Jiten C. Kalita

A transient higher order compact scheme for incompressible viscous flows on geometries beyond rectangular

In this paper, we propose an implicit high-order compact (HOC) finite-difference scheme for solving the two-dimensional (2D) unsteady Navier-Stokes (N-S) equations on irregular geometries on orthogonal grids. Our scheme is second order accurate in time and fourth order accurate in space. It is used to solve three pertinent fluid flow problems, namely, the flow decayed by viscosity, the lid-driven square cavity and the flow in a constricted channel. It is seen to efficiently capture both transient and steady-state solutions of the N-S equations with Dirichlet as well as Neumann boundary conditions. Apart from including the good features of HOC schemes, our formulation has the added advantage of capturing transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations. Detailed comparison data produced by the scheme for all the three test cases are provided and compared with analytical as well as established numerical results. Excellent comparison is obtained in all the cases.

A transformation-free HOC scheme for incompressible viscous flows past an impulsively started circular cylinder

In this paper, we present a higher order compact scheme for the unsteady two-dimensional (2D) Navier-Stokes equations on nonuniform polar grids specifically designed for the incompressible viscous flows past a circular cylinder. The scheme is second order accurate in time and at least third order accurate in space. The scheme very efficiently computes both unsteady and time-marching steady-state flow for a wide range of Reynolds numbers ( Re ) ranging from 10 to 9500 for the impulsively started cylinder. The robustness of the scheme is highlighted when it accurately captures the vortex shedding for moderate Re represented by the von Karman street and the so called α and s -phenomena for higher Re. Comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases, both qualitatively and quantitatively.

Higher Order Compact Simulation of Double-Diffusive Natural Convection in A Vertical Porous Annulus

Abstract 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.

Moffatt vortices in the lid-driven cavity flow

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.

Effects Of Clustering On The Simulation Of Incompressible Viscous Flows

Abstract In this paper, we investigate the effects of clustering on the simulation of incompressible viscous flows with special reference to the lid-driven cavity flow at Reynolds Number Re = 3200. Our study reveals that though clustering is helpful in resolving smaller scales, yet a bad choice of clustering parameters and grid sizes could lead to inaccurate solutions. Moreover we found that grid independence studies solely based on the the computed values of the flow variables at certain portions of the physical domain sometimes do not reflect the true accuracy of the solutions. However, judicious clubbing of the clustering parameter with the grid size not only produces computationally efficient and highly accurate numerical solutions, but also resolves the smaller scales accurately.

Finiteness of corner vortices

Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows, widely known as Moffatt vortices 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 addressing what might have gone wrong and pinpoint the shortcomings on the assumptions of the existing theorems on Moffatt vortices. We provide our own set of proofs for establishing the finiteness of the sequence of Moffatt vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a non zero 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. The centers of these vortices have been quantified by us as fixed points through Brouwer fixed-point theorem and boundary of a vortex as circle cell. With the aid of these new developments and making use of some existing theorems in topology along with some elementary concept of mathematical analysis, we provide several approaches to delve into this issue. All these approaches converge to the same conclusion that the sequence of Moffatt vortices cannot be infinite; in fact it is at most finite. Mathematics Subject Classification (2010). Primary 76D17; Secondary 37N10.

α-, β-phenomena in the post-symmetry break for the flow past a circular cylinder

In the existing literature, the so-called α- and β-phenomena have been reported only for the early stages for the flow past an impulsively started circular cylinder. The current study endeavours to explore the possible existence of these phenomena even in the later stages of the flow. The flow is computed using a recently developed compact finite difference method for the biharmonic form of the two-dimensional Navier-Stokes equations for a wide range of Reynolds numbers (Re). We establish that these secondary phenomena not only appear once the wake becomes asymmetric but also periodically during the post-vortex shedding period for Re = 1000. Further, the recently reported sub-α- and sub-β-phenomena for Re = 5000 at the tertiary level during the early stages of the flow could be identified even during the later stages of the flow as well. The formation of these tertiary structures has been explained through a detailed theoretical characterization of the topological aspects of the boundary layer separation....

A 4OEC scheme for the biharmonic steady Navier–Stokes equations in non-rectangular domains

Abstract Recently the biharmonic form of the Navier–Stokes (N–S) equations have been solved in various domains by using second order compact discretization. In this paper, we present a fourth order essentially compact (4OEC) finite difference scheme for the steady N–S equations in geometries beyond rectangular. As a further advancement to the earlier formulations on the classical biharmonic equation that were developed for Cartesian coordinate system, this scheme is capable of numerically solving the two-dimensional N–S equations using body fitted coordinate system. Despite the presence of extra derivative terms in the quasi-linear form of the biharmonic equation, our extended formulation continues to maintain its fourth order accuracy on a nine-point compact stencil. A spectral analysis on the scheme reveals its superior resolution properties. The formulation has been tested on fluid flow problems of varied complexities on different geometries which includes flow past an impulsively started circular cylinder and elliptic aerofoil with angles of attack. We present our numerical results and validate them with established numerical and experimental observations available in the literature; excellent comparison is obtained in all the cases.

Effect of boundary location on the steady flow past an impulsively started circular cylinder

In this paper, we study the effect of the outer boundary location for the flow past an impulsively started circular cylinder at low Reynolds numbers (Re). The assumed far field represented by the outer radius of the annular physical domain is thoroughly investigated along with that of the position of the cross-cut. We present numerical solutions for the steady-state Navier-Stokes equations that govern the incompressible viscous flows such as the one studied here. The simulations are carried out with the help of a recently developed fourth order accurate compact scheme. It is seen that length of the outer radius always affects the flow variables irrespective of the grid size and the cross-cut positioned at the left side of the cylinder yields more efficient computation than the one located on the right. Comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases, both qualitatively and quantitatively confirming the efficiency of the HOC scheme being used.

Moffatt eddies in the driven cavity: A quantification study by an HOC approach

Abstract In this paper, we establish Moffatt-likeness of the corner vortices in the lid-driven square cavity for Stokes flow and flow for moderate Reynolds numbers in the pre-asymptotic regime. The flow is computed by using an efficient transient Navier–Stokes (N–S) solver on compact non-uniform space grids. The quantification of the corner vortices in succession as Moffatt vortices follows from them following a fixed geometric ratio in sizes and intensities. In the process, we also provide more detailed benchmarking results for the corner vortices in the cavity flow. The accuracy of the scale resolution of the vortices has been verified by a novel approach to grid independence analysis. This approach utilizes the concept of adverse pressure gradients as a tool to validate that the separation zones in the neighborhood of the corners are consistent with the vortices obtained from our computed solution of the N–S equations.