Asymptotic Predictions Research Articles

Effect of surface temperature strips on the evolution of supersonic and hypersonic Mack modes: Asymptotic theory and numerical results

The hypersonic boundary-layer transition is affected crucially by surface imperfections such as heating or cooling sources. In order to quantify this effect we develop an asymptotic theory, which not only reveals the interaction mechanism between oncoming instability modes and surface heating/cooling strips, but also provides quantitatively the change of the transition onset due to the scattering effect. The asymptotic predictions agree favorably with harmonic linearized Navier-Stokes calculations and direct numerical simulations.

Internal shear layers in librating spherical shells: the case of periodic characteristic paths

Internal shear layers generated by the longitudinal libration of the inner core in a spherical shell rotating at a rate $\varOmega ^*$ are analysed asymptotically and numerically. The forcing frequency is chosen as $\sqrt {2}\varOmega ^*$ such that the layers issued from the inner core at the critical latitude in the form of concentrated conical beams draw a simple rectangular pattern in meridional cross-sections. The asymptotic structure of the internal shear layers is described by extending the self-similar solution known for open domains to closed domains where reflections on the boundaries occur. The periodic ray path ensures that the beams remain localised around it. Asymptotic solutions for both the main beam along the critical line and the weaker secondary beam perpendicular to it are obtained. The asymptotic predictions are compared with direct numerical results obtained for Ekman numbers as low as $E=10^{-10}$ . The agreement between the asymptotic predictions and numerical results improves as the Ekman number decreases. The asymptotic scalings in $E^{1/12}$ and $E^{1/4}$ for the amplitudes of the main and secondary beams, respectively, are recovered numerically. Since the self-similar solution is singular on the axis, a new local asymptotic solution is derived close to the axis and is also validated numerically. This study demonstrates that, in the limit of vanishing Ekman numbers and for particular frequencies, the main features of the flow generated by a librating inner core are obtained by propagating through the spherical shell the self-similar solution generated by the singularity at the critical latitude on the inner core.

Asymptotic theory of gas entrainment in a two-phase Couette flow

A gas film can be entrained when a liquid is forced to rapidly wet a substrate. In this work, we investigate the behavior of a two-phase flow in a Couette geometry, in which one plate is stationary and a gas film is entrained over the moving plate. We present an asymptotic theory of the selection of gas film thickness, based on lubrication approximations. It is found that the gas film thickness relies primarily on the curvature of the meniscus, which is determined by the balance between the capillary force and gravity. The influence of the plate speed on the gas film thickness follows the classical 2/3 power law analogous to liquid deposition, with a mild correction from the gas–liquid viscosity ratio. The asymptotic predictions agree well with the exact solutions of the lubrication equation.

Asymptotic study of linear instability in a viscoelastic pipe flow

It is recently found that viscoelastic pipe flows can be linearly unstable, leading to the possibility of a supercritical transition route, in contrast to Newtonian pipe flows. Such an instability is referred to as the centre mode, which was studied numerically by Chaudharyet al.(J. Fluid Mech., vol. 908, 2021, p. A11) based on an Oldroyd-B model. In this paper, we are interested in expanding the parameter space investigated and exploring the asymptotic scalings related to this centre instability in the Oldroyd-B viscoelastic pipe flow. It is found from the asymptotic analysis that the centre mode exhibits a three-layered asymptotic structure in the radial direction, a wall layer, a main layer and a central layer, which are driven by pure viscosity, axial and/or radial pressure gradient, and a combined effect of viscosity and elasticity, respectively. Depending on the relations of the control parameters, two regimes, the long-wavelength and short-wavelength centre instabilities, emerge, for which the central-layer thicknesses are of different orders of magnitude. Our large-Reynolds-number asymptotic predictions are compared to the numerical solutions of the original eigenvalue system, and favourable agreement is achieved, especially when the parameters approach their individual limits. In addition to revealing the dominant factors and their balances, the asymptotic analysis describes the instability system by reducing the number of control parameters, and furthermore explaining the collapse of the numerical results for different re-scalings.

Multiple-scales analysis of wave evolution in the presence of rigid vegetation

The study of free-surface flows over vegetative structures presents a challenging setting for theoretical, computational and experimental analysis. In this work, we develop a multiple-scales asymptotic framework for the evolution of free-surface waves over rigid vegetation and a slowly varying substrate. The analysis quantifies the balance between the competing effects of vegetation and shoaling, and provides a prediction of the amplitude as the wave approaches a coastline. Our analysis unifies and extends existing theories that study these effects individually. The asymptotic predictions are shown to provide good agreement with full numerical simulations (varying depth) and published experimental results (constant depth).

