Aditi Sengupta

Global spectral analysis of multi-level time integration schemes

An analysis is reported here for three-time level integration methods following the global spectral analysis (GSA) described in High Accuracy Computing Methods, T.K. Sengupta, Cambridge Univ. Press, USA. The focus is on the second order AdamsBashforth (AB2) and the extrapolation in time (EXT2) methods. Careful distinction is made for the first time step at t=0 by either Euler forward or four-stage, fourth order RungeKutta (RK4) time schemes. The latter is used to solve a benchmark aeroacoustic problem. Several one-dimensional wave propagation models are analyzed: pure advection and advection-diffusion equations. Various spatial discretizations are discussed, including Fourier spectral method. Attention is paid to the presence of physical and numerical modes as noted in the quadratic equation obtained from the difference equation for the model 1D convection equation. It is shown that AB2 method is less stable and accurate than EXT2 method, with respect to numerical dissipation and dispersion. This is true for the methods, in which the physical mode dominates over the numerical mode. Presented analysis provides useful guide to analyze any three-time level methods.

Roles of bulk viscosity on Rayleigh-Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts

Direct numerical simulations of Rayleigh-Taylor instability (RTI) between two air masses with a temperature difference of 70 K is presented using compressible Navier-Stokes formulation in a non-equilibrium thermodynamic framework. The two-dimensional flow is studied in an isolated box with non-periodic walls in both vertical and horizontal directions. The non-conducting interface separating the two air masses is impulsively removed at t = 0 (depicting a heaviside function). No external perturbation has been used at the interface to instigate the instability at the onset. Computations have been carried out for rectangular and square cross sections. The formulation is free of Boussinesq approximation commonly used in many Navier-Stokes formulations for RTI. Effect of Stokes’ hypothesis is quantified, by using models from acoustic attenuation measurement for the second coefficient of viscosity from two experiments. Effects of Stokes’ hypothesis on growth of mixing layer and evolution of total entropy for the Rayleigh-Taylor system are reported. The initial rate of growth is observed to be independent of Stokes’ hypothesis and the geometry of the box. Following this stage, growth rate is dependent on the geometry of the box and is sensitive to the model used. As a consequence of compressible formulation, we capture pressure wave-packets with associated reflection and rarefaction from the non-periodic walls. The pattern and frequency of reflections of pressure waves noted specifically at the initial stages are reflected in entropy variation of the system.

Non-equilibrium Thermodynamics of Rayleigh-Taylor instability

Here, the fundamental problem of Rayleigh–Taylor instability (RTI) is studied by direct numerical simulation (DNS), where the two air masses at different temperatures, kept apart initially by a non-conducting horizontal interface in a 2D box, are allowed to mix. Upon removal of the partition, mixing is controlled by RTI, apart from mutual mass, momentum, and energy transfer. To accentuate the instability, the top chamber is filled with the heavier (lower temperature) air, which rests atop the chamber containing lighter air. The partition is positioned initially at mid-height of the box. As the fluid dynamical system considered is completely isolated from outside, the DNS results obtained without using Boussinesq approximation will enable one to study non-equilibrium thermodynamics of a finite reservoir undergoing strong irreversible processes. The barrier is removed impulsively, triggering baroclinic instability by non-alignment of density, and pressure gradient by ambient disturbances via the sharp discontinuity at the interface. Adopted DNS method has dispersion relation preservation properties with neutral stability and does not require any external initial perturbations. The complete inhomogeneous problem with non-periodic, no-slip boundary conditions is studied by solving compressible Navier–Stokes equation, without the Boussinesq approximation. This is important as the temperature difference between the two air masses considered is high enough (\(\Delta T = 70\) K) to invalidate Boussinesq approximation. We discuss non-equilibrium thermodynamical aspects of RTI with the help of numerical results for density, vorticity, entropy, energy, and enstrophy.

Effects of Error on the Onset and Evolution of Rayleigh–Taylor Instability

Here we investigate effects of error in simulating Rayleigh Taylor instability (RTI). The error metrics are evaluated based on the correct spectral analysis of a model equation by Sengupta et al. (J Comput Phys 226:1211–1218, 2007) [13]. The geometry for RTI consists of a rectangular box with a partition at mid-height separating two volumes of air kept at a temperature difference of 70 K. This helps avoiding Boussinesq approximation and the present time-accurate computations for compressible Navier–Stokes equation (NSE) in 2D are reported. Computations for CFL numbers of 0.09 and 0.009 shows completely different onset of RTI, while the terminal mixed stage appears similar. The difference is traced to very insignificant difference in the value of numerical amplification factor for the two CFL numbers.

An enstrophy-based linear and nonlinear receptivity theory

In the present research, a new theory of instability based on enstrophy is presented for incompressible flows. Explaining instability through enstrophy is counter-intuitive, as it has been usually associated with dissipation for the Navier-Stokes equation (NSE). This developed theory is valid for both linear and nonlinear stages of disturbance growth. A previously developed nonlinear theory of incompressible flow instability based on total mechanical energy described in the work of Sengupta et al. [“Vortex-induced instability of an incompressible wall-bounded shear layer,” J. Fluid Mech. 493, 277–286 (2003)] is used to compare with the present enstrophy based theory. The developed equations for disturbance enstrophy and disturbance mechanical energy are derived from NSE without any simplifying assumptions, as compared to other classical linear/nonlinear theories. The theory is tested for bypass transition caused by free stream convecting vortex over a zero pressure gradient boundary layer. We explain the creation of smaller scales in the flow by a cascade of enstrophy, which creates rotationality, in general inhomogeneous flows. Linear and nonlinear versions of the theory help explain the vortex-induced instability problem under consideration.In the present research, a new theory of instability based on enstrophy is presented for incompressible flows. Explaining instability through enstrophy is counter-intuitive, as it has been usually associated with dissipation for the Navier-Stokes equation (NSE). This developed theory is valid for both linear and nonlinear stages of disturbance growth. A previously developed nonlinear theory of incompressible flow instability based on total mechanical energy described in the work of Sengupta et al. [“Vortex-induced instability of an incompressible wall-bounded shear layer,” J. Fluid Mech. 493, 277–286 (2003)] is used to compare with the present enstrophy based theory. The developed equations for disturbance enstrophy and disturbance mechanical energy are derived from NSE without any simplifying assumptions, as compared to other classical linear/nonlinear theories. The theory is tested for bypass transition caused by free stream convecting vortex over a zero pressure gradient boundary layer. We explain the ...

Non-linear instability analysis of the two-dimensional Navier-Stokes equation: The Taylor-Green vortex problem

An enstrophy-based non-linear instability analysis of the Navier-Stokes equation for two-dimensional (2D) flows is presented here, using the Taylor-Green vortex (TGV) problem as an example. This problem admits a time-dependent analytical solution as the base flow, whose instability is traced here. The numerical study of the evolution of the Taylor-Green vortices shows that the flow becomes turbulent, but an explanation for this transition has not been advanced so far. The deviation of the numerical solution from the analytical solution is studied here using a high accuracy compact scheme on a non-uniform grid (NUC6), with the fourth-order Runge-Kutta method. The stream function-vorticity (ψ, ω) formulation of the governing equations is solved here in a periodic square domain with four vortices at t = 0. Simulations performed at different Reynolds numbers reveal that numerical errors in computations induce a breakdown of symmetry and simultaneous fragmentation of vortices. It is shown that the actual physi...