Two-dimensional Euler Equations Research Articles

Numerical simulation of the interaction between a planar shock wave and a backward-facing triangular bubble containing gases with different Atwood numbers

The interaction between a shock wave and an interface delineating two gases engenders intricate flow physics, with particular attention drawn to the hydrodynamic instability due to its practical significance. Previous studies have primarily focused on elucidating different wave patterns and instabilities evolution at the interface during the initial phase of shock interaction with cylindrical or spherical bubbles. However, scant literature has shifted its focus toward exploring the long-term morphology of bubbles, especially those characterized by polygonal interfaces. Notably, the detailed examination of shock interaction with a polygonal interface, such as a triangular one with a constant incident angle, remains largely unexplored in existing literature. Recently, the longtime evolution of detailed flow structures across the interface of shock-forward-facing triangular bubbles was captured by Kundu et al. [“A study on dynamics of shock-accelerated forward-facing triangular bubbles at different Atwood numbers,” Phys. Fluids 36, 016110 (2024)] through numerical simulation. In this study, the dynamics of a shock-accelerated backward-facing triangular interface containing various gases, namely, Sulfur Hexafluoride, Refrigerant-22, Argon, Neon, and Helium, is studied for a shock Mach number of 1.21. Simulations were performed by solving the two-dimensional Euler equation using low-dissipative advection upwind splitting methods (AUSMD), in conjunction with a derived ninth-order upwind scheme and a four-stage third-order Runge–Kutta scheme for temporal integration. The development of Richtmyer–Meshkov (RM) and Kelvin–Helmholtz (K–H) instabilities at the interface, mixing, and normalized movements of backward-facing triangular bubbles is captured at different Atwood (At) numbers.

On the linear stability of the Lamb–Chaplygin dipole

The Lamb–Chaplygin dipole (Lamb, Hydrodynamics, 2nd edn, 1895, Cambridge University Press; Lamb, Hydrodynamics, 3rd edn, 1906, Cambridge University Press; Chaplygin, Trudy Otd. Fiz. Nauk Imper. Mosk. Obshch. Lyub. Estest., vol. 11, 1903, pp. 11–14) is one of the few closed-form relative equilibrium solutions of the two-dimensional (2-D) Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2-D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time integration of the 2-D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2-D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions.

On the composite waves of the two-dimensional pseudo-steady van der Waals gas satisfied Maxwell’s law

This paper is concerned with a model of phase transition, the Euler equations with van der Waals gas satisfied Maxwell’s law. We study the composite waves for the two-dimensional (2D) Euler equations for pseudo-steady supersonic flow with van der Waals gas satisfied Maxwell’s law around a sharp corner. According to the initial value of the specific volume and the properties of van der Waals gas, the composite waves at the sharp corner are constructed case by case. Recently, the centred wave solutions are constructed that include centered simple wave, contact discontinuity, composite wave (jump-fan, fan-jump or fan-jump-fan), etc. We mainly discuss the solutions involving expansive discontinuity wave in the present paper.

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

A study on dynamics of shock-accelerated forward-facing triangular bubbles at different Atwood numbers

The complexity of flow physics and the associated hydrodynamic instability arising out of interactions of a shock wave with forward and backward-facing triangular interfaces drew the attention of researchers around the globe. In earlier studies, many researchers focused on the formation of different wave patterns, the development of instabilities at the interface, and the flow morphology during the initial phase of shock wave interacting with light and heavier bubbles. However, limited studies are available in the literature on the interaction of shock with a polygonal interface. Furthermore, it is difficult to capture the complex flow physics of a polygonal interface accelerated by shock waves at later time instants. In the present study, the dynamics of shock-accelerated forward-facing triangular interface containing various gases, namely, sulfur hexafluoride, refrigerant-22, argon, neon, and helium, are examined numerically for a longer time duration for a shock Mach number of 1.21. The simulations were performed by solving two-dimensional Euler equations using a low-dissipative advection upstream splitting method algorithm coupled with a derived ninth-order upwind scheme and a four-stage third-order Runge–Kutta scheme. The numerical results demonstrated the influence of the Atwood number on vorticity generation, bubble deformation, mixing, and the development of Kelvin Helmholtz instabilities on the bubble interface up to long instants, which are not available in the literature. The Fourier spectra of the streamwise kinetic energy showed the distribution of energy in the larger and smaller scale vortical structures.

