Drag Regime Research Articles

Dusty gas with smoothed particle hydrodynamics - II. Implicit timestepping and astrophysical drag regimes

In a companion paper (Laibe & Price 2011b), we have presented an algorithm for simulating two-fluid gas and dust mixtures in Smoothed Particle Hydrodynamics (SPH). In this paper, we develop an implicit timestepping method that preserves the exact conservation of the both linear and angular momentum in the underlying SPH algorithm, but unlike previous schemes, allows the iterations to converge to arbitrary accuracy and is suited to the treatment of non- linear drag regimes. The algorithm presented in Paper I is also extended to deal with realistic astrophysical drag regimes, including both linear and non-linear Epstein and Stokes drag. The scheme is benchmarked against the test suite presented in Paper I, including i) the analytic solutions of the dustybox problem and ii) solutions of the dustywave, dustyshock, dustysedov and dustydisc obtained with explicit timestepping. We find that the implicit method is 1- 10 times faster than the explicit temporal integration when the ratio r between the the timestep and the drag stopping time is 1 < r < 1000.

Dusty gas with smoothed particle hydrodynamics - I. Algorithm and test suite

We present a new algorithm for simulating two-fluid gas and dust mixtures in Smoothed Particle Hydrodynamics (SPH), systematically addressing a number of key issues including the generalised SPH density estimate in multi-fluid systems, the consistent treatment of variable smoothing length terms, finite particle size, time step stability, thermal coupling terms and the choice of kernel and smoothing length used in the drag operator. We find that using double-hump shaped kernels improves the accuracy of the drag interpolation by a factor of several hundred compared to the use of standard SPH bell-shaped kernels, at no additional computational expense. In order to benchmark our algorithm, we have developed a comprehensive suite of standardised, simple test problems for gas and dust mixtures: dustybox, dustywave, dustyshock, dustysedov and dustydisc, the first three of which have known analytic solutions. We present the validation of our algorithm against all of these tests. In doing so, we show that the spatial resolution criterion \Delta < cs ts is a necessary condition in all gas+dust codes that becomes critical at high drag (i.e. small stopping time ts) in order to correctly predict the dynamics. Implicit timestepping and the implementation of realistic astrophysical drag regimes are addressed in a companion paper.

High strain-rate plastic flow in Al and Fe

Thin Fe and Al foils were ramp-compressed over several to tens of ns timescales to study the time-dependence associated with the onset of plastic flow. Peak stress states of 15–200 GPa were achieved through laser ramp-compression where the strain rate was varied, shot-to-shot, between 106 to 108 s−1. Our data combined with data from other dynamic compression platforms reveals a strong correlation between the peak elastic precursor stress, σE, and the strain rate at the onset of plastic flow, ɛ·p. In fcc Al, phonon drag dislocation flow dominates above ɛ·p~103s-1 and σE ∼ 0.03 GPa where σE scales as ɛ·p0.43. By contrast, the Al alloy 6061-T6 exhibits a relatively weak dependency of σE with ɛ·p up to strain rates of ∼107 s−1. Our Fe data, reveals a sharp increase in σE at ɛ·p&amp;gt;5×106s-1. This is consistent with a transition in plastic flow to a phonon drag regime.

Edge dislocation mobilities in bcc Fe obtained by molecular dynamics

In the traditional picture of plasticity in bcc metals, edge dislocations have been assumed to play a minor role due to their high mobility with respect to screw dislocations, which then control plastic flow. $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}{110}$ edge dislocations indeed fit this description, as it has been shown by way of numerous atomistic simulations. However, $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}{112}$ edge dislocations have been comparatively much less studied. The recent discovery of a possible regime where they move slowly via thermally activated kink-pair nucleation may have implications in the plastic behavior of bcc materials. Because dislocation mobilities are very difficult to measure experimentally, in this paper, we provide comprehensive mobility laws for both types of edge dislocations as a function of temperature and stress using molecular dynamics simulations. Our results confirm the existence of clearly delimited thermally activated and phonon drag dynamic regimes for $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}{112}$ edge dislocations and of a single viscous drag regime for their $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}{110}$ counterparts. We also provide an analysis to relate the difference in mobility to the dislocation core properties. Our fitted mobility laws may be used in dislocation dynamics simulations of plastic flow involving millions of segments.

