Recent zbMATH articles in MSC 76L05https://www.zbmath.org/atom/cc/76L052021-04-16T16:22:00+00:00WerkzeugUnique solvability in the Lavrent'ev-Bitsadze model for two problems of weakly supersonic symmetric flow with detached shock wave past a wedge.https://www.zbmath.org/1456.760722021-04-16T16:22:00+00:00"Moiseev, E. I."https://www.zbmath.org/authors/?q=ai:moiseev.evgeny-ivanovich"Shifrin, E. G."https://www.zbmath.org/authors/?q=ai:shifrin.eh-gSummary: We study two problems on planar steady-state weakly supersonic potential flows of an ideal perfect gas with detached shock wave. In the first problem, we consider the flow past a finite wedge in an unbounded stream; the second problem deals with the flow past an infinite wedge in a steady jet. The free-stream velocity is close to the speed of sound, and therefore, the entropy increments \(\Delta S(\psi )=O(\varepsilon^2)\) on the shock wave and the derivative of the entropy with respect to the stream function are disregarded. In the velocity hodograph plane, the cross-coupled sub- and supersonic flow behind a shock wave is described by a solution of a Tricomi type problem. On part of the boundary depicting the shock wave, the directional derivative condition is set for the stream function. It is proved that its direction is not tangent to the domain boundary. The uniqueness of the solution for the problems under consideration follows from the ``strong'' Hopf maximum principle for uniformly elliptic equations. Replacing the Chaplygin equation with the Lavrent'ev-Bitsadze equation leads to two Hilbert problems for analytic functions with piecewise constant boundary conditions. The solutions of the Hilbert problems are expressed using the Schwarz operator.Numerical and asymptotic solutions of generalised Burgers' equation.https://www.zbmath.org/1456.650772021-04-16T16:22:00+00:00"Schofield, J. M."https://www.zbmath.org/authors/?q=ai:schofield.j-m"Hammerton, P. W."https://www.zbmath.org/authors/?q=ai:hammerton.p-wSummary: The generalised Burgers' equation models the nonlinear evolution of acoustic disturbances subject to thermoviscous dissipation. When thermoviscous effects are small, asymptotic analysis predicts the development of a narrow shock region, which widens, leading eventually to a shock-free linear decay regime. The exact nature of the evolution differs subtly depending upon whether plane waves are considered, or cylindrical or spherical spreading waves. This paper focuses on the differences in asymptotic shock structure and validates the asymptotic predictions by comparison with numerical solutions. Precise expressions for the shock width and shock location are also obtained.A class of Lagrangian-Eulerian shock-capturing schemes for first-order hyperbolic problems with forcing terms.https://www.zbmath.org/1456.650562021-04-16T16:22:00+00:00"Abreu, E."https://www.zbmath.org/authors/?q=ai:abreu.eduardo|abreu.emerson-a-m|abreu.everton-m-c"Matos, V."https://www.zbmath.org/authors/?q=ai:matos.vitor|matos.vinicius-bitencourt"Pérez, J."https://www.zbmath.org/authors/?q=ai:perez.jorge-e|perez.j-jones|perez.jaime-castillo|perez-aparicio.jose-l|perez.jesus-hernando|perez.jose-philippe|perez.josep-m|perez.joaquin.1|perez.jose-n|perez.jose-miguel|perez.joaquin|perez.joe-j|perez.jezabel|perez.javier-j|perez.jerome|perez.jonathan|perez.jorge-a|perez.jose-antonio|perez.jesus-carretero|perez-jimenez.juan-de-dios|perez.jose-a-moreno|perez.juan-carlos|perez.jon|perez.juan-f|perez.jose-maria|perez.j-fernando|perez.juan-j|perez.johanna-camargo|perez.joshue|perez.jhon-jairo|perez.julien|perez.juan-antonio-navarro|perez.john|perez.juan-luis|perez.juan-manuel|perez.jose-luis|perez.jaime-a-moreno|perez.jesus-m|perez.jennifer|perez-garmendia.jose-luis|perez.joel"Rodríguez-Bermúdez, P."https://www.zbmath.org/authors/?q=ai:rodriguez-bermudez.pantersSummary: In this work, we develop an improved shock-capturing and high-resolution Lagrangian-Eulerian method for hyperbolic systems and balance laws. This is a new method to deal with discontinuous flux and complicated source terms having concentrations for a wide range of applications science and engineering, namely, 1D shallow-water equations, sedimentation processes, Geophysical flows in 2D, N-Wave models, and Riccati-type problems with forcing terms. We also include numerical simulations of a 1D two-phase flow model in porous media, with gravity and a nontrivial singular \(\delta\)-source term representing an injection point. Moreover, we present approximate solutions for 2D nonlinear systems (Compressible Euler Flows and Shallow-Water Equations) for distinct benchmark configurations available in the literature aiming to present convincing and robust numerical results. In addition, for the linear advection model in 1D and for a smooth solution of the nonlinear Burgers' problem, second order approximations were obtained. We also present a high-resolution approximation of the nonlinear non-convex Buckley-Leverett problem. Based on the work of