One-dimensional Hydrodynamic Equations Research Articles

Fundamental influence of irreversible stress–strain properties in solids on the validity of the ramp loading method

The widely used quasi-isentropic ramp loading technique relies heavily on back-calculation methods that convert the measured free-surface velocity profiles to the stress–density states inside the compressed sample. Existing back-calculation methods are based on one-dimensional isentropic hydrodynamic equations, which assume a well-defined functional relationship P(ρ) between the longitudinal stress and density throughout the entire flow field. However, this kind of idealized stress–density relation does not hold in general, because of the complexities introduced by structural phase transitions and/or elastic–plastic response. How and to what extent these standard back-calculation methods may be affected by such inherent complexities is still an unsettled question. Here, we present a close examination using large-scale molecular dynamics (MD) simulations that include the detailed physics of the irreversibly compressed solid samples. We back-calculate the stress–density relation from the MD-simulated rear surface velocity profiles and compare it directly against the stress–density trajectories measured from the MD simulation itself. Deviations exist in the cases studied here, and these turn out to be related to the irreversibility between compression and release. Rarefaction and compression waves are observed to propagate with different sound velocities in some parts of the flow field, violating the basic assumption of isentropic hydrodynamic models and thus leading to systematic back-calculation errors. In particular, the step-like feature of the P(ρ) curve corresponding to phase transition may be completely missed owing to these errors. This kind of mismatch between inherent properties of matter and the basic assumptions of isentropic hydrodynamics has a fundamental influence on how the ramp loading method can be applied.

Symmetry and scaling in one-dimensional compressible two-phase flow

Investigations of shock compression of heterogeneous materials often focus on the shock front width and overall profile. The number of experiments required to fully characterize the dynamic response of a material often belie the structure–property relationships governing these aspects of a shock wave. Recent observations measured a pronounced shock-front width on the order of 10 s of ns in particulate composites. Here, we focus on particulate composites with disparate densities and investigate whether the mechanical interactions between the phases are adequate to describe this emergent behavior. The analysis proceeds with a general Mie–Grüneisen equation of state for the matrix material, a general drag force law with general power-law scaling for the particle-matrix coupling of the phases, and a volume fraction-dependent viscosity. Lie group analysis is applied to one-dimensional hydrodynamic flow equations for the self-consistent interaction of particles embedded in a matrix material. The particle phase is characterized by a particle size and volume fraction. The Lie group analysis results in self-similar solutions reflecting the symmetries of the flow. The symmetries lead to well-defined scaling laws, which may be used to characterize the propagation of shock waves in particle composites. An example of the derived scaling laws for shock attenuation and rise time is shown for experimental data on shock-driven tungsten-loaded polymers. A key result of the Lie analysis is that there is a relationship between the exponents characterizing the form of the drag force and the exponent characterizing the shock velocity and its attenuation in a particulate composite. Comparison to recent experiments results in a single exponent that corresponds to a conventional drag force.

Modulational instability of ion-acoustic waves in a dense quantum plasma

Modulational instability of ion-acoustic (IA) waves is investigated in an unmagnetized quantum plasma by adopting an equation of state for Fermi electron gas. To this end, a basic set of one-dimensional quantum hydrodynamic equations is reduced to a nonlinear Schrödinger equation using the reductive perturbation method. Quantum corrections are described using a nondimensional quantum parameter H proportional to the quantum diffraction parameter. Based on normalized electron temperature and chemical potential, new quantum criteria of modulational instability for IA waves are obtained and analyzed. Moreover, formation of rogue waves due to the modulational instability of IA waves and propagation of modulated IA packets in the form of bright and dark envelope solitons are examined for two different types of astrophysical plasma regions: white dwarf stars and warm dense matter.