Space-time statistics of a linear dynamical energy cascade model

<abstract><p>A linear dynamical model for the development of the turbulent energy cascade was introduced in Apolinário et al. (J. Stat. Phys., <bold>186</bold>, 15 (2022)). This partial differential equation, randomly stirred by a forcing term which is smooth in space and delta-correlated in time, was shown to converge at infinite time towards a state of finite variance, without the aid of viscosity. Furthermore, the spatial profile of its solution gets rough, with the same regularity as a fractional Gaussian field. We here focus on the temporal behavior and derive explicit asymptotic predictions for the correlation function in time of this solution and observe that their regularity is not influenced by the spatial regularity of the problem, only by the correlation in time of the stirring contribution. We also show that the correlation in time of the solution depends on the position, contrary to its correlation in space at fixed times. We then investigate the influence of a forcing which is correlated in time on the spatial and time statistics of this equation. In this situation, while for small correlation times the homogeneous spatial statistics of the white-in-time case are recovered, for large correlation times homogeneity is broken, and a concentration around the origin of the system is observed in the velocity profiles. In other words, this fractional velocity field is a representation in one-dimension, through a linear dynamical model, of the self-similar velocity fields proposed by Kolmogorov in 1941, but only at fixed times, for a delta-correlated forcing, in which case the spatial statistics is homogeneous and rough, as expected of a turbulent velocity field. The regularity in time of turbulence, however, is not captured by this model.</p></abstract>

EFFECT OF THE COULOMB INTERACTION TO THE PROCESSES OF INELASTIC DEUTERON SCATTERING ON NUCLEI AND THE STRUCTURE OF DEUTERON

The available experimental data on the interaction of polarized and unpolarized deuterons with a large deuteron momentum transfer require their theoretical description and interpretation. At the same time, further experimental studies require theoretical predictions in turn. In particular, there is a need to develop a theoretical model that makes it possible to smoothly connect available experimental data obtained at low-energy region, at which the nucleon-meson structure of the deuteron appears, with data obtained at high-energy region. The latter are related to the structure of the deuteron at small distances. Approaches based on the quark description of the deuteron structure are more adequate for their interpretation. The creation of such a model is an urgent task of modern nuclear physics, since it will allow one to reliably compare the asymptotic predictions of perturbative quantum chromodynamics and other quark approaches with the available experimental data. This will also make it possible to develop the theoretical approaches necessary for planning future experimental studies of elastic and inelastic deuteron scattering at high energies. In the framework of the Glauber–Sitenko model, the contribution of the Coulomb interaction to the reaction cross section A(d,p) at high energies and zero proton angle is calculated. The effect of taking into account the Coulomb interaction for the cross section in a wide inter-nuclear region is evaluated. The present calculations are compared with experiment. One fines that the Coulomb interaction gives a sizable contribution in the observables only at a peak, where the proton momentum is near half of the deuteron momentum in the lab. frame. But it does not change the results in the high momentum region, where quark effects should be taken into account.

The oriented swap process and last passage percolation

AbstractWe present new probabilistic and combinatorial identities relating three random processes: the oriented swap process (OSP) on n particles, the corner growth process, and the last passage percolation (LPP) model. We prove one of the probabilistic identities, relating a random vector of LPP times to its dual, using the duality between the Robinson–Schensted–Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of “last swap times” in the OSP, is conjectural. We give a computer‐assisted proof of this identity for after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman–Greene correspondence. The conjectural identity provides precise finite‐n and asymptotic predictions on the distribution of the absorbing time of the OSP, thus conditionally solving an open problem posed by Angel, Holroyd, and Romik.

One-dimensional acoustic waves in air/water mixtures

Homogenisation theory is used to derive Wood’s formula for the speed at which a pressure wave can propagate in an air/water mixture. The smallness of the impedance ratio between air and water results in a dramatic lowering of the wave speed of the mixture and, when the air fraction is small, the possibility of regions of persistent ringing in the vicinity of the boundary where the pressure is applied. Only Goupillaud media are considered and, for the periodic case, the asymptotic predictions are validated against exact numerical calculations. An example in which the air/water layers vary randomly is also presented.

