Contact Wave Research Articles

Decay rate to contact discontinuities for the one-dimensional compressible Euler-Fourier system with a reacting mixture

In this paper, we investigate the nonlinear stability of contact waves to the Cauchy problem of the compressible Euler-Fourier system with a reacting mixture in one dimension under the non-zero mass condition. If the corresponding Riemann problem for the compressible Euler system admits a contact discontinuity solution, it is shown that the contact wave is nonlinearly stable, while the strength of the contact discontinuity and the initial perturbation are suitably small. Especially, we obtain the decay rate of contact waves by using anti-derivative methods and elaborated energy estimates.

Asymptotic stability of a composite wave to the one-dimensional compressible Navier-Stokes equations for a reacting mixture

This paper is concerned with the large time behavior of solutions for the Cauchy problem to the compressible Navier-Stokes equations for a reacting mixture with zero viscosity in one dimension. If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves and contact discontinuity, it is shown that the composition of two rarefaction waves with a viscous contact wave is asymptotically stable, while the strength of the composite wave and the initial perturbation are suitably small.

Asymptotic stability of the combination of a viscous contact wave with two rarefaction waves for 1-D Navier-Stokes equations under periodic perturbations

Considering the space-periodic perturbations, we prove the time-asymptotic stability of the composite wave of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in this paper. Perturbations of this kind oscillate in the far field and are not integrable. The key is to construct a suitable ansatz carrying the same oscillation as that of the initial data, but due to the degeneration of contact discontinuity, the construction is more subtle. A novel method for constructing robust ansatzes is presented. It allows the same weight function to be used on different variables and wave patterns while maintaining control over the errors. As a result, it is possible to apply this construction to contact discontinuities and composite waves. Lastly, we demonstrate that the Cauchy problem admits a unique global-in-time solution and the composite wave remains stable under space-periodic perturbations through the energy method.

Experimental analysis of dynamic behavior of elastic visco-plastic beam under repeated mass impacts

Repeated impact behaviors of an elastic visco-plastic steel beam struck by a hard sphere are experimentally examined in this paper. The interaction between impact bodies, behaviors of sub-impacts, wave propagations and dynamics of the beam are tested to get a better understanding into the mechanism of repeated impacts. At the four different impact locations on the beam, thirty repeated-mass-impact tests are conducted for each impact location under identical initial conditions except for the accumulated deformations. The effects of repeated impacts and impact location on the sub-impact behaviors, local contact deformations, wave propagations, high-frequency vibrations and long-term beam responses are analyzed. The sub-impact phenomenon is observed. The deterministic and random features of the influences of repeated impacts on the behaviors of sub-impacts are observed. The deterministic feature similar to the elastic shakedown phenomenon is found to occur for the first sub-impact behaviors, such as the kinetic coefficient of restitution, residual deformations and early wave propagations. The random feature is found to occur for the second sub-impact behaviors as well as the vibrational modes and long-term stress and displacement responses. The couplings of sub-impacts, wave propagation and high-frequency vibrations make the repeated impact behaviors of flexible structure much more complicated than single impact.

Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data

Abstract In this article, we study the large-time behavior of combination of the rarefaction waves with viscous contact wave for a one-dimensional compressible Navier-Stokes system whose transport coefficients depend on the temperature. It is shown that if the adiabatic exponent γ is suitably close to 1, the unique solution global in time to ideal polytropic gas exists and asymptotically tends toward the combination of a viscous contact wave with rarefaction waves under large initial perturbation. New and subtle analysis is developed to overcome difficulties due to the smallness of γ – 1 to derive heat kernel estimates. Moreover, our results extend the studies in a previous work [F. M. Huang, J. Li, and A. Matsumura, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 89–116].

