Euler Equations Research Articles

A Large Time Step Wave-Adding Simple Scheme for Unsteady Compressible Flows

In this paper, we construct a large time step wave-adding scheme (LTS-WA) for unsteady Euler equations in compressible flows with a much simpler strategy in the wave-adding procedure, named LTS-WAS. The new method does not need to calculate coefficients of contribution and also add the waves according to their types and families, which makes the scheme easy for coding and optimisation. The new scheme has a high speedup ratio, and its formulation is the same in scalar and system cases. The scheme is extended to 3D compressible flows using the dimension split method. Numerical experiments show that the new scheme maintains the advantages of the original LTS-WA scheme at large CFL numbers, and has a higher efficiency.

Vorticity Convergence from Boltzmann to 2D Incompressible Euler Equations Below Yudovich Class

Vorticity Convergence from Boltzmann to 2D Incompressible Euler Equations Below Yudovich Class

A mathematical model for flow field with condensation discontinuity: One‐dimensional Euler equations with singular sources

Condensation induces both heat addition and mass consumption effects on the carrier flow field, leading to complex wave patterns. In this study, we aim to elucidate the structures of flow field under the influence of condensation. To achieve this, we present a mathematical model, which is formulated using Euler equations with a singular source term. Given the difficulties posed by both weak solution theory and the Dal Maso–LeFloch–Murat (DLM) theory (J. Math. Pures Appl. 483‐548 (1995)) in defining solutions to the Riemann problem, we propose a novel approach. This involves the coupling of weak solutions within two subregions, taking into consideration the discontinuity of source term. Firstly, we identify two stationary discontinuities originating from the singular source term—namely, stationary waves and composite waves. Admissibility criterions are developed for these discontinuities to facilitate the selection of physically meaningful solutions. Secondly, we employ a double classical Riemann problems (CRPs) framework to analyze the structures of Riemann solution. Our analysis reveals that the Riemann solution may exhibit seven structures with stationary wave and four structures with composite wave. Finally, the proposed model and its theoritical results are applied to validate the structure of the flow field with condensation discontinuities. For the pure heat addition problem, we successfully demonstrate all wave patterns of the flow field, perfectly matching the outcomes of numerical experiments.

Grey prediction model based on Euler equations and its application in highway short-term traffic flow

Grey prediction model based on Euler equations and its application in highway short-term traffic flow

Continuous/smooth transonic solutions to the quasi-one-dimensional steady relativistic Euler equations

This paper establishes the existence and uniqueness of continuous/smooth Meyer type transonic solutions to the quasi-one-dimensional steady isentropic relativistic Euler equations. We use the high-order Taylor’s expansion of the velocity near the sonic point to overcome the difficulties of applying the implicit function theorem caused by the sonic degeneracy. It is verified that the flow in the De Laval nozzle accelerates from subsonic upstream and continuously/smoothly pass through the sonic surface at the throat and then becomes supersonic downstream. Moreover, the flow may have a zero or positive acceleration at the sonic surface, which depends on the geometric property of the nozzle at the throat.

Vortex splitting in two-dimensional fluids and non-neutral electron plasmas with smooth vorticity profiles

Vortex splitting in two-dimensional fluids and non-neutral electron plasmas with smooth vorticity profiles

The time‐asymptotic expansion for the compressible Euler equations with damping

AbstractIn 1992, Hsiao and Liu first showed that the solution to the compressible Euler equations with damping time‐asymptotically converges to the diffusion wave of the porous media equation. Geng et al. proposed a time‐asymptotic expansion around the diffusion wave , which is a better asymptotic profile than . In this paper, we rigorously justify the time‐asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressible Euler equations with damping is accurately characterized by the time‐asymptotic expansion.

