Contact Discontinuity Research Articles

The pulsar magnetosphere with machine learning: methodology

ABSTRACT In this study, we introduce a novel approach for deriving the solution of the ideal force-free steady-state pulsar magnetosphere in three dimensions. Our method involves partitioning the magnetosphere into the regions of closed and open field lines, and subsequently training two custom Physics Informed Neural Networks to generate the solution within each region. We periodically modify the shape of the boundary separating the two regions (the separatrix) to ensure pressure balance throughout. Our approach provides an effective way to handle mathematical contact discontinuities in force-free electrodynamics. We present preliminary results in axisymmetry, which underscore the significant potential of our method. Finally, we discuss the challenges and limitations encountered while working with Neural Networks, thus providing valuable insights from our experience.

Скоростные напоры и их соотношения за стационарными маховскими конфигурациями ударных волн, распространяющихся в потоке газа

We analyzed the parameters of gas streams, which form after the triple shock configurations occurring in Mach reflection with normal Mach shock (so called stationary Mach configuration). We suppose that this stationary Mach configuration moves in a counter flow with arbitrary velocity (and corresponding Mach number). Analyzing relations of the dynamic pressures across the contact discontinuity, which emanates from the triple point of the Mach reflection, we have shown that the streams after the triple shock configuration differs much in their translational action on surrounding objects.

Exact ideal magnetohydrodynamic Riemann solutions considering the strength of intermediate shocks

Exact magnetohydrodynamic (MHD) Riemann solutions are the basis of constructing numerical schemes and benchmarks for verifying the schemes. However, non-strict hyperbolicity and nonconvexity of MHD equations contribute to the appearance of intermediate shocks, causing low efficiency of existing exact solvers and high dependence on iterative initials. Utilizing the magnetic critical Mach number proposed in this paper, all possible intermediate shocks are analyzed, parameterized, and categorized. Moreover, the possible wave structures on one side of contact discontinuity are revealed to have 25 cases, and initial conditions are classified into three categories according to the coplanar properties. Based on our findings, a new exact MHD Riemann solver is built. The robustness has been significantly improved after avoiding considerable judgments and the dependence on iterative initials. The analysis of the exact MHD Riemann solution is carried out by the characteristic properties of MHD shocks in the parameterization, and it is found that a solution space exists with the highest dimension of two dimensions under the given initial condition. It is proposed to adopt the intensities of 2 → 3 intermediate shocks as the free parameters of solution space, which can completely express the degree of solution space freedom. Finally, two examples that possess the solution space are used as verifications. The physical properties of MHD equations show that the dominant factor for the solution space is the unique characteristic property of 2 → 3 intermediate shock: the existence of an additional free parameter with tangential symmetry simultaneously.

Shock stability of a novel flux splitting scheme

<abstract><p>This article introduced the HLL-CPS-T flux splitting scheme, which is characterized by low dissipation and robustness. A detailed theoretical analysis of the dissipation and shock stability of this scheme was provided. In comparison to Toro's TV flux splitting scheme, the HLL-CPS-T scheme not only exhibits accurate capture of contact discontinuity, but also demonstrates superior shock stability, as evidenced by its absence of 'carbuncle' phenomenon. Through an examination of the disturbance attenuation properties of physical quantities in the TV and HLL-CPS-T schemes, an inference was derived: The shock stability condition for an upwind method in the velocity perturbation was damped. Theoretical analysis was given to verify the reasonableness of this inference. Numerical experiments were carefully selected to test the robustness of the new splitting scheme.</p></abstract>

