Contact Wave Research Articles

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.

A simple FORCE-type centred scheme accurate for contact discontinuities: Application to compressible Euler flows

The FORCE-type centred schemes are simple and efficient and do not explicitly require the wave propagation information of the system to calculate the numerical flux. However, their poor resolution for contact discontinuities seriously affects their applications. In the present work, a simple FORCE-type centred scheme accurate for contact discontinuities is proposed and applied to calculations of compressible Euler flows. The missing contact wave of the original FORCE centred scheme is restored with an algebraic method and the resolution for contact discontinuities is further improved by using the boundary variation diminishing (BVD) algorithm to minimize the density jump in the numerical diffusion term of the original FORCE centred scheme. Numerical results of several one- and two-dimensional benchmark test problems fully demonstrate that the proposed centred scheme is capable of capturing contact discontinuities more sharply than the complete-wave HLLC upwind scheme. In addition, another advantage of the proposed centred scheme is that it is free from the carbuncle phenomenon which afflicts many contact-capturing upwind schemes (e.g. Roe and HLLC) in calculations of multidimensional flow problems involving strong shock waves.

A characteristic-featured shock wave indicator on unstructured grids based on training an artificial neuron

In a recent work Feng, Liu and Wang (2020) [10], we imbedded characteristic compressing into an artificial neuron (AN) to propose a shock wave indicator on uniform mesh. In this work, the indicator is developed to unstructured grid. To achieve that, we retrain an AN on 1D randomly perturbed mesh, two prior information, (a) eigenvalue variable and (b) side-weighted average, is used in data pre-processing for reducing the influence of mesh size and keeping AN structure simple. The output of AN is then modified into a generalized and explicable form, which is used as the present shock wave indicator. We show that the troubled-cells detected by the present indicator include discontinuities caused by compressing of characteristic curves. The present indicator is then extended to multi-dimensional unstructured grid through constructing side-weighted average of eigenvalue on each spatial dimension. Numerical results are presented to demonstrate the performance of the present indicator combined with slope limiter and artificial viscosity, respectively, on various unstructured grids, the results show that the present indicator can detect shock and contact waves with low noise, and improves the indicating efficiency as well, the present indicator provides an attractive alternative in detecting shock waves on arbitrary grids and can be combined with various discontinuity-processing techniques.

Design for additive manufacturing of variable dimension wave springs analyzed using experimental and finite element methods

Wave springs have better properties than helical springs in terms of load bearing capacity, mass, energy absorption efficiency, and stiffness. Currently, available wave springs have waves that do not merge with each other (non-contact) with uniform shape and dimensions, manufactured by traditional manufacturing processes using customized machines in which forming, twisting, and cutting processes take place simultaneously. This study aims to design and manufacture wave springs with waves that merge with each other by additive manufacturing (AM). Variable-dimension wave springs are complex in design and can only be manufactured by AM owing to its design freedom, but they have much higher performance in load bearing, stiffness, energy absorption, and energy release compared to both uniform-dimension and uniformly shaped wave springs. To accomplish wave springs manufactured by AM, ten different wave springs were designed and manufactured with variable and uniform dimensions, using a HP MultiJet Fusion printer, by keeping the mass and height of each design constant. Compression tests were carried out for up to ten cycles to study and compare the load bearing capacity, energy absorption, stiffness, and energy loss for all designs. The results showed that the stiffness of the spring increased in the contact wave springs, which ultimately led to more energy absorption in these designs. The taper-wave spring was the best among all designs for load bearing, and it had the highest energy loss capacity, which was validated by finite element analysis of each design.

Stability of Contact Discontinuity with General Perturbation for the Compressible Navier-Stokes Equations with Reaction Diffusion

In this article,for the compressible Navier-Stokes equations which have reaction diffusion, the stability of contact discontinuities is considered. The new characteristic for the flow is appearance of the divergence between energy gained and lost because of the reaction . In the energy equations,the term related to the mass fraction of the reactant leads to new technical problem. To solve this problem, in terms of the solutions,a new system should be set up. Using the anti-derivative method and the elaborated energy method, we obtain that as long as the general perturbation of the initial datum plane and the strength of the contact wave are properly small, the contact wave is nonlinear and stable. As a byproduct, we can establish the convergence velocity of contact wave.

