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The current study envisages to investigate numerically, probably for the first time, the combined effect of a circular obstacle and medium motion on the dynamics of a stable rotating spiral wave. A recently reconstructed spatially fourth and temporally second order accurate, implicit, unconditionally stable high order compact scheme has been employed to carry out simulations of the Oregonator model of excitable media. Apart from studying the effect of the stoichiometric parameter, we provide detailed comparison between the dynamics of spiral waves with and without the circular obstacles in the presence of straining effect. In the process, we also inspect the dynamics of rigidly rotating spiral waves without straining effect in presence of the circular obstacle. The presence of the obstacle was seen to trigger transition to non-periodic motion for a much lower strain rate.

All the previous studies on the cavity flow are confined to either the study of its 2D or its 3D configuration in isolation. In this study, we endeavour to gain some physical insight into the corner vortices from the perspective of the flow topology in the 2D vis a vis 3D driven cavity by employing some recent developments in the field of topological fluid dynamics. The computed flow is post-processed to identify critical points in the flow field leading to the prediction of separation, reattachment and vortical structures in the flow. The limit cycles in the plane of symmetry of the 3D flow representing the vortices are found to be stable ones. The Poincaré–Bendixson formula is used to validate the computed flow, i.e., the possible number of critical points in the 2D cavity identified by us from the computation. The topology of the corner vortices in actual 3D flow and its 2D idealization has also been compared in detail.

Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by diagnosing the assumptions of the existing theorems on such vortices. We provide our own set of proofs for establishing the finiteness of the sequence of corner vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a nonzero scale for the smallest vortex structure possible in incompressible viscous flows. We point out that the notion of infiniteness resulting from discrete self-similarity of the vortex structures is not physically feasible. Making use of some elementary concepts of mathematical analysis and our own construction of diametric disks, we conclude that the sequence of corner vortices is finite.

Studies on the $\\alpha $ - and $\\beta $ -phenomena, terms coined by Bouard & Coutanceau (J. Fluid Mech., vol. 101, 1980, pp. 583–607) for the flow past an impulsively started circular cylinder, have been confined only to the very early stages of the flow. In this paper, besides making a comprehensive in-depth analysis of these phenomena for a much longer period of time, we report the existence of some tertiary vortex phenomena for the first time, which we term the sub- $\\alpha $ - and sub- $\\beta $ -phenomena. The mechanism of unsteady flow separation at high Reynolds numbers for the flow past a circular cylinder developed in the last two decades has been used to understand these flow phenomena. The flow is computed using a recently developed compact finite difference method for the biharmonic form of the two-dimensional Navier–Stokes equations for the range of Reynolds number $500\\leq \\mathit{Re}\\leq 10\\hspace{0.167em} 000$ . We specifically choose $\\mathit{Re}= 5000$ to describe the interplay among the primary, secondary and tertiary vortices leading to these interesting vortex dynamics. We also report a $\\beta $ -like phenomenon which is very similar to the $\\beta $ -phenomenon, but slightly differs in details. We offer a new perception of the $\\alpha $ -phenomenon by defining its existence in a strong and weak sense along with a clearer characterization of the $\\beta $ -phenomenon. Apart from numerical computation, a detailed theoretical characterization using topological aspects of the boundary layer separation leading to the secondary and tertiary vortex phenomena has also been carried out. We compare our numerical results with established experimental and numerical results wherever available and an excellent match with the experimental results is obtained in all cases.

In incompressible viscous flows in a confined domain, vortices are known to form at the corners and in the vicinity of separation points. The existence of a sequence of vortices (known as Moffatt vortices) at the corner with diminishing size and rapidly decreasing intensity has been indicated by physical experiments as well as mathematical asymptotics. In this work, we establish the existence of Moffatt vortices for the flow in the famous Lid-driven square cavity at moderate Reynolds numbers by using an efficient Navier-Stokes solver on non-uniform space grids. We establish that Moffatt vortices in succession follow fixed geometric ratios in size and intensities for a particular Reynolds number. In order to eliminate the possibility of spurious solutions, we confirm the physical presence of the small scales by pressure gradient computation along the walls.

In this paper, we propose a high order compact scheme for the unsteady two-dimensional reaction diffusion equations which is second order accurate in time and fourth order accurate in space. The scheme is specifically designed to tackle problems in pattern formation arising frequently in Mathematical Biology, modeled by the Gray–Scott model. By converting the reaction–diffusion equations into a pure diffusion equation, we obtain an unconditionally stable convergent implicit scheme. Though originally designed to capture the patterns generated by the reaction–diffusion equations, the scheme serves a dual purpose by efficiently capturing the incompressible viscous flows governed by the unsteady Navier–Stokes equations with equal ease. To validate the scheme, it is firstly applied to the famous lid-driven square cavity flow problem and then to two problems on pattern formation. Our computed results are compared with existing numerical ones and excellent match is obtained in all the cases.

