Euler Equations Research Articles

A dissipative extension to ideal hydrodynamics

Abstract We present a formulation of special relativistic, dissipative hydrodynamics (SRDHD) derived from the well-established Müller-Israel-Stewart (MIS) formalism using an expansion in deviations from ideal behaviour. By re-summing the non-ideal terms, our approach extends the Euler equations of motion for an ideal fluid through a series of additional source terms that capture the effects of bulk viscosity, shear viscosity and heat flux. For efficiency these additional terms are built from purely spatial derivatives of the primitive fluid variables. The series expansion is parametrized by the dissipation strength and timescale coefficients, and is therefore rapidly convergent near the ideal limit. We show, using numerical simulations, that our model reproduces the dissipative fluid behaviour of other formulations. As our formulation is designed to avoid the numerical stiffness issues that arise in the traditional MIS formalism for fast relaxation timescales, it is roughly an order of magnitude faster than standard methods near the ideal limit.

Impact of a two-dimensional steep hill on wind turbine noise propagation

Abstract. Wind turbine noise propagation in a hilly terrain is studied through numerical simulation in different scenarios. Linearized Euler equations are solved in a moving frame that follows the wavefront, and wind turbine noise is modeled with an extended moving source. We employ large-eddy simulations to simulate the flow around the hill and the wind turbine. The sound pressure levels (SPLs) obtained for a wind turbine in front of a 2D hill and a wind turbine on a hilltop are compared to a baseline flat case. First, the source height and wind speed strongly affect sound propagation downwind. We find that topography influences the wake shape, inducing changes in the sound propagation that drastically modify the SPL downwind. Placing the turbine on the hilltop increases the average sound pressure level and amplitude modulation downwind. For the wind turbine placed upstream of a hill, a strong shielding effect is observed. But, because of the refraction by the wind gradient, levels are comparable with the baseline flat case just after the hill. Thus, considering how terrain topography alters the flow and wind turbine wake is essential to accurately predict wind turbine noise propagation.

Numerical Study of Shock Wave Interaction with V-Shaped Heavy/Light Interface

This paper investigates numerically the shock wave interaction with a V-shaped heavy/light interface. For numerical simulations, we choose six distinct vertex angles (θ=40∘,60∘,90∘,120∘,150∘, and 170∘), five distinct shock wave strengths (Ms=1.12,1.22,1.30,1.60, and 2.0), and three different Atwood numbers (At=−0.32,−0.77, and −0.87). A two-dimensional space of compressible two-component Euler equations are solved using a third-order modal discontinuous Galerkin approach for the simulations. The present findings demonstrate that the vertex angle has a crucial influence on the shock wave interaction with the V-shaped heavy/light interface. The vertex angle significantly affects the flow field, interface deformation, wave patterns, spike generation, and vorticity production. As the vertex angle decreases, the vorticity production becomes more dominant. A thorough analysis of the vertex angle effect identifies the factors that propel the creation of vorticity during the interaction phase. Notably, smaller vertex angles lead to stronger vorticity generation due to a steeper density gradient, while larger angles result in weaker, more dispersed vorticity and a less complex interaction. Moreover, kinetic energy and enstrophy both dramatically rise with decreasing vortex angles. A detailed analysis is also carried out to analyze the vertex angle effects on the temporal variations of interface features. Finally, the impacts of different Mach and Atwood numbers on the V-shaped interface are briefly presented.

An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension

An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension

Introduction of oblique plane waves through a forcing source term in Euler equations

The propagation of shock waves generated by a transonic flow at the tip of a propeller blade is numerically calculated in order to determine the pressure footprint on an aircraft fuselage. An academic case is first proposed to validate the methodology. An incoming signal is built up as oblique harmonic plane waves. The signal is introduced in the computational domain using a Gaussian volume forcing term in the conservation of mass and energy equations. The Euler equations are solved in two dimensions using finite-difference schemes with low dispersion and dissipation. Selective filtering has also been integrated in the algorithm to remove grid-to-grid oscillations. The numerical solution is compared to an analytical solution based on the tailored Green function. An illustration for a realistic open rotor is then considered. The pressure signal near the blade tip, determined from a preliminary RANS simulation, is introduced using a volume source in a solver of the Euler equations.