Gaussian Pulses over Random Topographies for the Linear Euler Equations

Gaussian Pulses over Random Topographies for the Linear Euler Equations

Fourth-Order Conservative Non-splitting Semi-Lagrangian Hermite WENO Schemes for Kinetic and Fluid Simulations

AbstractWe present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov–Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. The new HWENO reconstructions are meticulously designed to strike a delicate balance between curbing numerical oscillation and introducing excessive dissipation. Mass conservation naturally holds due to the weak formulation of the semi-Lagrangian discontinuous Galerkin method and the design of the HWENO reconstructions. We apply a positivity-preserving limiter to maintain the positivity of numerical solutions when needed. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

Nonlinear stability of entropy waves for the Euler equations

Nonlinear stability of entropy waves for the Euler equations

On Stability and Instability of $$C^{1,\alpha }$$ Singular Solutions to the 3D Euler and 2D Boussinesq Equations

On Stability and Instability of $$C^{1,\alpha }$$ Singular Solutions to the 3D Euler and 2D Boussinesq Equations

Euler and Homogeneous Difference Equations with Linear Coefficients

Euler and Homogeneous Difference Equations with Linear Coefficients

Finite-Time Attitude Control of Quadrotor Unmanned Aerial Vehicle with Disturbance and Actuator Saturation

This paper introduces a nonlinear dynamic inversion control algorithm designed to address unknown disturbances and actuator saturation issues in unmanned aerial vehicle (UAV) attitude control. The algorithm is based on a combination of finite-time disturbance observer and anti-saturation auxiliary system, which ensures the rapid convergence of attitude tracking error. Firstly, based on the Newton–Euler equations, this paper establishes a model of the attitude system for quadrotor UAVs, and this paper eliminates the small-angle flight assumption. Secondly, considering the actuator saturation problem, an anti-saturation auxiliary control system is designed to shorten the time when the control volume is in the saturation interval and achieve finite-time convergence of the attitude error. And then, to improve the robustness of the controller, this paper proposes a disturbance observer based on the finite-time stability theory, which achieves a continuous smooth output of the observation results by introducing a hyperbolic tangent function in the observer, so that the observation error can be converged to zero in a finite time. Finally, it is demonstrated by Simulink simulation that the attitude error and the observation error converge quickly to zero.

Total variation regularization analysis for inverse volatility option pricing problem

We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies.

Lagrangian dynamics and regularity of the spin Euler equation

We derive the spin Euler equation for ideal flows by applying the spherical Clebsch mapping. This equation is based on the spin vector, a unit vector field encoding vortex lines, instead of the velocity. The spin Euler equation enables a feasible Lagrangian study of fluid dynamics, as the isosurface of a spin-vector component is a vortex surface and material surface in ideal flows. We establish a non-blowup criterion for the spin Euler equation, suggesting that the Laplacian of the spin vector must diverge if the solution forms a singularity at some finite time. The direct numerical simulations (DNS) of three ideal flows – the vortex knot, the vortex link and the modified Taylor–Green flow – are conducted by solving the spin Euler equation. The evolution of the Lagrangian vortex surface illustrates that the regions with large vorticity are rapidly stretched into spiral sheets. The DNS result exhibits a pronounced double-exponential growth of the maximum norm of Laplacian of the spin vector, showing no evidence of the finite-time singularity formation if the double-exponential growth holds at later times. Moreover, the present criterion with Lagrangian nature appears to be more sensitive than the Beale–Kato–Majda criterion in detecting the flows that are incapable of producing finite-time singularities.

Numerical Investigation of Supersonic Flow over a Wedge by Solving 2D Euler Equations Utilizing the Steger–Warming Flux Vector Splitting (FVS) Scheme

Supersonic flow over a half-angle wedge (θ = 15°) with an upstream Mach number of 2.0 was investigated using 2D Euler equations where sea level conditions were considered. The investigation employed the Steger–Warming flux vector splitting (FVS) method executed in MATLAB 9.13.0 (R2022b) software. The study involved a meticulous comparison between theoretical calculations and numerical results. Particularly, the research emphasized the angle of oblique shock and downstream flow properties. A substantial iteration count of 2000 iteratively refined the outcomes, underscoring the role of advanced computational resources. Validation and comparative assessment were conducted to elucidate the superiority of the Steger–Warming flux vector splitting (FVS) scheme over existing methodologies. This research serves as a link between theoretical rigor and practical applications in high-speed aerospace design, enhancing the efficiency of aircraft components.