Numerical investigation of the depressurization performance in blowdown pipe using method of characteristics

The coolant leaks quickly after the break of the high-pressure pipeline system. The rapid pressure decrease and fluctuation will further affect the integrity of the pressure boundary and the effective heat removal of the reactor core in the early phase of blowdown. Generally, numerical simulation plays a vital role in the safety analysis of nuclear power plants (NPPs) under severe accidents like loss of coolant accident (LOCA) conditions. In this paper, one-dimensional two-phase hydrodynamic equations and essential closure models are established to describe the internal pipe flow firstly. Then, method of characteristics (MOC) realized by a self-programming code is introduced, which is different from the numerical scheme of system analysis code RELAP5 in the simplification and discretization. Finally, combined with the Edwards blowdown experiments, the two numerical solution methods are compared on modeling, sensitivity analysis, and calculation results under various working conditions. The influence of back pressure variation on the solution of thermal–hydraulic parameters has been considered during the blowdown process. The solution result of RELAP5 is highly related to the set discharge coefficients, which are usually given according to experience. However, the solution result of MOC can be enveloped by that of RELAP5 with different discharge coefficients. In addition, the results solved by MOC are closer and more conservative to the experimental values during the early phase of blowdown with low void fraction fluid. It lays a foundation for the further calculation and safety analysis of the complex loop system under LCOA conditions.

Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves

The recently proposed map [5] between the hydrodynamic equations governing the two-dimensional triangular cold-bosonic breathers [1] and the high-density zero-temperature triangular free-fermionic clouds, both trapped harmonically, perfectly explains the former phenomenon but leaves uninterpreted the nature of the initial (t=0) singularity. This singularity is a density discontinuity that leads, in the bosonic case, to an infinite force at the cloud edge. The map itself becomes invalid at times t<0t<0. A similar singularity appears at t = T/4t=T/4, where T is the period of the harmonic trap, with the Fermi-Bose map becoming invalid at t > T/4t>T/4. Here, we first map—using the scale invariance of the problem—the trapped motion to an untrapped one. Then we show that in the new representation, the solution [5] becomes, along a ray in the direction normal to one of the three edges of the initial cloud, a freely propagating one-dimensional shock wave of a class proposed by Damski in [7]. There, for a broad class of initial conditions, the one-dimensional hydrodynamic equations can be mapped to the inviscid Burgers’ equation, which is equivalent to a nonlinear transport equation. More specifically, under the Damski map, the t=0 singularity of the original problem becomes, verbatim, the initial condition for the wave catastrophe solution found by Chandrasekhar in 1943 [9]. At t=T/8t=T/8, our interpretation ceases to exist: at this instance, all three effectively one-dimensional shock waves emanating from each of the three sides of the initial triangle collide at the origin, and the 2D-1D correspondence between the solution of [5] and the Damski-Chandrasekhar shock wave becomes invalid.

Features of the application of one-dimensional equations of hydrodynamics for free-flow flows

The possibility of applying one-dimensional hydrodynamic equations of a nonstationary flow averaged over the cross-section of the channel during mathematical modeling of long waves-abruptly changing the movement of the water flow in channels is substantiated in the article. The characteristic dimensions of the length are much larger than the depth of the flow-the theory of shallow water. With this averaging, it is necessary to apply some hypotheses, the most important of which is the hypothesis of the distribution of pressure over the depth of the flow

Group-invariant solutions for one dimensional, inviscid hydrodynamics

In this paper, the results of the Lie group method carried out by Ovsiannikov are utilized to study the one-dimensional hydrodynamic equations governing compressible, inviscid fluid flow in the absence of heat conduction. One-parameter subgroups of the admissible R-parameter Lie group of point transformations of the system are applied to reduce the first-order, non-linear system of partial differential equations (PDE)s to a first-order system of ordinary differential equations (ODE)s. Closed-form solutions to the reduced ODE systems are subsequently determined using a linear velocity profile ansatz. These solutions are valid in one-dimensional (1D) planar, cylindrical and spherical geometries and are connected to solutions of the governing system of PDEs through an inverse map. The linear velocity type solutions were first considered by Sedov in 1953 and constitute a subclass of all possible similarity solutions of the compressible hydrodynamics equations. They further serve as illustrative examples of using solutions obtained for the reduced ODEs to find solutions of the associated system of PDEs. Consequently, the reduced systems of ODEs are provided in their entirety inviting additional solutions to be determined via alternative explicit or numerical solution techniques.