Sound transmission through an unbaffled plate

Commonly occurring complex structures can often be modelled as simple finite plates mounted without baffles. Thus, the objective here is to study the sound transmission loss of a simply-supported rectangular finite unbaffled plate impinged upon by an acoustic plane wave. The total pressure jump across the plate surface is expressed as the sum of the pressure jump due to pure diffraction, where the plate normal displacement is zero and the pressure jump due to plate radiation. These pressure jumps are computed using the definition of the free-space Green's function in the wavenumber domain and the Euler equation. Then, the coupled equation of motion is solved for the plate displacement. The method of stationary phase is used to derive the closed-form expression for the improper double integral that appears in the far-field acoustic pressure. Finally, the far-field transmitted power is numerically computed and the sound transmission loss of the unbaffled plate is compared with that of the baffled one.

On the collapse of the local Rayleigh condition for the hydrostatic Euler equations and the finite time blow-up of the semi-Lagrangian equations

On the collapse of the local Rayleigh condition for the hydrostatic Euler equations and the finite time blow-up of the semi-Lagrangian equations

Efficient modeling of atmospheric infrasound propagation with the Semi-Analytical-Finite-Element (SAFE) method

Downward refraction in the lower atmosphere, attributed to winds and temperature inversion, can enable infrasound to efficiently propagate over long distances inside acoustic waveguides. In this study, we use the semi-analytic finite element (SAFE) method to predict this effect in stratified, inhomogeneous, moving air for range-independent noise propagation over reflective half-planes. We present solutions for different temperature and velocity profiles and compare them to direct numerical solutions of the two-dimensional linearized Euler equations. The SAFE method separates the exact two-dimensional wave equation for a stratified, inhomogeneous, moving medium by expressing the pressure field as a sum of vertical eigenmodes propagating in the range direction. This simplifies the acoustic problem into a one-dimensional eigenvalue problem. As this problem is addressed with the finite-element method, the method can accommodate arbitrary wind and temperature profiles. For large, range-independent domains, the proposed procedure proves to be much more efficient than the tested direct numerical computations. Additionally, it converges against the exact solution of the acoustic problem if enough modes are included in the calculation. Therefore, the method offers a viable procedure for benchmarking other pressure field prediction techniques.

Flux reconstruction approach for long-time calculations of the linearized Euler equation

The Linearized Euler Equations (LEE) are one of the fundamental governing equations in acoustics research. Due to the relatively small magnitude of acoustic disturbances compared to the physical quantities of the background flow, ensuring highly precise solutions without dissipation and dispersion over time is essential. The Flux Reconstruction (FR) method, recognized as a high-order numerical method with flexible construction capabilities, has been extensively utilized in recent years for solving partial differential equations numerically. This paper introduces an offsetting numerical flux for one-dimensional LEE. Subsequently, it develops a flux reconstruction scheme with energy-conserving characteristics, referred to as the ECFR scheme, aimed at achieving high-precision numerical solutions for prolonged computations. The energy-conserving feature of the scheme is substantiated through theoretical derivation and further corroborated by numerical validation. Moreover, the impact of Mach numbers and the parameter of the offsetting flux on the scheme's performance is discussed. The stability of the scheme is also analyzed, with the limit of the CFL number being provided. The novel ECFR scheme facilitates obtaining high-precision acoustic numerical solutions, thereby providing strong support for the study of complex engineering problems such as aircraft, submarine, and automobile noise.

Discovering artificial viscosity models for discontinuous Galerkin approximation of conservation laws using physics-informed machine learning

Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models. We refer to the proposed approach as an “hybrid” approach which stands at the edge between supervised and unsupervised learning. More precisely, the proposed “hybrid” paradigm is not supervised in the classical sense as it does not utilize labeled data in the traditional way but relies on the intrinsic properties of the reference solution. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters.

