) contain mixed derivative terms such as ∂4φb. If such a grid or transformation is considered, and central difference is used for derivatives, equation (2.11) can be written as.

## Solution of algebraic systems

To solve the N-S equations representing incompressible viscous flows using the proposed scheme, we used the ψ-ζ formulations and applied an outside-inside iteration procedure. For both the vorticity and flow function equations, BiCGStab is used, which forms the inner iterations.

## Numerical Test Cases

### Test case 1: Linear convection-diffusion

The exact solution to this problem is given by. 2.21) Due to the presence of large gradients near the top and right walls (at x = 1 and y = 1) of the problem domain (see the numerical solution in Figure 2.2(b)), we generate the grid (see Figure 2.2(a) )) in such a way that points are clustered in those regions. Here φ,φF,φM stand for exact solution, the solution on a fine grid and the solution on a coarse grid with half the number of points in each direction respectively.

### Test case 2: 2D N-S equations with a constructed solution

It can be noted that the calculated scale deviates from the fourth-order scale for uniform meshes (λ= 0) is due to insufficient points in the high-gradient regions in the problem domain.

### Test case 3: Lid-driven cavity problem

It may be mentioned that the development of the second major vortex (second primary vortex) is a function of the aspect ratio (K). It is noted that the maximum size of the first major vortex decreases with the increase in aspect ratio.

## Conclusions

In this chapter, the extension of the proposed HOC scheme in the previous chapter for steady flow to transient has been carried out. 45] and Kalita et al. [66] developed some HOC schemes on non-uniform grids for 2D convection-diffusion equations.

## Basic formulations and Discretization Procedure

### Transformation of the Governing Equations

To validate the scheme, it is first applied to a viscosity-reduced flow problem with analytical solutions and then to a classical lid-driven cavity problem.

### Discretization in the transformed plane

It is seen that the expressions for the fourth-order derivative (see e.g. equation (3.15)) contain mixed derivative terms such as ∂4φb. The initial and boundary conditions for the ψ-ζ formulations of these equations can be easily derived from the exact solution.

### Test case 2: The lid-driven cavity problem

Then, the size of the vortex in the lower right corner increases compared to the vortex in the lower left corner (see figure at t=4.0 and 5.0). Then another vortex slowly develops at the top of the left wall (see figure at t=240.0).

## Conclusions

In this chapter we look at the flow in a symmetrical constricted channel with different forms of receding angles in two forms, namely (i) the forward one [94, 95]. We discuss in detail how the non-uniform channel geometry with a re-entry angle with different shapes affects the flow in a limiting way.

## Governing equations

The first is formed as a result of contraction, and the second is a result of expansion. We also present numerical results for the flow through symmetrical nonuniform rigid channel with varying degrees of constriction, either forward or backward, over a wide range of Reynolds numbers (1 ≤ Re ≤ 1000 and 1 ≤ Re ≤ 500 are studied for forward and backward constrictions respectively channel).

Boundary Conditions

## Mesh Structure

2τ where 2ri, 2ro are respectively the inlet and outlet heights of the confined channel and τ is a parameter that controls the smoothness as well as sharpness of the constriction: an increasing value of τ indicates a sharp angle. But a slight variation in the value of τ can drastically change the sharpness of the angle, as can be seen for τ = 0.9 and τ = 1.0.

## Discretization and Related Issues

The different shapes of the upper boundary due to the above fixed inlet and outlet radii, and different choices of degrees of constriction sharpness are shown in figure 4.3. NOTE: In addition to the mesh pitch and tolerance of the iterative procedure, the position of the upstream and downstream boundary affects the accuracy of the numerical solution.

## Results and Discussions

### General flow behavior study

In the following two subsections, we have presented results of numerical solutions of the N-S equations for unsteady flow in a nonuniform rigid channel with a forward or backward constriction with a smooth corner of increasing sharpness. A comprehensive set of model cases has been considered to study the numerical and physical aspects of the flow.