Testing analytical tidal propagation models of the one-dimensional hydrodynamic equations in Morocco's estuaries

ABSTRACTIn this paper, we tested two analytical models for tidal propagation in two estuaries, i.e. Sebou and Loukkos (Morocco): (1) is a hybrid analytical model, based on new approximation of the friction term in the momentum equation, tidal damping expression and river discharge effect; and (2) is a analytical model for the closed channel, i.e. with a closed landward boundary. For given topography, friction, and tidal amplitude at the seaward boundary, the velocity amplitude, the wave celerity, the tidal damping, and the phase lag along the estuaries axis are calculated. These models were calibrated and validated using observed data. A good-fit between models and measurements is obtained, as verified by statistical performance tests, which suggests that the predictive models can be used to obtain a first-order estimation of tidal dynamics and reflected wave along the estuaries axis.

Consistent initial conditions for the Saint-Venant equations in river network modeling

Abstract. Initial conditions for flows and depths (cross-sectional areas) throughout a river network are required for any time-marching (unsteady) solution of the one-dimensional (1-D) hydrodynamic Saint-Venant equations. For a river network modeled with several Strahler orders of tributaries, comprehensive and consistent synoptic data are typically lacking and synthetic starting conditions are needed. Because of underlying nonlinearity, poorly defined or inconsistent initial conditions can lead to convergence problems and long spin-up times in an unsteady solver. Two new approaches are defined and demonstrated herein for computing flows and cross-sectional areas (or depths). These methods can produce an initial condition data set that is consistent with modeled landscape runoff and river geometry boundary conditions at the initial time. These new methods are (1) the pseudo time-marching method (PTM) that iterates toward a steady-state initial condition using an unsteady Saint-Venant solver and (2) the steady-solution method (SSM) that makes use of graph theory for initial flow rates and solution of a steady-state 1-D momentum equation for the channel cross-sectional areas. The PTM is shown to be adequate for short river reaches but is significantly slower and has occasional non-convergent behavior for large river networks. The SSM approach is shown to provide a rapid solution of consistent initial conditions for both small and large networks, albeit with the requirement that additional code must be written rather than applying an existing unsteady Saint-Venant solver.

Self-similar solutions and converging shocks at the stellar surfaces

In this paper, Lie group of transformations method is used to investigate the self-similar solutions for the system of one-dimensional hydrodynamic equations describing the post-shock flow at the stellar surface. Necessary conditions for the existence of similarity solutions for strong shocks are obtained. The similarity exponent using Lie group of transformations and characteristic rule are obtained. Collapses of imploding cylindrical and spherical shocks are worked out in detail.

Solar Filament Longitudinal Oscillations along a Magnetic Field Tube with Two Dips

Abstract Large-amplitude longitudinal oscillations of solar filaments have been observed and explored for more than ten years. Previous studies are mainly based on the one-dimensional rigid flux tube model with a single magnetic dip. However, it has been noted that there might be two magnetic dips, and hence two threads, along one magnetic field line. Following previous work, we intend to investigate the kinematics of the filament longitudinal oscillations when two threads are magnetically connected, which is done by solving one-dimensional radiative hydrodynamic equations with the numerical code MPI-AMRVAC. Two different types of perturbations are considered, and the difference from previous works resulting from the interaction of the two filament threads is investigated. We find that even with the inclusion of the thread–thread interaction, the oscillation period is modified weakly, by at most 20% compared to the traditional pendulum model with one thread. However, the damping timescale is significantly affected by the thread–thread interaction. Hence, we should take it into account when applying the consistent seismology to the filaments where two threads are magnetically connected.

