Search Results (188)

The shallow water equations (SWEs) model fluid flow in rivers, coasts, and tsunamis. Their nonlinearity challenges analytical solutions. We present a numerical algorithm combining the finite integration method with Chebyshev polynomial expansion (FIM-CPE) to solve one- and two-dimensional SWEs. The method transforms partial differential equations into integral equations, approximates spatial terms via Chebyshev polynomials, and uses forward differences for time discretization. Validated on stationary lakes, dam breaks, and Gaussian pulses, the scheme achieved errors below ${10}^{-12}$ for water height and velocity, while conserving mass with volume deviations under ${10}^{-5}$. Comparisons showed superior shock-capturing versus finite difference methods. For two-dimensional cases, it accurately resolved wave interactions over complex topographies. Though limited to wet beds and small-scale two-dimensional problems, the method provides a robust simulation tool. Full article

An analytical study of the interaction of an ideal gas flow with a detonation wave was performed with account for the activation energy of chemical processes. Based on the modified Rankine-Hugoniot conditions, the effect of heat release on the limiting characteristics of detonation was analyzed. A dependence of the limiting value of the exponent Arrhenius number on the Mach number before the shock wave has been obtained. As the Mach number increases, the limiting value of the Arrhenius number decreases. An equation has been derived for determining the limiting value of the compression ratio in the shock wave. The effect of heat release intensity on the limiting compression ratio in a shock wave was elucidated. Also studied were effects of the Mach number and the Arrhenius number on the limiting compression ratio in a detonation wave. A condition for determining the critical value of the Arrhenius number necessary for the onset of detonation was obtained. Effects of the Mach number and the exponent of the Arrhenius number ${\mathrm{Ar}}_E$ on the critical value of the amplitude Arrhenius number ${\mathrm{Ar}}_A$ were discussed. The symmetry analysis of the gas flow parameters when passing through a detonation wave was performed. Full article

This study presents a one-dimensional solver of the shallow water equations designed for the wet-bed step Riemann problem. Nonlinear mass and momentum equations incorporating shock and rarefaction waves in a straight one-dimensional channel are expressed as a pair of equations that depend solely on local depth values either side of the step. These unified equations are uniquely designed for the four conditions involving shock and rarefaction waves that can occur in the Step Riemann Problem. The Levenberg–Marquardt method is used to solve these simplified nonlinear equations. Four verification tests are considered for shallow free surface flow in a wet-bed channel with a step. These cases involve two rarefactions, opposing shock-like hydraulic bores, and a rarefaction and shock-like bore. The numerical predictions are in close agreement with existing theory, demonstrating that the method is very effective at solving the wet-bed step Riemann problem. Full article

Apparently for the first time, shear shock wave fronts (shear shocks) are observed in a hyperfoam at the propagation of shear waves. The hyperfoam is modelled by the Ogden compressible hyperelastic potential. A possible appearance of the shear shocks may explain the kinetic and strain energy attenuation along with heat release at the propagation of shear waves in hyperfoams. The analysis is based on the Cauchy formalism for equations of motion, equations of energy balance, and FE analysis for solutions of the constructed nonlinear hyperbolic equation. Full article