Development of a carbuncle-free and low-dissipation Roe-type scheme: Applications to multidimensional Euler flows

The Roe scheme is known to have high resolution but to be afflicted by the notorious shock anomalies (such as the carbuncle phenomena) and the unphysical expansion shocks. This paper aims to develop a new Roe-type scheme which is robust for strong shock waves and low dissipative for contact waves and shear layers. Based on the mechanism analysis of shock instability, a dissipation-controlling strategy is adopted to adjust the entropy wave viscosity and shear wave viscosity of the transverse numerical flux in the numerical shock layer and the robustness of the Roe scheme is thus enhanced. The violation of entropy condition and the problem of expansion shocks are also fixed to broaden the scheme’s application scope. In addition, the THINC reconstruction procedure and a BVD algorithm are implemented to further improve the resolution for contact waves and shear layers. Numerical results of a series of classical one-, two- and three-dimensional test problems fully demonstrate that the proposed new Roe-type scheme has not only better robustness for strong shock waves but also higher resolution for contact waves and shear layers than the original Roe scheme.

Research on a Measurement Method for the Ocean Wave Field Based on Stereo Vision

The wave parameter is an important environmental input condition. Traditional contact wave measurement methods are unable to meet the requirements of high precision, non-contact, and ship wave field assessment. Alternatively, stereo vision technology can realize a non-contact and mobile form of measurement. However, this technology suffers from poor timeliness and adaptability. This paper proposes a comprehensive wave measurement method that is based on stereo vision, wherein the gridding of siftGPU is used to achieve the fast matching of large images. The whole algorithm can be run within 6 s and it guarantees more than 20,000 feature-matching logarithms. Furthermore, by utilizing the least squares method and sea surface wave surface theory, the sea surface base level can be calculated without control points, along with the inversion of the sea wave parameters (wave height, period, and wave direction) and error point fitting. The rationality and superiority of the algorithm were verified through multiple comparison experiments. Compared with the Richard Brancker Research (RBR) wave height meter, the measurement error of the wave height is less than 10%, the period error is less than 0.5 s, and the wave direction error is less than 10° with the proposed method.

Exact solution for Riemann problems of the shear shallow water model

The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.

Asymptotics Toward Viscous Contact Waves for Solutions of the Landau Equation

In the paper, we are concerned with the large time asymptotics toward the viscous contact waves for solutions of the Landau equation with physically realistic Coulomb interactions. Precisely, for the corresponding Cauchy problem in the spatially one-dimensional setting, we construct the unique global-in-time solution near a local Maxwellian whose fluid quantities are the viscous contact waves in the sense of hydrodynamics and also prove that the solution tends toward such local Maxwellian in large time. The result is proved by elaborate energy estimates and seems the first one on the dynamical stability of contact waves for the Landau equation. One key point of the proof is to introduce a new time-velocity weight function that includes an exponential factor of the form $\exp (q(t)\langle \xi\rangle^2)$ with $$ q(t):=q_1-q_2\int_0^tq_3(s)\,ds, $$ where $q_1$ and $q_2$ are given positive constants and $q_3(\cdot)$ is defined by the energy dissipation rate of the solution itself. The time derivative of such weight function is able to induce an extra quartic dissipation term for treating the large-velocity growth in the nonlinear estimates due to degeneration of the linearized Landau operator in the Coulomb case. Note that in our problem the explicit time-decay of solutions around contact waves is unavailable but no longer needed under the crucial use of the above weight function, which is different from the situation in [11, 14].

Investigating the Effect of Design Parameters on the Mechanical Performance of Contact Wave Springs Designed for Additive Manufacturing.