Internal shocks hydrodynamics: the collision of two cold shells in detail

ABSTRACT Emission in many astrophysical transients originates from a shocked fluid. A central engine typically produces an outflow with varying speeds, leading to internal collisions within the outflow at finite distances from the source. Each such collision produces a pair of forward and reverse shocks with the two shocked regions separated by a contact discontinuity (CD). As a useful approximation, we consider the head-on collision between two cold and uniform shells (a slower leading shell and a faster trailing shell) of finite radial width, and study the dynamics of shock propagation in planar geometry. We find significant differences between the forward and reverse shocks, in terms of their strength, internal energy production efficiency, and the time it takes for the shocks to sweep through the respective shells. We consider the subsequent propagation of rarefaction waves in the shocked regions and explore the cases where these waves can catch up with the shock fronts and thereby limit the internal energy dissipation. We demonstrate the importance of energy transfer from the trailing to leading shell through pdV work across the CD. We outline the parameter space regions relevant for models of different transients,e.g. Gamma-ray burst internal shock model, fast radio burst blast wave model, Giant flare due to magnetars, and superluminous supernovae ejecta. We find that the reverse shock likely dominates the internal energy production for many astrophysical transients.

Cell-centered Lagrangian Lax-Wendroff HLL hybrid scheme in 3D

This paper describes a cell-centered Lax-Wendroff type scheme with artificial dissipation for Lagrangian fluid dynamics on general unstructured 3D meshes. The method is an extension of a similar 2D approach to 3D. The dissipative terms are inspired by the finite volume HLL flux formulation and are used as artificial viscosity in the momentum equation and as artificial energy flux in the energy equation. Special treatment is formulated for contact discontinuities and mesh-aligned interfaces. The proposed second order method provides reasonable results for typical 3D hydrodynamic tests, such as Noh, Sedov, Kidder and spherical Sod problems.

The analytical solutions of the Riemann problem to 1-D non-ideal flow of dusty gas with external force

This work focuses on the analytical solution to the Riemann problem (RP) for a 1-D non-ideal flow of dusty gas with external force. Here it is presumed that external force is a continuous function of time. We explicitly obtain the elementary wave curves to 1-D non-ideal flow of dusty gas with external force and determine these wave curves in form of characteristics. Exhaustive calculations were performed for the elementary wave solutions, such as the rarefaction wave, shock wave, and contact discontinuity. We examine the influence of dust particles on density, velocity of flow, and shock speed and their implications on the solution of RP. Also, the implication of addition of external force is that all solutions are not self similar.

Exact Riemann solver for non-convex relativistic hydrodynamics

We present the general analytical solution of the Riemann problem (decay of a jump discontinuity) for non-convex relativistic hydrodynamics. In convex dynamics, an elementary nonlinear wave, i.e. a rarefaction or a shock, originates at the discontinuity and travels towards one of the initial states. Between the left and right waves, an equilibrium state appears represented by a contact discontinuity. The exact solution to the Riemann problem in convex relativistic hydrodynamics was first addressed by Martí & Müller (J. Fluid Mech., vol. 258, 1994, pp. 317–333). In non-convex dynamics, two sequences of elementary nonlinear waves move towards the left and right initial states. Solving the Riemann problem involves determining the types of wave developing and the equilibrium state where they coincide. The procedure consists of constructing the wave curves associated with the nonlinear waves in the pressure–velocity phase space, where the intersection of the wave curves indicates the equilibrium state. We describe the relation between the wave curves, the explicit formulas for their calculation, and the outline of the process for a correct derivation and representation of the waves in the spatial domain. We present examples of the exact solution of a Riemann problem that illustrate the complex phenomena of non-convex dynamics by using the phenomenological non-convex equation of state proposed by Ibáñez et al. (Mon. Not. R. Astron. Soc., vol. 476, 2017, pp. 1100–1110).

Slip partitioning and seismogenic stress buildup adjacent to the junction of the Lenglongling fault and the Tuolaishan fault: Insights from kinematically constrained numerical simulations

