Compressible Inviscid Research Articles

Kinetic Derivation of an Inviscid Compressible Leslie–Ericksen Equation for Rarified Calamitic Gases

Kinetic Derivation of an Inviscid Compressible Leslie–Ericksen Equation for Rarified Calamitic Gases

Numerical modelling of aerodynamic response to gusts and gust effect mitigation

Unmanned aerial vehicle (UAV) flights in urban environments are challenging due to the complex flow structures and elevated turbulence around buildings. Consequently, research has shifted towards investigating the impact of gusts on the aerodynamic stability and control of UAVs. This study focuses on enhancing gust numerical modelling capabilities to understand the aerodynamic response, specifically exploring gust mitigation strategies for UAVs operating in turbulent urban environments. The split-velocity method, originally designed for two-dimensional compressible inviscid flows, where the velocity components were decomposed into a prescribed gust velocity and the remaining velocity components, is extended to three-dimensional incompressible viscous flows. To facilitate effective gust mitigation techniques, a radial basis function is applied to the modified split-velocity method to numerically model wings in pitching motions under gust encounters. A novel strategy is proposed to correct the discretized gust velocities and ensure gust flux conservation, showing effective improvement to the numerical predictions. The computed results agreed well with the experimental data available in the public domain, confirming that wing pitch motion can effectively mitigate the effects of gusts.

Modal Discontinuous Galerkin Simulations of Richtmyer–Meshkov Instability at Backward-Triangular Bubbles: Insights and Analysis

This paper investigates the dynamics of Richtmyer–Meshkov instability (RMI) in shocked backward-triangular bubbles through numerical simulations. Two distinct gases, He and SF6, are used within the backward-triangular bubble, surrounded by N2 gas. Simulations are conducted at two distinct strengths of incident shock wave, including Ms=1.25 and 1.50. A third-order modal discontinuous Galerkin (DG) scheme is applied to simulate a physical conservation laws of two-component gas flows in compressible inviscid framework. Hierarchical Legendre modal polynomials are employed for spatial discretization in the DG platform. This scheme reduces the conservation laws into a semi-discrete set of ODEs in time, which is then solved using an explicit 3rd-order SSP Runge–Kutta scheme. The results reveal significant effects of bubble density and Mach numbers on the growth of RMI in the shocked backward-triangular bubble, a phenomenon not previously reported. These effects greatly influence flow patterns, leading to intricate wave formations, shock focusing, jet generation, and interface distortion. Additionally, a detailed analysis elucidates the mechanisms driving vorticity formation during the interaction process. The study also thoroughly examines these effects on the flow fields based on various integral quantities and interface characteristics.

Comparative analysis of compressible inviscid flow over symmetric and supercritical airfoil

Studying compressible inviscid flow over symmetric and supercritical airfoils is important because it is useful to understand and maximize aerodynamic performance in a range of industrial contexts. This study addresses the analysis of symmetric and supercritical airfoils for compressible inviscid flow for both time-dependent and stationary scenarios. The primary goal of these simulations’ is to look at how different velocity variations affected certain airfoil parameters. Air is utilized as the material above airfoils. The best fit for these simulations is the Spalart-Allmaras Turbulence model, which shows good agreement over a wider velocity range with published experimental data from other studies. The COMSOL Multiphysics software is used for all the analysis, for angle of attack 0°, temperature 288K, pressure 1bar and the airfoils that are constructed by importing airfoil data file in COMSOL in the form of excel file. On the comparison of compressible inviscid flow, the supercritical airfoil has better results on the basis of velocity magnitude, pressure, vorticity magnitude, shear rate and rotation rate. The results are converging faster in supercritical airfoil than symmetric airfoil. At a zero-degree angle of attack, velocity increases more on the suction side while decreases on the pressure side (bottom). The final solution error for supercritical airfoil is 1.6E-05, while final residual error for “velocity u pressure p” is 1.2E-05. For symmetric airfoil, final solution error is 0.15, although final residual error is 2.6. Based on the error plot, it decreases rapidly in case of supercritical with a range from 100 to 10-4. Additionally, the wall resolution is constructed for both cases i.e., symmetric airfoil and supercritical airfoil. The main result is that ,with the passage of time, the flow is becoming to stationary state. In the end, model validation via comparison with experimental data highlights the validity and applicability of the results.

