Karim Shariff

Toward Planetesimals: Dense Chondrule Clumps in the Protoplanetary Nebula

We outline a scenario that traces a direct path from freely floating nebula particles to the first 10-100 km sized bodies in the terrestrial planet region, producing planetesimals that have properties matching those of primitive meteorite parent bodies. We call this primary accretion. The scenario draws on elements of previous work and introduces a new critical threshold for planetesimal formation. We presume the nebula to be weakly turbulent, which leads to dense concentrations of aerodynamically size-sorted particles that have properties similar to those observed in chondrites. The fractional volume of the nebula occupied by these dense zones or clumps obeys a probability distribution as a function of their density, and the densest concentrations have particle mass densities that are 100 times that of the gas. However, even these densest clumps are prevented by gas pressure from undergoing gravitational instability in the traditional sense (on a dynamical timescale). While in this state of arrested development, they are susceptible to disruption by the ram pressure of the differentially orbiting nebula gas. However, self-gravity can preserve sufficiently large and dense clumps from ram pressure disruption, allowing their entrained particles to sediment gently but inexorably toward their centers, producing 10-100 km sandpile planetesimals. Localized radial pressure fluctuations in the nebula, as well as interactions between differentially moving dense clumps, will also play a role that must be accounted for in future studies. The scenario is readily extended from meteorite parent bodies to primary accretion throughout the solar system.

Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5

A direct numerical simulation of a supersonic turbulent boundary layer has been performed. We take advantage of a technique developed by Spalart for incompressible flow. In this technique, it is assumed that the boundary layer grows so slowly in the streamwise direction that the turbulence can be treated as approximately homogeneous in this direction. The slow growth is accounted for by a coordinate transformation and a multiple-scale analysis. The result is a modified system of equations, in which the flow is homogeneous in both the streamwise and spanwise directions, and which represents the state of the boundary layer at a given streamwise location. The equations are solved using a mixed Fourier and B-spline Galerkin method. Results are presented for a case having an adiabatic wall, a Mach number of M = 2.5, and a Reynolds number, based on momentum integral thickness and wall viscosity, of Re θ′ = 849. The Reynolds number based on momentum integral thickness and free-stream viscosity is Re θ = 1577. The results indicate that the Van Driest transformed velocity satisfies the incompressible scalings and a small logarithmic region is obtained. Both turbulence intensities and the Reynolds shear stress compare well with the incompressible simulations of Spalart when scaled by mean density. Pressure fluctuations are higher than in incompressible flow. Morkovins prediction that streamwise velocity and temperature fluctuations should be anti-correlated, which happens to be supported by compressible experiments, does not hold in the simulation. Instead, a relationship is found between the rates of turbulent heat and momentum transfer. The turbulent kinetic energy budget is computed and compared with the budgets from Spalarts incompressible simulations.

A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage

Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss & Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall & Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 – α 1 (β)/ Re s , for a large range of Re s . Here Re s is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then α 1 (β) varies weakly with core thickness β. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode ( n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n . The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted π/ n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial modes having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.

Instabilities of two-dimensional inviscid compressible vortices

The linear stability and subsequent nonlinear evolution and acoustic radiation of a planar inviscid compressible vortex is examined. Linear-stability analysis shows that vortices with smoother vorticity profiles than the Rankine vortex considered by Broadbent & Moore (1979) are also unstable. However, only neutrally stable waves are found for a Gaussian vorticity profile. The effects of entropy gradient are investigated and for the particular entropy profile chosen, positive average entropy gradient in the vortex core is destabilizing while the opposite is true for negative average entropy gradient. The linear initial-value problem is studied by finite-difference methods. It is found that these methods are capable of accurately computing the frequencies and weak growth rates of the normal modes. When the initial condition consists of random perturbations, the long-time behaviour is found to correspond to the most unstable normal mode in all cases. In particular, the Gaussian vortex has no algebraically growing modes. This procedure also reveals the existence of weakly decaying and neutrally stable waves rotating in the direction opposite to the vortex core, which were not observed previously. The nonlinear development of an elliptic-mode perturbation is studied by numerical solution of the Euler equations. The vortex elongates and forms shocklets; eventually, the core splits into two corotating vortices. The individual vortices then gradually move away from each other while their rate of rotation about their mid-point slowly decreases. The acoustic flux reaches a maximum at the time of fission and decreases as the vortices move apart.

A numerical experiment to determine whether surface shear-stress fluctuations are a true sound source

The sound generated due to a localized flow over a large (compared to the acoustic wavelength) plane no-slip wall is considered. It has been known since 1960 that for inviscid flow the pressure, while appearing to be a source of dipole sound in a formal solution to the Lighthill equation, is, in fact, not a true dipole source, but rather represents the surface reflection of volume quadrupoles. The subject of the present work—namely, whether a similar surface shear stress term constitutes a true source of dipole sound—has been controversial. Some have boldly assumed it to be a true source and have used it to calculate the noise in boundary-layer flows. Others have argued that, like the surface pressure, the surface shear stress is not a valid source of sound but rather represents a propagation effect. Here, a numerical experiment is undertaken to investigate the issue. A portion of an otherwise static wall is oscillated tangentially in an acoustically compact region to create shear stress fluctuations. The...