R modes and mutual friction in rapidly rotating superfluid neutron stars

We develop a new perturbative framework for studying the r-modes of rotating superfluid neutron stars. Our analysis accounts for the centrifugal deformation of the star, and considers the two-fluid dynamics at linear order in the perturbed velocities. Our main focus is on a simple model system where the total density profile is that of an $n=1$ polytrope. We derive a partially analytic solution for the superfluid analogue of the classical r-mode. This solution is used to analyse the relevance of the vortex mediated mutual friction damping, confirming that this dissipation mechanism is unlikely to suppress the gravitational-wave driven instability in rapidly spinning superfluid neutron stars. Our calculation of the superfluid r-modes is significantly simpler than previous approaches, because it decouples the r-mode from all other inertial modes of the system. This leads to the results being clearer, but it also means that we cannot comment on the relevance of potential avoided crossings (and associated "resonances") that may occur for particular parameter values. Our analysis of the mutual friction damping differs from previous studies in two important ways. Firstly, we incorporate realistic pairing gaps which means that the regions of superfluidity in the star's core vary with temperature. Secondly, we allow the mutual friction parameters to take the whole range of permissible values rather than focussing on a particular mechanism. Thus, we consider not only the weak drag regime, but also the strong drag regime where the fluid dynamics is significantly different.

Oscillations of dissipative superfluid neutron stars

We investigate the oscillations of slowly rotating superfluid stars, taking into account the vortex-mediated mutual friction force that is expected to be the main damping mechanism in mature neutron star cores. Working to linear order in the rotation of the star, we consider both the fundamental $f$-modes and the inertial $r$-modes. In the case of the (polar) $f$-modes, we work out an analytic approximation of the mode which allows us to write down a closed expression for the mutual friction damping time scale. The analytic result is in good agreement with previous numerical results obtained using an energy integral argument. We extend previous work by considering the full range of permissible values for the vortex drag, e.g. the friction between each individual vortex and the electron fluid. This leads to the first ever results for the $f$-mode in the strong drag regime. Our estimates provide useful insight into the dependence on, and relevance of, various equation of state parameters. In the case of the (axial) $r$-modes, we confirm the existence of two classes of modes. However, we demonstrate that only one of these sets remains purely axial in more realistic neutron star models. Our analysis lays the foundation for companion studies of the mutual friction damping of the $r$-modes at second order in the slow-rotation approximation, the first time evolutions for superfluid neutron star perturbations and also the first detailed attempt at studying the dynamics of superfluid neutron stars with both a relative rotation between the components and mutual friction.

Giant fluctuations of Coulomb drag

We have observed reproducible fluctuations of the Coulomb drag resistivity, originating from coherent scattering of electrons in two layers. The fluctuations are observed as functions of both the carrier density of each layer, and perpendicular magnetic field. The magnitude of the fluctuations is much larger than expected from the theory of ‘diffusive’ drag and the temperature dependence is stronger than theoretically predicted. We account for this enhancement by considering the ‘ballistic’ nature of the drag in our system. We also present results on the fluctuations at large magnetic fields, where the lowest Landau level is half filled so that coherent drag occurs between composite fermions. The magnitude of fluctuations is seen to be greatly enhanced compared with the small field case, although fluctuations of composite fermion drag show much better agreement with the theory developed for the ‘diffusive’ drag regime.

