Finite-sized Domain Research Articles

Theory and simulations of linear and nonlinear two-dimensional Rayleigh–Taylor dynamics with variable acceleration

Interfacial Rayleigh–Taylor mixing is crucial to describing important natural and engineering processes, such as exploding supernovae, laser micromachining, hot spots in inertial confinement fusion, and optical telecommunications. These require the characterization of the time dependence of the driving acceleration. We compare our theoretical formulation based on group theory foundations with interface-capturing numerical simulations for linear and nonlinear two-dimensional Rayleigh–Taylor instabilities in a finite-sized domain with time-varying acceleration over broad ranges of Atwood numbers and acceleration exponents. Detailed corroboration between theory and simulations is provided for this foundational case. Both demonstrate the strong interfacial nature of Rayleigh–Taylor instabilities, which suggests that practical flow fields can be reconstructed from the derived fluid potential using the proposed theory. A robust agreement is also obtained for the early and late-time evolution of the amplitudes of the bubble and spike, which demonstrate that the Rayleigh–Taylor flow can transition to the mixing regime even for a single-mode initial perturbation. Corroboration with experiments of high energy density plasmas motivated by studies of supernovae is also achieved. In addition, a long-standing puzzle in Rayleigh–Taylor dynamics on the interplay between the acceleration, the shear, and the interface morphology in the theory and simulations is resolved by accounting for finite viscosity of the fluids. The characterization of Rayleigh–Taylor instabilities as a highly interfacial phenomenon provides valuable insight into its multiscale nature, which enhances the design and understanding of numerous processes of practical interest.

On modelling of thermodynamic interaction of particles suspended in two-dimensional medium

Analytical description of temperature distribution in a medium with foreign inclusions is difficult due to the complicated geometry of the problem, so asymptotic and numerical methods are usually used to model thermodynamic processes in heterogeneous media. To be convinced in convergence of these methods the authors consider model problem about two identical round particles in infinite planar medium with temperature gradient which is constant at infinity. Authors refine multipole expansion of the solution obtained earlier by continuing it up to higher powers of small parameter, that is nondimensional radius of thermodynamically interacting particles. Numerical approach to the problem using ANSYS software is described; in particular, appropriate choice of approximate boundary conditions is discussed. Authors ascertain that replacement of infinite medium by finite-sized domain is important source of error in FEM. To find domain boundaries in multiple inclusions’ problem the authors develop “fictituous particle” method; according to it the cloud of particles far from the center of the cloud acts approximately as a single equivalent particle of greater size and so may be replaced by it. Basing on particular quantitative data the dependence of domain size that provides acceptable accuracy on thermal conductivities of medium and of particles is explored. Authors establish series of numerical experiments confirming convergence of multipole expansions method and FEM as well; proximity of their results is illustrated, too.

Scale-dependent Rayleigh–Taylor dynamics with variable acceleration in a finite-sized domain for three-dimensional flows

Three-dimensional Rayleigh–Taylor instability (RTI) with the time-varying acceleration in a finite domain is investigated in a systematic framework. The acceleration magnitude follows a power law in time with an exponent greater than −2. Applying the group theory, the instabilities are demonstrated considering the irreducible representations for observable periodic structures with a square symmetry in the plane normal to the acceleration. We derive the dynamical system and illustrate the universal form of the solutions in the linear and nonlinear regimes. The scale-dependent dynamics are shown to be single scale and multiscale in the two regimes, respectively. For the nonlinear regime solutions, fundamental scales are derived bridging the solutions in the finite- and infinite-sized domains. Special solutions for bubbles and spikes are identified from a one-parameter family of solutions. The effect of domain confinement is that the velocity and curvature decreases and shear increases as the domain size is reduced. The theory provides predictions for the flow field and demonstrates the interfacial behavior of RTI. Our results are in good agreement with the prior studies and also provide new benchmarks for experiments and simulations.

Interactive roles of geometrical distribution and geomechanical deformation of fracture networks in fluid flow through fractured geological media

In this study, the combined effects of geometrical distribution and geomechanical deformation of fracture networks on fluid flow through fractured geological media are investigated numerically. We consider a finite-sized model domain in which the geometry of fracture systems follows a power-law length scaling. The geomechanical response of the fractured rock is simulated using a hybrid finite-discrete element model, which can capture the deformation of intact rocks, the interaction of matrix blocks, the displacement of discrete fractures and the propagation of new cracks. Under far-field stress loading, the locally variable stress distribution in the fractured rock leads to a stress-dependent variable aperture field controlled by compression-induced closure and shear-induced dilatancy of rough fractures. The equivalent permeability of the deformed fractured rock is calculated by solving for the fracture-matrix flow considering the cubic relationship between fracture aperture and flow rate at each local fracture segment. We report that the geometrical connectivity of fracture networks plays a critical role in the hydromechanical processes in fractured rocks. A well-connected fracture system under a high stress ratio condition exhibits intense frictional sliding and large fracture dilation/opening, leading to greater rock mass permeability. However, a disconnected fracture network accommodates much less fracture shearing and opening, and has much lower bulk permeability. We further propose an analytical solution for the relationship between the equivalent permeability of fractured rocks and the connectivity metric (i.e. percolation parameter) of fracture networks, which yields an excellent match to the numerical results. We infer that fluid flow through a well-connected system is governed by traversing channels (forming an “in parallel” architecture) and thus equivalent permeability is sensitive to stress loading (due to stress-dependent fracture permeability), whilst fluid flow through a disconnected system is more ruled by matrix (linking isolated clusters “in series”) and has much less stress dependency.

