Two-dimensional Euler Equations Research Articles

Exact coherent structures in fully developed two-dimensional turbulence

This paper reports several new classes of unstable recurrent solutions of the two-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions are in many ways analogous to recurrent solutions of the Navier–Stokes equation which are often referred to as exact coherent structures. In particular, we find that recurrent solutions of the Euler equation are dynamically relevant: they faithfully reproduce large-scale flows in simulations of turbulence at very high Reynolds numbers. On the other hand, these solutions have a number of properties which distinguish them from their Navier–Stokes counterparts. First of all, recurrent solutions of the Euler equation come in infinite-dimensional continuous families. Second, solutions of different types are connected, e.g. an equilibrium can be smoothly continued to a travelling wave or a time-periodic state. Third, and most important, they are only weakly unstable and, as a result, fully developed turbulence mimics some of these solutions remarkably frequently and over unexpectedly long temporal intervals.

Mechanism of characteristic change of panel flutter caused by oblique shock impingement

Compared to the shock-free condition, the weak shock impingement stabilizes the flexible panel, while the strong shock impingement leads to the early onset of panel flutter with a significant increase in flutter amplitude and frequency. However, the reason for this change by shock impingement remains unclear. The current research examines the mechanism of this change by an in-house code where the von Kármán’s large deflection plate theory is coupled with two-dimensional Euler equations. Compared to the shock-free condition, the oblique shock impingement leads to the change of local dynamic pressure on the panel as well as the static pressure differential across the panel. The analysis on the influence of these changes indicates that, on the one hand, the average dynamic pressure on the panel becomes larger than the shock-free condition, accelerating the onset of panel flutter. On the other hand, the change of the static pressure differential across the panel alters the coupling characteristic between different natural frequencies (modes) of the panel structure. The dynamic response of panel flutter under shock impingements is dominated by the coupling between the second and third modes instead of the first two modes for panel flutter under the shock-free condition. The combined effect of these two changes leads to the change of flutter characteristics of the panel under shock impingement. These findings provide valuable insights into the mechanism of shock-induced panel flutter.

Investigating the flow characteristics and thermodynamic performance of curved detonation waves

Oblique detonation waves (ODWs) are induced by oblique shock waves (OSWs) and hold significant research value for air-breathing ramjet engines due to their high thermodynamic efficiency and specific impulse. In this research, detonation is induced using a curved shock wave (CSW) that provides greater flexibility compared to an OSW. Curved detonation waves (CDWs) are classified into two categories based on the curvature of the CSW: positive-curvature curved detonation waves (PC-CDWs) and negative-curvature curved detonation waves (NC-CDWs). We conduct a numerical investigation of the flow characteristics and potential thermodynamic performance in curved detonation waves (CDWs) by solving the two-dimensional multi-species Euler equations with a detailed hydrogen combustion mechanism. In contrast to ODWs, we observe continuous compression and expansion effects in the flow fields of PC-CDWs and NC-CDWs, respectively. Continuous compression shortens the initiation length of the detonation wave and intensifies the compression wave preceding the combustion front, whereas continuous expansion produces the opposite outcome. Changes in the curvature of the CSW result in alterations to the flow characteristics of the CDW. These flow characteristic changes are manifested in their thermodynamic and propulsion-related performance. More specifically, a lower overdrive of the detonation wave corresponds to a reduced loss in total pressure.

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics, both in time and space, which include the relaxation schemes by Jin and Xin. These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case. These kinetic models depend on a small parameter that can be seen as a “Knudsen” number. The method is asymptotic preserving in this Knudsen number. Also, the computational costs of the method are of the same order of a fully explicit scheme. This work is the extension of Abgrall et al. (2022) [3] to multi-dimensional systems. We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

Bulges at vortical outflows

This paper constructs steady solutions of the two-dimensional Euler equations corresponding to a line source of vortical fluid on the impermeable boundary of a quiescent flow. The nonlinear, free-boundary problem is solved by mapping the flow domain to the hodograph plane. A vortex dipole or, equivalently, a source–sink doublet is superposed on the source leading to flow patterns that model the ballooning outflows observed where rivers and straits discharge into the open ocean and in the rotating flow experiments and numerical simulations designed to reflect these observations.