Luttinger liquids with curvature: Density correlations and Coulomb drag effect

We consider the effect of the curvature in fermionic dispersion on the observable properties of Luttinger liquid (LL). We use the bosonization technique where the curvature is irrelevant perturbation, describing the decay of LL bosons (plasmon modes). When possible, we establish the correspondence between the bosonization and the fermionic approach. We analyze modifications in density correlation functions due to curvature at finite temperatures, T. The most important application of our approach is the analysis of the Coulomb drag by small momentum transfer between two LL, which is only possible due to curvature. Analyzing the a.c. transconductivity in the one-dimensional drag setup, we confirm the results by Pustilnik et al. for T-dependence of drag resistivity, R_{12} ~ T^2 at high and R_{12} ~ T^5 at low temperatures. The bosonization allows for treating both intra- and inter-wire electron-electron interactions in all orders, and we calculate exact prefactors in low-T drag regime. The crossover temperature between the two regimes is T_1 ~ E_F \Delta, with \Delta relative difference in plasmon velocities. We show that \Delta \neq 0 even for identical wires, due to lifting of degeneracy by interwire interaction, U_{12}, leading to crossover from R_{12} ~ U_{12}^2 T^2 to R_{12} \~ T^5/U_{12} at T ~ U_{12}.

Deformation behavior of an ultrafine-grained Al–Mg alloy produced by equal-channel angular pressing

Presented here is the deformation behavior of Al–1.5 wt% Mg alloy severely plastically deformed to equivalent pre-strains of 8 and 13 using the equal-channel angular pressing technique. The average subgrain size after severe plastic deformation was 280 and 230 nm respectively. Strain rate change and stress relaxation tests in the range 10 −4–10 −2 s −1 and 298–523 K were performed. The strain rate sensitivity of ultrafine-grained (UFG) Al–1.5Mg was enhanced and the peak strain rate sensitivity shifted to lower temperatures as compared with the coarse-grained (CG) alloy. The increased strain rate sensitivity is a direct consequence of the reduced activation volume. The increase in pre-strain from 8 to 13 has a small effect on both the microstructural refinement and the subsequent deformation behavior. With increasing temperature the UFG material softens compared with the CG material. This demarcation has been clearly shown on a strain rate by temperature plot. Refinement of grain results in an enhanced solute drag regime, primarily due to the decreased activation energy of diffusion.

Flow Patterns and Drag in Near-Critical Flow over Isolated Orography

Abstract Since early manned space flight orographically forced cloud patterns have been described in terms of the single isolated shock structure of shallow-water flow or, equivalently, compressible fluid flow. Some of these observations show, behind an initial “bow wave,” a series of almost parallel wave crests. This paper considers the simplest extension of shallow-water theory that retains not only nonlinear steepening of waves but includes departures from hydrostatic balance, and thus wave dispersion, showing that the single shocks of shallow-water theory are transformed into multiple parallel finite-amplitude wave crests. The context of the discussion is the forced Kadomtsev–Petviashvili equation from classical ship wave dynamics, which plays the same role in two-dimensional near-critical fluid flow as the more familiar Korteweg–de Vries equation in one-dimensional flow. The drag and flow regimes in near-critical flow over isolated orography are described in terms of the three governing parameters of the flow: the deviation of the flow speed from critical, the strength of nonhydrostatic effects, and the strength of orographic forcing.

Dislocation motion in BCC metals by molecular dynamics

We performed molecular dynamics (MD) simulation of edge dislocations in BCC metal Mo with 1/2〈1 1 1〉 Burgers vector gliding on (1 1 0) plane using the Finnis–Sinclair N-body empirical potential and periodic boundary conditions (PBC). The profile of dislocation line, extracted from atomic displacements during MD simulation, suggests a dominant mechanism of double-kink nucleation, without appreciable kink migration. This observation is consistent with further simulations in which dislocations with pre-existing kinks are observed to move at the same velocity as the initially straight dislocations. Our results show a linear stress dependence of the dislocation velocity and a decrease of mobility with increasing temperature, both features interpreted as signifying that the simulations pertain to the viscous phonon drag regime of dislocation mobility.

