Two-dimensional Euler Equations Research Articles

On the rarefaction waves of the two-dimensional compressible Euler equations for magnetohydrodynamics

The expansion of a wedge of magnetic fluid into vacuum is studied in this paper. The magnetic fluid away from the sharp corner of a wedge expands into the vacuum as two plane-symmetric rarefaction waves, and the problem can be reduced to the interaction of these two rarefaction waves. In order to determine the flow in the interaction zone, we formulate a Goursat problem for the two-dimensional, self-similar Euler equations of magnetohydrodynamic. This system is of mixed type, and the type at each point is determined by the local fluid velocity and the local magneto-acoustic speed. We establish that the system is uniformly hyperbolic in the interaction zone when the half-angle of the wedge is less than some angle [Formula: see text], while the existence of a global classical solution to the Goursat problem is proven by a method of characteristic decomposition.

A Characteristic Mapping method for the two-dimensional incompressible Euler equations

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure which allows for the decomposition of a long-time deformation into short-time submaps. This leads to a numerical scheme that achieves exponential resolution in linear time. Error estimates are provided and conservation properties are analysed. The computational efficiency and the high precision of the method are illustrated in the vortex merger, four-modes and random flow problems. Comparisons with the Cauchy-Lagrangian method [3] are also presented.

High-order point-value enhanced finite volume method for two-dimensional hyperbolic equations on unstructured meshes

Presently, an approximate delta function (ADF) is defined in multi-dimensional space in the form of a finite-order polynomial that holds identical integral properties to the Dirac delta function when used in conjunction with a finite-order polynomial integrand spanning a finite domain. By using this ADF over a two-dimensional unstructured mesh, a compact high-order point-value enhanced finite volume method (PFV) is constructed. The corresponding scheme is capable of storing and updating cell-averaged values inside each element as well as the unknown quantities and their derivatives (when needed) on vertex points. The sharing of vertex information with surrounding elements reduces the number of degrees of freedom relative to other compact and high-order methods of comparable convergence rate. To ensure conservation, cell-averaged values are updated using an identical approach to the one traditionally followed in the finite volume method. To verify the performance of the PFV scheme at successive orders, its accuracy and stability are vetted using benchmark problems associated with the two-dimensional wave and Euler equations encompassing a total of six standard test cases.

Non uniform Rotating Vortices and Periodic Orbits for the Two-Dimensional Euler Equations

This paper concerns the study of some special ordered structures in turbulent flows. In particular, a systematic and relevant methodology is proposed to construct non trivial and non radial rotating vortices with non necessarily uniform densities and with different m-fold symmetries, $$m\geqq 1$$ . In particular, a complete study is provided for the truncated quadratic density $$(A|x|^2+B){\mathbf{1}}_{{\mathbb {D}}}(x)$$ , with $${\mathbb {D}}$$ the unit disc. We exhibit different behaviors with respect to the coefficients A and B describing the rarefaction of bifurcating curves.

On A Posteriori Estimation of the Approximation Error Norm for an Ensemble of Independent Solutions

An ensemble of independent numerical solutions makes it possible to construct a hypersphere around an approximate solution that contains the true solution. The analysis is based on some geometry considerations, such as the triangle inequality and the measure concentration in spaces of large dimensions. As a result, a nonintrusive postprocessor providing error estimation on an ensemble of solutions can be constructed. Some numerical tests for the two-dimensional compressible Euler equations are given to demonstrate the properties of such postprocessing.

Numerical study on the shock/combustion interaction of oblique detonation waves

To determine the shock/combustion interaction structure of a wedge-induced oblique detonation wave, high-order numerical simulations solving the two-dimensional reactive Euler equations embedded with a two-step chemical kinetic model have been performed. In this study, various dimensionless heat release amounts ranging from 40.0 to 52.5 are selected to investigate the flow configuration. The computational results show that four types of typical shock/combustion interactions, namely, Type VI, I, V and IV (following the classification of shock/shock interaction proposed by Edney), can be observed. The detailed structures and characteristics of the interaction types are presented and shown to be different than the classic shock/shock interaction. Additional insight into the transition principles of different shock/combustion interactions is illustrated through shock polar analysis.

