Anurag Dipankar

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.

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.

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.

Symmetrized compact scheme for receptivity study of 2D transitional channel flow

In many physical problems, one must maintain symmetry of numerical schemes - the symmetry arising due to geometry and/or physical conditions defining the flow. The direct simulation of transitional plane Poiseuille flow presents one such case, that motivated the present study. It has been shown here, using the method of analyzing non-periodic problems in Sengupta et al. (2003), that compact and other high accuracy schemes display strong directionality of the algorithm and such methods cannot be used for direct simulation of the physical flow. This prevalent, but unacceptable situation for DNS is rectified in the present work and a new scheme is introduced that prevents this asymmetry from contaminating results. The simulated transitional flow past 2D channel here, using the new scheme, are in agreement with experimental results and other recent views of sub-critical instabilities.

Subcritical instability on the attachment-line of an infinite swept wing

The leading-edge contamination (LEC) problem of an infinite swept wing is shown here as vortex-induced instability. The governing equation for receptivity is presented for LEC in terms of disturbance energy based on the Navier-Stokes equation. The unperturbed shear layer given by the swept Hiemenz boundary-layer solution is two-dimensional and an exact solution of incompressible the Navier-Stokes equation. Thus, the LEC problem is solved numerically by solving the full two-dimensional Navier-Stokes equation. The contamination at the attachment-line is shown by solving a receptivity to a convecting vortex moving outside the attachment-line boundary layer, which triggers subcritical spatio-temporal instability