Tapan K. Sengupta

Error dynamics

The propagation of a signal in a continuous medium and the associated evolution of error is of prime importance in many applications of applied physics. There have been many efforts in analyzing error dynamics, using method attributed to von Neumann [1,2], that is readily applied for linear equations and in quasilinearized form for non-linear equations. The main assumption for linear problems is that the error and the signal follow the same dynamics. While this appears intuitively correct, the main aim behind this work is to show that this is not correct for discrete computing due to dispersion or phase error or when the numerical method is not strictly neutrally stable. We demonstrate the above with the help of the linear advection equation. For the analysis of space-time discretization schemes, the linear advection equation as a model that represents many flows and wave phenomena is used,

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.

Suppression of vortex shedding behind a circular cylinder by another control cylinder at low Reynolds numbers

Vortex shedding behind a cylinder can be controlled by placing another small cylinder behind it, at low Reynolds numbers. This has been demonstrated experimentally by Strykowski & Sreenivasan ( J. Fluid Mech . vol. 218, 1990, p. 74). These authors also provided preliminary numerical results, modelling the control cylinder by the innovative application of boundary conditions on some selective nodes. There are no other computational and theoretical studies that have explored the physical mechanism. In the present work, using an over-set grid method, we report and verify numerically the experimental results for flow past a pair of cylinders. Apart from providing an accurate solution of the Navier–Stokes equation, we also employ an energy-based receptivity analysis method to discuss some aspects of the physical mechanism behind vortex shedding and its control. These results are compared with the flow picture developed using a dynamical system approach based on the proper orthogonal decomposition (POD) technique.

Further improvement and analysis of CCD scheme: Dissipation discretization and de-aliasing properties

In this paper, we further analyze a combined compact difference (CCD) scheme proposed recently [T.K. Sengupta, V. Lakshmanan, V.V.S.N. Vijay, A new combined stable and dispersion relation preserving compact scheme for non-periodic problems, J. Comput. Phys. 228 (8) (2009) 3048-3071] for its dissipation discretization properties to show that its superiority also helps in controlling aliasing error for a benchmark internal flow. However, application of the same CCD method to study the receptivity of a boundary layer experiencing adverse pressure gradient is not successful. This is traced to the nature of the equilibrium flow where the better dissipation property is not helpful in the inviscid part of the flow, where the aliasing problems continue to persist. A further modification is proposed to the CCD method here to solve complex physical problems requiring information on higher order disturbance quantities - as in problems of flow receptivity and instability.

A new combined stable and dispersion relation preserving compact scheme for non-periodic problems

A new compact scheme is presented for computing wave propagation problems and Navier-Stokes equation. A combined compact difference scheme is developed for non-periodic problems (called NCCD henceforth) that simultaneously evaluates first and second derivatives, improving an existing combined compact difference (CCD) scheme. Following the methodologies in Sengupta et al. [T.K. Sengupta, S.K. Sircar, A. Dipankar, High accuracy schemes for DNS and acoustics, J. Sci. Comput. 26 (2) (2006) 151-193], stability and dispersion relation preservation (DRP) property analysis is performed here for general CCD schemes for the first time, emphasizing their utility in uni- and bi-directional wave propagation problems - that is relevant to acoustic wave propagation problems. We highlight: (a) specific points in parameter space those give rise to least phase and dispersion errors for non-periodic wave problems; (b) the solution error of CCD/NCCD schemes in solving Stommel Ocean model (an elliptic p.d.e.) and (c) the effectiveness of the NCCD scheme in solving Navier-Stokes equation for the benchmark lid-driven cavity problem at high Reynolds numbers, showing that the present method is capable of providing very accurate solution using far fewer points as compared to existing solutions in the literature.