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

Generalization to a wider class of entropy split methods for compressible ideal MHD

The high order entropy split methods of Sjögreen & Yee (2019, 2021), by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function (Harten, 1983; Gerritsen and Olsson, 1996; Yee et al., 2000). Their formulation have been proven entropy conserving and stable by taking advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, dispersion relation-preserving (DRP) (Bogey and Bailly, 2002; Brambley, 2016; Johansson, 2004) or Padé (compact) spatial discretizations (Hirsh, 1975) with summation-by-parts (SBP) operators (Strand, 1994). These high order entropy split methods not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation. The present work employs a new approach to obtain a wider class of high order entropy split methods that do not have the homogeneous property. The new method consists of a two-point numerical flux portion and a non-conservative portion in such a way that entropy conservation holds without requiring the homogeneity property of the compressible inviscid flux. For high order classical spatial central, DRP or Padé spatial discretizations, this new approach can be proven to be entropy conservative while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. More importantly, we extend this new approach to derive a new entropy split method for the equations of ideal MHD. Representative test cases are illustrated with comparison among existing methods of the same high order of accuracy.

Rendering Maxwell Equations into the Compressible Inviscid Fluid Dynamics Form

Maxwell equations governing electromagnetic effects are being shown to be equivalent to the compressible inviscid Navier–Stokes equations applicable in fluid dynamics and representing conservation of mass and linear momentum. The latter applies subject to a generalized Beltrami condition to be satisfied by the magnetic field. This equivalence indicates that the compressible inviscid Navier–Stokes equations are Lorentz invariant as they derive directly from the Lorentz-invariant Maxwell equations subject to the same Beltrami condition, provided the pressure wave propagates at the speed of light, i.e., vo=co. In addition, the derivation and results provide support for the claim that electromagnetic potentials have physical significance as demonstrated by Aharonov–Bohm effect, and are not only a convenient mathematical formulation.

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases.

Dispersion of axisymmetric longitudinal waves propagating in an inhomogeneous prestressed hollow cylinder immersed in an inviscid compressible fluid

The paper investigates the dispersion of axisymmetric waves in a cylinder with inhomogeneous initial stresses caused by the hydrostatic pressure of the compressible inviscid fluid in which this cylinder is immersed. In the context of this study, the motion of the cylinder is described by the three-dimensional linearized equations for elastic waves in bodies with initial stresses, but the flow of the fluid is described by the linearized Navier-Stokes equations for compressible (barotropic) inviscid fluids. The initial stresses in the cylinder, due to the action of hydrostatic pressure on the outer surface of the cylinder, are determined according to the well-known Lame formulas of classical linear elasticity theory. The corresponding eigenvalue problem for the dispersion of axisymmetric waves propagating in the hydro-elastic system under consideration is solved using the discrete-analytic method. The dispersion equation is obtained and solved numerically. Concrete numerical investigations are carried out for three pairs of materials: Soft Rubber + Glycerin, Lucite + Water and Steel + Water. The dispersion curves obtained for these pairs of materials are presented and discussed for different values of the intensity of the external pressure. In particular, it is found that the inhomogeneous initial stresses in the cylinder cause zero group velocity (ZGV) points to appear on the dispersion curves of the lowest modes. However, the ZGV points (or modes) appear on the dispersion curves of the higher modes as a result of the contact of the hollow cylinder with the fluid.

Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.

Investigation of aspect ratio effects on flow characteristics and vorticity generation in shock-induced rectangular bubble

In shock-induced Richtmyer–Meshkov instability, the polygonal bubbles, including the rectangular interface can offer favorable circumstances for shock refraction research. Also, the aspect ratio is crucial in characterizing the physics behind this instability. Therefore, the aspect ratio effects on the flow characteristics and vorticity generation in the shock-induced rectangular light gas bubbles are investigated numerically in this study. This investigation is performed for a wide range of aspect ratios varying from 0.25 to 3.0. These effects are highlighted in the interface morphology, wave paradigms, vorticity generation, enstrophy evolution, and a qualitative examination of interface deformation. For simulating two-component gas flows, a high-order explicit modal discontinuous Galerkin approach is employed to solve a two-dimensional system of compressible inviscid Euler equations. Existing experimental results for shock-induced square and cylindrical light bubbles are used to validate the numerical model and solver. The results indicate that the effects of aspect ratio are critical in characterizing the interface morphology during the interaction between the shock wave and rectangular bubble. The interface morphology evolves faster as the aspect ratio increases, and the process of transverse jet formation is observed to be significantly different. The aspect ratio effects result in a substantial change in the flow field with bubble deformation, vortex development, vorticity generation, and transverse jet formation. The effects of aspect ratio are further investigated in detail through the mechanism of vorticity generation and enstrophy evolution. Finally, these effects on shock trajectories and interface characteristics over time are thoroughly explored.