A nonlocal shallow-water model with the weak Coriolis and equatorial undercurrent effects

We derive a quasilinear nonlocal shallow-water model equation for moderate-amplitude equatorial waves with effects of the weak Coriolis force and equatorial undercurrent. This mathematical modeling is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and rotational Euler equations in the equatorial region. Moreover, we investigate the influences and interactions of the weak Coriolis force caused by the Earth rotation, vorticity and nonlocal higher nonlinearities on the wave-breaking phenomena. In certain cases, by applying the method of characteristics and extremal property of the solution to the Riccati-type differential inequality, we demonstrate that wave breaking phenomena occur only depending on a shape of wave initially in the local spacial point.

Numerical Simulation of Detonation Combustion of Kerosene Vapors in an Expanding Nozzle

The initiation and stabilization of detonation combustion of kerosene vapor in a supersonic air flow entering an expanding axisymmetric nozzle with a coaxial central body is numerically studied. Calculations are based on a reduced kinetic model of combustion, which includes 68 reactions for 44 components. Enthalpy and entropy of components are determined using approximating polynomials from the NASA database. A hydrodynamic model is based on two-dimensional unsteady Euler equations for an axisymmetric flow of a multicomponent reacting gas. Calculations are performed using the Godunov finite-difference scheme and its $$\beta$$ -modification of improved accuracy on smooth solutions. The flow parameters that ensure the steady detonation combustion of kerosene vapor in the nozzle under consideration are determined. The detonation combustion of kerosene has higher thrust than hydrogen combustion, but is noticeably inferior with regard to specific thrust. The calculations are performed on the “Lomonosov" supercomputer at the Lomonosov Moscow State University.

A compact subcell WENO limiting strategy using immediate neighbours for Runge-Kutta discontinuous Galerkin methods

A compact subcell weighted essentially non oscillatory (CSWENO) limiter is proposed for the solution of hyperbolic conservation laws with discontinuous Galerkin Method which uses only the immediate neighbours of a given cell. These neighbours are divided into the required stencil for WENO reconstruction and an existing WENO limiting strategy is used. Accuracy tests and results for one-dimensional and two-dimensional Burgers' equation and one-dimensional and two-dimensional Euler equations for Cartesian meshes are presented using this limiter. Comparisons with the parent WENO limiter are provided wherever appropriate and the performance of the current limiter is found to be slightly better than the parent WENO limiter for higher orders.

Numerical study on transition structures of oblique detonations with expansion wave from finite-length cowl

The oblique detonation wave (ODW) triggered by a semi-infinite cowl or wedge has been widely studied, but the effects of finite-length cowls require further clarification to enhance their applicability. Based on two-dimensional inviscid Euler equations and a two-step induction–reaction model, two structures with smooth and abrupt transitions are simulated, and their interactions with the cowl-induced expansion wave are investigated. The expansion waves located downstream and far away from the initiation zone do not affect the macroscopic structures of the ODW. However, when the expansion waves move upstream, the structures evolve into a decoupled shock and a reactive front. Alongside findings from previous studies, these results indicate that the initiation mechanism, rather than the transition type, determines the near-quenching evolution of the structure. Moreover, given the same parameter settings for the chemical and inflow gas dynamics, these numerical results show that the abrupt transition evolves into a smooth transition under an expansion wave disturbance. This demonstrates that the local wave interaction plays a key role in the ODW structures.

Invariant Measures for Stochastic Damped 2D Euler Equations

We study the two-dimensional Euler equations, damped by a linear term and driven by an additive noise. The existence of weak solutions has already been studied; pathwise uniqueness is known for solutions that have vorticity in $L^\infty$. In this paper, we prove the Markov property and then the existence of an invariant measure in the space $L^\infty$ by means of a Krylov-Bogoliubov's type method, working with the weak$\star$ and the bounded weak$\star$ topologies in $L^\infty$.