A. Harten, we derive a convergent Lagrangian-Eulerian scheme that is total variation diminishing and second-order accurate, away from local extrema and discontinuous data. Additionally, using a suitable Kružkov's entropy definition, introduced by K. H. Karlsen and J. D. Towers, we can verify that our improved Lagrangian-Eulerian scheme converges to the unique entropy solution for conservation laws with a discontinuous space-time dependent flux. A key hallmark of our method is the dynamic tracking forward of the no-flow curves, which are locally conservative and preserve the natural setting of weak entropic solutions related to hyperbolic problems that are not reversible systems in general. In the end, we have a general procedure to construct a class of Lagrangian-Eulerian schemes to deal with hyperbolic problems with or without forcing terms. The proposed scheme is free of Riemann problem solutions and no adaptive space-time discretizations are needed. The numerical experiments verify the efficiency and accuracy of our new Lagrangian-Eulerian method.A monotonicity-preserving higher-order accurate finite-volume method for Kapila's two-fluid flow model.https://www.zbmath.org/1456.760772021-04-16T16:22:00+00:00"de Böck, R."https://www.zbmath.org/authors/?q=ai:de-bock.r"Tijsseling, A. S."https://www.zbmath.org/authors/?q=ai:tijsseling.arris-s"Koren, B."https://www.zbmath.org/authors/?q=ai:koren.barrySummary: In preparation of the study of liquefied natural gas (LNG) sloshing in ships and vehicles, we model and numerically analyze compressible two-fluid flow. We consider a five-equation two-fluid flow model, assuming velocity and pressure continuity across two-fluid interfaces, with a separate equation to track the interfaces. The system of partial differential equations is hyperbolic and quasi-conservative. It is discretized in space with a tailor-made third-order accurate finite-volume method, employing an HLLC approximate Riemann solver. The third-order accuracy is obtained through spatial reconstruction with a limiter function, for which a novel formulation is presented. The non-homogeneous term is handled in a way consistent with the HLLC treatment of the convection operator. We study the one-dimensional case of a liquid column impacting onto a gas pocket entrapped at a solid wall. It mimics the impact of a breaking wave in an LNG containment system, where a gas pocket is entrapped at the tank wall below the wave crest. Furthermore, the impact of a shock wave on a gas bubble containing the heavy gas R22, immersed in air, is simulated in two dimensions and compared with experimental results. The numerical scheme is shown to be higher-order accurate in space and capable of capturing the important characteristics of compressible two-fluid flow.Nonlinear hyperbolic waves in relativistic gases of massive particles with Synge energy.https://www.zbmath.org/1456.761642021-04-16T16:22:00+00:00"Ruggeri, Tommaso"https://www.zbmath.org/authors/?q=ai:ruggeri.tommaso"Xiao, Qinghua"https://www.zbmath.org/authors/?q=ai:xiao.qinghua"Zhao, Huijiang"https://www.zbmath.org/authors/?q=ai:zhao.huijiangSummary: In this article, we study some fundamental properties of nonlinear waves and the Riemann problem of Euler's relativistic system when the constitutive equation for energy is that of Synge for a monatomic rarefied gas or its generalization for diatomic gas. These constitutive equations are the only ones compatible with the relativistic kinetic theory for massive particles in the whole range from the classical to the ultra-relativistic regime. They involve modified Bessel functions of the second kind and this makes Euler's relativistic system rather complex. Based on delicate estimates of the Bessel functions, we prove: (i) a limit on the speed of sound of \(1/\sqrt{3}\) times the speed of light (which a fortiori implies subluminality, that is causality), (ii) the genuine non-linearity of the acoustic waves, (iii) the compatibility of Rankine-Hugoniot relations with the second law of thermodynamics (entropy growth through all Lax shocks), and (iv) the unique resolvability of the initial value problem of Riemann (if we include the possibility of vacuum as in the non-relativistic context).On traveling waves in compressible Euler equations with thermal conductivity.https://www.zbmath.org/1456.350702021-04-16T16:22:00+00:00"Thanh, Mai Duc"https://www.zbmath.org/authors/?q=ai:mai-duc-thanh."Vinh, Duong Xuan"https://www.zbmath.org/authors/?q=ai:vinh.duong-xuanSummary: Heat conduction plays an important role in fluid dynamics. However, the modeling of thermal conductivity involves higher order derivatives which causes a tough obstacle for the study of traveling waves. In this work, we propose a modified term for the thermal conductivity coefficient in viscous-capillary compressible Euler equations. By approximation, which is crucial in any mathematical modeling, the heat conduction may be assumed to depend only on the specific volume. Then, we