Events during transition to turbulence either follow modal or non-modal routes, or combinations of the two. Here, we report a computational investigation of strong freestream excitation caused by a train of convecting vortices. For this TCV excitation, we show a strong interaction of modal and non-modal components causing a spectacular growth of disturbances. We propose this as the mechanism for the severe encounters due to convective vortical disturbances on the underlying shear layer.

The Navier-Stokes equation for a compressible Newtonian fluid is reassessed, in view of recent investigation results on the foundations of aeroacoustics. In the process, the role and necessity of the so-called Stokes' hypothesis is reconsidered, which in its current practice omits the bulk viscosity. In contrast, the relevance of a new generalized kinematic viscosity is proposed, along with a new methodology that will enable estimation of second coefficient of viscosity completely obviating the necessity of making the Stokes' hypothesis. This opens up the vista of a new formulation for the Navier-Stokes equation applicable for both the subjects of fluid mechanics and aeroacoustics in a unified approach.

Two-dimensional (2D) Taylor–Green vortex (TGV) dynamics is investigated here by performing a direct numerical simulation (DNS) from the receptivity to decayed turbulence stage, with the specific intention of finding the explicit role played by round-off error. The governing Navier–Stokes equations are solved using the stream function–vorticity formulation by removing errors caused by spatial discretization, adopting the Fourier spectral method, while controlling temporal discretization error with different time steps used for the four stage, fourth order Runge–Kutta method. The resultant error dynamics are predominantly governed by the round-off error for which we have reported here results comparing double and quadruple precision computations. To our knowledge, such a study of performing quadruple precision DNS has not been reported before. The effects of round-off errors on receptivity to turbulence of fluid flows have been demonstrated by performing extremely high-accuracy simulations of the benchmark 2D TGV problem by controlling all other forms of numerical errors, barring the round-off error. The choice of the 2D TGV problem is explained by highlighting the effects of round-off error. The quadruple precision results demonstrate the qualitatively different route from the receptivity to the decayed turbulence stages compared to double precision computations—specifically the identification of the receptivity stage, which is a novel feature. The presented research is relevant for physical instabilities, emphasizing the effects of precision accurately.

The present work provides a time-resolved benchmark dataset for simulating supercritical, unsteady flow inside a square lid-driven cavity (LDC). To explain the role of fidelity of the computations and that of the numerical error in triggering instabilities, the flow is computed with three time-steps. This effort is substantiated with the use of an error dynamics equation, developed to model the physical processes in the Navier-Stokes equation (NSE), using global spectral analysis (GSA) of a model convection-diffusion equation. The error is found to be diffusion-dominated for this particular class of flows. There exists lack of consensus regarding the critical value of Reynolds number (Re) beyond which flow becomes unsteady for the square LDC. The study aims to bridge this gap by devoting a detailed discussion on the onset of unsteadiness by simulating LDC for Re in the range of 7900 to 8900. Furthermore, the first Hopf bifurcation that is typically observed for LDC is quantified, laying the foundation for development of data-driven machine learning alternatives to expensive high fidelity simulations.

A comprehensive assessment is reported for the Fourier pseudo-spectral methods employing second and fourth order Runge-Kutta, RK2 and RK4 respectively, methods for time integration using global spectral analysis (GSA) in a two-dimensional (2D) framework. These methods are often used for direct numerical simulation (DNS) of homogeneous isotropic turbulence (HIT) problem to understand turbulence generation. Recent analyses for the linear one-dimensional (1D) convection and convection-diffusion equations using GSA revealed instability problems for RK2 time integration. This is conclusively demonstrated in the present paper using GSA for a model 2D linear convection-diffusion equation employing RK2 and RK4 time integration strategies for the Fourier pseudo-spectral methods. GSA confirms issues for RK2 time integration, while significantly superior accuracy is recorded for the RK4 method for DNS of transitional and turbulent flows. Furthermore, we demonstrate excellent dispersion error control and the absence of upstream propagating q-waves for the first time, a property of the adopted pseudo-spectral methods. Finally, a long time integration of Navier-Stokes equations for the 2D Taylor-Green vortex (TGV) problem using RK4 scheme is presented highlighting the importance of GSA by comparing the long term dynamics of (4 X 4)- and (8 X 8)- initial configuration of cells. Despite the complex intermediate dynamics following a primary instability, a unique, universal equilibrium (2 X 1)-cell configuration is eventually established. This is consistent with the enstrophy transport equation for 2D flows.