Additive manufacturing (AM) enables design freedom to fabricate functionally graded wave springs designed by varying design parameters, which are not possible in traditional manufacturing. AM also enables optimization of the wave spring design for specific load-bearing requirements. Existing wave springs are manufactured by metal with constant dimensions (width and thickness of the strip, diameter) using customized traditional machines in which design variations are almost impossible. This study aims to investigate the effect of wave height, the overlap between the two consecutive coils, and the number of waves per coil on the mechanical properties, for example, load-bearing capacity, stiffness, and energy absorption of contact wave springs. Two designs, that is, rectangular and variable thickness wave springs, were chosen and the design of experiment was devised using Minitab software, resulting in 24 samples. HP MultiJet Fusion (MJF) printer was used to manufacture the samples for performing uniaxial compression tests up to 10 cycles and 90% of the compressible distance to study the variation in mechanical properties due to changes in parameters. Experimental and simulation results showed that variable thickness wave springs have better load bearing, stiffness, and energy absorption compared with the rectangular counterparts. In addition to that, the number of waves per coil and the overlap are directly proportional to the load-bearing capacity as well as stiffness of the wave springs, while the constant wave height is responsible for more uniformly distributed stresses throughout the coils. Load-bearing capacity was increased by 62%, stiffness by 37%, and energy absorption by 20% once the number of waves per coil is increased from 5 to 6 in rectangular wave springs. Overall, the parametric variations significantly affect the performance of wave springs; thus, designers can choose the optimized values of investigated parameters to design customized wave springs for specific applications as per load/stiffness requirements.

Design and performance evaluation of multifunctional midsole using functionally gradient wave springs produced using multijet fusion additive manufacturing process

Additive manufacturing (AM) is getting more attention and is considered a better choice of fabrication in almost every industry due to its unlimited design freedom, optimization, light-weight and customization. Currently, manufacturing constraints, limited design freedom and customization issues restrict manufacturers from designing specific shoes for each particular application e.g., walking, running, hiking, sports, etc. However, AM can design and fabricate a multifunctional, usable single shoe-midsole for all aforementioned applications. This study aims to design a multifunctional shoe midsole incorporated with functionally gradient wave springs (FGWS) at the critical areas of foot pressure (heel, forefoot and toe) measured using the F-scan system (insole plantar pressure measurement system). The non-critical areas were designed by a gradient cellular structure with graded unit cells as per load requirement. The designed midsole with FGWS was printed by an HP MultiJet Fusion printer using PA 12 (polyamide). A LISA printer (selective laser sintering) using TPU (thermoplastic polyurethane) material was also used to check its manufacturability. Compression testing, each spring up to 80% of its compressible distance studied the load-bearing capacity, energy absorption, stiffness and cushioning properties. The load-bearing capacity of FGWS was decreased and energy absorption was enhanced by compressing each spring with EVA (Ethylene-vinyl acetate) foam while traditionally manufactured non-contact metal wave springs showed a lower load-bearing capacity than AM (contact) wave springs. Moreover, the fatigue properties of both were compared after 100 cycles of compression, which found a good response of AM wave springs. Experimental compression results revealed a gradual smooth response of load against compression, i.e., cushioning and more energy absorption, validated by finite element analysis (FEA) with very little deviation.

Decay rates to viscous contact wave for the one-dimensional compressible Navier–Stokes equations with radiation

In this paper, we consider the stability and decay rate of viscous contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with radiation under the zero mass condition. we consider the case that the constitutive relations for the pressure p, the specific internal energy e, the specific volume v and the absolute temperature θ are given by p=Rθ/v+aθ4/3, e=Cvθ+avθ4, with R>0, Cv>0, and a>0 being the perfect gas constant, the specific heat and the radiation constant, respectively.Our main idea is to use the smallness of the strength of the contact wave to control the possible growth of its solutions induced by the nonlinearity of the system and interactions between the solutions and the contact wave. The key point in our analysis is to obtain the uniform convergence rate of a contact wave pattern (1+t)−58ln12(2+t) by using a new inequality (see Lemma 2.3) and a weighted characteristic energy estimate.