Understanding fault kinematics plays a crucial role in estimating seismic hazards and linking the relationship between seismogenic stress buildup and tectonic deformation. The northwestern segment of the Qilian-Haiyuan fault zone (QHFZ) serves as a natural laboratory for investigating how the oblique block motion decomposes into a complex fault system. In this study, we constructed a 3D FEM model incorporating the active faults as contact discontinuities to obtain a continuous numerical image of fault slip rates. Our findings revealed that deformation partitioning within the context of oblique block convergence generates a spatially heterogeneous distribution of fault slip rates. The inferred strike-slip rates of the Tuolaishan fault and the Lenglongling fault (LLLF) range from approximately 2 to 4 mm/yr and 4 to 7 mm/yr, respectively. The Lenglongling fault's slip behavior exhibits a spatial transition, with a dominant strike-slip component dominating in the western part and gradually changing to a combination of reverse faulting and strike-slip in the eastern segment. The Northern Lenglongling fault (NLLLF), characterized by a listric shape, merges with the LLLF in the deep crust, resulting in the formation of a positive flower structure that enhances the accumulation of compressional seismogenic stress. The region with high shear strain rates within this area has experienced significant earthquakes, including the 2022 Menyuan MS 6.9 earthquake, the 2016 Menyuan MS 6.4 earthquake, and the 1986 Menyuan MS 6.4 earthquake. Our numerical study gained valuable insights into the relationship between slip partitioning, fault structures, and regional stress.

A shock-stable numerical scheme accurate for contact discontinuities: Applications to 3D compressible flows

The low-dissipation Roe scheme is a popular shock capturing scheme but the shock instability, such as the infamous carbuncle phenomenon, seriously affects its application in high Mach number flows. Although considerable research has been undertaken in two dimensions, mechanism analysis of the three-dimensional shock instability is relatively scarce. Numerical simulations of the Sedov blast wave indicate that the shock instability is more prominent in three dimensions, so it has theoretical and practical significance to analyze and heal the numerical instability. The mechanism for three-dimensional shock instability has been thoroughly studied by means of the linearized stability analysis and the dissipation-controlling approach which introduces the required transverse dissipation in the numerical shock layer and is adopted to enhance the Roe scheme’s shock stability. Furthermore, the unphysical expansion shock resulting from the violation of entropy condition is eliminated by a simple modification of numerical signal speed. For the improvement of the resolution for contact discontinuities and shear waves, an algebraic method which combines the THINC reconstruction scheme and the BVD algorithm is employed to minimize the density difference in the numerical diffusion term. A series of benchmark numerical experiments fully demonstrate that the proposed scheme is endowed with excellent robustness against the shock instability and high accuracy for contact discontinuities and shear waves.

Исследование течения, создаваемого скользящим разрядом, с помощью методов компьютерного зрения

A quantitative study has been made of the flow with shock waves generated in air by a sliding surface discharge lasting less than one microsecond. The high-speed flow was visualized using the shadowgraph method, the process was recorded at a rate of 124 000 frames/s, the exposure time was 1 μs. The aim of this work is to study the dynamics of the two discontinuities: the cylindrical shock wave and the contact surface generated by the discharge. Each experiment allowed several hundred images to be taken of a short-lived gas-dynamic process lasting up to 1 ms. A YOLOv8 convolutional neural network was trained and used to determine the positions of the discontinuities. A data set of 984 markups was labeled. The model on the mAP50 metric achieved 0.887 and the mAP50-95 was 0.557. The model was used to automatically measure the vertical dimensions of the contact discontinuity. It expands at times up to 0.4 - 0.8 ms to a vertical size of 5 - 11 mm. The x-t plots and the velocities of the cylindrical shock waves were measured. It is shown that up to 1 ms after the discharge, the flow development is due to the blast wind motion behind the shock wave. It is shown that the use of computer vision can significantly speed up the analysis of high-speed flow visualizations and the extraction of quantitative information.

A polarization study of jets interacting with turbulent magnetic fields

ABSTRACT We investigate the effect of the jet’s immediate surroundings on the non-thermal synchrotron emission and its polarization properties. The ambient medium is equipped with a turbulent magnetic field, which is compressed and amplified by the jets as they progress. This leads to high polarization at the forward shock surface. The randomness in the magnetic polarities of the external fields in the shocked ambient medium (SAM) results in vector cancellation of the polarized components from the jet, thereby causing depolarization of the radiation from the cocoon. We find that due to the slow decay of the fields in the SAM, such depolarization by the fields with large correlation lengths is more prominent when compared to the small-scale fields. Also, the low-power jets, which have magnetic fields comparable in strength to those in the SAM, are more severely affected by the SAM’s depolarizing effect, than the high-power ones. The turbulent backflows in the cocoon, as well as the shearing of fields near the contact discontinuity, strengthen the poloidal component in the jet. This causes internal depolarization due to the cancellation of the orthogonally polarized components along the line of sight as the field transitions from ordered toroidal to poloidal. The synchrotron maps display high-emission filaments in the cocoon with magnetic fields aligned along them. The kink instability leads to the wiggling motion of the jet’s spine, resulting in hotspot complexes in low-power sources.