Accurately simulating wave breaking is crucial for modeling hydrodynamics over steep coral reef slopes, yet it remains a challenge for Boussinesq-type models like FUNWAVE-TVD. The model’s standard hybrid breaking mechanism, triggered by a fixed free surface elevation-to-depth ratio ($\eta /d>0.8$), often lacks physical sensitivity to local slope and wave conditions prevalent in reef environments and suffers from inaccuracies associated with using η as a direct proxy for wave height (H). This study introduces and validates a novel, enhanced hybrid breaking module within FUNWAVE-TVD, specifically designed to overcome these limitations on steep slopes. The core novelty lies in the synergistic implementation of two key components: (1) replacing the fixed threshold with a dynamic, physically-based criterion derived from the Modified Goda formula (MGO) by Rattanapitikon and Shibayama, which calculates the breaking wave height ($H_b$) based on local depth, slope, and deep-water wavelength; and (2) developing and applying a practical method, using the wave vertical asymmetry relationship proposed by Yu and Li, to dynamically convert the calculated $H_b$ into an equivalent breaking surface elevation threshold (${\eta }_b$). This derived dynamic threshold (${\eta }_b/d$) is then used to trigger the model’s existing switch from Boussinesq to Nonlinear Shallow Water Equations (NSWE), allowing for energy dissipation via shock-capturing while retaining the physical basis of the MGO criterion. The performance of this enhanced module was rigorously evaluated against five laboratory experiments of regular waves breaking on impermeable slopes ranging from mild (1:10) to extremely steep (1:1), contrasting results with the original FUNWAVE-TVD. The modified model demonstrates significantly improved accuracy (model skill increases ranging from 10.16% to 42.49%) compared to the original model for breaking location and wave height prediction on steeper slopes ($m\ge 1:6$). Conversely, tests on the 1:1 slope confirmed the inherent limitations of the MGO criterion itself under surging breaker conditions ($m\ge 1:2.3$), highlighting the applicability range. This work provides a validated methodology for incorporating slope-aware, dynamic breaking criteria effectively into hybrid Boussinesq models, offering a more robust tool for simulating wave processes on steep reef topographies. Full article

The design of heat and drag reduction systems for hypersonic vehicles has garnered widespread global attention. In this study, the Navier–Stokes equations and the SST k-ω turbulence model are employed to establish a simulation model for heat and drag reduction induced by a forward-facing cavity. The numerical methods are validated using existing experimental results. The oscillation characteristics of the bow shock wave at the head and the shock inside the cavity in hypersonic flows are investigated. The heat and drag reduction mechanisms of the forward-facing cavity are discussed. The effects of the diameter and depth of the cavity on drag and heat reduction are comprehensively analyzed. The obtained results show that a reduction in drag and heat is achieved when a forward-facing cavity is added to the vehicle. The main reasons for this heat reduction are the cold ring mechanism and the energy conversion mechanism. The size of the cold ring is significantly affected by the cavity diameter, whereas the energy conversion mechanism is more sensitive to variations in diameter. The maximum reduction in heat load is 2.2%, and the maximum reduction in the Stanton number is 25.3%. Increases in both diameter and depth enhance drag reduction, achieving an average drag reduction of approximately 1.65%. Full article

This paper employs the similarity method to derive self-similar solutions for a model of a spherical shock wave. The discussion involves the propagation of a spherical shock wave within an ideal gas experiencing self-gravitating effects, utilizing the equation of motion and shock conditions. The expressions of infinitesimal generators within the Lie group of transformations contain arbitrary constants, leading to the identification of four potential solution cases. Among these, we focus on two cases, one following a power-law and the other an exponential law. The outcomes discussed are contingent on the variation of flow variables behind the shock, visually represented through graphical illustrations. Full article

In the present study, abilities of various macroscopic models (Navier–Stokes–Fourier, Burnett, original and regularized Grad’s 13-moment equations) in predicting the nonequilibrium molecular velocity distribution are examined. The results of the local distribution function reconstruction from flow macroparameters for the models considered are compared with each other and with the reference solution. Two different flows are considered: normal shock wave and stationary regular reflection of oblique shock waves. The Direct Simulation Monte Carlo method is used to obtain the reference solution and the flow macroparameters required for the distribution function reconstruction. All models under consideration predict the distribution function in the upstream low-density region rather poorly, with strong oscillations and unphysical negative values (especially regularized Grad’s 13-moment equations). In the high-density downstream region, the shape of the reference distribution is close to equilibrium, and all macroscopic models predict it rather accurately. Full article