can derive a \(2\times 2\) system of first-order differential equations for traveling waves of the given model, whose equilibria can be shown to admit a stable-saddle connection for 1-shocks and a saddle-stable connection for 3-shocks. This establishes the existence of a traveling wave of the viscous-capillary Euler equations with the presence of a modified thermal conductivity effect.Rapid distortion theory analysis on the interaction between homogeneous turbulence and a planar shock wave.https://www.zbmath.org/1456.760572021-04-16T16:22:00+00:00"Kitamura, T."https://www.zbmath.org/authors/?q=ai:kitamura.takuya|kitamura.takayuki|kitamura.tomonori|kitamura.tazuko|kitamura.toshiaki|kitamura.tomotaka|kitamura.tadashi|kitamura.toyoyuki|kitamura.takashi"Nagata, K."https://www.zbmath.org/authors/?q=ai:nagata.kuniichi|nagata.kenji|nagata.koji.1|nagata.kensuke|nagata.kazuhiro|nagata.ken-ichi|nagata.kouji|nagata.keitaro|nagata.kiyoshi"Sakai, Y."https://www.zbmath.org/authors/?q=ai:sakai.yosuke|sakai.yuta|sakai.yasuyuki|sakai.yuichi|sakai.yoshifumi|sakai.yutaka|sakai.yasuhiro|sakai.yoshinori|sakai.yuusuke|sakai.yoshikiyo|sakai.yukiko|sakai.yuki|sakai.yoshihiko|sakai.yusuke|sakai.yuzuru|sakai.yuho|sakai.yasuhiko|sakai.yuji|sakai.yuma|sakai.yoshio"Sasoh, A."https://www.zbmath.org/authors/?q=ai:sasoh.akihiro"Ito, Y."https://www.zbmath.org/authors/?q=ai:ito.yuta|ito.yoshinori|ito.yoshimichi|ito.yosuke|ito.yusaku|ito.yukio|ito.yoshihiro|ito.yasuhiko|ito.yoshifusa|ito.yuji|ito.yu|ito.yutaka|ito.yoshiyasu|ito.yuto|ito.yusuke.1|ito.yasumasa|ito.yuki|ito.youji|ito.yasuaki|ito.yoshifumi|ito.yasushi|ito.youichi|ito.yukari|ito.yuuki|ito.yohei|ito.yoshihiko.1|ito.yoshikazuSummary: The interactions between homogeneous turbulence and a planar shock wave are analytically investigated using rapid distortion theory (RDT). Analytical solutions in the solenoidal modes are obtained. Qualitative answers to unsolved questions in a report by \textit{Y. Andreopoulos} et al. [Shock wave -- turbulence interactions'', Ann. Rev. Fluid Mech. 524, 309--345 (2000; \url{doi:10.1146/annurev.fluid.32.1.309})] are provided within the linear theoretical framework. The results show that the turbulence kinetic energy (TKE) is increased after interaction with a shock wave and that the contributions to the amplification can be interpreted primarily as the combined effect of shock-induced compression, which is a direct consequence of the Rankine-Hugoniot relation, and the nonlinear effect, which is an indirect consequence of the Rankine-Hugoniot relation via the perturbation manner. For initial homogeneous axisymmetric turbulence, the amplification of the TKE depends on the initial degree of anisotropy. Furthermore, the increase in energy at high wavenumbers is confirmed by the one-dimensional spectra. The enstrophy is also increased; its increase is more significant than that of the TKE because of the significant increase in enstrophy at high wavenumbers. The vorticity components perpendicular to the shock-induced compressed direction are amplified more than the parallel vorticity component. These results strongly suggest that a high resolution is needed to obtain accurate results for the turbulence-shock-wave interaction. The integral length scales \((L)\) and the Taylor microscales \((\lambda)\) are decreased for most cases after the interaction. However, \(L_{22,3}(=\,L_{33,2})\) and \(\lambda_{22,3}(=\,\lambda_{33,2})\) are amplified. Here, the subscripts 2 and 3 indicate the perpendicular components relative to the shock-induced compressed direction. The dissipation length and TKE dissipation rate are amplified.Modeling of Richtmyer-Meshkov instability development using the discontinuous Galerkin method and local-adaptive meshes.https://www.zbmath.org/1456.760542021-04-16T16:22:00+00:00"Zhalnin, R. V."https://www.zbmath.org/authors/?q=ai:zhalnin.ruslan-viktorovich"Masyagin, V. F."https://www.zbmath.org/authors/?q=ai:masyagin.viktor-fedorovich"Peskova, E. E."https://www.zbmath.org/authors/?q=ai:peskova.elizaveta-evgenevna"Tishkin, V. F."https://www.zbmath.org/authors/?q=ai:tishkin.vladimir-fedorovichSummary: The article presents a numerical algorithm for solving equations of multicomponent gas dynamics using the discontinuous Galerkin method on local-adaptive grids. The numerical algorithm uses a data structure and a dynamic local grid adaptation algorithm from the p4est library. We use Lax-Friedrichs-Rusanov numerical and HLLC flows. To get rid of non-physical oscillations, the Barth-Jespersen limiter is applied. As a result of the study, a numerical simulation of the development of the Richtmyer-Meshkov instability was carried out, the results obtained were compared with experimental results and known numerical solutions of this problem. It is concluded that the calculated and experimental data are in good agreement. In the future, it is expected to study this process using a model that takes into account the phenomena of viscosity and thermal conductivity.