On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws

We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-like interface terms for multi-block upwind SBP methods applied to nonlinear conservation laws. We investigate the behavior of the upwind SBP methods for flux vector splittings of varying complexity on Cartesian as well as unstructured curvilinear multi-block meshes. Moreover, we analyze the local linear/energy stability of these methods following Gassner, Svärd, and Hindenlang (2022, doi:10.1007/s10915-021-01720-8). Finally, we investigate the robustness of upwind SBP methods for challenging examples of shock-free flows of the compressible Euler equations such as a Kelvin-Helmholtz instability and the inviscid Taylor-Green vortex.

A two-fluid multiphase flow model coupled with the population balance equations for the flow of a multidispersed particle system

SummaryWe present a two-fluid multiphase flow model that can be applied to the flow of polydisperse particle systems. In the model, the flow of polydisperse particle systems with different N particle sizes is considered only as two phases of fluid and particle phases, without considering the individual N phases, and the motion of particles with different diameters is represented by the motion characteristics of fluid and granular phases. The equations for the granular phase are obtained by averaging the Euler equations for the particles of different particle sizes. And the velocity of particles of different particle sizes is calculated by an explicit approximation, similar to determining the slip velocity in a mixed model of multiphase flow without solving the Eulerian momentum equation. Also, the conservation equation of particles of different particle sizes was obtained by adding the source term to the population balance equation, which is caused by the difference in the particle velocity of the particles with different particle sizes. The proposed method was compared with the results of experimental studies presented in the previous study to verify the validity of the method. The method has the advantage of not only visualizing the particle size fraction distribution but also of being stable in calculation and not increasing the computational effort significantly in studying the flow of polydisperse particle systems with particle size distribution.

Longitudinal wave propagation in FG rods under impact force

A well-known fact is that one-dimensional wave analysis is the theoretical basis of the famous Hopkinson bar dynamic testing technique. The current one-dimensional wave theory is mostly confined to the slender rods of isotropic materials. It is not easy to obtain an analytical solution to the wave equation of an anisotropic rod. In this work, rods with both elastic modulus and density graded in the length direction are presented and analyzed. The one-dimensional variable coefficient wave equation corresponding to the functionally graded rod is constructed and converted into a second-order variable coefficient partial differential equation using the Laplace approach. Then, the details of solving the partial differential equation of the second-order variable coefficients are given. It is worth noting that here we construct a variable coefficient equation that satisfies the form of Euler's equation. It is still difficult to obtain analytical solutions for other equations that do not satisfy this form. Subsequently, studies of the wave propagation characteristics of rods with different graded configurations are carried out. The theoretical results show that the wave propagation behavior and post-impact vibration of the rod are significantly influenced by the graded configuration. It is possible to adjust not only the impact response at the end but also the impact response in the middle of the rod by optimizing the design of the rod's graded configuration.

A High-Order Well-Balanced Discontinuous Galerkin Method for Hyperbolic Balance Laws Based on the Gauss-Lobatto Quadrature Rules

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests.

A Shifted Boundary Method for the compressible Euler equations

The Shifted Boundary Method (SBM) is applied to compressible Euler flows, with and without shock discontinuities. The SBM belongs to the class of unfitted (or immersed, or embedded) finite element methods and avoids integration over cut cells (and the associated implementation/stability issues) by reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. We specifically discuss the advantages the proposed method offers in avoiding spurious numerical artifacts in two scenarios: (a) when curved boundaries are represented by body-fitted polygonal approximations and (b) when the Kutta condition needs to be imposed in immersed simulations of airfoils. An extensive suite of numerical tests is included.