Numerical investigation of shock Mach number effects on Richtmyer–Meshkov instability in a heavy square bubble

This study conducts a two-dimensional numerical investigation of incident shock Mach number effects on the Richtmyer–Meshkov (RM) instability after a shock wave impulsively drives a heavy square bubble. The square bubble is composed of SF6 gas, which is surrounded by nitrogen gas. Five different shock Mach numbers are considered: Ms=1.12, 1.22, 1.4, 1.7, and 2.1. Two-dimensional compressible Euler equations for two-component gas flows are simulated with a high-order modal discontinuous Galerkin solver. For validation, the numerical results are compared with the existing experimental results and are found to be in good agreement. The present results reveal that the incident shock Mach number is critical in the growth of RM instability in a heavy square bubble. The shock Mach number affects flow morphology significantly, resulting in complicated wave patterns, shock focusing, jet creation, bubble deformation, and vorticity generation. The convergent shape of the bubble causes shock focusing, which creates a local high-pressure zone and, as a result, an outward jet generation. It is found that the bubble deforms differently with increasing shock Mach number, and the different shock Mach numbers alter the distance between the shock focusing position and the downstream bubble interface. The effects of shock Mach numbers are investigated in depth through physical phenomena such as vorticity production, kinetic energy, and enstrophy. Finally, the shock Mach number impacts on the time-variations of the shock trajectories and interface structure are thoroughly investigated.

Sonic boom propagation over real topography

The effect of elevation variation on sonic boom reflection is investigated using real terrain data. To this end, the full two-dimensional Euler equations are solved using finite-difference time-domain techniques. Numerical simulations are performed for two ground profiles of more than 10 km long, extracted from topographical data of hilly regions, and for two boom waves, a classical N-wave, and a low-boom wave. For both ground profiles, topography affects the reflected boom significantly. Wavefront folding due to terrain depression is notably highlighted. For the ground profile with mild slopes, the time signals of the acoustic pressure at the ground are, however, only slightly modified compared to the flat reference case, and the associated noise levels differ by less than 1 dB. With steep slopes, the contribution due to wavefront folding has a large amplitude at the ground. This results in an amplification of the noise levels: a 3 dB increase occurs at 1% of the positions along the ground surface, and a maximum of 5–6 dB is reached near the terrain depressions. These conclusions are valid for the N-wave and low-boom wave.

High-Order Cut-Cell Discontinuous Galerkin Difference Discretization

We present a high-order cut-cell method based on the discontinuous Galerkin difference (DGD) discretization. We leverage the inherent properties of the DGD basis functions to construct a cut-cell discretization that does not require special treatment to mitigate the small-cell problem. The paper describes how the DGD discretization can be constructed from an existing discontinuous Galerkin (DG) discretization, and we highlight differences between the DG and DGD methods. By performing condition-number studies on one- and two-dimensional model problems, we demonstrate that cut-cell DGD discretization remains well conditioned even when the cut-cell volume is orders of magnitude smaller than neighboring cells. We verify the high-order accuracy of the discretization by solving the two-dimensional steady-state Euler equations.

Maximum-principle-preserving high-order discontinuous Galerkin methods for incompressible Euler equations on overlapping meshes

In this paper, we construct a new local discontinuous Galerkin (LDG) algorithm to solve the incompressible Euler equation in two space dimensions on overlapping meshes. This method solves the vorticity, velocity field and potential function on different meshes. Different from the traditional LDG method, the overlapping meshes used in this paper make the velocity to be continuous along the interfaces of the primitive meshes. Therefore, the upwind fluxes can be applied. We derive two sufficient conditions to obtain the maximum principle of vorticity. The first one is the divergence-free numerical approximation of the velocity field. This condition further grants that the scheme of the vorticity equation keeps constant solutions. The second one is to preserve the positivity of the numerical vorticity. We select suitable time step sizes to construct positive numerical cell averages of the vorticity provided the vorticity in the previous time step is positive. Then a slope limiter can be applied to enforce the positivity of the numerical approximation of the vorticity. Thanks to the above two conditions, we can arbitrarily add constants to the vorticity function and construct high-order MPP LDG methods on overlapping meshes for the two-dimensional incompressible Euler equation in the vorticity stream function formulation. Numerical tests will be given to demonstrate the good performance of the proposed method.

Shock capturing using discontinuous Galerkin method and overset grids for two-dimensional Euler equations

A new procedure to capture shocks which confines the shock capturing algorithms to overset grids is proposed and demonstrated for steady state flows containing straight and bow shocks using a discontinuous Galerkin solver for two dimensional Euler equations. Preliminary results for capturing moving straight shocks are also presented. Troubled cell data, obtained using a coarse grid, is used to determine the approximate location of the shock. A refined and structured overset grid aligned to the shock is constructed based on this information. The solver is run again with the coarse grid solution as the initial condition using a shock capturing overset grid to give a solution with the shock aligned to the grid line. Solutions obtained with a coarse grid alone, overset grid aligned to the shock, and one not aligned to the shock are used to demonstrate the efficacy of the scheme for problems having analytical solutions. Results for supersonic flow over a ramp, shock reflecting off a flat plate, the supersonic flow over a circular cylinder, and a simple Mach reflection are presented. For the flow over a circular cylinder, the results obtained are validated using an existing analytical method for calculating the shock offset distance. Since there is a manual component in the construction of overset grids, as a first step towards eliminating it, a moving normal shock is also captured accurately using an overset grid at the location of the shock with an arbitrary Lagrangian-Eulerian discontinuous Galerkin method.