Physics-informed neural networks for parametric compressible Euler equations

The numerical approximation of solutions to the compressible Euler and Navier–Stokes equations is a crucial but challenging task with relevance in various fields of science and engineering. Recently, methods from deep learning have been successfully employed for solving partial differential equations by incorporating the equations into a loss function that is minimized during the training of a neural network. This approach yields a so-called physics-informed neural network. It is not based upon classical discretizations, such as finite-volume or finite-element schemes, and can even address parametric problems in a straightforward manner. This has raised the question, whether physics-informed neural networks may be a viable alternative to conventional methods for computational fluid dynamics. In this article we introduce an adaptive artificial viscosity reduction procedure for physics-informed neural networks enabling approximate parametric solutions for forward problems governed by the stationary two-dimensional Euler equations in sub- and supersonic conditions. To the best of our knowledge, this is the first time that the concept of artificial viscosity in physics-informed neural networks is successfully applied to a complex system of conservation laws in more than one dimension. Moreover, we highlight the unique ability of this method to solve forward problems in a continuous parameter space. The presented methodology takes the next step of bringing physics-informed neural networks closer towards realistic compressible flow applications.

Numerical study of detonation propagation in a semi-confined nonuniform layer bound by an inert gas

This study investigates detonation propagation through a stoichiometric hydrogen-oxygen layer stratified above an inert gas, i.e., argon, with a diffuse interface at atmospheric pressure. A simple model capable of replicating the detonation inert gas diffuse layer interaction was developed that simulates previously conducted experiments. Detonation propagation is modeled by solving the two-dimensional reactive Euler equations for a calorically perfect gas with single-step Arrhenius chemistry. The acoustic impedance gradient associated with the argon diffusion into the hydrogen-oxygen results in a nonuniform initial density. Because of the perfect gas assumption, for a uniform initial pressure, as in the experiment, results in an initial temperature that is artificially nonuniform. The activation energy is adjusted for the imposed temperature nonuniformity in order to yield the correct ZND reaction zone length for the local argon dilution. Numerical soot foils are compared to experiments. Detonation propagation is possible due to the generation of triple points along the diffuse layer interface. These triple points, that generate new detonation cells, form from inflections in the reaction zones that develop because of the local increases in detonation strength.

Об использовании канонического разложения для визуализации результатов параметрических расчетов

The approximation of the tensor appearing at a discretization of the multidimensional function is considered from the viewpoint of storing and treating of the results of parametric computations obtained in computational aerogasdynamics. The new algorithm for the computation of the canonical decomposition using gradient descent and approximately decomposable goal functional is described. This algorithm applies the random set of points on the hyperplane orthogonal to the computed core of the canonical decomposition (“umbrella”) that ensures its flexible application for an approximation of the tensors with a priori unknown rank and may be naturally transferred on such tensor decomposition as the tensor train. The results of the numerical tests are presented for the model six-dimensional functions and for an ensemble of the numerical solutions for the two-dimensional Euler equations. These equations describe the flow of the compressible gas with two crossing shock waves. The Mach number and angles of the flow deflection serve as the flow parameters. The results are provided for the dimensionality 3 (simple numerical solution) and 4 (the ensemble of the numerical solutions in dependence on the Mach number).

Accelerated initiation of oblique detonation induced by disturbance in detonative zone

In the real flow for high altitude, the initiation mechanism of the oblique detonation involves the coupling of complex wave structures and combustion waves, which may result in the phenomenon of extinction. The approach of stable initiation is one of the key factors that restricting the oblique detonation engine from theory to practice. The wave structure, initiation characteristics and characteristic parameters of the flow field are analyzed, the two-dimensional Euler equation considering the detailed chemical reactions of multi-component are solved. First, the approaches of blunt bump and transverse jet are combined used to shorten the detonation distance of the oblique detonation. Results show that these methods can promote the accelerated initiation of the oblique detonation effectively, and the detonation distance can be shortened more than 90%. There are two wave structures induced by the blunt bump: weak coupled and strong coupled, adding the transverse jet promotes the transition between these two wave structures. Then, the shape of the bump and the characteristic parameters of the transverse jet are optimized. The streamline-shape bump can eliminate the recirculation zone formed after the circle bump and the ellipse bump. The transverse jet will produce an oblique detonation wave at the front of the jet position, which changes the wave structure and initiation mode of the oblique detonation wave. It is expected to reveal the accelerated initiation mechanism of oblique detonation under the induction zone disturbance and complex flow environment, deepen the understanding of the detonation law of oblique detonation wave under real flow conditions, and provide a scientific basic for the development of the combustor of the oblique detonation engine.