Dynamic Transition of Unsteady Supersonic Flow From Mach to Regular Reflection Over a Moving Wedge

Abstract The design of supersonic and hypersonic air-breathing vehicles is influenced by the transition between the Mach reflection (MR) and regular reflection (RR) phenomena. The purpose of this study is to investigate the dynamic transition of unsteady supersonic flow from MR to RR over a two-dimensional wedge numerically. The trailing edge of the wedge moves downstream along the x-direction with a velocity, V(t) at a freestream Mach number of 3. An unsteady compressible inviscid flow solver is used to simulate the phenomenon. The Arbitrary Lagrangian–Eulerian (ALE) technique is applied to deform the mesh during the wedge motion. The dynamic transition from MR to RR is defined by two criteria, the sonic and the von Neumann. Moreover, the lag in the dynamic transition from the steady-state condition is studied using various nondimensional angular velocities, κ, in the range of [0.1-2]. The lag effect in the shock system is remarkable at the high values of κ greater than 1.5. Furthermore, the dynamic transition from MR to RR happens below the dual solution domain (DSD) because the shock is upstream curved during the wedge motion.

Local well-posedness of the free-boundary problem in compressible resistive magnetohydrodynamics

We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand-Shkoller to construct the approximation solutions. One of the key observations is that the Christodoulou-Lindblad type elliptic estimate together with magnetic diffusion not only gives the common control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.

Three-dimensional third-order gas-kinetic scheme on hybrid unstructured meshes for Euler and Navier–Stokes equations

In this paper, a third-order gas-kinetic scheme is developed on the three-dimensional hybrid unstructured meshes for the compressible inviscid and viscous flows. In the classical weighted essentially non-oscillatory (WENO) scheme, the high-order spatial accuracy is achieved by the non-linear combination of lower order polynomials. However, for the hybrid unstructured meshes, the procedures, including the selection of candidate stencils and calculation of linear weights, become extremely complicated, especially for three-dimensional problems. To overcome the drawbacks, a third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A general strategy is given for the selection of big stencil and sub-stencils for hybrid meshes, and the topologically independent linear weights are used in the spatial reconstruction. A unified interpolation is also used for the volume integral of different types of control volumes, as well as the flux integration over different cell interfaces. Incorporate with the two-stage fourth-order temporal discretization, the explicit high-order gas-kinetic schemes are developed for unsteady problems. With the lower–upper symmetric Gauss–Seidel (LU-SGS) methods and Jacobi iteration, the implicit high-order gas-kinetic schemes are developed for steady problems. To accelerate the computation, both explicit and implicit scheme is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). Especially, for implicit parallel computation, Jacobi iteration is used. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.

ChatGPT FOR PROGRAMMING NUMERICAL METHODS

ChatGPT is a large language model recently released by the OpenAI company. In this technical report, we explore for the first time the capability of ChatGPT for programming numerical algorithms. Specifically, we examine the capability of GhatGPT for generating codes for numerical algorithms in different programming languages, for debugging and improving written codes by users, for completing missed parts of numerical codes, rewriting available codes in other programming languages, and for parallelizing serial codes. Additionally, we assess if ChatGPT can recognize if given codes are written by humans or machines. To reach this goal, we consider a variety of mathematical problems such as the Poisson equation, the diffusion equation, the incompressible Navier-Stokes equations, compressible inviscid flow, eigenvalue problems, solving linear systems of equations, storing sparse matrices, etc. Furthermore, we exemplify scientific machine learning such as physics-informed neural networks and convolutional neural networks with applications to computational physics. Through these examples, we investigate the successes, failures, and challenges of ChatGPT. Examples of failures are producing singular matrices, operations on arrays with incompatible sizes, programming interruption for relatively long codes, etc. Our outcomes suggest that ChatGPT can successfully program numerical algorithms in different programming languages, but certain limitations and challenges exist that require further improvement of this machine learning model.

Plasma formation in ambient fluid from hypervelocity impacts

The generation of plasma from hypervelocity impacts is an active research topic due to its important science and engineering ramifications in various applications. Previous studies have mainly focused on the ionization of the solid materials that constitute the projectile and the target. In this letter, we consider impact events that occur in a fluid (e.g., gas) medium, and present a multiphysics computational modeling approach and associated analysis to predict the behavior of the dynamic fluid–solid interaction that causes the surrounding fluid to ionize. The proposed computational framework is applied to a specific case involving a system of three interacting domains: a copper rod projectile impacting onto a soda lime glass target in a neon gas environment. The impact velocity is varied between 3km/s and 6km/s in different simulations. The computational model couples the compressible inviscid Navier–Stokes equations with the Saha ionization equations. The three material interfaces formed among the projectile, the target, and the ambient gas are tracked implicitly by solving two level set equations that share the same velocity field. The mass, momentum, and energy fluxes across the interfaces are computed using the FInite Volume method with Exact two-material Riemann problems (FIVER). The simulation result reveals a region of neon gas with high velocity, temperature, pressure, and mass density, formed in the early stage of the impact mainly due to the hypersonic compression of the fluid between the projectile and the target. For impact velocities higher than 4km/s, ionization is predicted in this region.