Theory of Microphase Segregation in ABA Triblock Comb-Shaped Copolymers: Lamellar Mesophase

A theory describing lamellar mesophases formed in the melt of ABA triblock copolymers with strongly incompatible comblike blocks is developed using a combination of a strong-stretching self-consistent field (SS-SCF) analytical approach and numerical self-consistent field calculations based on the Scheutjens-Fleer (SF-SCF) methods. The structural and thermodynamic properties of the lamellae are analyzed as a function of their architectural parameters, i.e., grafting density and polymerization degree of the main and side chains in the comb-shaped blocks. In particular, we distinguish between the loops and bridges formed by the middle comb-shaped block and demonstrate how a fraction of the bridges connecting the neighboring AB interfaces of the lamellar layers and shear modulus of the mesophase are mediated by the architectural parameters of the comblike blocks. In particular, we predict an increase in the shear modulus upon replacement of the linear ABA triblock copolymers by comblike ones with the same composition. The asymptotic analytical predictions of the theory are complemented by the results of numerical modeling.

A review of the second normal-stress difference; its importance in various flows, measurement techniques, results for various complex fluids and theoretical predictions

Shear flow is ubiquitous. Not only is it arguably the most widely-used deformation type to characterise complex fluids in rheological studies but also, in practice, the deformation most likely to occur in the great majority of flows, e.g. involving fluid transport through pipes or conduits. In steady simple shear flow the rheological properties of a complex fluid are completely characterised in just three material functions; the variation with shear rate of the shear viscosity and the so-called first and second normal-stress differences. Despite requiring only three material functions to be completely characterised, most shear-flow rheological characterisations are usually restricted simply to the shear viscosity and, at best, the variation of the first normal-stress difference N1 with shear rate. The second normal-stress difference N2 remains very much neglected. For dilute polymer solutions where this quantity may be negligibly small in comparison to the first normal-stress difference, such neglect is justified but for a whole range of complex fluids – indeed even polymer solutions outside of the dilute regime and especially melts – it is not clear that N2 may be safely disregarded. Indeed, in this review article we spotlight a number of flows where second normal-stress differences are of importance and potentially major consequence. Following this attention, we review the many experimental techniques which have been proposed for its measurement and survey the available literature for measurements of this quantity for various complex fluids including the aforementioned polymeric solutions, melts, liquid crystals, dense non-Brownian suspensions (both with Newtonian and complex fluid bases), semi-dilute wormlike micellar fluids and magnetorheological fluids. Theoretical predictions for N2 from various commonly-used continuum constitutive equations – primarily from the polymer literature – are also given and their asymptotic predictions at low and high shear rates compared. Finally, we end with a brief summary and outlook.

An asymptotic theory of the roughness impact on inviscid Mack modes in supersonic/hypersonic boundary layers

Abstract

Transient Analysis of the Bias-Compensated LMS Algorithm

In most supervised adaptive filtering settings, only the additive noise of the reference signal is taken into account. However, in many practical situations the excitation data is also immersed in noise, which leads to a bias in the estimation procedure. In order to mitigate such issue, adaptive algorithms with bias compensation schemes have been proposed. This paper advances for the first time a stochastic model that predicts the average and mean-square learning behavior of the bias-compensated least mean squares algorithm in the transient region. Asymptotic predictions can also be obtained as a result of the devised analysis. Tracking capabilities and the impact of employing sub-optimal length adaptive filter are also considered, without restricting the input signal to be neither white nor Gaussian. Results indicate that the proposed analysis reveals accurate agreement with simulation results.

Diffusion in arrays of obstacles: beyond homogenization

We revisit the classical problem of diffusion of a scalar (or heat) released in a two-dimensional medium with an embedded periodic array of impermeable obstacles such as perforations. Homogenization theory provides a coarse-grained description of the scalar at large times and predicts that it diffuses with a certain effective diffusivity, so the concentration is approximately Gaussian. We improve on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the concentration through a rate function obtained by solving a family of eigenvalue problems. We focus on cylindrical obstacles and on the dense limit, when the obstacles occupy a large area fraction and non-Gaussianity is most marked. We derive an asymptotic approximation for the rate function in this limit, valid uniformly over a wide range of distances. We use finite-element implementations to solve the eigenvalue problems yielding the rate function for arbitrary obstacle area fractions and an elliptic boundary-value problem arising in the asymptotics calculation. Comparison between numerical results and asymptotic predictions confirms the validity of the latter.