Characteristics of cellular structure of detonation waves propagating in annular channels

This study investigates the characteristics of stable and unstable cells and wavefronts of detonation waves propagating in annular channels with different inner radii and channel widths using two-dimensional Euler equations along with a two-step induction-exothermic reaction kinetics. The results reveal that the effect of annular channels on the detonation cell structure depends on both the inner radius and channel width. To quantify this effect, a parameter σ is introduced, representing the ratio of the inner and outer radii of the channel. We have discovered that for values of the parameter σ exceeding a critical value σs, the detonation wavefront demonstrates characteristics similar to those observed in a straight channel scenario. On the contrary, when σ is below σs, the wavefront becomes distorted, potentially leading to Mach reflection as σ decreases further to another critical value σm. Additionally, the interaction among expansion waves induced by the inner walls leads to an augmented induced length and the potential occurrence of localized decoupling of the detonation wave, particularly for unstable detonation waves. However, it is worth noting that the re-initiation of the detonation wave may be triggered by the formation of hotspots resulting from the interaction between transverse shock waves and the detonation wave. This study aims to characterize the propagation characteristics of detonation waves within annular channels, with the objective of providing valuable insights for the design and optimization of annular chamber configurations in systems involving detonation.

Entropic Regularization of the Discontinuous Galerkin Method for Two-Dimensional Euler Equations in Triangulated Domains

Entropic Regularization of the Discontinuous Galerkin Method for Two-Dimensional Euler Equations in Triangulated Domains

Data-driven stochastic spectral modeling for coarsening of the two-dimensional Euler equations on the sphere

A resolution-independent data-driven subgrid-scale model in coarsened fluid descriptions is proposed. The method enables the inclusion of high-fidelity data into the coarsened flow model, thereby enabling accurate simulations also with the coarser representation. The small-scale model is introduced at the level of the Fourier coefficients of the coarsened numerical solution. It is designed to reproduce the kinetic energy spectra observed in high-fidelity data of the same system. The approach is based on a control feedback term reminiscent of continuous data assimilation implemented using nudging (Newtonian relaxation). The method relies solely on the availability of high-fidelity data from a statistically steady state. No assumptions are made regarding the adopted discretization method or the selected coarser resolution. The performance of the method is assessed for the two-dimensional Euler equations on the sphere for coarsening factors of 8 and 16 times. Applying the method at these significantly coarser resolutions yields good results for the mean and variance of the Fourier coefficients and leads to improvements in the empirical probability density functions of the attained vorticity values. Stable and accurate large-scale dynamics can be simulated over long integration times and are illustrated by capturing long-time vortex trajectories.

Exact coherent structures in fully developed two-dimensional turbulence

This paper reports several new classes of unstable recurrent solutions of the two-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions are in many ways analogous to recurrent solutions of the Navier–Stokes equation which are often referred to as exact coherent structures. In particular, we find that recurrent solutions of the Euler equation are dynamically relevant: they faithfully reproduce large-scale flows in simulations of turbulence at very high Reynolds numbers. On the other hand, these solutions have a number of properties which distinguish them from their Navier–Stokes counterparts. First of all, recurrent solutions of the Euler equation come in infinite-dimensional continuous families. Second, solutions of different types are connected, e.g. an equilibrium can be smoothly continued to a travelling wave or a time-periodic state. Third, and most important, they are only weakly unstable and, as a result, fully developed turbulence mimics some of these solutions remarkably frequently and over unexpectedly long temporal intervals.

Mechanism of characteristic change of panel flutter caused by oblique shock impingement

Compared to the shock-free condition, the weak shock impingement stabilizes the flexible panel, while the strong shock impingement leads to the early onset of panel flutter with a significant increase in flutter amplitude and frequency. However, the reason for this change by shock impingement remains unclear. The current research examines the mechanism of this change by an in-house code where the von Kármán’s large deflection plate theory is coupled with two-dimensional Euler equations. Compared to the shock-free condition, the oblique shock impingement leads to the change of local dynamic pressure on the panel as well as the static pressure differential across the panel. The analysis on the influence of these changes indicates that, on the one hand, the average dynamic pressure on the panel becomes larger than the shock-free condition, accelerating the onset of panel flutter. On the other hand, the change of the static pressure differential across the panel alters the coupling characteristic between different natural frequencies (modes) of the panel structure. The dynamic response of panel flutter under shock impingements is dominated by the coupling between the second and third modes instead of the first two modes for panel flutter under the shock-free condition. The combined effect of these two changes leads to the change of flutter characteristics of the panel under shock impingement. These findings provide valuable insights into the mechanism of shock-induced panel flutter.