In numerical simulations of underwater explosions, inaccuracies in the parameters of the Jones–Wilkins–Lee (JWL) equation of state often result in significant deviations between predicted shock wave pressure peaks or bubble pulsation periods and experimental or empirical results. To achieve the precise forecasting of underwater explosion loads, a corrected method for adjusting the initial conditions of explosives is proposed. This method regulates explosion loads by correcting the initial density and initial internal energy per unit mass of the explosive, offering a straightforward implementation and easy extension to complex scenarios. In addition, the accuracy and feasibility of the proposed method were validated through comparisons with experimental data and empirical formulas from international studies. The numerical framework employs the Runge–Kutta Discontinuous Galerkin (RKDG) method to solve the one-dimensional Euler equations. The spatial discretization of the Euler domain is achieved using the discontinuous Galerkin (DG) method, while temporal discretization utilizes a third-order Runge–Kutta (RK) method. The results demonstrate that the proposed correction method effectively compensates for load discrepancies caused by inaccuracies in the JWL equation of state parameters. After correction, the maximum error in the shock wave pressure peak is reduced to less than 4.5%, and the maximum error in the bubble pulsation period remains below 1.9%. Full article

This study numerically explores the initiation of detonation through shock wave reflection and focusing on a 90-degree wedge in varying mixtures of hydrogen–air. The simulations were conducted using the ddtFoam code, an integral part of the OpenFOAM open-source Computational Fluid Dynamics (CFD) package of density-based code for solving the unsteady, compressible Navier–Stokes equations. The simulation results unveil three potential outcomes in the corner post-reflection: deflagrative ignition in the corner, deflagrative ignition with intermediate transient phases leading to a delayed transition to detonation in the trailing combustion zone close to the apex of the wedge, and ignition with an immediate transition to detonation, resulting in the formation of a detonation wave in the corner tip. In the experimental investigation, the transition velocity for the stoichiometric mixture stood at approximately 719 m/s. In contrast, the numerical simulation indicated a transition velocity of 664 m/s for the same stoichiometric mixture, reflecting a 5.5% decrease in velocity. Such an underestimation level of 5–8% by the simulation results was observed for mixtures of 25–45% H₂ in air. Full article

This article theoretically investigates the interaction of a normal shock wave in a flow with chemical reactions under high-temperature conditions. The main novelty of the work is that the thermal effect of chemical reactions is modeled as a function of the temperature. A modified Rankine–Hugoniot model for a shock wave in a flow with chemical reactions has been developed. It is shown that for an exothermic reaction the pressure jump increases with increasing Arrhenius numbers. This is due to the additional energy introduced into the flow as heat is released during the chemical reaction. For endothermic reactions, the opposite trend is observed. The change in the speed of the adiabatic gas flow as it passes through a normal shock wave depending on the type of chemical reaction is clarified. The study provides comparisons between the results of the analytical and numerical solutions of the modified Hugoniot adiabatic equations. Full article

Miscible gas flooding improves oil displacement through mass exchange between oil and gas phases. It is one of the most efficient enhanced oil recovery methods for intermediate density oil reservoirs. In this work, analytical solutions for saturation, concentration and pressure are derived for oil displacement by a partially miscible gas injection at a constant rate. The mathematical model considers two-phase, three-component fluid flow in a one-dimensional homogeneous reservoir initially saturated by a single oil phase. Phase saturations and component concentrations are described by a $2\times 2$ hyperbolic system of partial differential equations, which is solved by the method of characteristics. Once this Goursat–Riemann problem is solved, the pressure drop between two points in the porous media is obtained by the integration of Darcy’s law. The solution of this problem may present three different fluid regions depending on the rock–fluid parameters: a single-phase gas region near the injection point, followed by a two-phase region where mass transfer takes place and a single-phase oil region. We considered the single-phase gas and the two-phase gas/oil regions as incompressible, while the single-phase oil region may be incompressible or slightly compressible. The solutions derived in this work are applied for a specific set of rock and fluid properties. For this data set, the two-phase region displays rarefaction waves, shock waves and constant states. The pressure behavior depends on the physical model (incompressible, compressible and finite or infinite porous media). In all cases, the injection pressure is the result of the sum of two terms: one represents the effect of the mobility contrast between phases and the other represents the single-phase oil solution. The solutions obtained in this work are compared to an equivalent immiscible solution, which shows that the miscible displacement is more efficient. Full article