Artificial Compressibility with Riemann Solvers: Convergence of Limiters on Unstructured Meshes

Free-surface flows and other variable density incompressible flows have numerous important applications in engineering.One way such flows can be modelled is to extend established numerical methods for compressible flows to incompressible flows using the method of artificial compressibility. Artificial compressibility introduces a pseudo-time derivative for pressure and, in each real-time step, the solution advances in pseudo-time until convergence to an incompressible limit - a fundamentally different approach than SIMPLE, PISO, and PIMPLE, the standard methods used in OpenFOAM. Although the artificial compressibility method is widespread in the literature, its application to free-surface flows is not. In this paper, we apply the method to variable density flows on 3D unstructured meshes for the first time, implementing a Godunov-type scheme with MUSCL reconstruction and Riemann solvers, where the free surface gets captured automatically by the contact wave in the Riemann solver. The critical problem in this implementation lies in the slope limiters used in the MUSCL reconstruction step. It is well-known that slope limiters can inhibit convergence to steady state on unstructured meshes; the problem is exacerbated here as convergence in pseudo-time is required not just once, but at every real-time step. We compare the limited gradient schemes included in OpenFOAM with an improved limiter from the literature, testing the solver against dam-break and hydrostatic pressure benchmarks. This work opens OpenFOAM up to the method of artificial compressibility, breaking the mould of PIMPLE and harnessing high-resolution shock-capturing schemes that are easier to parallelise.

An effective all-speed Riemann solver with self-similar internal structure for Euler system

The HLLI Riemann solver proposed by Balsara is a universal scheme which is capable of resolving full family of intermediate waves with minimal dissipation for hyperbolic conservation law. However, asymptotic analyses demonstrate that such a scheme is incapable of obtaining physical solutions at low speeds due to excessive numerical dissipations of the velocity-difference terms in the momentum equations. To accommodate this deficiency, we employ the construction of the HLLEM scheme to separate shear waves which contain velocity difference in the tangential direction from contact waves and modify the quantity of the velocity difference. In addition, a function κ is adopted to damp the shear waves in the vicinity of the shockwaves to avoid shock instability phenomena. With these improvements, proper dissipation can be obtained at all speeds without suffering from the appearance of shock anomalies. Finally, a series of numerical experiments illustrate that the novel HLLIM scheme we propose is effective and robust at all speeds. Therefore, analyses in this study could provide reference for the improvement of the HLLI solver for the other hyperbolic conservation law in cases the eigenvalues vary sharply.

High-order methods for diffuse-interface models in compressible multi-medium flows: A review

The diffuse interface models, part of the family of the front capturing methods, provide an efficient and robust framework for the simulation of multi-species flows. They allow the integration of additional physical phenomena of increasing complexity while ensuring discrete conservation of mass, momentum, and energy. The main drawback brought by the adoption of these models consists of the interface smearing, increasing with the simulation time, therefore, requiring a counteraction through the introduction of sharpening terms and a careful selection of the discretization level. In recent years, the diffuse interface models have been solved using several numerical frameworks including finite volume, discontinuous Galerkin, and hybrid lattice Boltzmann method, in conjunction with shock and contact wave capturing schemes. The present review aims to present the recent advancements of high-order accuracy schemes with the capability of solving discontinuities without the introduction of numerical instabilities and to put them in perspective for the solution of multi-species flows with the diffuse interface method.

A novel high-order low-dissipation TENO-THINC scheme for hyperbolic conservation laws

The high-order low-dissipation shock-capturing scheme is important for compressible fluid simulations, in particular for cases where both shock and turbulence present. However, the competing requirements for resolving the small-scale turbulence structures with low dissipation and capturing the discontinuities sharply render it non-trivial for numerical development. In this work, based on a novel parameter-free discontinuity-detection criterion, a new shock-capturing framework is proposed by combining the standard TENO (targeted essentially non-oscillatory) scheme for smooth regions with the non-polynomial based THINC (tangent of hyperbola for INterface capturing) reconstruction for non-smooth discontinuities. The resulting hybrid scheme retains the low-dissipation property of TENO scheme for smooth flow scales and the discontinuity-resolving capability of THINC reconstruction for shock and contact waves. A set of benchmark simulations involving a wide range of length scales demonstrates that the new TENO-THINC scheme performs significantly better than the standard TENO scheme without the necessity of parameter tuning case by case, and thus is promising for more complex compressible flow predictions where the excessive numerical dissipation of existing schemes prevents the targeted flow structures from being properly resolved. Moreover, the present results serve as the state-of-the-art references for future numerical method development.