Vortex-induced instability of an incompressible wall-bounded shear layer

The unsteady separated flow produced by a finite-core vortex on a plane shear layer is studied here as a vortex-induced instability. The mechanism of such an interaction, where the distance between the wall and the vortex is many times the local boundary layer thickness, is shown here by flow visualization and the solution of the unsteady Navier-Stokes equation. A new theory is proposed here, which is generic to the Navier-Stokes equation without any assumptions, that is based on growth of disturbance energy in time. A dynamical systems approach based on the proper orthogonal decomposition technique is used to provide a quantitative measure

A Comparative Study of Time Advancement Methods for Solving Navier–Stokes Equations

A qualitative and quantitative study is made for choosing time advancement strategies for solving time dependent equations accurately. A single step, low order Euler time integration method is compared with Adams–Bashforth, a second order accurate time integration strategy for the solution of one dimensional wave equation. With the help of the exact solution, it is shown that the presence of the computational mode in Adams–Bashforth scheme leads to erroneous results, if the solution contains high frequency components. This is tested for the solution of incompressible Navier–Stokes equation for uniform flow past a rapidly rotating circular cylinder. This flow suffers intermittent temporal instabilities implying presence of high frequencies. Such instabilities have been noted earlier in experiments and high accuracy computations for similar flow parameters. This test problem shows that second order Adams– Bashforth time integration is not suitable for DNS.

Onset of asymmetry : Flow past circular and elliptic cylinders

A computational study of the development of two-dimensional unsteady viscous incompressible flow around a circular cylinder and elliptic cylinders is undertaken at a Reynolds number of 10,000. A higher-order upwind scheme is used to solve the Navier-Stokes equations by the finite difference method in order to study the onset of computed asymmetry around bluff bodies. For the computed cases the ellipses develop asymmetry much earlier than the circular cylinder. The receptivity of the computed flows in the presence of discrete roughness and surface vibration is studied. Finally, the role of discrete roughness in triggering asymmetry for flow past a circular cylinder is studied and compared with flow visualization experiments at Re = 10,000

Optimal time advancing dispersion relation preserving schemes

In this paper we examine the constrained optimization of explicit Runge-Kutta (RK) schemes coupled with central spatial discretization schemes to solve the one-dimensional convection equation. The constraints are defined with respect to the correct error propagation equation which goes beyond the traditional von Neumann analysis developed in Sengupta et al. [T.K. Sengupta, A. Dipankar, P. Sagaut, Error dynamics: beyond von Neumann analysis, J. Comput. Phys. 226 (2007) 1211-1218]. The efficiency of these optimal schemes is demonstrated for the one-dimensional convection problem and also by solving the Navier-Stokes equations for a two-dimensional lid-driven cavity (LDC) problem. For the LDC problem, results for Re=1000 are compared with results using spectral methods in Botella and Peyret [O. Botella, R. Peyret, Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids 27 (1998) 421-433] to calibrate the method in solving the steady state problem. We also report the results of the same flow at Re=10,000 and compare them with some recent results to establish the correctness and accuracy of the scheme for solving unsteady flow problems. Finally, we also compare our results for a wave-packet propagation problem with another method developed for computational aeroacoustics.

A new compact scheme for parallel computing using domain decomposition

Direct numerical simulation (DNS) of complex flows require solving the problem on parallel machines using high accuracy schemes. Compact schemes provide very high spectral resolution, while satisfying the physical dispersion relation numerically. However, as shown here, compact schemes also display bias in the direction of convection - often producing numerical instability near the inflow and severely damping the solution, always near the outflow. This does not allow its use for parallel computing using domain decomposition and solving the problem in parallel in different sub-domains. To avoid this, in all reported parallel computations with compact schemes the full domain is treated integrally, while using parallel Thomas algorithm (PTA) or parallel diagonal dominant (PDD) algorithm in different processors with resultant latencies and inefficiencies. For domain decomposition methods using compact scheme in each sub-domain independently, a new class of compact schemes is proposed and specific strategies are developed to remove remaining problems of parallel computing. This is calibrated here for parallel computing by solving one-dimensional wave equation by domain decomposition method. We also provide the error norm with respect to the wavelength of the propagated wave-packet. Next, the advantage of the new compact scheme, on a parallel framework, has been shown by solving three-dimensional unsteady Navier-Stokes equations for flow past a cone-cylinder configuration at a Mach number of 4. Additionally, a test case is conducted on the advection of a vortex for a subsonic case to provide an estimate for the error and parallel efficiency of the method using the proposed compact scheme in multiple processors.