Phase-Space and Dissipation Studies of the Dancing Wing Nut

In this paper, we address the nonlinear dynamics of Dzhanibekov’s dancing wing nut and the related intermediate axes theorem governing the motion of a tossed tennis racket. Drawing first on the comprehensive investigations by Murakami, Rios and Impelluso [J. Appl. Mech. 83 (2016), https://doi./org/10.1115/1.4034318] we extend the exploration of the wing nut’s spherical phase portrait by incorporating basins of attraction and their heteroclinic connections. We then turn our attention to the significant damping studies of Español, Thachuk and de la Torre [Eur. J. Mech. A Solids 103, 105184 (2024)] who derive from first principles a generalized stochastic version of the Euler equations which can incorporate both oriental realignment (which they call oriental diffusion) and viscoelastic effects. Applied to the wing nut dancing, they showed that the former is the more significant effect and as the authors say “Had Dzhanibekov observed his wing nut for a sufficiently long period, he would have found it spinning around the major axis. In other words, precession relaxation kills the Dzhanibekov effect in the long run.”

Geometric analysis of the generalized surface quasi-geostrophic equations

AbstractWe investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach

The inviscid incompressible limit of Kelvin–Helmholtz instability for plasmas

Introduction: The Kelvin–Helmholtz Instability (KHI) is an interface instability that develops between two fluids or plasmas flowing with a common shear layer. KHI occurs in astrophysical jets, solar atmosphere, solar flows, cometary tails, planetary magnetospheres. Two applications of interest, encompassing both space and fusion applications, drive this study: KHI formation at the outer flanks of the Earth’s magnetosphere and KHI growth from non-uniform laser heating in magnetized direct-drive implosion experiments. Here, we study 2D KHI with or without a magnetic field parallel to the flow. We use both the GAMERA code, which solves the compressible Euler equations, and the STRATOSPEC code, which solves the Navier-Stokes equations under the Boussinesq approximation, coupled with the magnetic field dynamics. GAMERA is a global three-dimensional MHD code with high-order reconstruction in arbitrary nonorthogonal curvilinear coordinates, which is developed for a large range of astrophysical applications. STRATOSPEC is a three-dimensional pseudo-spectral code with an accuracy of infinite order (no numerical diffusion). Magnetized KHI is a canonical case for benchmarking hydrocode simulations with extended MHD options.Methods: An objective is to assess whether or not, and under which conditions, the incompressibility hypothesis allows to describe a dynamic compressible system. For comparing both codes, we reach the inviscid incompressible regime, by decreasing the Mach number in GAMERA, and viscosity and diffusion in STRATOSPEC. Here, we specifically investigate both single-mode and multi-mode initial perturbations, either with or without magnetic field parallel to the flow. The method relies on comparisons of the density fields, 1D profiles of physical quantities averaged along the flow direction, and scale-by-scale spectral densities. We also address the triggering, formation and damping of filamentary structures under varying Mach number or Atwood number, with or without a parallel magnetic field.Results: Comparisons show very satisfactory results between the two codes. The vortices dynamics is well reproduced, along with the breaking or damping of small-scale structures. We end with the extraction of growth rates of magnetized KHI from the compressible regime to the incompressible limit in the linear regime assessing the effects of compressibility under increasing magnetic field.Discussion: The observed differences between the two codes are explained either from diffusion or non-Boussinesq effects.

Implicit interpolation method for immersed boundary methods

Immersed boundary (IB) methods have been successfully implemented for different applications. This paper focuses on the immersed boundary implementation for two different governing equations, namely the diffusion equation and Euler equations, using a bi-linear interpolation for the implementation of the boundary condition. The concept of implicit interpolation is introduced which eradicates the problems faced with the explicit interpolation in which it is required to move away from the boundary in the fluid domain in order to complete the interpolation stencil.