This study aims to investigate the effects of implicit numerical excitation on the receptivity of flow inside a square lid-driven cavity (LDC) leading to bifurcation and instability for a fixed (257 × 257) grid with different temporal resolutions via the solution of the Navier-Stokes equation. Computational results have been provided showing the flow dynamics of the LDC problem as explained with a time series at a representative point near the top corner of the cavity at (0.95, 0.95) for supercritical Reynolds numbers with respect to the bifurcation phenomenon by lowering the time step. As the accuracy of numerical methods plays a vital role in capturing the dynamics at different Reynolds numbers, this vortex-dominated flow is explained for bifurcation and instability. We propose this as a benchmark problem for the direct numerical simulation (DNS) and for machine learning (ML) of fluid flow that will lead to efficient ML algorithms and an understanding of flow receptivity, instability, and transition by DNS.

In: "Three-dimensional direct numerical simulation (DNS) of Rayleigh-Taylor instability (RTI) trigerred by acoustic excitation – Sengupta et al. 34,054108 (2022)" the receptivity of RTI to pressure pulses have been established. It has also been shown that at the onset of RTI these pulses are one-dimensional and the dissipation of the pressure pulses are governed by a dissipative wave equation. The propagation of these infrasonic to ultrasonic pressure pulses have been studied theoretically and numerically by a high fidelity numerical procedure in the physical plane. The numerical results are consistent with the theoretical analysis and the DNS of RTI noted above. The properties of pulse propagation in a quiescent dissipative ambience have been theoretically obtained from the linearized compressible Navier-Stokes equation, without Stokes' hypothesis. This analysis is extended here for a special class of excitation, with combination of wavenumbers and circular frequencies for which the phase shift results in an imposed time period is integral multiple of π, and the signal amplification is by a real factor. Here, the governing partial differential equation (PDE) for the free-field propagation of pulses is solved by the Bromwich contour integral method in the spectral plane. This method, for an input Gaussian pulse excited at a fixed frequency, is the so-called signal problem. Responses for the specific phase shifts integral multiple of π can reinforce each other due to the phase coherence. It is shown that these combinations occur at a fixed wavenumber, with higher frequencies attenuated more in such a sequence.

Direct numerical simulation (DNS) of Rayleigh-Taylor instability (RTI) has been reported in Sengupta, et al. (2022) [35] using more than 4 billion points with initial time steps of 7.69 x10(-8)s. The set-up consists of quiescent heavy (cold) fluid atop lighter (hot) fluid initially separated by an insulated partition. Removal of this partition creates acoustic pulses in the directions normal to the interface, which provides the receptivity route of RTI to transient acoustic pulses spread over several mega-Hz frequencies, far in excess of ultrasonic frequencies. The DNS and its analysis reveal the pulses to be severely attenuated, which cannot be explained by the classical wave equation. In the present research, after demonstrating the features of the DNS results, we report in detail, the theoretical development of a wave equation incorporating losses based on the Navier-Stokes equation without Stokes' hypothesis for a quiescent ambience. Apart from the physical properties of this altered wave equation, the numerical solution of the same is obtained. The governing partial differential equation (PDE) for the propagation of disturbances in the spectral plane, provides the dispersion relation between wavenumber and circular frequency in the dissipative medium accounting for viscous losses. The presented analysis provides physical properties using the global spectral analysis (GSA). This shows the far-field perturbation to propagate either as attenuated waves or strictly in a diffusive manner depending upon the wavenumber. The PDE is solved numerically by a high-accuracy compact scheme and the four-stage, Runge-Kutta scheme for time advancement. The computed solution is shown to match not only with the developed theoretical analysis but also explains the DNS results for RTI at early times when the computed flow field truly represents the quiescent ambience.

