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. Then some more complicated flows are analyzed, where the biharmonic formulation of the Navier-Stokes (N-S) equations is used for the first time.
General Background
2 Introduction Finite difference methods are one of the simplest and oldest methods for the numerical solution of differential equations. Finite difference approximations to derivatives are mainly based on Taylor series expansion of variables at nodal points.
Motivation
Moreover, only a few HOC schemes have been developed so far for problems in gas dynamics (compressible flows), in particular for the Euler equations. The application of HOC schemes to these problems presents challenges and is full of many interesting possibilities.
Literature Review
To date, several HOC schemes have been proposed for the convection-diffusion equation, most of which deal very well with the two-dimensional (2D) incompressible NS equations. However, compact, higher-order schemes for the more general convection-diffusion-reaction equation are relatively scarce.
Objectives
These schemes were further trimmed down to solve the transient two-dimensional convection-diffusion-reaction equations, which are then applied to some of the well-known model problems of pattern formation in mathematical biology. This has subsequently been applied to some of the well-known 1D problems on Gas Dynamics, and excellent comparison with the existing results has been observed.
The Work
To test the efficiency of the proposed schemes, they are applied to unsteady three-dimensional convection-diffusion reaction problems with both constant and variable convective reaction coefficients and the results obtained are found to be in excellent agreement with the analytical solutions. To delve deeper into the fundamental aspects of the proposed scheme, dispersion-dissipation analysis and Fourier stability analysis are provided.
Organization of the Thesis
In this work, we will essentially follow two different approaches, ψ-ω and ψ-v formulation of the N-S equations. In most cases, numerical simulation of fluid flow problems has been performed by non-dimensionalizing the governing equation.
HOC scheme for the streamfunction-vorticity formulation
Note that all the derivatives and cross derivatives appearing in the RHS of equation (2.14) have O(h2) approximations. By substituting this type of O(h4) approximations for all the derivatives appearing in equation (2.14), we can arrive at the following fourth-order accurate HOC scheme for equation (2.1).
Compact scheme for the streamfunction-velocity formulation . 19
This formulation is known as the stream function and velocity formulation or the biharmonic formulation of the 2D Navier-Stokes equations. 56] to perform a global two-dimensional (2D) stability analysis of a two-sided 2D staggered cavity flow driven by cover 1.
The Problem
The Numerical Scheme
To discretize the governing equations, we use the HOC scheme developed by Kalita et al. The non-stationary two-dimensional convection-diffusion equation for the transport variable φ in a domain can be written as
Approximation of the vorticity boundary conditions
The next step is now to solve equation (3.19); since the coefficient matrix is not diagonally dominant in general, conventional solvers such as Gauss-Seidel cannot be used. For the inner iterations, the calculations were stopped when the norm of the residual vector r =B−Aφ(φ is either ψ or ω), which originates from equation (3.19) fell below 10−6.
The Algorithm
Steady state was assumed to be reached when the maximum error ω between two consecutive external time steps of iteration was less than the tolerance limit of 0.5×10−6. In cases where steady state is attainable, a time-marching strategy is adopted until a steady state is reached.
Validation of the code
Antiparallel motion
The corresponding vorticity contours for the same range of Reynolds number shown in Figure 3.3, where the solid and dashed lines correspond to positive and negative values of the vorticity. These graphs also confirm the symmetric pattern of the flow for the range of Reynolds numbers considered here.
Parallel motion
In Table 3.3 we present the strengths and locations of the tertiary vortices in the range 400≤Re≤3,200. 42 Global 2D stability analysis of the staggered cavity Table 3.3: Locations and vorticity values in the centers of tertiary vortices - parallel motion.
Hydrodynamic Stability Analysis
48 Global 2D Stability Analysis of Staggered Cavity Table 3.6: Critical parameters for antiparallel motion where the first Hopf bifurcation occurs. 52 Global 2D Staggered Cavity Stability Analysis Table 3.8: Parallel motion: Reynolds number depends on the eigenvalue closest to the imaginary axis (211×211 grid).
Conclusion
It is found that for the antiparallel motion, the flow loses its stability at a relatively low Eq at a much higher value 4075.482 for the parallel motion. For the unsteady flow, to study the asymptotic behavior of the flow, calculations were continued until a non-dimensional time t = 10,000 was reached.
Introduction
The Numerical Scheme
2fi,j (4.8) The velocities u and v can be obtained from equation (4.3) using fourth-order central difference approximations. 4(vi+1,j−vi−1,j) (4.10) Using forward differences for the time derivative, an implicit finite difference approximation to equation (4.6) can be written as.
Hydrodynamic stability analysis
The boundary conditions for the steady equation depend on the problem at hand, while the stability equations follow homogeneous boundary conditions. First, we use (4.8) to solve the stability equation (4.15) and obtain the value of ψ, which is then substituted into the stability equation (4.16) to find ˜ψ and λ.
Numerical examples
Simple lid-driven cavity flow on a square
The stability analysis of the simple LDC current was performed using two different grids with dimensions 129×129 and 501×501. It can be easily seen from the table that our estimated value of the critical Reynolds number is 8025.9.