A New Method for Wet-Dry Front Treatment in Outburst Flood Simulation

When utilizing a finite volume method to predict outburst flood evolution in real geometry, the processing of wet-dry front and dry cells is an important step. In this paper, we propose a new approach to process wet-dry front and dry cells, including four steps: (1) estimating intercell properties; (2) modifying interface elevation; (3) calculating dry cell elevations by averaging intercell elevations; and (4) changing the value of the first term of slope limiter based on geometry in dry cells. The Harten, Lax, and van Leer with the contact wave restored (HLLC) scheme was implemented to calculate the flux. By combining the MUSCL (Monotone Upstream–centred Scheme for Conservation Laws)-Hancock method with the minmod slope limiter, we achieved second-order accuracy in space and time. This approach is able to keep the conservation property (C-property) and the mass conservation of complex bed geometry. The results of numerical tests in this study are consistent with experimental data, which verifies the effectiveness of the new approach. This method could be applied to acquire wetting and drying processes during flood evolution on structured meshes. Furthermore, a new settlement introduces few modification steps, so it could be easily applied to matrix calculations. The new method proposed in this study can facilitate the simulation of flood routing in real terrain.

The Riemann Problem with Phase Transitions for Fluid Flows in a Nozzle

Phase transitions in a model of van der Waals fluid flows in a nozzle with discontinuous cross-sectional area are investigated. The model admits a physically unstable elliptic region which causes phase transitions, and a jump in cross-sectional area which causes stationary contact waves. Compositions between waves in different characteristic fields are constructed differently in subsonic or supersonic regions. A new kind of waves is introduced which helps establish the existence of solutions of the Riemann problem. There are regions in which the Riemann problem admits a unique solution. Resonant phenomena occur when shock waves associated with different characteristic fields propagate with the same speed. Multiple solutions are also observed even in a strictly hyperbolic region not only by the modeling, but also by the type of a van der Waals fluid.

Reacting Flow Simulations of a Dual Throat-Dual Fuel Thruster

Numerical simulation of the supersonic turbulent reacting flow field in a dual throat- dual fuel rocket thrust chamber is presented. Future single stage to orbit and high lift space transportation missions aspire a reliable, efficient, and cost-effective propulsion systems. The dual throat-dual fuel concept is a simple altitude compensating propulsion alternative with reusable possibilities. Turbulent reacting supersonic flow field emanating from independent thrust chambers needs to be resolved for a better understanding of the flow structures and design modifications for the performance improvement. The operation of a dual throat nozzle brings about a unique shock train in reacting supersonic flow. Two-dimensional axis-symmetric compressible reacting flow field has been solved using HLLC (Harten, Lax, van Leer, with Contact wave) scheme based finite volume Riemann solver with multi-step finite rate chemistry model for hydrocarbon/hydrogen-oxygen combustion. The computational procedure has been validated with experimental data for species distribution of a coaxial supersonic combustor. Chemical species distribution in the supersonic free shear layer is analyzed in detail to explore the nature of active reaction zones in the flow field.

Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model

<p style='text-indent:20px;'>The phenomena of concentration and cavitation for the Riemann problem of the Baer-Nunziato (BN) two-phase flow model has been investigated in this paper. By using the characteristic analysis method, the formation of <inline-formula><tex-math id="M1">\begin{document}$ \delta- $\end{document}</tex-math></inline-formula>waves and vacuum states are obtained as the pressure for both phases vanish in the BN model. The solid contact wave is carefully dealt. The comparison with the solutions of pressureless two-phase model shows that, two shock waves tend to a <inline-formula><tex-math id="M2">\begin{document}$ \delta- $\end{document}</tex-math></inline-formula>shock solution, and two rarefaction waves tend to a two contact discontinuity solution when the solid contact discontinuity is involved. Moreover, the detailed Riemann solutions for two-phase flow model are given as the double pressure parameters vanish. This may contribute to the design of numerical schemes in the future research.</p>

Asymptotic Stability of the Superposition of Viscous Contact Wave with Rarefaction Waves for the Compressible Navier--Stokes--Maxwell Equations