Terahertz Small-Signal Response of Field-Effect Transistor Channels

In this paper, we present an analytical approach for the study of the small-signal response of nanometric field-effect transistor at terahertz frequencies. One-dimensional hydrodynamic equations coupled with a pseudo-two-dimensional Poisson equation are derived at first-order to obtain the small-signal admittance matrix describing the transistor. This matrix is then used to model the loaded device and establish its voltage amplification. The admittances and voltage amplification spectra exhibit sharp resonances associated with the plasma modes of the channel. The influences of the device geometry, operating temperature and connected load are investigated.

First and second sound in a two-dimensional harmonically trapped Bose gas across the Berezinskii–Kosterlitz–Thouless transition

We theoretically investigate first and second sound of a two-dimensional (2D) atomic Bose gas in harmonic traps by solving Landau’s two-fluid hydrodynamic equations. For an isotropic trap, we find that first and second sound modes become degenerate at certain temperatures and exhibit typical avoided crossings in mode frequencies. At these temperatures, second sound has significant density fluctuation due to its hybridization with first sound and has a divergent mode frequency towards the Berezinskii–Kosterlitz–Thouless (BKT) transition. For a highly anisotropic trap, we derive the simplified one-dimensional hydrodynamic equations and discuss the sound-wave propagation along the weakly confined direction. Due to the universal jump of the superfluid density inherent to the BKT transition, we show that the first sound velocity exhibits a kink across the transition. These predictions might be readily examined in current experimental setups for 2D dilute Bose gases with a sufficiently large number of atoms, where the finite-size effect due to harmonic traps is relatively weak.

First and second sound of a unitary Fermi gas in highly elongated harmonic traps

Using a variational approach, we present the full solutions of the simplified one-dimensional two-fluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonic potential, which was recently derived by G. Bertaina and coworkers [Phys. Rev. Lett. 105, 150402 (2010)]. We calculate the discretized mode frequencies of first and second sound along the weak axial trapping potential, as a function of the temperature and the form of superfluid density. We show that the density fluctuations in second-sound modes, due to their coupling to first-sound modes, are large enough to be measured in current experimental setups such as that exploited by M. K. Tey et al. at the University of Innsbruck [Phys. Rev. Lett. 110, 055303 (2013)]. Owing to the sensitivity of second sounds on the form of superfluid density, the high precision of the measured second-sound frequencies may provide us a promising way to accurately determine the superfluid density of a unitary Fermi gas, which so far has remained elusive.

The Investigation of Plasma Produced by Intense Nanosecond Laser Ablation in Vacuum Under External Magnetic Field Using a Two-Stage Model

In this paper a two-stage physics-based model has been applied to study the evolution of plasma produced by high-intensity nanosecond laser ablation in vacuum under external magnetic field. In the early stage (Stage I), the laser-induced plasma generation and its short-term evolution are described through one-dimensional (1D) hydrodynamic equations. An equation of state (EOS) that can cover the density and temperature range in the whole physical domain has been applied to supplement the hydrodynamic equations. In the later stage (Stage II), the plasma long-term evolution is simulated by solving 2D gas dynamic equations. The two-stage model can predict the spatial distributions and temporal evolutions of plasma temperature, density, velocity, and other parameters. The model is used to study and discuss the effects of external magnetic field on the plasma evolution. It provides a useful tool for related fundamental studies and practical applications.