Simulating laser-fluid coupling and laser-induced cavitation using embedded boundary and level set methods

A computational method for simulating thermal cavitation induced by long-pulsed laser is presented. This method accounts for the absorption of laser light by a liquid, the formation of vapor bubbles due to localized heating, and the dynamics of the bubbles and the surrounding liquid. The physical model combines the Euler equations for a compressible inviscid two-phase fluid flow, a reduced form of the radiative transfer equation for laser radiation, and a local thermodynamic model of vaporization. The Euler equations are solved using the FInite Volume method with Exact two-phase Riemann solvers (FIVER). Following this method, numerical fluxes across phase boundaries are computed by constructing and solving one-dimensional bimaterial Riemann problems. The paper focuses on numerical methods for coupling the laser and fluid governing equations and tracking the vapor bubbles. An embedded boundary finite volume method is proposed to solve the laser radiation equation on the same mesh created for the Euler equations, which usually does not resolve the boundary and propagation directions of the laser beam. To impose boundary conditions, ghost nodes outside the laser domain are populated by mirroring and interpolation techniques. The existence and uniqueness of solution are proved for the two-dimensional case, leveraging the special geometry of the laser domain. The order of accuracy of the method is also proved, and verified using numerical tests. A method of latent heat reservoir is proposed to predict the onset of vaporization, which accounts for the accumulation and release of latent heat. A unique challenge associated with long-pulsed laser is that the dynamics of vapor bubbles is driven not only by the inertia of the bubble nuclei, but also by the continuation of vaporization. In this work, the localized level set method is employed to track the bubble surface, and a method of local correction and reinitialization is proposed to account for continuous phase transitions. Several numerical tests are presented to verify the convergence of these methods. Two realistic simulations of laser-induced cavitation are presented at the end, showing that the computational method is able to capture the key phenomena in these events, including non-spherical bubble expansion, shock waves, and the “Moses effect”.

Comprehensive analysis of entropy conservation property of non-dissipative schemes for compressible flows: KEEP scheme redefined

A theoretical analysis of the entropy conservation properties is conducted to explain the different behaviors of the non-dissipative finite-difference spatial discretization schemes, such as the kinetic-energy and entropy preserving (KEEP) schemes. The analysis is conducted based on the spatially-discretized entropy-evolution equation derived from the Euler equations with retaining the discrete-level strictness. The present analysis shows that the analytical relations (ARs) employed by the KEEP schemes eliminate some terms of the discretized entropy-evolution equation and help simplify the equation, while the ARs are not sufficient to explain the entropy-conservation property. Therefore, the entropy error is decomposed into several terms, and the behaviors of those decomposed error terms are observed in the compressible inviscid Taylor–Green vortex test case. The results of the test case show that the terms containing the velocity difference between two grid points cause significant entropy conservation error and result in the different entropy conservation properties of the tested non-dissipative schemes. Furthermore, the KEEP schemes are redefined based on the present entropy-error analysis. In the redefined KEEP schemes, the formulation does not contain logarithmic means, and the strictness of entropy conservation can be adjusted easily by the truncation order of the Maclaurin expansions. Finally, the present entropy-error analysis and the redefined KEEP schemes are extended to the generalized curvilinear coordinates.

Ideal free flows of optimal foragers: Vertical migrations in the ocean

We consider the collective motion of animals in time-varying environments, using as a case the diel vertical migration of marine copepods. The animals are distributed in space such that each animal moves optimally, seeking regions which offer high growth rates and low mortalities, subject to costs on excessive movements as well as being in regions with high densities of conspecifics. The model applies to repeated scenarios such as diel or seasonal patterns, where the animals are aware of both current and future environmental conditions. We show that this problem can be viewed as a differential game of mean field type, and that the evolutionary stable solution, i.e., the Nash equilibrium, is characterized by partial differential equations, which govern the distributions and migration velocities of animals. These equations have similarities to equations that appear in the fluid dynamics, specifically the Euler equations for compressible inviscid fluids. If the environment is constant, the ideal free distribution emerges as an equilibrium. We illustrate the theory with a numerical example of vertical migrations of oceanic copepods, where animals are attracted to nutrient-rich surface waters while repulsed from light during daytime due to the presence of visual predators, aiming to reduce both proximity to conspecifics and swimming efforts. For this case, we show that optimal movements are diel vertical migrations in qualitative agreement with observations of copepods.