This study investigates the long-time vorticity dynamics of the multi-cellular configurations of the twodimensional (2D) Taylor-Green vortex (TGV). The pseudo-spectral method is used to solve the incompressible Navier-Stokes equation to analyze the evolution of TGV arrays. The focus is on understanding vortex interactions leading to vortex filamentation and stripping (forward cascade) during primary instability; merger and reconnection (inverse cascade) among the TGV vortical cells subsequently. Here, consideration of multiple cells avoids imposing symmetries at the smallest periodic length scale, and thereby affecting disturbance growth. The initial condition is taken from the analytic solution of the TGV, and Fourier spectral method is employed to track the interactions of the initial doubly-periodic vortices. The full sequence of evolution from one equilibrium state to another for the TGV is not addressed before, as reported here to fill this gap for multiple TGV cells in both directions. By studying various vortical interactions in the ensemble, here we report the enstrophy and energy spectra for different number of TGV cells. This is crucial in understanding the very long-time evolution process, at post-critical Reynolds numbers for the 2D TGV problem in the same physical domain, (0 <= (x, y) <= 4 pi) having (4 x 4) and (6 x 6) cells. Reported results show the evolution of these vortical cells from original configurations to finally a (1 x 1) vortical cells - the universal state not demonstrated before.

The classical Lax–Wendroff (LW) method employs explicit second order central difference schemes to solve partial differential equations. Recently, the authors in Sengupta et al. (2023) have identified numerical parameter ranges in performing large eddy simulation (LES) using this classical LW method. In a bid to improve these numerical parameter ranges for high fidelity computations, a new non-uniform grid based compact scheme for LW method (NUCLW) is developed here. This new method uses sixth order non-uniform compact scheme (NUC6) for the first and higher order derivatives. Reported global spectral analysis confirms substantial improvement in the resolution and accuracy of NUCLW scheme over the classical LW scheme. A significant advantage of the present scheme is its ability to compute solutions for non-uniform, structured grids, and that does not suffer degradation in accuracy. The potential of the NUCLW scheme for high accuracy computations is demonstrated by solving 2D incompressible Navier–Stokes equations for the benchmark unsteady flow inside a square lid driven cavity problem for a subcritical Reynolds number of 3200 and a post-critical Reynolds number of 10,000 and simulating the instability of the Taylor–Green vortex problem. These demonstrate the ability of the NUCLW scheme to solve the Navier–Stokes equations for high fidelity simulations such as LES/DNS with enhanced ranges of numerical parameters with respect to enhanced accuracy and faster computations.

The study presents a comprehensive numerical investigation of the Kelvin-Helmholtz Rayleigh-Taylor Instability (KHRTI) onset using highly resolved peta-scale direct numerical simulations by solving the compressible Navier-Stokes equations (NSE). The numerical framework incorporates a three-dimensional (3D) cuboidal domain with differential heating applied to two air streams, fostering the development of the KHRTI. A novel numerical methodology with selective mesh refinement near critical regions is employed with the help of a non-uniform compact scheme to capture small-scale phenomena accurately. Analysis of pressure disturbances during early KHRTI stages reveal distinct wave propagation patterns influenced by Rayleigh-Taylor (RT) and Kelvin-Helmholtz (KH) mechanisms. Enstrophy dynamics are quantified through the compressible enstrophy transport equation (CETE), highlighting dominant contributions from viscous stresses during early receptivity stages. The study provides insights into KHRTI evolution, shedding light on shear-buoyancy-driven instabilities and their implications for transition to turbulence.

