Nathaniel S. Barlow

Mayer Sampling Monte Carlo calculation of virial coefficients on graphics processors

Virial coefficients for various molecular models are calculated up to B 8 using the Mayer Sampling Monte Carlo method and implemented on a graphics processing unit (GPU). The execution time and performance of these calculations is compared with equivalent computations done on a CPU. The speedup between virial coefficient computations on a CPU (w/optimized C code) and a GPU (w/CUDA) is roughly two orders of magnitude. We report values of B 6, B 7, and B 8 of the Lennard-Jones (LJ) model, as computed on the GPU, for temperatures T = 0.6 to 40 (in LJ units).

Transience to instability in a liquid sheet

Series solutions are found which describe the evolution to absolute and convective instability in an inviscid liquid sheet flowing in a quiescent ambient gas and subject to a localized perturbation. These solutions are used to validate asymptotic stability predictions for sinuous and varicose disturbances. We show how recent disagreements in growth predictions stem from assumptions made when arriving at the Fourier integral response. Certain initial conditions eliminate or reduce the order of singularities in the Fourier integral. If a Gaussian perturbation is applied to both the position and velocity of a sheet when the Weber number is less than one, we observe absolutely unstable sinuous waves which grow like t 1/3 .I f only the position is perturbed, we find that the sheet is stable and decays like t −2/3 at the origin. Furthermore, if both the position and velocity of a sheet are perturbed in the absence of ambient gas, we observe a new phenomenon in which sinuous waves neither grow nor decay and varicose waves grow like t 1/2 with a convective instability. A brief history of the fluid dynamics of liquid sheets and its applications is given in Lin & Jiang (2003). Here, we mention only the work that is directly relevant to this study. As shown in the analysis of Rayleigh (1896), there are two linearly independent wave modes of a liquid sheet. The sinuous mode moves the two free surfaces of a sheet in phase. The varicose mode symmetrically moves the free surfaces in opposite directions. These modes were later confirmed in the experiments of Taylor (1959). The onset of wave instability was analysed by Squire (1953) through the use of classical temporal stability theory. The classical theory predicts, for finite Q = ρg/ρl (ρg and ρl being, respectively, the gas and liquid densities), that the sinuous wave is only unstable if the Weber number is greater than one. The Weber number is defined as We = ρlU 2 h0/S, where U is the liquid velocity in the sheet, h0 is the half-sheet thickness and S is the interfacial tension. The experiments of Brown (1961) indicated instability for We <1, which seemed to contradict the classical theory. Around this time, a new stability theory was being developed to study the complex roots of a dispersion relationship and to take into consideration the possibility of both spatial and temporal growth (Sturrock 1958). Sturrock introduced a method for determining the nature of spatio-temporal growth, and established that the mapping between frequency and wavenumber in their corresponding complex

Communication: Analytic continuation of the virial series through the critical point using parametric approximants

The mathematical structure imposed by the thermodynamic critical point motivates an approximant that synthesizes two theoretically sound equations of state: the parametric and the virial. The former is constructed to describe the critical region, incorporating all scaling laws; the latter is an expansion about zero density, developed from molecular considerations. The approximant is shown to yield an equation of state capable of accurately describing properties over a large portion of the thermodynamic parameter space, far greater than that covered by each treatment alone.

An asymptotically consistent approximant for the equatorial bending angle of light due to Kerr black holes

An accurate closed-form expression is provided to predict the bending angle of light as a function of impact parameter for equatorial orbits around Kerr black holes of arbitrary spin. This expression is constructed by assuring that the weak- and strong-deflection limits are explicitly satisfied while maintaining accuracy at intermediate values of impact parameter via the method of asymptotic approximants (Barlow et al, 2016 Q. J. Mech. Appl. Math., doi=10.1093/qjmam/hbw014). To this end, the strong deflection limit for a prograde orbit around an extremal black hole is examined, and the full non-vanishing asymptotic behavior is determined. The derived approximant may be an attractive alternative to computationally expensive elliptical integrals used in black hole simulations.

Accurate closed-form trajectories of light around a Kerr black hole using asymptotic approximants

Highly accurate closed-form expressions that describe the full trajectory of photons propagating in the equatorial plane of a Kerr black hole are obtained using asymptotic approximants. This work extends a prior study of the overall bending angle for photons (Barlow, et al. 2017, Class. Quantum Grav., 34, 135017). The expressions obtained provide accurate trajectory predictions for arbitrary spin and impact parameters, and provide significant time advantages compared with numerical evaluation of the elliptic integrals that describe photon trajectories. To construct approximants, asymptotic expansions for photon deflection are required in various limits. To this end, complete expansions are derived for the azimuthal angle as a function of radial distance from the black hole in the far-distance and closest-approach (pericenter) limits, and new coefficients are reported for the bending angle in the weak-field limit (large impact parameter).

Visualizing the wake of aquatic swimmers

Fish-like swimming is fascinating not only for its fundamental scientific value but also for engineering biomimetically inspired vehicles. Discovering physical principles behind the evolution of different aquatic swimmers can drastically improve the design of such vehicles. We are interested in the evolution of different caudal fin profiles (shape) because it is hypothesized that most of the thrust force is generated by the caudal fin. In fact, the caudal fin shape varies from hemocercal in mackerel to almost trapezoidal in trout and heterocercal in sharks. We investigate if such shape differences have hydrodynamic implications using numerical simulations. The equations governing the fluid motion are solved in the non-inertial reference frame attached to the fish center of mass (COM) via the curvilinear/immersed boundary method (CURVIB), which is capable of carrying out direct numerical simulation of flows with complex moving boundaries. The motion of the fish body is prescribed based on carangiform kinematics while the motion of the COM is calculated based on the fluid forces on the fish body through the fluid-structure interaction algorithm of Borazjani et. al. (2008) [3]. The reader is referred to Borazjani & Sotiropoulos (2010) [4] for the details of the method. For self-propelled simulations, the virtual swimmers start to undulate in an initially stagnant fluid and the swimming speed is determined based on the forces on the fish body. Therefore, physical parameters based on the swimming velocity change as the swimmer accelerates until the quasi-steady state is reached. The computational domain and time step for the self-propelled fish body simulations in the free stream is a cuboid with dimensions 2LxLx7L, which is discretized with 5.5 million grid nodes. The domain width 2L and height L are more than ten times the mackerel width 0.2L and height 0.1L, respectively. The fish is placed 1.5L from the inlet plane in the axial direction and centered in the transverse and the vertical directions. The simulations are partly run on our in-house computing cluster, Nami, with a total of 448 computing cores distributed across 28 nodes, each node containing a 2x8 Magny-Cours core (AMD 2.0 GHz). The memory available is 2GB RAM/core, 896GB total and the nodes are connected through QDR Infiniband. Some of the simulations were run on the dual-quad core nodes in the u2 cluster at CCR; these are also connected through QDR Infiniband. The simulations generate velocity field data in VTK format [5], allowing one to apply ParaViews tetrahedralize algorithm [2] to the 5.5 million point data set. The result is shown in Figure 1 for a swimming mackerel, where volume rendered points are colored by the magnitude of the velocity field. The domain has been truncated in the vicinity of the fish and an appropriate colormap has been chosen to emphasis dynamics local to the fish. For each time-step and viewing angle, the tetrahedralize algorithm is applied. A single frame takes (at least) 10 minutes to render on an Intel dual-quad core node (w/ 24GB RAM). An animation of 95 frames was generated in a batch job using off-screen rendering. The animation can be downloaded at [1].