An accurate, robust and efficient convection-pressure flux splitting scheme for compressible Euler flows

The popular Roe's flux difference splitting scheme has high resolution for various types of discontinuities but it fails to satisfy the entropy condition and will encounter different forms of instability while calculating shock waves, such as the post-shock oscillation of the slowly moving shock waves and the carbuncle phenomenon of the multidimensional strong shock waves. In the present work, a simple flux difference splitting scheme based on the idea of convection-pressure splitting is proposed. With application of the Toro-Vázquez splitting procedure, the flux of the 3D Euler equations is split into the convection part and the pressure part. Due to having a complete set of linearly independent eigenvectors, the pressure subsystem can be solved by the conventional flux difference splitting method. The weakly hyperbolic convection subsystem, however, does not have a complete set of linearly independent eigenvectors and thus is solved by the flux difference splitting scheme based on the generalized eigenvectors [1]. In addition, a strategy that combines the THINC reconstruction with the BVD algorithm is employed on the numerical dissipation term of the convection flux to further improve the resolution for contact discontinuities. A comprehensive range of numerical simulations for classical 1D, 2D and 3D benchmark test problems demonstrate the good performance of the proposed scheme.

Inverse asymptotic treatment: Capturing discontinuities in fluid flows via equation modification

A major challenge in developing accurate and robust numerical solutions to multi-physics problems is to correctly model evolving discontinuities in field quantities. These manifest themselves as interfaces between different phases in multi-phase flows, or as shock and contact discontinuities in (either single- or multi-phase) compressible flows. A plethora of bespoke discretization schemes have been developed to capture both types of discontinuities in physics-based simulations. However, when a quick response is required to rapidly emerging challenges such as the need to design novel hypersonic vehicles, the complexity of such implementations impedes a swift transition from problem formulation to computation. This is exacerbated by the need to compose multiple interacting physics, which may cause specialized schemes to break unexpectedly as governing equations need to be changed and/or added. We introduce “inverse asymptotic treatment” (IAT) as a unified framework for capturing discontinuities in fluid flows that enables building directly computable models based on standard, off-the-shelf numerics. By capturing discontinuities through modifications at the level of the governing equations, rather than via reliance on specialized discretization schemes, IAT can seemlessly handle additional physics and thus makes it easier for novice end users to quickly obtain numerical results for a variety of multi-physics scenarios. This strategy also facilitates the use of a “multi-physics” compiler that automates the conversion of the modified PDEs to numerical source code readable by popular computational frameworks like OpenFOAM. We outline IAT in the context of phase-field modeling of two-phase incompressible flows, and then demonstrate its generality by showing how localized artificial diffusivity (LAD) methods for single-phase compressible flows can be viewed as instances of IAT. Through the real-world example of a laminar hypersonic compression corner, we illustrate IAT’s ability to, within a span of just a few months, generate a directly computable model whose wall metrics predictions for sufficiently small corner angles come close to that of NASA’s state-of-the-art VULCAN-CFD solver. Finally, we propose a novel LAD approach via “reverse-engineered” PDE modifications, inspired by total variation diminishing (TVD) flux limiters, to eliminate the problem-dependent parameter tuning that plagues traditional LAD. Through canonical numerical tests, we demonstrate that this “limiter-inspired” LAD approach, when combined with second-order central differencing, can robustly and accurately model compressible flows.