Quantum ion-acoustic solitary waves: The effect of exchange correlation

Quantum ion-acoustic solitary waves are studied by considering the effects of exchange and correlation for the electrons. Starting from one-dimensional quantum hydrodynamic equations, including the term of exchange correlation for electrons, we obtain a model in which two dimensionless parameters appear (in addition to the parameter measuring the quantum diffraction) measuring the exchange and the correlation. A new deformed Korteweg-de Vries equation is derived. The effect of exchange and correlation is reflected in the phase speed as well as in the nonlinear and dispersion terms. Its solution shows that the exchange-correlation effects modify the amplitude as well as the width of the weak solitary waves. In the arbitrary amplitude regime, and as may be expected, a pseudopotential analysis shows that the exchange-correlation effects may change the nature (compressive or rarefactive) of the quantum ion-acoustic solitary waves. Our results complement and give new insight into the previously published work on this problem.

Long-range dynamic correlations in a compressible viscous fluid confined in a circular tube

A set of simplified one-dimensional hydrodynamic equations proposed for the description of transient small amplitude flow of a compressible viscous fluid in a linear channel is tested by comparison with exact results for a circular tube obtained from the linearized Navier-Stokes equations. There is a qualitative agreement between the approximate and exact results, but for the simplified calculation the amplitude of the algebraic long-time tail is a factor of 2 too small.

Modeling and experimental verification of plasmas induced by high-power nanosecond laser-aluminum interactions in air

It has been generally believed in literature that in nanosecond laser ablation, the condensed substrate phase contributes mass to the plasma plume through surface evaporation across the sharp interface between the condensed phase and the vapor or plasma phase. However, this will not be true when laser intensity is sufficiently high. In this case, the target temperature can be greater than the critical temperature, so that the sharp interface between the condensed and gaseous phases disappears and is smeared into a macroscopic transition layer. The substrate should contribute mass to the plasma region mainly through hydrodynamic expansion instead of surface evaporation. Based on this physical mechanism, a numerical model has been developed by solving the one-dimensional hydrodynamic equations over the entire physical domain supplemented by wide-range equations of state. It has been found that model predictions have good agreements with experimental measurement for plasma front location, temperature, and electron number density. This has provided further evidence (at least in the indirect sense), besides the above theoretical analysis, that for nanosecond laser metal ablation in air at sufficiently high intensity, the dominant physical mechanism for mass transfer from the condensed phase to the plasma plume is hydrodynamic expansion instead of surface evaporation. The developed and verified numerical model provides useful means for the investigation of nanosecond laser-induced plasma at high intensities.

A simple two-stage model for the formation and expansion of the plasma induced by high intensity nanosecond laser metal ablation in vacuum

A simple two-stage model for the formation and expansion of the plasma induced by high intensity nanosecond laser metal ablation in vacuum has been developed. To simulate the first stage of plasma formation and initial expansion within the laser pulse, the one-dimensional hydrodynamic equations are supplemented by a wide-range equation of state and solved numerically. For the second stage of plasma expansion after the end of laser pulse, a three-dimensional semi-analytical model is applied, the input of which comes from the output of the model in the first stage. In this way, a self-closed calculation is realized. This two-stage model is easy to apply and has very small computational cost. Although simplified and approximate, its predictions, however, are reasonably accurate, which has been shown by the good agreements between simulations and experimental measurements for the plasma front location, temperature and electron number density.

A one-dimensional hydrodynamic model for pressures induced near the coating-water interface during laser shock peening

In laser shock peening (LSP) under a water-confinement regime, laser-matter interaction near the coating-water interface can induce very high pressures in the order of gigapascal, which can impart compressive residual stresses into metal workpieces to improve fatigue and corrosion properties. However, self-closed models with spatial distribution considerations for the induced pressures near the coating-water interface in LSP are rarely reported in literature. In this paper, a self-closed model is developed by numerically solving the one-dimensional hydrodynamic equations, supplemented with appropriate equations of state of water and the coating material. The model can produce the one-dimensional spatial distributions of the material responses near the water-coating interface in LSP. The model-predicted pressures have been compared with experimental measurements under a variety of conditions typical for LSP, and good agreements have been found for both the transient pressure history and the peak pressure magnitude.