Nonlinear stability of diffusive contact wave for a chemotaxis model

We consider a 2×2 system of hyperbolic-parabolic balance laws. Our system is the converted form under inverse Hopf-Cole transformation of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We study Cauchy problem when the Cauchy data are near a diffusive contact wave. The contact wave connects two different end-states as x→±∞, reflecting the situation when the logarithmic singularity plays an intrinsic role in the original chemotaxis model. We establish global existence of solution and study time asymptotic behavior of the solution. Consequently, we obtain nonlinear stability of the diffusive contact wave. Our result shows a significant difference when comparing our model to Euler equations with damping. In our case, there exists a secondary wave in the asymptotic ansatz. Therefore, the solution to Cauchy problem converges to the diffusive contact wave slower than in the case of Euler equations with damping. Besides its own physical relevance, our model is a prototype of a general system of hyperbolic-parabolic balance laws. Our results shed light on the future study of nonlinear stability of elementary waves for a general system.

Grid-characteristic numerical method for medical ultrasound

Grid-characteristic method (GCM) is a fast and reliable numerical method that allows to model wave effects in viscoelastic media with high accuracy, including surface and contact waves. This research is dedicated to the application of GCM to the problem of medical ultrasound. Calculations for High-Intensity Focused Ultrasound (HIFU) were performed on 3D model statements for homogenous and inhomogeneous media, and a qualitative correspondence with experimental data was achieved. Numerical results include estimation of consumed energy (based on Maxwell viscosity model), velocity vector and stress tensor components. Various material parameters were considered, including relaxation time and inclusions of different types.

Asymptotic stability of viscous contact wave for the inflow problem of the heat-conductive ideal gas without viscosity

This paper is devoted to studying the inflow problem governed by the non-viscous and heat-conductive gas dynamic system in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact wave. The contact discontinuity in the linearly degenerate field is less stable, and the dissipative mechanism for non-viscous systems is also weaker than that of viscous systems, these all make the problem more challenging. We used the weighted energy estimates to overcome those difficulties. Some technical discussions were created carefully by taking good advantage of properties of the supersonic region and the viscous contact wave.

Numerical Simulation of a High-Speed Impact of Metal Plates Using a Three-Fluid Model

The process of wave formation at the contact boundary of colliding metal plates is a fundamental basis of explosive welding technology. In this case, the metals are in a pseudo-liquid state at the initial stages of the process, and from a mathematical point of view, a wave formation process can be described by compressible multiphase models. The work is devoted to the development of a three-fluid mathematical model based on the Baer–Nunziato system of equations and a corresponding numerical algorithm based on the HLL and HLLC methods, stiff pressure, and velocity relaxation procedures for simulation of the high-speed impact of metal plates in a one-dimensional statement. The problem of collision of a lead plate at a speed of 500 m/s with a resting steel plate was simulated using the developed model and algorithm. The problem statement corresponded to full-scale experiments, with lead, steel, and ambient air as three phases. The arrival times of shock waves at the free boundaries of the plates and rarefaction waves at the contact boundary of the plates, as well as the acceleration of the contact boundary after the passage of rarefaction waves through it, were estimated. For the case of a 3-mm-thick steel plate and a 2-mm-thick lead plate, the simulated time of the rarefaction wave arrival at the contact boundary constituted 1.05 μs, and it was in good agreement with the experimental value 1.1 μs. The developed numerical approach can be extended to the multidimensional case for modeling the instability of the contact boundary and wave formation in the oblique collision of plates in the Eulerian formalism.