GRAVITATIONAL INSTABILITY OF SOLIDS ASSISTED BY GAS DRAG: SLOWING BY TURBULENT MASS DIFFUSIVITY

The Goldreich & Ward (axisymmetric) gravitational instability of a razor thin particle layer occurs when the Toomre parameter Q T ≡ c pΩ0/πGΣp 1, sufficiently long waves were always unstable. Youdin carried out a detailed analysis and showed that the instability allows chondrule-sized (~1 mm) particles to undergo radial clumping with reasonable growth times even in the presence of a moderate amount of turbulent stirring. The analysis of Youdin includes the role of turbulence in setting the thickness of the dust layer and in creating a turbulent particle pressure in the momentum equation. However, he ignores the effect of turbulent mass diffusivity on the disturbance wave. Here, we show that including this effect reduces the growth rate significantly, by an amount that depends on the level of turbulence, and reduces the maximum intensity of turbulence the instability can withstand by 1-3 orders of magnitude. The instability is viable only when turbulence is extremely weak and the solid to gas surface density of the particle layer is considerably enhanced over minimum-mass-nebula values. A simple mechanistic explanation of the instability shows how the azimuthal component of drag promotes instability while the radial component hinders it. A gravito-diffusive overstability is also possible but never realized in the nebula models.

Dynamical systems analysis of fluid transport in time-periodic vortex ring flows

It is known that the stable and unstable manifolds of dynamical systems theory provide a powerful tool for understanding Lagrangian aspects of time-periodic flows. In this work we consider two time-periodic vortex ring flows. The first is a vortex ring with an elliptical core. The manifolds provide information about entrainment and detrainment of irrotational fluid into and out of the volume transported with the ring. The likeness of the manifolds with features observed in flow visualization experiments of turbulent vortex rings suggests that a similar process might be at play. However, what precise modes of unsteadiness are responsible for stirring in a turbulent vortex ring is left as an open question. The second situation is that of two leapfrogging rings. The unstable manifold shows striking agreement with even the fine features of smoke visualization photographs, suggesting that fluid elements in the vicinity of the manifold are drawn out along it and begin to reveal its structure. We suggest that interpretations of these photographs that argue for complex vorticity dynamics ought to be reconsidered. Recently, theoretical and computational tools have been developed to locate structures analogous to stable and unstable manifolds in aperiodic, or finite-time systems. The usefulness of these analogs is demonstrated, using vortex ring flows as an example, in the paper by Shadden, Dabiri, and Marsden [Phys. Fluids 18, 047105 (2006)].

B-spline Methods in Fluid Dynamics

Basis splines (B-splines) are basis functions for piecewise polynomials having a high level of derivative continuity. They possess attractive properties for complex flow simulations: they have compact support, provide a straightforward handling of boundary conditions and grid nonuniformities, yield numerical schemes with high resolving power, and the order of accuracy is a mere input parameter. This paper reviews progress made in the development and application of B-spline numerical methods to computational fluid dynamics. Basic approximation properties of B-spline schemes are discussed, and their relationship with conventional numerical methods is reviewed. Some fundamental developments towards spline methods in complex geometries are covered. These include local interpolation methods, fast solution algorithms on Cartesian grids, block-structured discretization and compatible pressure bases for the Navier-Stokes equations. Finally, application of some of these techniques to the computation of viscous incompressible flows is presented.

Numerical Study of Wind-Tunnel Sidewall Effects on Circulation Control Airfoil Flows

Two- and three-dimensional numerical simulations are performed of the flow around a circulation control airfoil (using a Coanda jet blowing over a rounded trailing edge) placed in a rectangular wind-tunnel test section. The airfoil model spans the entire tunnel and the span-to-chord ratio of the model is 3.26. The objective of this numerical study, in which we solve the compressible Reynolds-averaged Navier―Stokes equations in a time-resolved manner (but the solutions eventually converge to steady states), is to investigate the physical mechanisms of wind-tunnel sidewall effects on the flow, especially in the midspan region. The three-dimensional simulations predict that the Coanda jet flow is quasi-two-dimensional until the flow separates from the trailing edge of the airfoil; however, the spanwise ends of this Coanda jet sheet then three-dimensionally roll up on the side walls of the wind tunnel to form two large streamwise vortices downstream. Careful comparisons between the two- and three-dimensional simulations reveal that the wind-tunnel stream goes below the airfoil more in the three-dimensional cases than in the two-dimensional cases due to the presence of these two streamwise vortices downstream. This results in smaller lift and larger drag being produced at the midspan of the airfoil in the three-dimensional cases than in the two-dimensional cases.

A contour dynamics algorithm for axisymmetric flow

The method of contour dynamics, developed for two-dimensional vortex patches by Zabusky et al. [N.J. Zabusky, M.H. Hughes, K.V. Roberts, Contour dynamics for the Euler equations in two-dimensions, J. Comp. Phys. 30 (1979) 96-106] is extended to vortex rings in which the vorticity distribution varies linearly with normal distance from the symmetry axis. The method tracks the motion of the boundaries of the vorticity regions and hence reduces the dimensionality of the problem by one. We discuss the formulation and implementation of the scheme, verify its accuracy and convergence, and present illustrative examples.