Investigating the flow characteristics and thermodynamic performance of curved detonation waves

Oblique detonation waves (ODWs) are induced by oblique shock waves (OSWs) and hold significant research value for air-breathing ramjet engines due to their high thermodynamic efficiency and specific impulse. In this research, detonation is induced using a curved shock wave (CSW) that provides greater flexibility compared to an OSW. Curved detonation waves (CDWs) are classified into two categories based on the curvature of the CSW: positive-curvature curved detonation waves (PC-CDWs) and negative-curvature curved detonation waves (NC-CDWs). We conduct a numerical investigation of the flow characteristics and potential thermodynamic performance in curved detonation waves (CDWs) by solving the two-dimensional multi-species Euler equations with a detailed hydrogen combustion mechanism. In contrast to ODWs, we observe continuous compression and expansion effects in the flow fields of PC-CDWs and NC-CDWs, respectively. Continuous compression shortens the initiation length of the detonation wave and intensifies the compression wave preceding the combustion front, whereas continuous expansion produces the opposite outcome. Changes in the curvature of the CSW result in alterations to the flow characteristics of the CDW. These flow characteristic changes are manifested in their thermodynamic and propulsion-related performance. More specifically, a lower overdrive of the detonation wave corresponds to a reduced loss in total pressure.

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

AbstractWe present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics, both in time and space, which include the relaxation schemes by Jin and Xin. These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case. These kinetic models depend on a small parameter that can be seen as a “Knudsen” number. The method is asymptotic preserving in this Knudsen number. Also, the computational costs of the method are of the same order of a fully explicit scheme. This work is the extension of Abgrall et al. (2022) [3] to multi-dimensional systems. We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

Bulges at vortical outflows

This paper constructs steady solutions of the two-dimensional Euler equations corresponding to a line source of vortical fluid on the impermeable boundary of a quiescent flow. The nonlinear, free-boundary problem is solved by mapping the flow domain to the hodograph plane. A vortex dipole or, equivalently, a source–sink doublet is superposed on the source leading to flow patterns that model the ballooning outflows observed where rivers and straits discharge into the open ocean and in the rotating flow experiments and numerical simulations designed to reflect these observations.

Numerical investigation of shock Mach number effects on Richtmyer–Meshkov instability in a heavy square bubble

This study conducts a two-dimensional numerical investigation of incident shock Mach number effects on the Richtmyer–Meshkov (RM) instability after a shock wave impulsively drives a heavy square bubble. The square bubble is composed of SF6 gas, which is surrounded by nitrogen gas. Five different shock Mach numbers are considered: Ms=1.12, 1.22, 1.4, 1.7, and 2.1. Two-dimensional compressible Euler equations for two-component gas flows are simulated with a high-order modal discontinuous Galerkin solver. For validation, the numerical results are compared with the existing experimental results and are found to be in good agreement. The present results reveal that the incident shock Mach number is critical in the growth of RM instability in a heavy square bubble. The shock Mach number affects flow morphology significantly, resulting in complicated wave patterns, shock focusing, jet creation, bubble deformation, and vorticity generation. The convergent shape of the bubble causes shock focusing, which creates a local high-pressure zone and, as a result, an outward jet generation. It is found that the bubble deforms differently with increasing shock Mach number, and the different shock Mach numbers alter the distance between the shock focusing position and the downstream bubble interface. The effects of shock Mach numbers are investigated in depth through physical phenomena such as vorticity production, kinetic energy, and enstrophy. Finally, the shock Mach number impacts on the time-variations of the shock trajectories and interface structure are thoroughly investigated.

Sonic boom propagation over real topography

The effect of elevation variation on sonic boom reflection is investigated using real terrain data. To this end, the full two-dimensional Euler equations are solved using finite-difference time-domain techniques. Numerical simulations are performed for two ground profiles of more than 10 km long, extracted from topographical data of hilly regions, and for two boom waves, a classical N-wave, and a low-boom wave. For both ground profiles, topography affects the reflected boom significantly. Wavefront folding due to terrain depression is notably highlighted. For the ground profile with mild slopes, the time signals of the acoustic pressure at the ground are, however, only slightly modified compared to the flat reference case, and the associated noise levels differ by less than 1 dB. With steep slopes, the contribution due to wavefront folding has a large amplitude at the ground. This results in an amplification of the noise levels: a 3 dB increase occurs at 1% of the positions along the ground surface, and a maximum of 5–6 dB is reached near the terrain depressions. These conclusions are valid for the N-wave and low-boom wave.

High-Order Cut-Cell Discontinuous Galerkin Difference Discretization

We present a high-order cut-cell method based on the discontinuous Galerkin difference (DGD) discretization. We leverage the inherent properties of the DGD basis functions to construct a cut-cell discretization that does not require special treatment to mitigate the small-cell problem. The paper describes how the DGD discretization can be constructed from an existing discontinuous Galerkin (DG) discretization, and we highlight differences between the DG and DGD methods. By performing condition-number studies on one- and two-dimensional model problems, we demonstrate that cut-cell DGD discretization remains well conditioned even when the cut-cell volume is orders of magnitude smaller than neighboring cells. We verify the high-order accuracy of the discretization by solving the two-dimensional steady-state Euler equations.