An efficient HLL-based scheme for capturing contact-discontinuity in scalar transport by shallow water flow

An efficient numerical scheme is developed for coupled hydrodynamic and scalar transport systems to guarantee the conservation and the positivity-preserving properties for water depth and scalar concentration. A second-order well-balanced positivity-preserving central-upwind scheme based on the finite volume method is adopted to discretize both the Saint-Venant system and the advective fluxes in the scalar transport system. In particular, an anti-diffusion modification is augmented in an ad hoc manner to the Harten–Lax–van Leer approximate Riemann solver for the scalar transport system, with the aim of significantly reducing the numerical diffusion near contact discontinuities. The proposed scheme is validated through seven numerical experiments, wherein the advection and diffusion processes of scalar transport are considered either separately or in combination. Accuracy is measured based on the difference between the numerical results and analytical solutions, and the performance of the proposed scheme is assessed by comparing it with that of the existing scheme. Convergence analysis confirms that the proposed scheme is accurate to the second order. The stationary steady state with wet-dry fronts is simulated, which verifies that the proposed scheme can preserve the exact C-property. The results for the pure diffusion case reveal that the original scheme has limitations in precisely predicting scalar diffusion when the diffusion coefficient is small. By contrast, the proposed scheme accurately approximates scalar diffusion, even at low diffusivity. The results under flow conditions with various Froude and Péclet numbers confirm that the proposed scheme is more accurate than the existing scheme in solving scalar transport problems over a wide range of shallow water flows.

Existence of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics

AbstractWe establish the local existence and uniqueness of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two‐phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well‐posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably‐chosen nonlinear approximate problems for the two‐phase compressible viscous non‐resistive MHD.

The perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms.

On the Riemann problem and interaction of elementary waves for two‐layered blood flow model through arteries

In this paper, we focus on the Riemann problem for two‐layered blood flow model, which is represented by a system of quasi‐linear hyperbolic partial differential equations (PDEs) derived from the Euler equations by vertical averaging across each layer. We consider the Riemann problem with varying velocities and equal constant density through arteries. For instance, the flow layer close to the wall of vessel has a slower average speed than the layer far from the vessel because of the viscous effect of the blood vessel. We first establish the existence and uniqueness of the corresponding Riemann solution by a thorough investigation of the properties of elementary waves, namely, shock wave, rarefaction wave, and contact discontinuity wave. Further, we extensively analyze the elementary wave interaction between rarefaction wave and shock wave with contact discontinuity and rarefaction wave and shock wave. The global structure of the Riemann solutions after each wave interaction is explicitly constructed and graphically illustrated towards the end.

Dynamic scaling design strategy of discontinuous bolted rotor systems based on bidirectional parameter mapping

Rods fastened rotor system are widely used in heavy-duty gas turbines. The contact nonlinearity and discontinuity induced by bolted rotor structures, make it difficult to design a scaled rotor system accurately. Thus, a dynamic scaling design strategy for discontinuous bolted rotor systems is proposed in this paper. An improved continuous modeling method considering rods deformation based on equivalent material layer (EML) is introduced to reflect the weakening effect of the rotor stiffness with bolted structure, and the parameter mapping relationships between the continuous and the discontinuous structure are constructed. The scaling factors of each element of the rotor system are derived by the similarity theory. Through this approach, a discontinuous rotor system can be scaled, demonstrating a high level of consistency with the prototype in terms of natural frequencies, critical speeds as well as frequency response. In addition, modal experiments under different preloads and dynamic acceleration experiment are carried out to verify of the proposed modeling method for discontinuous rotor structures. The testing and simulation results prove that the introduced equivalent material layer reflects the contact effect of prototype accurately. Compared with the experimental results, the maximum deviation of the first two natural frequencies is 4.06%, and the maximum deviation of the critical speeds is 1.36%, demonstrating the potential of small size, low-cost scaled models for predicting the complex dynamic characteristics of the prototype rotor system. Overall, the proposed scaling design strategy provides practical guidance for the design and manufacture of the discontinuous scaled rotor systems, which has broad application in testing, fault prediction and dynamic design of prototypes.

A numerical extension of the spatially-filtered Euler equations for contact discontinuities

A numerical extension of the spatially-filtered Euler equations for contact discontinuities