Asymptotic Stability of the Superposition of Viscous Contact Wave with Rarefaction Waves for the Compressible Navier--Stokes--Maxwell Equations

Simulation of Mosul Dam Break Using Finite Volume Method

Mosul dam is an earth-fill embankment located north of Iraq on the Tigris River forming a reservoir with 11.11 km3 water storage capacity which is the largest dam in the country. The dam is built on a rock bed foundation, in which the dissolution process is dynamic in the zone where gypsum and anhydrite layers present. During the construction development seepage locations were found in the dam foundation and the grouting process is in progress until now to control this problem. Therefore, the possibility of the Mosul dam break is highlighted by previous studies. In this research, a FORTRAN code based on the finite volume method is modified to solve the two-dimensional shallow water equations and simulating the Mosul dam break. The computational domain discretized using unstructured triangular mesh. The solver applied Harten lax van leer with contact (HLLC) wave approximate Riemann solver to calculate the cell interface fluxes, and the semi-implicit scheme employed to solve the friction source term. The numerical scheme applied to two benchmark test cases, and the results showed that the presented model was robust and accurate especially in handling wet/dry beds, mixed flow regimes, discontinuities, negative water depths, and complex topography. The results of this study demonstrate that flood waves may reach the center of Mosul city in < 6 h and water depth may rise to 34 m after 7 h of Mosul dam breaking. Finally, the simulation results of the Mosul dam break were used to prepare an emergency action plan.

Transient contact-impact behavior for passive walking of compliant bipedal robots

When bipedal robots walk, the compliance of local contact zone and whole structure usually causes high amplitude contact force and wave propagation excited by impact. The object of this paper is to analyze the transient contact-impact behavior during the walking of compliant bipedal robots (CBR) using a completely flexible body model (CFBM) and obtain its fine contact modes. The applicability of the traditional method based on completely rigid body model (CRBM) to the compliant bipedal robots is also discussed. In CFBM, the structural deformation field and inertial field are discretized by finite element theory, and the penalty function method is adopted to account for the unilateral contact constraint. The dynamic equations, strain and stress equations for the CFBM are derived. The contact forces, impact-induced wave propagation and total mechanical energy of the CBR are calculated. The results show that the relative contact motions between the foot and the slope experience two phases: (1) macroscopic oblique impact phase (duration time is 0.1∼0.2 s) and (2) rolling phase (duration time is 0.6∼0.7 s). Phase 1 consists of a series of fast normal contact-separation switches and tangential stick–slip switches. Phase 2 means there are continuous normal compression and tangential stick. Since the traditional method neglects the first phase, it is not comprehensive for investigating the CBR with impact-induced waves and vibration, especially for the case of lower coefficient of friction (COF) μ. In addition, a ‘dynamic self-locking’ phenomenon is found as μ>0.9, which will lead to the walking unstable.

Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations

<p style='text-indent:20px;'>This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [<xref ref-type="bibr" rid="b6">6</xref>], since the rarefaction wave is a nonlinear expansive wave, while the contact discontinuity wave is a linearly degenerate diffusive wave. So we need to take good advantage of properties of the viscous contact discontinuity wave instead. Moreover, series of tricky argument on the boundary is done carefully based on the construction and the properties of the viscous contact discontinuity wave for the radiative Euler equations. Our result shows that radiation contributes to the stabilization effect for the supersonic inflow problem.

A stable hybrid Roe scheme on triangular grids

AbstractNumerical shock instability is a common problem for shock‐capturing methods that try to resolve contact and shear waves with minimal diffusion. Most flux‐difference splitting and the AUSM family of schemes produce the carbuncle phenomenon on both structured and unstructured grids. The original Roe scheme is well known to generate shock anomalies and can lead to nonentropic weak solutions to the Euler equations. A simple and robust approach for healing these numerical instabilities is to apply the hybrid technique incorporated with an efficient weighting switch function to control the amount of dissipation in the vicinity of shock waves. This article proposes a simple, robust, and accurate hybrid Roe scheme (Roe+scheme) by hybridizing the Roe scheme and the modified AUSMV+scheme. A new normalized pressure/density‐based weighting switch function is proposed and applied to the scheme to minimize the numerical dissipation and maintain the robustness of the hybridization. The linearized discrete analysis is performed to evaluate the proposed scheme according to the perturbation damping mechanism of an odd–even decoupling problem. The resulting recursive equations indicate that the hybridized mechanism damps all perturbations effectively. Finally, several numerical examples demonstrated that the Roe+scheme provides an accurate, robust, and carbuncle‐free solution on both structured and unstructured triangular grids.