Studies on the$\\alpha $- and$\\beta $-phenomena, terms coined by Bouard & Coutanceau ( J. Fluid Mech. , vol. 101, 1980, pp. 583–607) for the flow past an impulsively started circular cylinder, have been confined only to the very early stages of the flow. In this paper, besides making a comprehensive in-depth analysis of these phenomena for a much longer period of time, we report the existence of some tertiary vortex phenomena for the first time, which we term the sub-$\\alpha $- and sub-$\\beta $-phenomena. The mechanism of unsteady flow separation at high Reynolds numbers for the flow past a circular cylinder developed in the last two decades has been used to understand these flow phenomena. The flow is computed using a recently developed compact finite difference method for the biharmonic form of the two-dimensional Navier–Stokes equations for the range of Reynolds number$500\\leq \\mathit{Re}\\leq 10\\hspace{0.167em} 000$. We specifically choose$\\mathit{Re}= 5000$to describe the interplay among the primary, secondary and tertiary vortices leading to these interesting vortex dynamics. We also report a$\\beta $-like phenomenon which is very similar to the$\\beta $-phenomenon, but slightly differs in details. We offer a new perception of the$\\alpha $-phenomenon by defining its existence in a strong and weak sense along with a clearer characterization of the$\\beta $-phenomenon. Apart from numerical computation, a detailed theoretical characterization using topological aspects of the boundary layer separation leading to the secondary and tertiary vortex phenomena has also been carried out. We compare our numerical results with established experimental and numerical results wherever available and an excellent match with the experimental results is obtained in all cases.

In this work, we present an optimization strategy for implementing the BiCGStab iterative solver on graphic processing units (GPU) for computing incompressible viscous flows governed by the unsteady Navier–Stokes (N–S) equations on a CUDA platform. A recently developed ψ –v formulation is used to discretize the biharmonic form of the N–S equation. Special emphasis was given on optimizing the matrix-vector multiplication and the dot product kernels in GPU as the bulk of the computational efforts was seen to be occupied by these two kernels. The parallel code was implemented on the benchmark problem of lid-driven square cavity flow and remarkable speed-up up to 20 times was achieved on finer grids. The GPU implementation enabled us to compute the flow on extremely fine grids and very small scales were resolved with remarkable accuracy.

In the last few decades, the Higher Order Compact (HOC) finite difference schemes are gaining momentum in the fluid dynamics community because of their high accuracy and advantages associated with compact difference stencils. However, in most of the cases, their application is seen to be limited solely to the computation of fluid flows and in some cases to problems of heat transfer only. The present work is perhaps first in the direction where an HOC algorithm has been extended to a problem of combined heat and mass transfer. A recently developed higher-order compact (HOC) scheme of fourth order spatial and second order temporal accuracy is employed to carry out simulation of double-diffusive natural convection in a vertical porous annulus between two concentric cylinders maintained at constant temperatures and concentrations. Flow is investigated in the regime -50≤N≤50, 1 ≤A≤10, 1≤κ≤50, 0≤Ra≤1000 and 1≤Le≤500, where N, A, κ, Ra and Le are the buoyancy ratio, aspect ratio, radius ratio, thermal Rayleigh number and the Lewis number respectively. Comparison is made with established numerical results and very good comparison is obtained on relatively coarser grids.

In this study, we investigate the rectilinear displacement and deformation of a highly viscous, miscible circular blob influenced by a less viscous fluid within a homogeneous porous medium featuring physically realistic no-flux boundaries. We utilize a fourth-order accurate compact finite difference scheme for the spatial discretization of the nonlinear partial differential equations that govern this phenomenon. The resulting semi-discrete equations are then integrated using the second-order Crank-Nicolson (CN) method. We conduct numerical simulations for a Péclet number (\\(Pe 3000\\)) and a log-mobility ratio \\(0 R 7\\), which reveal three distinct pattern formations: comet-shape, lump-shape, and viscous fingering instability. Our results demonstrate that the deformation, spreading, and mixing of the blob vary non-ideally with both \\(Pe\\) and \\(R\\), a behavior attributed to the blob's initial curvature. Consequently, enhanced mixing can be achieved at intermediate values of \\(Pe\\) and \\(R\\), suggesting the existence of an optimal mixing condition. These findings have significant implications for fields such as oil recovery, CO\\(_2\\) sequestration, pollution remediation, and chromatography separation.