SOLUTIONS OF A SYSTEM OF TWO-DIMENSIONAL EULER EQUATIONS AND STATIONARY STRUCTURES IN AN IDEAL FLUID

SOLUTIONS OF A SYSTEM OF TWO-DIMENSIONAL EULER EQUATIONS AND STATIONARY STRUCTURES IN AN IDEAL FLUID

Mach cutoff phenomenon of sonic boom in temperature inversion layers

The Mach cutoff of sonic boom occurs when shock waves propagate from supersonic to subsonic regions in a stratified atmosphere. In this study, numerical simulations are performed to clarify Mach cutoff behaviors in atmospheric temperature inversion layers, which have not yet been investigated. The two-dimensional Euler equations with a gravitational source term are solved in a horizontally stratified atmosphere. Surface and non-surface temperature inversion layers include three and four supersonic–subsonic mixing layers, respectively. Computations are made to analyze the propagation characteristics of a conventional N-shaped waveform. The results show that multiple upward and downward cutoffs occur owing to atmospheric temperature inversions, with multiple outgoing and evanescent waves, and multiple shock wave focusing. The upward and downward cutoffs act similarly. The positive- and negative-pressure regions are inversed when a cutoff occurs twice; thus, each cutoff shifts a phase by π/2, as expressed in the linear theory of caustics. A waveform transition occurs owing to wave splitting in a cutoff region, in addition to the phase shift. For surface temperature inversion, upward cutoffs make the waves between the cutoff and ground surfaces persist, resulting in rumbling. For non-surface temperature inversion, upward and downward cutoffs occur alternately, and the waves between them persist as well. Therefore, multiple cutoffs in temperature inversion layers play an important role in sonic boom evaluation during low supersonic flight.

Numerical study on the scattering of acoustic waves by a compact vortex

A new family of compact vortex models is developed and taken as base vortical flows to numerically study the acoustic scattering by solving the two-dimensional Euler equations in the time domain with high-order accurate finite-difference methods and nonreflecting boundary conditions. The computations of scattered fields with very small amplitude are found to be in excellent agreement with a benchmark provided by previous studies. Simulations for the scattering from a Taylor vortex reveal that the amplitude of the scattered fields is strongly influenced by two dimensionless quantities, the vortex strength Mv based on the maximal velocity of the vortex, and the acoustic length-scale ratio λ/R defined as the acoustic wavelength relative to the vortex core size. To have a deep understanding of the roles played by these two quantities, another significant quantity used for describing quantitatively the total amount of scattering, namely, scattered sound power, is introduced. Thereupon, on the basis of a global analysis of scale effects of these two dimensionless quantities on the scattered sound power, the scattering defined in a physical coordinate system with Mv and λ/R is divided into three domains, long-wave domain, resonance domain, and geometrical-acoustics domain. For each domain, we examine the influence of Mv and λ/R in detail and derive the explicit scaling laws involved in the strength of the scattered field and these two dimensionless quantities separately. Furthermore, the computations for the scattering from a high-order compact vortex are conducted at a wide range of Mv and λ/R and compared with the results from the Taylor vortex in each domain to gain some insights into the acoustic scattering by a compact vortex.

Two-dimensional pseudo-steady supersonic isothermal flow around a sharp corner

The problem of the two-dimensional pseudo-steady supersonic isothermal flow around a sharp corner expanding into vacuum is studied in this article. It is worth noting that the gas will diffuse to infinity, that is, there is no vacuum boundary in this work. The essence of this problem is the interaction of the centered simple wave with the planar rarefaction wave, which can be solved by the Goursat problem for the two-dimensional self-similar Euler equations. Through the methods of characteristic decompositions and invariant regions, we establish a priori C1 estimates and the hyperbolicity of the solution in the wave interaction domain. The global existence of solution to the gas expansion problem is obtained constructively.

Rotation-dominant three-scale limit of the Cauchy problem to the inviscid rotating stratified Boussinesq equations

In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions.