The Flux Reconstruction Method with Lax–Wendroff Type Temporal Discretization for Hyperbolic Conservation Laws

In this paper, we develop a Lax–Wendroff type time discretization method for high order Flux Reconstruction scheme to solve hyperbolic conservation laws. Through Cauchy–Kowalewski procedure, the resulting Lax–Wendroff Flux Reconstruction (LWFR) scheme is an alternative spatial–temporal coupling method to the popular Runge–Kutta Flux Reconstruction (RKFR) scheme. LWFR is a one-step explicit high order discontinuous finite element method and its discretization procedure is more compact and effective for certain problems than that of RKFR. Furthermore, aiming at accurate simulation of discontinuity, we propose a robust local artificial viscosity formulation of LWFR for the first time. A collection of successful numerical experiments show that LWFR can give essentially non-oscillatory and sharp solutions for discontinuity, and maintain designed order of accuracy for smooth regions, both in one-dimensional and two-dimensional Euler equations. In conclusion, LWFR scheme is cost effective and accuracy-preserving for certain problems and can be an attractive candidate to solve hyperbolic conservation laws.

A modified HLLEM scheme and shock stability analysis

Reliable numerical simulations for hypersonic flows require an accurate, robust and efficient numerical scheme. The low-dissipation shock-capturing methods often suffer various forms of shock wave instabilities when used to simulate hypersonic flow problems numerically. For the two-dimensional(2D) inviscid compressible Euler equations, the stability analysis of the low-dissipation HLLEM scheme is conducted. The odd and even perturbations are added to the initial state in the streamwise direction and the transverse direction respectively, and the evolution equations of perturbations are deduced to explore the mechanism of instability inherent in the HLLEM scheme. The results of stability analysis show that the perturbations of density and shear velocity in the flux transverse to the shock wave front are undamped. Due to the symmetry, the 2D Sedov blast wave problem is computed to prove the multidimensionality of the shock instability. In the one-dimensional case which is free from the instability, the undamped property of density perturbation is also existent but no shear velocity is found. The conclusion can be drawn as follows: the shock instability of HLLEM scheme is triggered by the perturbation growth of shear velocity in the flux transverse to the shock wave front. Based on the conclusion of stability analysis, the instability of HLLEM scheme is cured by adding the shear viscosity to the transverse flux. In order to avoid affecting the resolution of the shear layer due to the introduction of too high shear viscosity, two functions to detect the shock wave and the subsonic regimes are defined, so that the shear viscosity is only added to the transverse flux in the subsonic regime of the shock layer, while the rest of numerical fluxes are still computed by the original HLLEM scheme. The results of stability analysis and some challenging numerical test problems show that the modified HLLEM scheme not only retains the merits of the original HLLEM, such as, resolving contact discontinuity and shear wave accurately, but also has greatly improved its robustness, inhibiting the unstable phenomena from occurring effectively when computing the strong shock wave problems.

Two-front solutions of the SQG equation and its generalizations

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0<\alpha \le 2$. Special cases are the two-dimensional incompressible Euler equations ($\alpha = 2$) and the surface quasi-geostrophic (SQG) equations ($\alpha = 1$). We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-in-time existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1<\alpha\le 2$, and small, smooth solutions in the parameter regime $0<\alpha\le 1$.