The design and analysis of numerical methods are usually guided by the following: (a) von Neumann analysis using Fourier series expansion of unknowns, (b) the modified differential equation approach, and (c) a more generalized approach that analyzes numerical methods globally, using Fourier–Laplace transform to treat the total or disturbance quantities in terms of waves. This is termed as the global spectral analysis (GSA). GSA can easily handle non-periodic problems, by invoking wave properties of the field through the correct numerical dispersion relation, which is central to the design and analysis. This has transcended dimensionality of the problem, while incorporating various physical processes e.g. by studying convection, diffusion and reaction as the prototypical elements involved in defining the physics of the problem. Although this is used for fluid dynamical problems, it can also explain many multi-physics and multi-scale problems. This review describes this powerful tool of scientific computing, with new results originating from GSA: (i) providing a common framework to analyze both hyperbolic and dispersive wave problems; (ii) analyze numerical methods by comparing physical and numerical dispersion relation, which leads to the new class of dispersion relation preserving (DRP) schemes; (iii) developing error dynamics as a distinct tool, identifying sources of numerical errors involving both the truncation and round-off error. Such studies of error dynamics provide the epistemic tool of analysis rather than an aleatoric tool, which depends on uncertainty quantification for high performance computing (HPC). One of the central themes of GSA covers the recent advances in understanding numerical phenomenon like focusing, which defied analysis so far. An application of GSA shown here for the objective evaluation of the so-called DNS by pseudo-spectral method for spatial discretization along with time integration by two-stage Runge–Kutta method is performed. GSA clearly shows that this should not qualify as DNS for multiple reasons. A new design of HPC methods for peta- and exa-flop computing tools necessary for parallel computing by compact schemes are also described.

The vorticity dynamics of the two-dimensional (2D) Taylor-Green vortex (TGV) problem is investigated in its multi-cellular configuration by solving the incompressible Navier-Stokes equation for long time intervals using a pseudo-spectral method. This helps follow the vorticity dynamics of periodic free shear layer flows by solving an extremely accurate algorithm to explain vortex interactions that lead to vortex stripping (forward cascade), merger, and reconnection (inverse cascade) during various stages of evolution of periodic arrangements of a large number of TGV vortical cells. This latter aspect has been adopted so as not to be affected by the periodicity constraints of a single periodic cell and the various imposed symmetries that attenuate disturbance growth. The analytic solution of the TGV provides the initial condition and the spatially accurate Fourier spectral method enables one to track the first instability of the initial doubly periodic vortices. Despite a plethora of studies following the primary instability to relate it with transition to turbulence and the subsequent decay of turbulence in the literature, the topic of bifurcation sequence for periodic TGV is rare, and that is one of the main aims of the present research. Instead of restricting one's attention on a single periodic TGV cell, here it is purposely reported for multiple cells of the TGV in both directions, without invoking any asymmetries extraneously. For such an ensemble, one can study various vortical interactions giving rise to atypical energy spectra, a topic that has also been seldom addressed to distinguish between successive instabilities that can upon a conjecture, lead to transition and subsequent relaminarization, versus the bifurcation sequences leading from one equilibrium state to subsequent ones. The present study shows the dominance of the latter for 2D TGV at post-critical Reynolds number.

Here, the perturbation equation for a dissipative medium is derived from the first principle from the linearized compressible Navier-Stokes equation without Stokes's hypothesis. The dispersion relations of this generic governing equation are obtained for one and three-dimensional perturbations, which exhibit both the dispersive and dissipative nature of the perturbations traveling in a dissipative medium, depending upon the length scale. We specifically provide a theoretical cut-off wave number above which the perturbation equation represents diffusive and dissipative nature. Such behavior has not been reported before, as per the knowledge of the authors.

The perturbation equation for aeroacoustics has been derived in a dissipative medium from the linearized compressible Navier-Stokes equation without any assumption, by expressing the same in spectral plane as in Continuum perturbation field in quiescent ambience: Common foundation of flows and acoustics Sengupta et al., Phys. Fluids,35, 056111 (2023). The governing partial differential equation (PDE) for the free-field propagation of the disturbances in the spectral plane provides the dispersion relation between wavenumber and circular frequency in the dissipative medium, as characterized by a nondimensional diffusion number. Here, the implications of the dispersion relation of the perturbation field in the quiescent medium are probed for different orders of magnitude of the generalized kinematic viscosity, across large ranges of the wavenumber and the circular frequency. The adopted global spectral analysis helps not only classify the PDE into parabolic and hyperbolic types, but also explain the existence of a critical wavenumber depending on space-time scales.