This paper describes a constitutive model for progressive damage in carbon fibre-reinforced composites (CFRPs), developed in the framework of thermodynamics and coupled with a vector equation of state. This made the constitutive model capable of modelling shock wave propagation within orthotropic materials. Damage is incorporated in the model by using reduction in the principal material stiffness based on the effective stress concept and the hypothesis of strain energy equivalence. Damage evolution was defined in terms of a modified Tuler–Bucher criteria. The constitutive model was implemented into Lawrence Livermore National Laboratory (LLNL) DYNA3D nonlinear hydrocode. Simulation results were validated against post-impact experimental data of spherical projectile impact on an aerospace-grade woven CFRP composite panel. Two plate thicknesses were considered and a range of impact velocities above the ballistic limit of the plates, ranging from 194 m/s to 1219 m/s. Other than for the size of the delamination zone in the minor material direction, the discrepancy between the experiments and numerical results for damage and delamination in the CFRP target plates was within 8%. Full article

In this study, the effects of pressure anisotropy and viscosity on the propagation of shock waves in spin-polarized degenerate quantum magnetoplasma are studied under the influence of the streaming energy of ion beams. The effects of different suitable plasma parameters on the shock wave’s potential profile are studied using the steady state solution of the Zakharov–Kuznetsov–Burgers (Z–K–B) equation, as well as the numerical simulation of the governing non-linear Z–K–B equation. First-order analysis of the non-linear wave propagation depicted a new beam-induced stable mode whose Mach number may be subsonic or supersonic depending on the anisotropic pressure combination in the presence of different spin density polarization ratios. This is the first observation of this new beam-induced stable mode in ion beam plasma, apart from the other existing modes of ion beam plasma systems, namely, the fast beam mode, the slow beam mode, the inherent ion acoustic mode, and the coupled mode, which also has unique propagation characteristics compared to the other modes. The spin density polarization ratio of spin-up and spin-down electrons have an unprecedented effect on the polarity and the direction of propagation of different shock wave modes in such plasma systems. Apart from the spin effect, anisotropic pressure combinations, as well as the viscosity of ions and ion beams, also play an outstanding role in controlling the nature of propagation of shock waves, especially in the newly detected beam-induced stable mode, and depending on the viscosity parameters of ions and ion beams, both oscillatory and monotonic shock waves can propagate in such plasma. Full article

Chaplygin gas magnetohydrodynamics Euler equations with active terms are often used to study physical phenomena in the universe, which is of great significance for exploring unknown fields. This article mainly studies the limited behavior of Riemann solutions of Chaplygin gas magnetohydrodynamics Euler equations with active terms using methods such as the characteristic line method and the Lax–Friedrichs method. In cases where only the magnetic field disappears, it was found, using the characteristic line method, that the solution converges to Chaplygin gas magnetohydrodynamics Euler equations with active terms. Additionally, we have identified the cause of the generation of $\delta$ shock waves. When pressure and magnetic induction disappear simultaneously, the reasons for the generation of $\delta$ shock waves and vacuum solutions are found. In the Discussion section, the Lax–Friedrichs method was used for numerical experiments to simulate the occurrence of phenomena such as mass concentration and vacuum. The innovation of this article lies in the construction of the Riemann problem for Chaplygin gas MHD Euler equations with active terms, as well as the study of the scenario where the magnetic field and pressure gradually weaken and approach zero. Finally, numerical experiments are used to verify the theoretical results. Compared with previous work, this article not only focuses on theoretical derivation, but also applies numerical simulation, especially simulating the characteristic lines of physical planes. This achievement provides a powerful tool for studying more complex Riemann problems. Full article