A hybrid numerical method for interfacial fluid flow with soluble surfactant

We address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the physically representative limit of large bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a transition layer near the interface in which the surfactant concentration varies rapidly, and large gradients at the interface must be resolved accurately to evaluate the exchange of surfactant between the interface and bulk flow. We use the slenderness of the layer to develop a fast and accurate ‘hybrid’ numerical method that incorporates a separate, singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the interfacial free boundary problem. The accuracy and efficiency of the method is assessed by comparison with a more ‘traditional’ numerical approach that uses finite differences on a curvilinear coordinate system exterior to the bubble, without the separate transition layer reduction. The traditional method implemented here features a novel fast calculation of fluid velocity off the interface.     

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin’s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system’s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix–vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.     

A low-dissipation and time-accurate method for compressible multi-component flow with variable specific heat ratios

A low-dissipation method for calculating multi-component gas dynamics flows with variable specific heat ratio that is capable of accurately simulating flows which contain both high- and low-Mach number features is proposed. The technique combines features from the double-flux multi-component model, nonlinear error-controlled WENO, adaptive TVD slope limiters, rotated Riemann solvers, and adaptive mesh refinement to obtain a method that is both robust and accurate. Success of the technique is demonstrated using an extensive series of numerical experiments including premixed deflagrations, Chapman–Jouget detonations, re-shocked Richtmyer–Meshkov instability, shock-wave and hydrogen gas column interaction, and multi-dimensional detonations. This technique is relatively straight-forward to implement using an existing compressible Navier–Stokes solver based on Godunov’s method.     

A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows

We extend [Shravan K. Veerapaneni, Denis Gueyffier, Denis Zorin, George Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, Journal of Computational Physics 228(7) (2009) 2334–2353] to the case of three-dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous flows. Although the main components of the algorithm are similar in spirit to the 2D case—spectral approximation in space, semi-implicit time-stepping scheme—the main differences are that the bending and viscous force require new analysis, the linearization for the semi-implicit schemes must be rederived, a fully implicit scheme must be used for the toroidal topology to eliminate a CFL-type restriction and a novel numerical scheme for the evaluation of the 3D Stokes single layer potential on an axisymmetric surface is necessary to speed up the calculations. By introducing these novel components, we obtain a time-scheme that experimentally is unconditionally stable, has low cost per time step, and is third-order accurate in time. We present numerical results to analyze the cost and convergence rates of the scheme. To verify the solver, we compare it to a constrained variational approach to compute equilibrium shapes that does not involve interactions with a viscous fluid. To illustrate the applicability of method, we consider a few vesicle-flow interaction problems: the sedimentation of a vesicle, interactions of one and three vesicles with a background Poiseuille flow.     

The effect of two-dimensional shear flow on the stability of a crystal interface in a supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of twodimensional shear flow on the stability of a crystal interface in the supercooled melt of a pure substance.Similar to the three-dimensional shear flow as described in our previous paper,the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude.However,compared with the case of the Laplace equation for a steady-state thermal diffusion field,due to the existence of time partial derivatives of the temperature fields in the diffusion equation the absolute value of the gradients of the temperature fields increases,therefore destabilizing the interface.The circular interface is more unstable than in the case of Laplace equation without time partial derivatives.The critical stability radius of the crystal interface increases with shearing rate increasing.The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

The effect of two-dimensional of a crystal interface in shear flow on the stability a supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of twodimensional shear flow on the stability of a crystal interface in the supercooled melt of a pure substance. Similar to the three-dimensional shear flow as described in our previous paper, the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude. However, compared with the case of the Laplace equation for a steady-state thermal diffusion field, due to the existence of time partial derivatives of the temperature fields in the diffusion equation the absolute value of the gradients of the temperature fields increases, therefore destabilizing the interface. The circular interface is more unstable than in the case of Laplace equation without time partial derivatives. The critical stability radius of the crystal interface increases with shearing rate increasing. The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

The effect of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt of pure substance. Similar to the three-dimensional shear flow as described in our previous paper, the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude. However, compared with the case of Laplace equation for steady state thermal diffusion field, due to the existence of time partial derivatives of the temperature fields in diffusion equation the absolute value of the gradients of the temperature fields increases, therefore destabilizing the interface. The circular interface is more unstable than in the case of Laplace equation without time partial derivatives. The critical stability radius of the crystal interface increases with shearing rate increasing. The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

Efficient evaluation of Casimir force in arbitrary three-dimensional geometries by integral equation methods

In this Letter, we generalized the surface integral equation method for the evaluation of Casimir force in arbitrary three-dimensional geometries. Similar to the two-dimensional case, the evaluation of the mean Maxwell stress tensor is cast into solving a series of three-dimensional scattering problems. The formulation and solution of the three-dimensional scattering problems are well-studied in classical computational electromagnetics. This Letter demonstrates that this quantum electrodynamic phenomenon can be studied using the knowledge and techniques of classical electrodynamics.     

A note on pressure accuracy in immersed boundary method for Stokes flow

In this short note, we provide a simplified one-dimensional analysis and two-dimensional numerical experiments to predict that the overall accuracy for the pressure or indicator function in immersed boundary calculations is first-order accurate in L₁ norm, half-order accurate in L₂ norm, but has O(1) error in L_∞ norm. Despite the pressure has O(1) error near the interface, the velocity field still has the first-order convergence in immersed boundary calculations.     

Distribution function for large velocities of a two-dimensional gas under shear flow

The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degreek⩾4 diverge for shear rates larger than a critical valuea _c ^(k) , which behaves for largek asa _c ^(k) ∼k ⁻¹. This divergence is consistent with an algebraic tail of the formf(V) ∼V ^−4-σ(a), where σ is a decreasing function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.     

