Euler Equations Research Articles

The impact of modulational instability on coastal wave forecasting using quadratic models

Coastal wave forecasting over large spatial scales is essential for many applications (e.g., coastal safety assessments, coastal management and developments, etc.). This demand explains the necessity for accurate yet effective models. A well-known efficient modelling approach is the quadratic approach (often referred to as frequency-domain models, nonlinear mild-slope models, amplitude models, etc.). The efficiency of this approach stems from a significant modelling reduction of the original governing equations (e.g., Euler equations). Most significantly, the description of wave nonlinearity essentially collapses into a single mode coupling term determined by the quadratic interaction coefficients. As a result, it is expected that the efficiency achieved by the quadratic approach is accompanied by a decrease in prediction accuracy. In order to gain further insight into the predictive capabilities of this modelling approach, this study examines six different quadratic formulations, three of which are of the Boussinesq type and the other three are referred to as fully dispersive. It is found that while the Boussinesq formulations reliably predict the evolution of coastal waves, the predictions by the fully dispersive formulations tend to be affected by false developments of modulational instability. Consequently, the predicted wave fields by the fully dispersive formulations are characterized by unexpectedly strong modulations of the sea-swell part and associated unexpected infragravity response. The impact of the modulational instability on wave prediction based on the quadratic approach is further demonstrated using existing laboratory results of bichromatic and irregular wave conditions.

A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations

AbstractWe propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low‐order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two‐step technique: computation of a tentative flux, here, with a centered approximation of the velocity, and limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation (with the velocity, one of its component, the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi‐implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi‐implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.

Cell-centered indirect Arbitrary Lagrangian-Eulerian numerical strategy for solving 3D gas dynamics equations

Solving the Euler equations under the Lagrangian formalism enables to simulate various complex engineering applications. However, the use of this formalism can lead to significant mesh deformations as the mesh follows the fluid velocity. The mesh quality may be considerably deteriorated requiring a regularization procedure. In the present document, it is shown that the ideas presented by J. Yao (2013)[68] and J. Yao and D. Stillman (2016) [30] may be used efficiently in a 3D hydrodynamics code to perform reliable regularization steps. The flexibility of the methodology in addition to its simplicity, since it only relies on trivial geometrical considerations, opens the way for various extensions. The remapping step then considered, uses the geometrical splitting procedure of the Lagrangian phase to perform effective projections. This coherence ensures the compactness of the overall algorithm. At last but not least, a 3D Flux-Corrected-Remapping method is presented. This yields a particularly robust remapping algorithm while also leading the way for higher order projection extensions.

Improved well-balanced AWENO schemes with hydrostatic reconstruction for the Euler equations under gravitational fields

The Euler equations under gravitational fields allow the hydrostatic equilibrium states, which requires that the numerical scheme of the system should also have this characteristic. In our previous work, a well-balanced finite difference conservative AWENO scheme has been constructed to preserve the isothermal equilibrium state accurately (Fu et al., Appl. Numer. Math., 180:1-15, 2022). However, the scheme fails to maintain more complicated hydrostatic equilibrium states, such as the isentropic equilibrium state. In this study, the improved well-balanced AWENO schemes with hydrostatic reconstruction are designed to maintain the general equilibrium states numerically. The key technologies to achieve the well-balanced property are rewriting the source term into an appropriate form based on the equilibrium equation, acting on the conserved variables with an affine-invariant WENO or linearized WENO interpolation operator, employing hydrostatic reconstruction methods to create auxiliary conserved variables, and modifying the numerical flux with auxiliary conserved variables and higher-order conserved terms based on pressure. The proposed well-balanced schemes maintain the equilibrium solution near the machine precision and capture small perturbations of the equilibrium state without spurious oscillations even at the coarse mesh. Furthermore, they can be applicable to the general hydrostatic equilibrium more than the isothermal equilibrium state. Several numerical experiments are tested to show that the developed well-balanced schemes perform well in maintaining equilibrium states and capturing small perturbations.

Spontaneous Stochasticity Amplifies Even Thermal Noise to the Largest Scales of Turbulence in a Few Eddy Turnover Times.