A cooled turbine airfoil performance prediction method with two-dimensional CFD computation and loss models

Abstract The role of turbine blade cooling and coolants are significant factors in modern gas turbine aerodynamic design. This paper presents an effective and rapid airfoil design method based on CFD computation of the S1 surface and the existing loss correlations. The method can assess the coolant mixing loss by identifying each cooling hole separately and obtain the overall mainflow aerodynamic loss for cooled airfoil. The CFD computation code of the S1 surface is powered by a two-dimensional Euler equation, which is inviscid. Typical Kacker–Okapuu empirical correlations are then used to assess the airfoil friction loss, trailing edge loss, and shock loss. A novel form of the Hartsel model for coolant mixing loss is developed and employed in the CFD codes. In the reformed model, the mixing loss coefficient is directly associated with the blowing ratio and the total pressure/temperature ratio of mainstream-to-coolant, making it more convenient than the original model in the airfoil design process. Based on a transonic turbine vane airfoil, the influences of the film outflow location and outflow Mach number on the coolant mixing loss are investigated using the above prediction method and the cascade blowing test.

Spectral difference method with a posteriori limiting: application to the Euler equations in one and two space dimensions

ABSTRACTWe present a new numerical scheme which combines the spectral difference (SD) method up to arbitrary high order with a-posteriori limiting using the classical MUSCL-Hancock scheme as fallback scheme. It delivers very accurate solutions in smooth regions of the flow while capturing sharp discontinuities without spurious oscillations. We exploit the strict equivalence between the SD scheme and a finite-volume scheme based on the SD control volumes to enable a straightforward limiting strategy. At the end of each stage of our high-order time-integration ADER (Arbitrary high order using Derivatives) scheme, we check if the high-order solution is admissible under a number of numerical and physical criteria. If not, we replace the high-order fluxes of the troubled cells by fluxes from our robust second-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) fallback scheme. We apply our method to a suite of test problems for the one-dimensional and two-dimensional Euler equations. We demonstrate that this combination of SD and ADER provides a virtually arbitrary high order of accuracy while at the same time preserving good sub-element shock capturing capabilities.

Acoustic emission due to the interaction between shock and instability waves in two-dimensional supersonic jet flows

An analytical model is developed to study the sound produced by the interaction between shock and instability waves in two-dimensional supersonic jet flows. The jet is considered to be of vortex-sheet type and two-dimensional Euler equations are linearized to determine the governing equations for shock and instability waves and their interaction. Pack's model is used to describe shock waves, while instability waves are calculated using spatial stability analysis. The interaction between shock and instability waves can be solved analytically by performing Fourier transform and subsequently using the method of steepest descent. Sound produced by the interaction between the instability wave and a single shock cell is studied first, after which that due to a number of cells follows. We find that the model developed in this study can correctly predict the frequencies of the fundamental screech tone and its first and second harmonics. We show that the predicted sound directivity, even from a single shock cell, is in good agreement with experimental data. In particular, this model shows the strongest noise emission close to the upstream direction but the emitted noise starts to rapidly decay as the observer angle approaches $180^\circ$ , which is in accordance with experimental results; this suggests that the effective noise from a single shock cell is far from of the monopole type as assumed in the classical Powell's model. We find that the noise directivity is very sensitive to the local growth rate of the instability waves and the noise is generated primarily through the Mach wave mechanism.

Influence of different interface component distributions on Richtmyer-Meshkov instability

In this paper, the Richtmyer-Meshkov instability is studied numerically by using the high-resolution Roe scheme based on the two-dimensional unsteady Euler equation, which is caused by the interaction between shock wave and the helium circular light gas cylinder with different component distributions. The numerical results are used to further discuss the deformation process of the gas cylinder and the wave structure of the flow field, and also to quantitatively analyze the characteristic dimensions (length, height and central axial width) of the gas cylinder, the time-dependent volume compression ratio of the cylinder. In addition, the flow mechanism of shock-driven interface gas mixing is analyzed from multiple perspectives by combining the flow field pressure, velocity, circulation and gas mixing rate. Then the effects of different initial component distribution conditions on interface instability are investigated. The results show that when the diffusion interface transforms into the sharp interface, the reflection coefficient gradually increases on both sides of interface. When the incident shock wave interacts with the cylinder, the transmission of the shock wave will transform from conventional transmission into unconventional transmission. At the same time, the reflected shock wave is gradually strengthened and the transmitted shock wave is gradually weakened, which leads the Richtmyer-Meshkov instability to be strengthened. Moreover, the Atwood numbers on both sides of the interface also increase as the diffusion interface transforms into the sharp interface, which leads the Rayleigh-Taylor instability and the Kelvin-Helmholtz instability to be strengthened. Therefore, the increase of instability will cause the circulation to increase, resulting in the increase of the growth rate of gas mixing rate.

On a degenerate mixed-type boundary value problem for the two-dimensional self-similar Euler equations

On a degenerate mixed-type boundary value problem for the two-dimensional self-similar Euler equations