Tunable traveling wave properties in one-dimensional chains composed from hollow cylinders: From compression to rarefaction waves

We report a hollow cylindrical granular system supporting highly tunable compression and rarefaction waves by varying particle thickness ratios. Soliton-like and mitigating waves are observed in the proposed granular chain with strain hardening and softening contact behaviors, respectively. Moreover, validated finite element (FE) models describing the contact behaviors and wave dynamics are adopted and specific relation between the contact behavior (i.e., nonlinear exponent) and geometric parameter (i.e., thickness ratio) is analyzed. The tunability of a one-dimensional hollow cylindrical granular chain by carrying out parametric studies for governing factors is demonstrated, including the impact velocity, impactor-to-particle mass ratio, particle thickness ratio, and particle number. An optimal impact mitigating efficiency can be obtained within a quite short chain, which is exceedingly favorable for the design and construction of impact mitigation systems. Results provide a novel design and methodology for various wave tuning and mitigation purposes.

Asymptotic Stability of a Viscous Contact Wave for the One-Dimensional Compressible Navier-Stokes Equations for a Reacting Mixture

We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L2-energy methods.

Evolution of Contact and Weak Discontinuity Waves in Two Phase Drift Flux Model

In this paper, we consider two-phase drift flux model and give a full symmetry group classification of the governing quasilinear system of partial differential equations (PDEs). An exact solution is obtained by mapping the governing system to an analogous system of PDEs, where the flow variable exhibits space-time dependence. The drift flux model, which is a strictly hyperbolic system, is shown to have a linearly degenerated characteristic field, therefore, the associated wave is a contact discontinuity. The transport equations for the evolution of the contact and weak discontinuity waves are studied explicitly and some intriguing observations on the behavior of these elementary waves are obtained.

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.

Decay rate to contact discontinuities for the 1-D compressible Navier-Stokes system

We revisit the classical work of Huang-Matsumura-Xin [9] for contact wave of the one-dimensional compressible Navier-Stokes system under the zero mass condition. The large-time asymptotic stability of a contact wave pattern with a uniform convergence rate (1+t)−14 was proved by Huang-Matsumura-Xin. In this paper, Huang-Matsumura-Xin's convergence rate is improved to (1+t)−58ln12⁡(2+t) by using a new Poincaré type inequality and a detailed energy analysis. Our work is motivated by a problem arising in [9].

High-accurate and robust conservative remapping combining polynomial and hyperbolic tangent reconstructions

In this article we present a 1D single-material conservative remapping method that relies on high accurate reconstructions: polynomial (P4,P1 with slope limiter) and non-linear hyperbolic tangent (THINC) representations. Such remapping procedure is intended to be used pairwise with a cell-centered Lagrangian scheme along with a rezone strategy to build a so-called indirect Arbitrary-Lagrangian-Eulerian scheme. Most of practically used Lagrangian schemes are second-order accurate. The goal of this work is to handle with accuracy contact using THINC reconstructions. At the same time, the smooth part of the solution is dealt with quartic polynomials, resulting locally in fifth order accurate remapping method. To ensure robustness, TVD-like reconstructions (P1 with slope limiter) are employed otherwise. A simple feature tracking algorithm is designed to assign a reconstruction type per cell (P4,P1limor THINC). This tracking algorithm is based on the nature of the contact waves which are traveling at the fluid velocity, while the shocks are compressive and detectable by following a change of cell volumes. Numerical results assess the behavior of such a remapping method on pure remapping problems of a scalar quantity and in the context of the full hydrodynamics equations. The associated indirect cell-centered ALE numerical scheme is run and produces numerical results that are presented to assess the extreme accuracy gained by such a remapping procedure employing a mix of reconstruction types.