How predictable are turbulent flows? Here, we use theoretical estimates and shell model simulations to argue that Eulerian spontaneous stochasticity, a manifestation of the nonuniqueness of the solutions to the Euler equation that is conjectured to occur in Navier-Stokes turbulence at high Reynolds numbers, leads to universal statistics at finite times, not just at infinite time as for standard chaos. These universal statistics are predictable, even though individual flow realizations are not. Any small-scale noise vanishing slowly enough with increasing Reynolds number can trigger spontaneous stochasticity, and here we show that thermal noise alone, in the absence of any larger disturbances, would suffice. If confirmed for Navier-Stokes turbulence, our findings would imply that intrinsic stochasticity of turbulent fluid motions at all scales can be triggered even by unavoidable molecular noise, with implications for modeling in engineering, climate, astrophysics, and cosmology.

Analyzing the spatial motion of a rigid body subjected to constant body-fixed torques and gyrostatic moment

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body’s middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions’ extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications.

Multi-species kinetic-fluid coupling for high-energy density simulations

Many physics problems are subject to a mix of continuum and non-equilibrium flows or conditions where one transition into the other, as a function of time and/or space. Numerically, such flows could be described with a purely kinetic method which, in the limit of small particle mean free paths, reproduces a continuum regime. However, from a computational perspective, such approaches are usually very expensive or simply not feasible, even on modern computational architectures. A solution is the construction of hybrid methods which connect the kinetic and hydrodynamic system of evolution equations. Such coupling usually requires specific boundary conditions between non-equilibrium and continuum regimes which come with their own numerical challenges and can introduce unphysical effects. Here, we present a hybrid model which uses a buffer region to transition from a kinetic into a fluid description following the original work of Degond et al. [1]. In the buffer region, both, the kinetic and hydrodynamic equations are solved simultaneously while being coupled via a so-called transition function. The latter also ensures a smooth conversion from the coupled model to either the kinetic or continuum approach at the interfaces of the buffer region. With that, the method avoids the need to find direct interface boundary conditions and allows one to localize the use of a high dimensional kinetic model only where it is needed. In our work, we extend the original method of Degond to flows with multiple particle species in 3D. We derive the coupled equations for the multispecies Vlasov-Bhatnagar-Gross-Krook (VBGK) model and its limiting Euler or Navier-Stokes hydrodynamic equations. To validate our model numerically, we simulate a Sod shock problem and study the effect of kinetic multi-species mixing in the preheat phase of a high energy-density plasma physics experiment.

Vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg equations with slip boundary condition

In this paper, we investigate the vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg (NSK) equations in a three-dimensional horizontally periodic strip domain, in which the velocity of the fluid is supplemented with slip boundary condition and the gradient of density with Dirichlet boundary condition on the boundary. We prove that there exists an positive constant T0 independent on the capillarity and viscosity coefficients, such that the incompressible NSK equations have a unique strong solution on [0,T0] and the solution is uniformly bounded in H3. Based on the uniform estimates, we further give the convergence rate in H1 from the solutions of the incompressible NSK equations to the solution of the inhomogeneous incompressible Euler equations as the capillarity and viscosity coefficients go to zero simultaneously.

Symmetries of the Large Scale Structures of the Universe as a Phenomenology of a Fractal Turbulence: The Role of the Plasma Component

We present a new perspective on the symmetries that govern the formation of large-scale structures across the Universe, particularly focusing on the transition from the seeds of galaxy clusters to the seeds of galaxies themselves. We address two main features of cosmological fluid dynamics pertaining to both the linear and non-linear regimes. The linear dynamics of cosmological perturbations within the Hubble horizon is characterized by the Jeans length, which separates stable configurations from unstable fluctuations due to the gravitational effect on sufficiently large (and therefore, massive enough) overdensities. On the other hand, the non-linear dynamics of the cosmological fluid is associated with a turbulent behavior once the Reynolds numbers reach a sufficiently high level. This turbulent regime leads to energy dissipation across smaller and smaller scales, resulting in a fractal distribution of eddies throughout physical space. The proposed scenario suggests that the spatial scale of eddy formation is associated with the Jeans length of various levels of fragmentation from an original large-scale structure. By focusing on the fragmentation of galaxy cluster seeds versus galaxy seeds, we arrived at a phenomenological law that links the ratio of the two structure densities to the number of galaxies in each cluster and to the Hausdorff number of the Universe matter distribution. Finally, we introduced a primordial magnetic field and studied its influence on the Jeans length dynamics. The resulting anisotropic behavior of the density contrast led us to infer that the main features of the turbulence could be reduced to a 2D Euler equation. Numerical simulations showed that the two lowest wavenumbers contained the major energy contribution of the spectrum.