A Particle Filter for Stochastic Advection by Lie Transport: A Case Study for the Damped and Forced Incompressible Two-Dimensional Euler Equation

In this work, we combine a stochastic model reduction with a particle filter augmented with tempering and jittering, and apply the combined algorithm to a damped and forced incompressible two-dimensional Euler dynamics defined on a simply connected bounded domain. We show that using the combined algorithm, we are able to assimilate data from a reference system state (the “truth'') modeled by a highly resolved numerical solution of the flow that has roughly $3.1 \times 10^6$ degrees of freedom, into a stochastic system having two orders of magnitude less degrees of freedom, which is able to approximate the true state reasonably accurately for five large-scale eddy turnover times, using modest computational hardware. The model reduction is performed through the introduction of a stochastic advection by Lie transport (SALT) model as the signal on a coarser resolution. The SALT approach was introduced as a general theory using a geometric mechanics framework from Holm [Proc. A, 471 (2015)]. This work follows on the numerical implementation for SALT presented by Cotter et al. [SIAM Multiscale Model. Simul., 17 (2019), pp. 192--232] for the flow in consideration. The model reduction is substantial: the reduced SALT model has $4.9 \times 10^4$ degrees of freedom. Results from reliability tests on the assimilated system are also presented.

Diffraction of cellular detonation wave over a cylindrical convex wall

Numerical simulations were performed to study diffraction of a cellular detonation wave over a 90∘ cylindrical convex wall. The dynamics of this diffraction phenomenon was described by the two-dimensional reactive Euler equations with a detailed hydrogen/oxygen chemistry model and solved numerically by using the adaptive mesh refinement code AMROC. The present numerical observations indicate that the continuous variation trend of area divergence ratio caused by the curved convex wall can make detonation transverse wave reflect on the wall and sustain the coupling of the diffracted detonation wave for a certain distance. The reflection of the transverse waves can also enhance the energy to promote the re-intiation of the diffracted detonation. As the wall curvature radius increases to a certain value, the detonation wave can even propagate stably without decoupling in the expansion area. The critical wall curvature radius for re-initiation is found to have negative correlation with the contained detonation cell number in the exit, while the critical radius for stable propagation is mainly determined by the initial pressure. The critical re-initiation criterion for the detonation diffraction is further explored by introducing two different re-initiation mechanisms and considering the width of an “equivalent exit” calculated by the Chester-Chisnell-Whitham theory to satisfy the re-initiation condition. Meanwhile the critical criterion of stable propagation is described quantitatively by applying the Dn−κ theory to confirm the relation between the propagation velocity and the curvature of the diffracted detonation wave front, and calculating the minimum curvature radius of the curved detonation wave front theoretically. The accuracy of the criterions are verified by compared with the numerical results.

A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics

The incompressible 2D Euler equations on a sphere constitute a fundamental model in hydrodynamics. The long-time behaviour of solutions is largely unknown; statistical mechanics predicts a steady vorticity configuration, but detailed numerical results in the literature contradict this theory, yielding instead persistent unsteadiness. Such numerical results were obtained using artificial hyperviscosity to account for the cascade of enstrophy into smaller scales. Hyperviscosity, however, destroys the underlying geometry of the phase flow (such as conservation of Casimir functions), and therefore might affect the qualitative long-time behaviour. Here we develop an efficient numerical method for long-time simulations that preserve the geometric features of the exact flow, in particular conservation of Casimirs. Long-time simulations on a non-rotating sphere then reveal three possible outcomes for generic initial conditions: the formation of either 2, 3, or 4 coherent vortex structures. These numerical results contradict the statistical mechanics theory and show that previous numerical results, suggesting 4 coherent vortex structures as the generic behaviour, display only a special case. Through integrability theory for point vortex dynamics on the sphere we present a theoretical model which describes the mechanism by which the three observed regimes appear. We show that there is a correlation between a first integral $\gamma$ (the ratio of total angular momentum and the square root of enstrophy) and the long-time behaviour: $\gamma$ small, intermediate, and large yields most likely 4, 3, or 2 coherent vortex formations. Our findings thus suggest that the likely long-time behaviour can be predicted from the first integral $\gamma$.

Rotating equilibria of vortex sheets

We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann–Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is real-valued and also vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain distributional sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations.

On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data

We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class i.e. it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method.

Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel T×R. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν−1/3, ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schrödinger operators, combined with a hypocoercivity argument to handle the viscous case.