Early- and late-time evolution of Rayleigh–Taylor instability in a finite-sized domain by means of group theory analysis

We have developed a theoretical analysis to systematically study the early and late-time evolution of the Rayleigh–Taylor instability in a finite-sized spatial domain. The nonlinear dynamics of fluids with similar and contrasting densities are considered for two-dimensional flows driven by sustained acceleration. The flows are periodic in the plane normal to the direction of acceleration and have no external mass sources. Group theory analysis is applied to accurately account for the mode coupling. Asymptotic linear and nonlinear solutions are found to describe the interfacial dynamics far from and near the boundaries. The influence of the size of the domain on the diagnostic parameters of the flow is identified. In particular, it is shown that in a finite-sized domain the flow is slower compared to the spatially extended case. The direct link between the multiplicity of solutions and the interfacial shear is explored. It is suggested that the interfacial shear function acts as a natural parameter to the family of nonlinear asymptotic solutions.

Mathematical computational grains for direct numerical simulations of nanocomposites with a large number of nano-inclusions, using parallel computations

In this study, computational grains (CGs) are developed for micromechanical modelling of heterogeneous materials with nanoscale inhomogeneities, considering the interface stress effect. Each two-dimensional CG, which is a virtual or mathematically defined finite-sized geometrical domain of a polygonal shape, can include a circular elastic nano inclusion. In the present model, along the outer-boundary of each CG an inter-CG compatible displacement field is assumed, while independent Trefftz trial functions are assumed as displacement fields inside the matrix and the inclusion within each CG. Complex potentials scaled by characteristic lengths are used to derive the Trefftz trial displacement fields in the matrix as well as the inclusion. The stress jump across the matrix/inclusion interface is described by the generalized Young–Laplace equation, which is enforced in a weak sense by Lagrange multipliers in a newly-developed boundary-only-type multi-field boundary variational principle. A parallel algorithm is introduced to further accelerate the computation when modelling an RVE containing a large number of nano-inclusions. Numerical examples for problems of a single, multiple, and a large number of nanoscale inhomogeneities are given to demonstrate the validity and the power of the currently developed CG model for nanomechanics.

Finite Element Modeling of Fringe Fields in Wedge Diffraction Problem

The finite element method is applied to the modeling of fringe currents and fields in a diffraction problem, where a perfectly conducting wedge is illuminated by a line source. A spatial superposition approach is employed to compute the fringe currents. The locally conformal perfectly matched layer approach is used to truncate the infinitely long conducting structure in a finite-sized domain. MATLAB codes are developed, and some numerical examples are demonstrated. The results are compared to those of the physical theory of diffraction and the method of moments.

Non-steady-state heat conduction in composite walls.

The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but neither temperature nor heat flux is prescribed there. Instead, the physical assumptions of their continuity at the interfaces are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. We indicate also how to extend the solution method to the setting of one finite-sized domain surrounded on both sides by semi-infinite domains, and on that of three finite-sized domains.

Mesoglobule Morphologies of Amphiphilic Polymers

A statistical theory of finite-sized aggregates and domain structures in dilute solutions of amphiphilic macromolecules (homopolymers or copolymers) is developed. A minimalist model involving, essentially, two types of chemical groups (insoluble H and soluble P) is studied in the regime where polymer tends to phase separate, forming rather dense amorphous particles with low surface energy. The surface tension and the elastic bending moduli of the polymer/solvent interface are obtained and are related to the molecular parameters. The Gaussian modulus κG is predicted to be negative; typically |κG| is smaller than the mean bending modulus κ2. It is shown that the condensed polymer phase can remain dense, homogeneous, and stable with respect to microphase separation even as the surface tension decreases down to zero (or below it). Stable large aggregates of well-defined size are predicted in this regime. The size of aggregates and their shape depend on surface tension γ, the spontaneous bending modulus κ1, and the ratio κG/κ2. We show that bicontinuous morphologies (rather than colloidally stable polymer mesoglobules) are thermodynamically favorable for small |κG|. We present a quantitative argument showing that the gyroid structure is more favorable than other bicontinuous morphologies (primitive cubic and double diamond). The γ−κ1 phase diagrams showing the regions of stability of different morphologies are obtained. Two types of spherical mesoglobules are predicted, namely equilibrium and metastable globules whose sizes are shown to be essentially different: they depend on κ1 in qualitatively different ways. Recent experimental data on mesoglobule formation in solutions of thermosensitive amphiphilic polymers are discussed in the theoretical light.