Shock-driven three-fluid mixing with various chevron interface configurations

When a shock wave crosses a density interface, the Richtmyer–Meshkov instability causes perturbations to grow. Richtmyer–Meshkov instabilities arise from the deposition of vorticity from the misaligned density and pressure gradients at the shock front. In many engineering applications, microscopic surface roughness will grow into multi-mode perturbations, inducing mixing between the fluid on either side of an initial interface. Applications often have multiple interfaces, some of which are close enough to interact in the later stages of instability growth. In this study, we numerically investigate the mixing of a three-layer system with periodic zigzag (or chevron) interfaces, calculating the dependence of the width and mass of mixed material on properties such as the shock timing, chevron amplitude, multi-mode perturbation spectrum, density ratio, and shock mach number. The multi-mode case is also compared with a single-mode perturbation. The Flash hydrodynamic code is used to solve the Euler equations in three dimensions with adaptive grid refinement. Key results include a significant increase in mixed mass when changing from a single-mode to a multi-mode perturbation on one of the interfaces. The mixed width is mainly sensitive to the density ratio and chevron amplitude, whereas the mixed mass also depends on the multi-mode spectrum. Steeper initial perturbation spectra have lower mixed mass at early times but a greater mixed mass after the reflected shock transits back across the layer.

Research on compressor cascade flow field modeling method based on finite volume flux-informed neural network

For compressor cascade flow field modeling, there exists strong velocity shear in the leading edge separation flow, boundary layer, and wake, which leads to increased modeling errors. To improve the accuracy of the flow field modeling method, this paper introduces the concept of numerical flux from the finite volume method into the loss function to implement Euler equation physics-informed learning, and a finite volume flux-informed neural network (FVFI-net) is constructed. Selecting a high-load, large-turning-angle compressor cascade as the study object, a comparative analysis is conducted on the advantages and disadvantages of purely data-driven, weak physical constraint, and finite volume flux-informed methods in compressor cascade flow field modeling. The study found that compared to purely data-driven and weak physical constraint methods, FVFI-net can reduce the average error of aerodynamic parameters in the flow field by approximately 45.6% and 29.5%, respectively, at a 0° angle of attack. For the flow separation problem occurring at the suction side leading edge and the blade wake area caused by a 5° angle of attack, FVFI-net can effectively reduce modeling errors near the leading edge, in the wake region, and near the periodic boundaries, thus reducing the average error of the aerodynamic parameters of the flow field by about 49.2%and 31.3%, respectively, compared to pure data-driven and weak physical constraint methods.

The Non-zonal Rossby–Haurwitz Solutions of the 2D Euler Equations on a Rotating Ellipsoid

The Non-zonal Rossby–Haurwitz Solutions of the 2D Euler Equations on a Rotating Ellipsoid

Concentration and cavitation in the Riemann solutions to the Umami Chaplygin Euler equations

The concentration phenomena in fluid dynamics can be mathematically described by delta-shocks. With the introduction of flux-function, the Riemann problem for the Euler system with Umami Chaplygin gas equation of state is discussed. What Umami Chaplygin gas means is that the fluid obeys the pressure–density relation where the pressure is negative and is a new generalization of Chaplygin gas. The solutions with six kinds of structures are constructed. Unlike the Chaplygin gas, the delta-shock occurs in solutions, even though the system is strictly hyperbolic and two characteristic fields are genuinely nonlinear. The generalized Rankine–Hugoniot relation and entropy condition for delta-shock are clarified. Additionally, the phenomena of concentration and cavitation and the formation of delta-shocks and vacuum states in solutions are identified and analyzed as the Umami Chaplygin gas pressure and flux-function vanish simultaneously. It is proved that as the pressure and flux-function drop to zero, any solution consisting of two shocks tends to the delta-shock solution of the pressureless Euler system, and any solution consisting of two rarefaction waves tends to the vacuum Riemann solution of the pressureless Euler system. Finally, some numerical results exhibiting the processes of formation of delta-shocks and vacuum states are presented.

Time periodic solutions for the 2D Euler equation near Taylor-Couette flow

Time periodic solutions for the 2D Euler equation near Taylor-Couette flow