A positivity preserving and oscillation-free entropy stable discontinuous Galerkin scheme for the reactive Euler equations

The reactive Euler equations are a basic model for fluid flows with chemical reactions. In this work, we construct a high order positivity preserving and oscillation-free entropy stable discontinuous Galerkin (DG) scheme for the reactive Euler equations. The main ingredients of the scheme include (i) entropy preserving and entropy stable fluxes to achieve entropy stability, (ii) artificial damping terms to restrain spurious oscillations near the shocks, and (iii) positivity preserving limiters to guarantee the positivity of solutions. These ingredients are compatible with each other so that our scheme simultaneously enjoys the properties of entropy stable, oscillation-free and positivity preserving. Another distinctive feature of our scheme is that it is entropy stable for both the thermodynamic and mathematical entropies. Numerical experiments validate the designed high convergence orders of the scheme and demonstrate its good performances for discontinuous problems.

Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics

The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method is fully discrete and has a compact stencil in space and time. An important component of Active Flux methods is the evolution formula for the update of the point values. A previously proposed exact evolution formula for acoustics is reviewed and used to construct an Active Flux method for the two-dimensional Maxwell’s equation. Furthermore, the method of bicharacteristics is discussed as a methodology for the derivation of truly multidimensional approximative evolution operators that can be used for the evolution of point values in Active Flux methods. We study accuracy and stability of the resulting methods for acoustics and compare with the Active Flux method that uses the exact evolution operator. Finally, we used the method of bicharacteristics to derive Cartesian grid Active Flux methods for the linearised and nonlinear Euler equations. Numerous test computations illustrate the performance of these new Active Flux methods.

Solving Euler equations with gradient-weighted multi-input high-dimensional feature neural network

The study found that it is difficult to capture the solutions at the shock wave and discontinuity surfaces when solving Euler equations using physics informed neural network. Thus, this paper proposes an improved neural network based on adaptive weights for multi-input high-dimensional features to solve the Euler equations. First, adaptive weights regarding the velocity are added to the control equation of each residual to train the shock wave region better. Then, more residual training points are used in regions with initial time discontinuities to improve the training efficiency. The problem that the sigmoid activation function is more prone to gradient pathologies than tanh in the training process is also analyzed to show that the Euler equations can be better solved using tanh. Numerical experiments verify that even though the solution process becomes complicated, it outperforms the original physics informed neural network in terms of computational efficiency and computational accuracy and can better portray the physical phenomena of Euler equations.

Physics-informed neural networks for transonic flow around a cylinder with high Reynolds number

The physics-informed neural network (PINN) method is extended to learn and predict compressible steady-state aerodynamic flows with a high Reynolds number. To better learn the thin boundary layer, the sampling distance function and hard boundary condition are explicitly introduced into the input and output layers of the deep neural network, respectively. A gradient weight factor is considered in the loss function to implement the PINN methods based on the Reynolds averaged Navier–Stokes (RANS) and Euler equations, respectively, denoted as PINN–RANS and PINN–Euler. Taking a transonic flow around a cylinder as an example, these PINN methods are first verified for the ability to learn complex flows and then are applied to predict the global flow based on a part of physical data. When predicting the global flow based on velocity data in local key regions, the PINN–RANS method can always accurately predict the global flow field including the boundary layer and wake, while the PINN–Euler method can accurately predict the inviscid region. When predicting the subsonic and transonic flows under different freestream Mach numbers (Ma∞= 0.3–0.7), the flow fields predicted by both methods avoid the inconsistency with the real physical phenomena of the pure data-driven method. The PINN–RANS method is insufficient in shock identification capabilities. Since the PINN–Euler method does not need the second derivative, the training time of PINN–Euler is only 1/3 times that of PINN–RANS at the same sampling point and deep neural network.

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

Airfoil Design Framework for Optimized Boundary-Layer Integral Parameters

An airfoil design framework is introduced in which boundary-layer integral parameters serve as the driving design mechanism. The method consists of generating a parameterized pressure distribution capable of producing the desired boundary-layer characteristics for inverse design use. Additionally, by deduction from the Squire–Young theory, the method allows for the determination of the pressure distribution that results in the minimum theoretical drag. To assess this design framework, several airfoils were developed based on the mission requirements of the RQ-4B Global Hawk aircraft. Numerical results obtained using a viscous-inviscid solver of the integral boundary layer and Euler equations showed that the optimized airfoils achieved profile drag reductions of 9.06 and 6.00%, respectively, for α=0° and L/Dmax design points. A validation experimental campaign was also performed using the optimized CA5427-72 airfoil. The acquired data produced the expected pressure distribution characteristics and aerodynamic performance improvements, typifying the efficacy of the design framework.

Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry

We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler–Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier–Stokes–Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}, and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum (ρ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =0$$\\end{document}) and far-field (ρ=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =\\infty $$\\end{document}) are carefully analyzed. Owing to the generality of the pressure law, only the Wloc-1,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{-1,p}_{\ extrm{loc}}$$\\end{document}-compactness of weak entropy dissipation measures with p∈[1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in [1,2)$$\\end{document} can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.

Numerical study of the scattering of acoustic waves by an elliptic vortex.

The scattering of the acoustic waves generated by a monopolar source propagating through a two-dimensional elliptic vortex, fixed or convected by a uniform flow, is studied by solving the Linearized Euler Equations in Cartesian coordinates using the Discontinuous Galerkin Method. For a fixed vortex position, the number, amplitudes, and angular spreads of the acoustic interference beams resulting from the sound scattering are found to significantly depend on the orientation of the vortex major axis with respect to the direction of the incident waves and on the vortex maximum tangential velocity. In particular, additional interference beams are obtained at large observation angles for a more elliptical vortex. For a convected elliptic vortex, the interference beams are curved as the angle between the incident acoustic wave and the vortex major axis varies when the vortex travels in the downstream direction. As expected, the scattering of the acoustic waves leads to spectral broadening in this case. Moreover, the widths and the frequencies of the lateral lobes obtained in the spectra on both sides of the peak at the source frequency are different for elliptic and round vortices.

Variational principles for the hydrodynamics of the classical one-component plasma

Hydrodynamic equations for a one-component plasma are derived as a unification of the Euler equations with long-range Coulomb interaction. By using a variational principle, these equations self-consistently unify thermodynamics, dispersion laws, nonlinear motion, and conservation laws. In the moderate and strong coupling limits, it is argued that these equations work down to the length scale of the interparticle spacing. The use of a variational principle also ensures that closure is achieved self-consistently. Hydrodynamic equations are evaluated in both the Eulerian frame, where the fluid variables depend on the position in the laboratory, and the Lagrangian frame, where they depend on the position in some reference state, such as the initial position. Each frame has its advantages and our final theory combines elements of both. The properties of longitudinal and transverse dispersion laws are calculated for the hydrodynamic equations. A simple step function approximation for the pair distribution function enables simple calculations that reveal the structure of the equations of motion. The obtained dispersion laws are compared to molecular dynamics simulations and the theory of quasilocalized charge approximation. The action, which gives excellent agreement for both longitudinal and transverse dispersion laws for a wide range of coupling strengths, is elucidated. Agreement with numerical experiments shows that such a hydrodynamic approach can be used to accurately describe a one-component plasma at very small length scales comparable to the average interparticle spacing. The validity of this approach suggests considering nonlinear flows and other systems with long-range interactions in the future.

Interactions between elementary waves and weak discontinuity in two-layer blood flow through artery

Blood flow through arterial circulation can be characterized by fluid transport in flexible tubes and modeled mathematically using the conservation of mass and momentum. A one-dimensional model for two-layer blood flow with different blood velocities and the same constant density in each layer derived from the Euler equations of gas dynamics by taking the vertical average across each layer. This work presents interactions of elementary waves with a weak discontinuity for the quasilinear 3 × 3 system of conservation laws governing the two-layer blood flow in arteries. Exploiting elementary waves as a single-parameter curve, we study the Riemann solution uniquely and consequently establish the condition on initial data for the existence of a solution to the Riemann problem. Furthermore, we discuss the evolution of weak discontinuity waves and subsequently derive their amplitudes; in what follows, we investigate the interactions of weak discontinuity with contact discontinuity and shocks. Finally, a series of numerical tests have been performed to understand the impact of shock strength and the initial data on the amplitudes of reflected and transmitted waves and the jumps in shock acceleration.

Testing in smoothed GMM quantile models with an application to quantile Euler equation

Within the framework of smoothed generalized method of moments for quantile models, including instrumental variables quantile regression as a special case, a distance metric hypothesis test and an overidentification test are proposed. The asymptotoic validity including the consistency of the distance metric hypothesis test are established. Simulations show reasonable finite-sample performance of both proposed tests in terms of size and power. In an empirical example, the proposed tests are applied to test the discount factor and elasticity of intertemporal substitution parameters in quantile Euler equations and overall the quantile Euler equation itself.