Two-sided cross lid-driven cavity flow
80 A Biharmonic Approach for the Stability Analysis of Incompressible Viscous Flows Table 4.4: Locations and vorticity values in the centers of primary, secondary and tertiary vortices - transverse lid driven cavity flow. From the table it is clear that for the cross LDC flow the Rec value lies.
Flow past an inclined square cylinder
At the entrance of the computational domain, a Dirichlet boundary condition u = U∞, v = 0 is used, while at the exit, we have used a convective boundary condition such that ∂φ. We show the evolution of the drag and lift coefficients for Re= 40.7, 50 and 60 in Figure 4.18, which again depicts the eventual periodic flow patterns.
Conclusion
On the other hand, for the flow in the inter-LDC problem, our approach is perhaps the first attempt to understand the global 2D stability for this problem. This approach, due to the computational economy of the generalized eigenvalue problem, can be very useful in analyzing the global stability of more complex fluid flows.
Introduction
It is noted that these schemes well address problems governed by the transient 3D convection-diffusion reaction equations, including the 3D incompressible NS equations.
Basic formulation and numerical procedure
- Constant reactive and convective coefficients
- Variable reactive and convective coefficients
- The transient form of the convection-diffusion-reaction
- Implementation on N-S equations
If the convection coefficients p, q and r and the reactive coefficient e are constants, the derivatives of p, q, r and e appearing in equation (5.4) vanish. We thus have equation (5.9) as the HOC finite-difference approximation to the transient 3D convection-diffusion-reaction equation with fourth-order spatial accuracy.
Fundamental studies
Convergence analysis
Now, the proof of the above lemma follows from the definition of the inner product and the Cauchy-Schwarz inequality. 110 HOC schemes for the transient 3D convection-diffusion-reaction equations. 5.38) which ultimately shows that, according to the principle of mathematical induction, we have
Dispersion-dissipation analysis
Amplitude and phase errors can now be measured with the help ofkr and ki respectively. To analyze the distribution and dispersion errors of our proposed scheme, in figure 5.2, we have shown the plots of kr and ki against β2 and β respectively, keeping R = 1.
Stability analysis
Numerical test cases
- Problem 1
- Problem 2
- Problem 3
- Problem 4 : The cubical lid-driven cavity flow
However, for the 3D lid-driven cubic cavity, only a meager amount of numerical results are available in the existing literature. From these graphs it can be seen that for the cubic cavity the flow motion is reduced even at low Reynolds numbers, unlike the square cavity.
Conclusion
Introduction
With the advent of these models, there has been a tremendous growth in pattern design research. In recent years, there has been a rapid rise in the field of mathematical biology, and it quite prominently includes the study of pattern formation.
Code validation
138 Compact higher-order simulation of model formation in Mathematical Biology Table 6.2: Convergence result of the code validation problem at times= 1.0 on a 101×101 grid. Code validation: 139 Table 6.3: Error and convergence rate of the numerical scheme at times= 1s for the code validation problem.
Numerical Test Cases
The Gierer-Meinhardt model problem: Formation of spikes140
The initial conditions are given by the homogeneous steady state of the system and the boundary conditions are assumed to be periodic. It is remarkable that the cubic interaction produces stripe-like patterns, while the quadratic interaction favors dots [25].
Gray-Scott model problem (Pearson’s form)
In Figure 6.7, some labyrinth-type patterns were observed, while in Figure 6.8, we have shown some point-like patterns obtained from our numerical results. The new spots are clearly visible in Figure 6.9(c) and the full repetition of these new spots can be seen in Figure 6.9(d).
Conclusion
Introduction
In this chapter, we proposed a new family of HOC schemes for the Euler equations that are spatially accurate to the fourth order, and one of the schemes has a second-order temporal accuracy of 1. This results in a family of compact higher-order schemes for the one-dimensional (1D) Euler equations of gas dynamics.
Numerical Scheme
Now if △t is the time step, using equation (7.10) and applying forward differentiation for (∂φ)/(∂t) we get the following HOC approximation for the unstable equation (7.4). This gives us a family of compact fourth-order finite difference schemes for the unsteady 1D convective equation.
Numerical test cases
The 1D Converging-Diverging Nozzle Flow (de Laval
In the convergent part of the nozzle, the flow is locally subsonic, while in the divergent part of the nozzle it is supersonic. The governing equations of this problem can be derived from the integral form of the continuity, momentum and energy equations as given in [53].
Conclusion
Unlike the earlier proposed HOC schemes, the numerical results obtained by the present HOC schemes show almost no smearing over all the discontinuities. We will also highlight the scope for future work from the current study.
Concluding remarks
Richardson's extrapolation and Lagrange interpolation of the data were also performed which confirmed the theoretical rate of convergence and justified the critical Reynolds number. Extensive studies by means of phase level and spectral density analysis have also been carried out and results are presented for a wide range of Reynolds numbers to study the nature of the flow.
Scope for future work
