skip to main content

Title: Numerical study of non-uniqueness for 2D compressible isentropic Euler equations.
In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi- dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Journal of computational physics
Page Range / eLocation ID:
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.

    more » « less
  2. Whether the 3D incompressible Navier–Stokes equations can develop a finite time sin- gularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompress- ible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Poten- tial singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equa- tions develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 107. We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and neg- ative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya– Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of Lqt Lxp norm of the velocity with 3/p + 2/q ≤ 1. Our numerical results for the cases of (p,q) = (4,8), (6,4), (9,3) and (p,q) = (∞,2) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of (p,q) = (3,∞) is more difficult to verify numerically due to the extremely slow growth rate in the L3 norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global L3 norm of the velocity grows very slowly, the localized version of the L 3 norm of the velocity experiences rapid dynamic growth relative to the localized L 3 norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations. 
    more » « less
  3. Abstract

    The dynamical core that predicts the three‐dimensional vorticity rather than the momentum, which is called Vector‐Vorticity Model (VVM), is implemented on a cubed sphere. Its horizontal coordinate system is not restricted to orthogonal, while the vertical coordinate is orthogonal to the horizontal surface. Accordingly, all the governing equations of the VVM, which are originally developed with Cartesian coordinates, are rewritten in terms of general curvilinear coordinates. The local coordinates on each cube surface are constructed with the gnomonic equiangular projection. Using global channel domains, the VVM on the cubed sphere has been evaluated by (1) advecting a passive tracer with a bell‐shaped initial perturbation along an east‐west latitude circle and along a north‐south meridional circle and (2) simulating the evolution of barotropic and baroclinic instabilities. The simulated results with the cubed‐sphere grids are compared to analytic solutions or those with the regular longitude‐latitude grids. The convergence with increasing spatial resolution is also quantified using standard error norms. The comparison shows that the solutions with the cubed‐sphere grids are quite reasonable for both linear and nonlinear problems when high resolutions are used. With coarse resolution, degeneracy appears in the solutions of the nonlinear problems such as spurious wave growth; however, it is effectively reduced with increased resolution. Based on the encouraging results in this study, we intend to use this model as the cloud‐resolving component in a global Quasi‐Three‐Dimensional Multiscale Modeling Framework.

    more » « less
  4. In 1983, Antontsev, Kazhikhov, and Monakhov published a proof of the existence and uniqueness of solutions to the 3D Euler equations in which on certain inflow boundary components fluid is forced into the domain while on other outflow components fluid is drawn out of the domain. A key tool they used was the linearized Euler equations in vorticity form. We extend their result on the linearized problem to multiply connected domains and establish compatibility conditions on the initial data that allow higher regularity solutions. 
    more » « less
  5. Inviscid spatial Landau damping is studied experimentally for the case of oscillatory motion of a two-dimensional vortex about its elliptical equilibrium in the presence of an applied strain flow. The experiments are performed using electron plasmas in a Penning–Malmberg trap. They exploit the isomorphism between the two-dimensional Euler equations for an ideal fluid and the drift-Poisson equations for the plasma, where plasma density is the analog of vorticity. Perturbed elliptical vortex states are created using [Formula: see text] strain flows, which are generated by applying voltages to electrodes surrounding the plasma. Measurements of spatial Landau damping (also called critical-layer damping) are in agreement with previous studies in the absence of an applied strain, where the damping is due to a resonance between the local fluid motion and the vortex oscillations. Interestingly, the damping rate does not change significantly over a wide range of applied strain rates. This can be accurately predicted from the initial vorticity profile, even though the resonant frequency is reduced substantially due to the applied strain. For higher amplitude perturbations, nonlinear trapping oscillations also exhibit behavior similar to the strain-free case. In principle, higher-order effects of the applied strain, such as separatrix crossing of peripheral vorticity and interactions with harmonics of the fundamental resonance, are expected to change the damping rate. However, this occurs only for conditions that are not realized in the experiments described here. Vortex-in-cell simulations are used to investigate the possible roles of these effects. 
    more » « less