Emily S. C. Ching

Beyond all orders: Singular perturbations in a mapping

SummaryWe consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parameterε. Asε → 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for smallε the heteroclinic connection breaks up and that the splitting between its components scales withε likeεγexp[-β/ε]. We estimateβ using the singularities of theε → 0+ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior of the functions in the complex plane, are supported by our numerical calculations.

Energy dependence of impact fragmentation of long glass rods

We have performed an experimental study of impact fragmentation with a focus on the dependence on the energy input. Long glass rods were dropped horizontally onto the ground from seven different heights. We find that the energy dependence is better characterized by studying the differential mass distribution rather than the cumulative mass distribution. For lower dropping heights, the differential mass distribution is well approximated by one power law while for higher heights, it has to be represented by two power laws. Moreover, the power-law exponent for small mass fragments increases and approaches an asymptotic value as the dropping height is increased.

Thermal boundary layer equation for turbulent Rayleigh-Bénard convection

We report a new thermal boundary layer equation for turbulent Rayleigh-Bénard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr≳1 and (b) Pr≫1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol), respectively.

Statistics of the locally averaged thermal dissipation rate in turbulent Rayleigh–Bénard convection

From the measured thermal dissipation rate in turbulent Rayleigh–Benard convection in a cylindrical cell, we construct a locally averaged thermal dissipation rate χ fτ by averaging over a time interval τ. We study how the statistical moments ⟨(χ fτ) p ⟩ depend on τ at various locations along the vertical axis of the convection cell. We find that ⟨(χ fτ) p ⟩ exhibits good scaling in τ, of about a decade long, with scaling exponents μ(p) for p = 1–6. For Rayleigh number (Ra) around 8×109, the scaling range is 1.4–21 s at the cell center and 4–21 s at the bottom plate. The dissipative and turnover times are about 0.8 s and 35 s respectively, while the timescale corresponding to the local Bolgiano scale is estimated to be about 31 s at the cell center and 3.5 s at the bottom plate. On the basis of several assumptions, we derive theoretical predictions for μ(p) at the different locations. The measured values of μ(p) are presented and shown to be in good agreement with our theoretical predictions.

Locally averaged thermal dissipation rate in turbulent thermal convection: A decomposition into contributions from different temperature gradient components

temperature gradient components in the x, y, and z directions and systematically study their statistics and scaling properties. It is found that the moments of i r,t exhibit good scaling in , i.e., i p i p , for all three components and for p up to 6. The obtained exponents i p at three representative locations in the convection cell are explained by a phenomenological model, which combines the effects of velocity statistics and geometric shape of the most dissipative structures in turbulent convection.

The break-up of a heteroclinic connection in a volume preserving mapping

Abstract We consider a family of three-dimensional, volume preserving maps depending on a small parameter e. As e→0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for small e the heteroclinic connection breaks up and that the splitting between its components scales with e like eγ exp(-β/e). We estimate β using the singularities of the e→0+ heteroclinic orbit in the complex plane. We then estimate γ using linearization about orbits in the complex plane. While these estimates are not proven, they are well supported by our numerical calculations. The work described here is a special case of the theory derived by Amick et al. which applies to q-dimensional volume preserving mappings.

Additive equivalence in turbulent drag reduction by flexible and rodlike polymers.

We address the additive equivalence discovered by Virk and co-workers: drag reduction affected by flexible and rigid rodlike polymers added to turbulent wall-bounded flows is limited from above by a very similar maximum drag reduction (MDR) asymptote. Considering the equations of motion of rodlike polymers in wall-bounded turbulent ensembles, we show that although the microscopic mechanism of attaining the MDR is very different, the macroscopic theory is isomorphic, rationalizing the interesting experimental observations.

Passive scalar conditional statistics in a model of random advection

We study numerically a model of random advection of a passive scalar by an incompressible velocity field of different prescribed statistics. Our focus is on the conditional statistics of the passive scalar and specifically on two conditional averages: the averages of the time derivative squared and the second time derivative of the scalar when its fluctuation is at a given value. We find that these two conditional averages can be quite well approximated by polynomials whose coefficients can be expressed in terms of scalar moments and correlations of the scalar with its time derivatives. With the fitted polynomials for the conditional averages, analytical forms for the probability density function (pdf) of the scalar are obtained. The variation of the coefficients with the parameters of the model result in a change in the pdf. Three different kinds of velocity statistics, (i) Gaussian, (ii) exponential, and (iii) triangular, are studied, and the same qualitative results are found demonstrating that the one...

Energy dependence of mass distributions in fragmentation

We study fragmentation numerically using a simple model in which an object is taken to be a set of particles that interact pairwisely via a Lennard–Jones potential while the effect of the fragmentation-induced forces is represented by some initial velocities assigned to the particles. The motion of the particles, which is given by Newtons laws, is followed by molecular dynamics calculations. As time evolves, the particles form clusters which are identified as fragments. The steady-state cumulative distribution of the fragment masses is studied and found to have an effective power-law region. The power-law exponent increases with the energy given to the particles by the fragmentation-induced forces. This result is confirmed by experiments.

Reconstructing links in directed networks from noisy dynamics.

We address the long-standing challenge of how to reconstruct links in directed networks from measurements, and present a general method that makes use of a noise-induced relation between network structure and both the time-lagged covariance of measurements taken at two different times and the covariance of measurements taken at the same time. When the coupling functions have certain additional properties, we can further reconstruct the weights of the links.