Experimental Verification of Two New Theories Predicting Temperature-Dependent Viscosity Effects on the Forced Convection in a Porous Channel

Experimental Verification of Two New Theories Predicting Temperature-Dependent Viscosity Effects on the Forced Convection in a Porous Channel

The Effect of Rotation and Surface Friction on Orographic Drag

Abstract A numerical, hydrostatic model is used to investigate the form and magnitude of the pressure drag created by 3D elliptical mountains of various heights (h) and aspect ratios (R) in flows characterized by uniform upstream velocity (U) and stability (N). Three series of simulations, corresponding to increasing degrees of realism, are performed: (i) without rotation and surface friction; (ii) with rotation, but no surface friction; (iii) with rotation and surface friction. For the simulations with rotation, the Coriolis parameter has a typical midlatitude value and the upstream flow is geostrophically balanced. The surface friction is introduced by the use of a typical roughness length. For low values of the nondimensional height (Nh/U), the pressure drag is reduced by the effect of rotation, in agreement with well-known results of linear theory. This seems to be valid until Nh/U ∼ 1.4, that is, in the high drag regime. On the other hand, for large values of Nh/U, that is, in the blocked flow regime...

The Brownian Motion of Dust Particles in the Solar Nebula: An Experimental Approach to the Problem of Pre-planetary Dust Aggregation

Laboratory experiments were performed to study the Brownian motion of μm-sized dust grains and small aggregates under pre-planetary nebula conditions, i.e., in a thin gas atmosphere (Epstein drag regime), in the ballistic limit, and under microgravity conditions. The results of these experiments, i.e., the grain diffusivities, are in quantitative agreement with theoretical predictions for single spherical grains. Deviations from particle sphericity, i.e., in our case aggregates consisting of monodisperse spherical grains, cause only minor deviations between the Epstein drag formula for spheres and our experimental results for equal particle cross section. Thus, we find a quantitative agreement of our measurements with the Epstein relationD∝ 1/σabetween grain diffusivity and geometrical (aerodynamic) cross section.The results of our investigations can be used for the calculation of the gas–grain stopping time τf= ϵ (m/σa) (1/ρgvm) which, in turn, is an important grain characteristic for the calculation of pre-planetary dust aggregation. Here,mand σaare the mass and the aerodynamic, i.e., geometric cross section of the grain, ρgandvmare the mass density of the gas and the mean thermal velocity of the gas molecules, and ϵ is a proportionality factor which we determined to be ϵ = 0.68 ± 0.10. The gas–grain stopping time describes the strength of grain coupling to a given gas motion and its value determines relative velocties and, hence, collision frequencies between dust grains due to sedimentation, drag-induced orbital decay, and gas turbulence in the solar nebula.

New equations for the terminal velocity of precipitation droplets in a standard atmosphere

New equations are presented which give the terminal velocity of water droplets in three physically distinct drag regimes explicitly as functions of equivalent drop radius and height in a standard atmosphere. It is shown that at h = 0 , the expressions give results which agree with Gunn and Kinzer's terminal velocity measurements with an overall divergence less than the measurement uncertainty of the data. Results are also compared with the semi-empirical formulations of Beard and shown to be in good agreement at all drop sizes 0.25 r0 3500 μm and heights -1.5 h 5 km in the atmosphere.

Dynamic restoration mechanisms in Al-5.8 At. Pct Mg deformed to large strains in the solute drag regime

An Al-5.8 at. pct Mg (5.2 wt pct Mg) alloy was deformed in torsion within the solute drag regime to various strains, up to the failure strain of 10.8. Optical microscopy (OM) and transmission electron microscopy (TEM) were used to analyze the evolution of the microstructure and to determine the dynamic restoration mechanism. Transmission electron microscopy revealed that subgrain formation is sluggish but that subgrains eventually (e ≈ 1) fill the grains. The “steady-state” subgrain size (λ ≈ 6 μm) and misorientation angle (θ ≈ 1.6 deg) are reached by e ≈ 2. These observations confirm that subgrains eventually form during deformation in the solute drag regime, though they do not appear to significantly influence the strength. At low strains, nearly all of the boundaries form by dislocation reaction and are low angle (θ < 10 deg). At a strain of 10.8, however, the boundary misorientation histogram is bimodal, with nearly 25 pct of the boundaries having high angles due to their ancestry in the original grain boundaries. This is consistent with OM observations of the elongation and thinning of the original grains as they spiral around the torsion axis. No evidence was found fordiscontinuous dynamic recrystallization, a repeating process in which strain-free grains nucleate, grow, deform, and give rise to new nuclei. It is concluded that dynamic recovery in the solute drag regime gives rise togeometric dynamic recrystallization in a manner very similar to that already established for pure aluminum, suggesting that geometric dynamic recrystallization may occur generally in materials with a high stacking-fault energy (SFE) deformed to large strains.

