Two-dimensional Euler Equations Research Articles

Second order conservative Lagrangian DG schemes for compressible flow and their application in preserving spherical symmetry in two-dimensional cylindrical geometry

In this paper, we construct a class of second-order cell-centered Lagrangian discontinuous Galerkin (DG) schemes for the two-dimensional compressible Euler equations on quadrilateral meshes. This Lagrangian DG scheme is based on the physical coordinates rather than the fixed reference coordinates, hence it does not require studying the evolution of the Jacobian matrix for the flow mapping between the different coordinates. The conserved variables are solved directly, and the scheme can preserve the conservation property for mass, momentum and total energy. The strong stability preserving (SSP) Runge-Kutta (RK) method is used for the time discretization. Furthermore, there are two main contributions. Firstly, differently from the previous work, we design a new Lagrangian DG scheme which is truly second-order accurate for all the variables such as density, momentum, total energy, pressure and velocity, while the similar DG schemes in the literature may lose second-order accuracy for certain variables, as shown in numerical experiments. Secondly, as an extension and application, we develop a particular Lagrangian DG scheme in the cylindrical geometry, which is designed to be able to preserve one-dimensional spherical symmetry for all the linear polynomials in two-dimensional cylindrical coordinates when computed on an equal-angle-zoned initial grid. The distinguished feature is that it can maintain both the spherical symmetry and conservation properties, which is very important for many applications such as implosion problems. A series of numerical experiments in the two-dimensional Cartesian and cylindrical coordinates are given to demonstrate the good performance of the Lagrangian DG schemes in terms of accuracy, symmetry and non-oscillation.

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

The stabilizing effect of temperature and magnetic field on a 2D magnetic Bénard fluids

In this paper we study the stability of a special magnetic Bénard system near equilibrium, where there exists Laplacian magnetic diffusion and temperature damping but the velocity equation involves no dissipation. Without any velocity dissipation, the fluid velocity is governed by the two-dimensional incompressible Euler equation, whose solution can grow rapidly in time. However, when the fluid is coupled with the magnetic field and temperature through the magnetic Bénard system, we show that the solution is stable. Our results mathematically illustrate that the magnetic field and temperature have the effect of enhancing dissipation and contribute to stabilize the fluid.

Unconditionally optimal high-order weighted compact nonlinear schemes with sharing function for Euler equations

Weighted compact nonlinear schemes (WCNS) are a type of high-order shock-capturing finite difference schemes commonly used in various applications. Their spatial discretizations involve a nonlinear interpolation step and a linear difference step. However, the nonlinear interpolation step requires significantly more computational resources compared to the linear difference step. Therefore, simplifying the interpolation step is an effective way to improve the efficiency of these schemes. In this paper, we propose a new approach to construct WCNS schemes based on the sharing function for Euler equations. This approach uses a set of common nonlinear interpolation weights for different components, resulting in a significant improvement in efficiency compared to the original WCNS schemes that use different sets of weights for each component. To ensure accuracy at critical points of any orders, we employ nonlinear weights that guarantee unconditionally optimal high order. Additionally, this new approach may reduce oscillations caused by non-characteristic interpolation. Furthermore, we also develop a new shock detector using the sharing function, enabling us to employ characteristic interpolation for nonsmooth regions and linear component-wise interpolation for the rest. We validate the proposed schemes based on the sharing function through numerical examples of one- and two-dimensional Euler equations, demonstrating their effectiveness in terms of shock-capturing capability and efficiency.

Eulerian and Lagrangian Stability in Zeitlin’s Model of Hydrodynamics

The two-dimensional (2-D) Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations. The Zeitlin model is a finite-dimensional analogue of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin’s model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, we give here two results. First, convergence of the sectional curvature in the Euler–Zeitlin equations on the Lie algebra su(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {su}(N)$$\\end{document} to that of the Euler equations on the sphere. Second, L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The results allow geometric conclusions about Zeitlin’s model to be transferred to Euler’s equations and vice versa, which could expedite the ultimate aim: to characterize the generic long-time behaviour of perfect 2-D fluids.

Flow characteristics and propulsive performance of oblique detonation waves induced by a transverse jet

The initiation of oblique detonation waves (ODWs) is a key component of the successful application of oblique detonation wave engines (ODWEs). This paper numerically investigates the initiation of ODWs under the active control of a transverse jet by solving the two-dimensional multi-species Euler equations, focusing on the morphology of the flow fields as well as the relationship between the flow structures and propulsive performance using the concept of thrust potential. Active jet control significantly shortens the initiation length of the ODW. The results reveal that the jet-induced flow field shows four typical patterns depending on the jet momentum flux ratio and wedge angle: shock-induced combustion, a type I pattern, a type II pattern, and a type I-II pattern. For the jet-induced ODW flow field, the propulsive performance declines as the momentum flux ratio increases when the wedge angle is certain. The larger the wedge angle, the greater the magnitude of the decline. The thrust of the flow field consists of two main components: the thrust generated by the mixture that passes first through the oblique shock wave and then through the detonation wave, and the thrust generated by the mixture that passes directly through the ODW front. Since the ODW upstream front has a larger wave angle, the flow loss of the mixture passes through the ODW upstream front is higher and the thrust potential is lower. This work could guide the active control of the initiation of ODWs at low flow losses.

Development of Node-centered finite-volume diffusion method on triangle mesh for detonation propagation simulation in insensitive high explosives

This study presents a numerical simulation of detonation waves in insensitive high explosives (IHE). A direct numerical simulation (DNS) method of detonation waves propagation was developed. It solves two-dimensional reactive Euler equations by using a semi-discrete node-centered finite-volume (NCFV) scheme on triangle mesh. Employing ZND model analytical solution as the initial condition, the upper and lower boundary conditions were designed as local sonic equilibrium conditions. The DNS method was validated using steady detonation wave propagation experimental results for tri-amino-tri-nitro-benzene (TATB) based explosives. The two-dimensional steady detonation propagation of the circular arc experiments was completed using a non-embedded technique (electric and optical fibre probe velocimetry). The comparison results demonstrate that the numerical method can provide a good prediction of the pseudo-steady-state detonation wave front propagation and the angular speed.

Extensions of the symmetry algebra and Lax representations for the two-dimensional Euler equation

We find the twisted extensions of the symmetry algebra of the 2D Euler equation in the vorticity form and use them to construct new Lax representation for this equation. Then we generalize this result by considering the transformation Lie–Rinehart algebras generated by finite-dimensional subalgebras of the symmetry algebra and derive a family of Lax representations for the Euler equation. The family depends on functional parameters and contains a non-removable spectral parameter.

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

In this work, we enhance the fifth-order Weighted Essentially Non-Oscillatory (WENO) shock-capturing scheme by integrating deep learning techniques. We improve the established WENO algorithm by training a compact neural network to dynamically adjust the smoothness indicators within the WENO scheme. This modification boosts the accuracy of the numerical results, particularly in proximity to abrupt shocks. Notably, our approach eliminates the need for additional post-processing steps, distinguishing it from previous deep learning-based methods. We substantiate the superiority of our new approach through the examination of multiple examples from the literature concerning the two-dimensional Euler equations of gas dynamics. Through a thorough investigation of these test problems, encompassing various shocks and rarefaction waves, our novel technique consistently outperforms the traditional fifth-order WENO scheme. This superiority is especially evident in cases where numerical solutions exhibit excessive diffusion or overshoot around shocks.

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.