Glassy dynamics of a binary Voronoi fluid: a mode-coupling analysis

The binary Voronoi mixture is a fluid model whose interactions are derived from the Voronoi–Laguerre tessellation of the configurations of the system. The resulting interactions are local and many-body. Here we perform molecular-dynamics (MD) simulations of an equimolar mixture that is weakly polydisperse and additive. For the first time we study the structural relaxation of this mixture in the supercooled-liquid regime. From the simulations we determine the time- and temperature-dependent coherent and incoherent scattering functions for a large range of wave vectors, as well as the mean-square displacements of both particle species. We perform a detailed analysis of the dynamics by comparing the MD results with the first-principles-based idealized mode-coupling theory (MCT). To this end, we employ two approaches: fits to the asymptotic predictions of the theory, and fit-parameter-free binary MCT calculations based on static-structure-factor input from the simulations. We find that many-body interactions of the Voronoi mixture do not lead to strong qualitative differences relative to similar analyses carried out for simple liquids with pair-wise interactions. For instance, the fits give an exponent parameter λ ≈ 0.746 comparable to typical values found for simple liquids, the wavevector dependence of the Kohlrausch relaxation time is in good qualitative agreement with literature results for polydisperse hard spheres, and the MCT calculations based on static input overestimate the critical temperature, albeit only by a factor of about 1.2. This overestimation appears to be weak relative to other well-studied supercooled-liquid models such as the binary Kob–Andersen Lennard-Jones mixture. Overall, the agreement between MCT and simulation suggests that it is possible to predict several microscopic dynamic properties with qualitative, and in some cases near-quantitative, accuracy based solely on static two-point structural correlations, even though the system itself is inherently governed by many-body interactions.

Small-Peclet-number approximation for stellar turbulent mixing zones.

The purpose of this work is to derive a small turbulent Péclet-small turbulent Mach number approximation for hydroradiative turbulent mixing zones encountered in stellar interiors where the radiative conductivity can overwhelms the turbulent transport. To this end, we proceed to an asymptotic analysis and determine orders of magnitude for the fluctuating temperature and pressure, as well as closed expressions for the fluctuating conduction and velocity divergence. The latter is used to extend a Reynolds stress model to the small-Péclet regime. Three-dimensional direct numerical simulations of radiative Rayleigh-Taylor turbulent mixing zones are performed, first, to validate the asymptotic predictions and, second, to validate their use in the Reynolds stress model.

Dispersion of inertial particles in cellular flows in the small-Stokes, large-Péclet regime

Abstract

A heat and mass transfer problem for the dissolution of an alumina particle in a cryolite bath

We investigate the spherically symmetric dissolution of an initially cold alumina particle in a bath of molten cryolite. The cryolite initially freezes on the particle, forming a shell that must melt before the particle can dissolve. We derive asymptotic solutions valid in the limits of small-superheat and of small Stefan number. In the small-superheat limit, the evolution of the boundary exhibits a two-scale behaviour. In the small Stefan number limit, we find that the behaviour of a particle could be limited by either the dissolution (in the case where the temperature differences are small) or by heat transfer (when the latent heat is large and the temperature gradients are large). Our asymptotic predictions are validated by a front fixing numerical scheme that we initiate using the early-time asymptotics.

On the thin-film asymptotics of surface tension driven microfluidics

Abstract

Nonequilibrium kinetics of excess defect generation and dynamic scaling in the Ising spin chain under slow cooling.

We investigate the relaxation dynamics of a one-dimensional Ising chain via Glauber kinetic Monte Carlo simulations, when the system is cooled slowly from infinite temperature to zero temperature with different cooling protocols. The main quantity of interest is the excess defect density that represents the total defect density minus the equilibrium defect density at varying temperatures. We find that, for three classes of cooling protocols, the time dependence of the excess defect density for various cooling speed shows a dynamic scaling behavior that largely encompasses the Kibble-Zurek mechanism as well as Krapivsky's calculation of the final defect density at zero temperature. We also find distinct features in the behavior of the dynamic scaling when the temperature approaches in a power-law fashion to zero temperature and the excess defect density reaches a peak at finite temperature, where the scaling of the excess defect density at its peak and that at zero temperature exhibits different asymptotic behavior due to different logarithmic corrections. This characteristic behavior can be attributed to the exponential divergence of the relaxation time near zero temperature. While the qualitative theoretical predictions by Krapivsky on the asymptotic exponents are confirmed, the asymptotic predictions for amplitudes overestimate by up to 40% the simulation results.