### Flow in a channel with forward constriction

It shows the existence of several high-gradient regions around the throat in the flow domain. Detachment and reattachment points for different Res are shown in Table 4.3 for τ = 0.9 and τ = 1.0, respectively.

### Flow in a channel with backward constriction

From this table it can be seen that in each model the center of the vortex moves downstream with the increase of Re. This is due to the fact that after the throat, the upstream flux in the forward constricted channel decreases significantly downstream due to contraction.

## Conclusions

In many channel flow systems, the development of the recirculation region due to a sudden non-uniform large-scale expansion of the flow transitions (i.e. flow phenomena in an asymmetric channel with backward pinching) plays an important role, which is of practical and theoretical interest. The need for a better understanding of the effects of expansion ratios and corner sharpness on the asymmetric back-tightening channel is indicated.

## The problem

The present study in combination with our previous work focused on the transient flow analysis for the asymmetric channel and the robustness of our proposed scheme in section 3. We present the mesh distributions in the physical plane and the computational plane as in Figure 5.2(b) and (c) respectively.

Discretization and Related Issues

## Results and Discussions

### Expansion ratio 1:2

It can also be seen that for a fixed value of τ, asReincreases, the length of the separation region increases. For Figure 5.7 (c0), the minimum shear stress value of the wall is at the cutoff points for allRes, which could be due to the severe sharpness of the corner.

Expansion ratio 1:4

## Conclusions

It is well known that hemodynamics plays an important role in the genesis and progression of arterial diseases. Flow in a dilatable channel provides an idealization of flow through an aneurysmal vessel.

## Numerical procedure

### The problem

We have focused on numerical solutions of the flow characteristics in different lateral dilated channels and also on those in symmetric non-uniform rigid channels with different degrees of dilation, as there is limited literature available on these flow geometries. The symmetry conditions are imposed along the horizontal centerline of the symmetric extended channel, which is read as .

Mesh structure

## Results and discussions

### Lateral dilated channel

We can see that for all models the changes in wall shear stress values mostly occur around the neck of the expanded region. We see that for a wide range of aperture length (about −4.0≤ x≤4.0), two counter-rotating vortices develop, which is not observed in previous models.

### Symmetric dilated channel

It should be mentioned that the velocity decreases in the region around the maximum area of the expanded channel section. In figure 6.16 we have shown the axial velocity profile at the maximum cross section (x= 0) of the dilated channel.

## Conclusions

Since the flow in a constricted tube is an ideal case of the flow through a constricted vessel, the theoretical study of the flow in a constricted tube can provide some basic information and predictions about the flow through a constricted artery. It is very difficult to find an analytical solution to the problem of fluid flows through a constricted pipe.

## Numerical procedure

### The problem

Therefore, in the present study, we have considered the flow as Newtonian and the pipe as rigid. Keeping the physics of the flow unchanged, this singularity has been carefully treated.

### Flow geometry and Mesh structure

We experimented numerically and, accordingly, we found the minimum inlet distance from the throat of the constricted tube, which is z ≈ −10, so that the stenosis effect at the inlet is negligible. And to get a fully developed flow at the outlet for Reynolds numbers up to Re= 1000 we experimented numerically and found a common minimum distance from the throat as z ≈ 30.

## Results and discussions

In figure 7.6, the streamline contours for Re= 200 are presented for different area reductions with the same stricture lengths (−1≤ l ≤ 1). The variation of the axial velocity along the centerline is clear from figure 7.8 for different degrees of area reductions.

## Conclusions

Some prominent observations of the flow field in the backward constricted channel compared to the same in the forward constricted channel are as follows. One of the important aspects when investigating the flow problem of dilated channels is the effects of the degree of dilation.

## Scope for future work

In view of the successful implementation of our schemes for a wide range of problems with irregular geometric settings, these schemes can be treated as the most general HOC schemes for incompressible viscous flows. However, the formulations of the proposed HOC schemes presented herein are limited to orthogonal grids only.