Explicit second order central difference (CD2) based Lax-Wendroff (LW) method is evaluated for large eddy simulation here, using global spectral analysis (GSA) for its ability to compute fluid flows up to a specified accuracy. While the LW method has been analyzed for numerical stability in the past, rigorous quantification for performing accurate simulations including error dynamics has not been reported before. This is addressed here by determining optimal numerical parameters for prescribed accuracies based on the GSA of 2D convection-diffusion equation (CDE), which is the main focus of the present research. First, two LW methods are reported based on the application of the method for convection and both convection and diffusion operators, respectively, with the latter also having third derivative dispersive and fourth order diffusion terms. Using GSA, the optimal scheme is determined by comparing the two schemes for accuracy. The accuracy of GSA for wave propagation is also demonstrated by the validation of the q-waves determined from the analysis with the numerical simulations of 2D CDE. Finally, motivated by a one-to-one correspondence of the Navier-Stokes equation with the linear CDE established in "Effects of numerical anti-diffusion in closed unsteady flows governed by two-dimensional Navier-Stokes equation-(Suman et al., 2020) ", optimal simulation parameters are determined for the accuracy in representing physical diffusion of the scheme and validated using simulations of the canonical lid-driven cavity problem for a sub-and a super-critical Reynolds number.

In this study, we investigate the onset process of Rayleigh-Taylor Instability (RTI), which arises when two fluids of different densities mix upon removing the non-conductive interface separating them. Prior research [1 ] demonstrated this process to be initiated by infra and ultra-sonic acoustic waves, achieved by solving the compressible Navier-Stokes equation using 4.19 billion grid points and a refined timestep of 7.69E-8 s. The interface was set at y∗ = 0.5L, following Read’s (1984) experiments [2]. We expand upon this work by relocating the interface to y∗ = 0.3L. To create the density discrepancy, air masses of distinct temperatures are initially divided by an impermeable partition. Upon impulsive removal at t = 0, the resulting instability is calculated using a parallel algorithm with error-free sub-domain closure to examine the compressible enstrophy transport equation when modifying the interface location. The high-performance computing (HPC) algorithm employs exceptional resolution to accurately capture acoustic disturbances and flow instability. We report on the parallel performance and efficiency of the HPC with the specialized sub-domain boundary closure, utilizing up to 262k cores.

Having developed the perturbation equation for a dissipative quiescent medium for planar propagation using the linearized compressible Navier-Stokes equation without the Stokes' hypothesis \cite{arxiv2023}, here the same is extended where a uniform mean flow is present in the ambiance to explore the propagation properties for the Doppler effect.

Compact schemes are often preferred in performing scientific computing for their superior spectral resolution. Error-free parallelization of a compact scheme is a challenging task due to the requirement of additional closures at the inter-processor boundaries. Here, sources of the error due to sub-domain boundary closures for the compact schemes are analyzed with global spectral analysis. A high-accuracy parallel computing strategy devised in “ A high-accuracy preserving parallel algorithm for compact schemes for DNS. ACM Trans. Parallel Comput. 7, 4, 1-32 (2020)” systematically eliminates error due to parallelization and does not require overlapping points at the sub-domain boundaries. This closure is applicable for any compact scheme and is termed here as non-overlapping high-accuracy parallel (NOHAP) sub-domain boundary closure. In the present work, the advantages of the NOHAP closure are shown with the model convection equation and by solving the compressible Navier–Stokes equation for three-dimensional Rayleigh–Taylor instability simulations involving multiphysics dynamics and high Reynolds number flow past a natural laminar flow airfoil using a body-conforming curvilinear coordinate system. Linear scalability of the NOHAP closure is shown for the large-scale simulations using up to 19,200 processors.

Here, the perturbation equation for a dissipative medium is derived from the first principles for the linearized compressible Navier–Stokes equation without Stokes' hypothesis. Dispersion relations of this generic governing equation are obtained, which exhibits both the dispersive and dissipative nature of perturbations traveling in a dissipative medium, depending upon the length scale. We specifically provide a theoretical cutoff wave number above which the perturbation equation represents diffusive and dissipative nature of the quiescent flow. It is shown that perturbation equations for pressure and velocity retain the same form in one-dimension, but it is not the same for multi-dimensional perturbation fields. Such behavior has not been reported before, as per the knowledge of the authors.