An Experimental Investigation of Forces Induced on Cylinders by Random Oscillatory Flow

Results are presented from experiments investigating hydrodynamic forces induced on fixed circular cylinders by random oscillatory flows. The oscillatory flow was generated in a water tunnel and the inline and transverse forces induced on the cylinders were measured simultaneously. Analysis of the measured forces shows that significant differences exist when comparing drag and inertia coefficients to those measured under sinusoidal flow conditions, particularly in the inertia/ drag regime of the Keulegan-Carpenter number. The comparison of results for random versus sinusoidal flow conditions generated in the same experimental apparatus quantifies the difference in the physical mechanisms, specifically the vortex shedding and pairing, which influence the induced forces. A study is also made of the dependence of measured force power spectral densities on the statistical Keulegan-Carpenter number, Reynold’s number and on the spectral width of the flow velocity auto-power spectrum.

Quantitative examination of the thermoelectric power of n-type Si in the phonon drag regime

A quantitative comparison between theory and experiment is presented for the thermoelectric power of n-type Si at low temperatures, where crystal boundary scattering of phonons is predominant. The expression for the Seebeck coefficient is obtained by simultaneously solving the coupled Boltzmann equations for electrons and phonons. The thermoelectric power is measured from samples of two different donor concentrations (ND=8×1014 cm−3 and 7×1015 cm−3) at temperatures ranging from 25 to 200 K, where the thickness is varied by etching the samples (370–73 μm). The phonon mean free path for crystal boundary scattering, determined from experimental thermopower, agrees well with that gained from the theory of thermal conductivity corrected for the phonon velocities of interest. An additional measurement of the thermal conductivity yields the mean free path for phonon–phonon scattering, which is about two orders of magnitude less than that obtained from thermopower. This difference confirms the influence of the different parts of the phonon spectrum involved in the calculation of the appropriate transport parameters.

Pulsating Downslope Windstorms

Abstract The flow configurations that obtain in several severe downslope windstorm events generated over isolated topography are studied using a two-dimensional nonlinear anelastic model. A new high resolution simulation of the 11 January 1972 windstorm in Boulder, Colorado, constructed using a very large model domain, is shown to qualitatively reproduce the strong, quasi-periodic, 5–15 min transience in surface wind speed to the lee of the topography that was actually observed during this event. It is demonstrated that this transience is caused by the continuous generation of strong pulses of enhanced surface wind on the lee slope, which thereafter propagate downstream with individually constant speeds. An identical phenomenon is shown to be characteristic of the high drag regime in severe downslope windstorms simulated in flows characterized by upstream profiles having constant wind and stability. This newly discovered pulsation phenomenon is therefore a generic property of flows induced by the breaking...

Turbulence in multiphase flow

We establish in this paper the foundations of a two-field turbulent flow model that includes two turbulent fields. The case of dispersed particles in an incompressible carrier fluid is treated here, but the very presence of these two fields allows for the generalization of the model to the instability-induced turbulent mixing of two materials. This model describes both cases of turbulent mass diffusion and small drag regime, “wave-like” interpenetration of the two components. It also includes the damping of the turbulence due to the presence of the particles. In addition, a theoretical derivation of the drag-induced decay of the large-scale turbulence kinetic energy is proposed as another mechanism specific to turbulent multiphase flow.