Analytical solution of navier-stokes flow of a viscous compressible fluid

An analytical procedure is proposed to study the flow of viscous compressible continuous fluids.     

Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain,a boundary integral method

We consider numerical algorithms for the simulation of hydrodynamics of two-dimensional vesicles suspended in a viscous Stokesian fluid. The motion of vesicles is governed by the interplay between hydrodynamic and elastic forces. Continuum models of vesicles use a two-phase fluid system with interfacial forces that include tension (to maintain local “surface” inextensibility) and bending. Vesicle flows are challenging to simulate. On the one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives in the bending term. On the other hand, implicit time-stepping schemes can be expensive because they require the solution of a set of nonlinear equations at each time step.     

A Fictitious Domain,parallel numerical method for rigid particulate flows

In this paper, we develop a Fictitious Domain, parallel numerical method for the Direct Numerical Simulation of the flow of rigid particles in an incompressible viscous Newtonian fluid. A Simultaneous Directions Implicit algorithm is employed which gives the model a high level of parallelization. The projection of the fluid velocity onto rigid motion on the particles is based on a fast computational technique which relies on the conservation of linear and angular momenta. Numerical results are presented which confirm the ability of the proposed method to simulate the sedimentation of one and many particles; the parallel efficiency of the algorithm is also assessed.     

A front-tracking/ghost-fluid method for fluid interfaces in compressible flows

A front-tracking/ghost-fluid method is introduced for simulations of fluid interfaces in compressible flows. The new method captures fluid interfaces using explicit front-tracking and defines interface conditions with the ghost-fluid method. Several examples of multiphase flow simulations, including a shock–bubble interaction, the Richtmyer–Meshkov instability, the Rayleigh–Taylor instability, the collapse of an air bubble in water and the breakup of a water drop in air, using the Euler or the Navier–Stokes equations, are performed in order to demonstrate the accuracy and capability of the new method. The computational results are compared with experiments and earlier computational studies. The results show that the new method can simulate interface dynamics accurately, including the effect of surface tension. Results for compressible gas–water systems show that the new method can be used for simulations of fluid interface with large density differences.     

Similarity solutions for two-dimensional magnetogasdynamic boundary layer flow in a transverse magnetic field

Summary The conditions for the existence of the similarity solution of a two-dimensional boundary layer flow past a plane wall of an electrically conducting gas, in the presence of a transverse magnetic field at small magnetic Reynolds number, are studied. It is found that, in general, the similarity requirements involve as many as eleven parameters out of which eight are the same as these of nonmagnetic gasdynamic boundary layer flow. The solution is obtained for power law velocity distribution, in Illingworth variables, at the outer edge of the boundary layer and for the corresponding variable magnetic field. The Prandtl number of the fluid is taken as unity and the magnetic-field effects are confined to the boundary layer only.

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Riassunto Si studiano le condizioni per l’esistenza della soluzione di somiglianza di un flusso nello stato di confine bidimensionale oltre una parete piana di un gas che conduce electtricità, in presenza di un campo magnetico trasverso con piccolo numero magnetico di Reynolds. Si trova che, in generale, le esigenze di somiglianza coinvolgono undici parametri, otto dei quali sono identici a quelli del flusso della strato di confine non magnetico e gas dinamico. La soluzione è ottenuta per la distribuzione di velocità della legge di potenza, in variabili di Illingworth, nel bordo esterno dello strato limite e per il corrispondente campo magnetico variabile. Il numero di Prandtl del fluido è preso uguale a uno e gli effetti del campo magnetico sono confinati solo allo strato limite.

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Резюме Исследуются условия существования подобного решения для двумерного течения в пограничиом слое вдоль плосокой стенки газа, проводящего электричество, в присутствии поперечного магнитного поля при малых магнитных числах Рейнольдса. Получено, что в общем случае требования подобия включают одиннадцать параметров, восемь из которых являются соответствующими параметрами для немагнитного газодинамического течения в пограничном слое. Ппределено решение в случае степенного закона распределения по скоростям на внешнем краю пограничного слоя и для соотвтствуюего переменного магнитного поля. Число Прандтля для жидкости считается равным единице и эффекты магнитного поля ограничиваются только пограничным слоем.

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Towards oscillation-free implementation of the immersed boundary method with spectral-like methods

It is known that, when the immersed boundary method (IBM) is implemented within spectral-like methods, the Gibbs oscillation seriously deteriorates the calculation of derivatives near the body surface. In this paper, a radial basis function (RBF) based smoothing technique is proposed with the intention of eliminating or efficiently reducing the Gibbs oscillation without affecting the flow field outside the body. Based on this technique, a combined IBM/spectral scheme is developed to solve the incompressible Navier–Stokes equations. Numerical simulations of flow through a periodic lattice of cylinders of various cross sections are performed. The results demonstrate that the proposed methodology is able to give accurate and nearly oscillation-free numerical solutions of incompressible viscous flows.     

一种新的模拟渗流运动的数值方法

根据格子Boltzmann方法及相关理论，建立了一个新的模拟渗流运动的数值模型，所得模型没有在边界上采取相应平均措施，同时还避免了一些非物理副产品的出现-实例计算数值结果与精确解符合较好，证明模型可靠- 关键词： 渗流 格子Boltzmann方法 数值模型     

Allowance for the imperfection of optical elements and inaccuracy in the initial adjustment of the rotating phase plate in a stokes polarimeter

For a Stokes polarimeter with a rotating phase plate, formulas were obtained for calculating the polarization characteristics and their systematic errors caused by the departure of the anisotropy parameters of the optical elements used from ideal ones and also by the deflection of the initial position of the phase plate "fast" axis from the analyzer transmission axis. The manifestation of the mentioned factors in the systematic errors of determining the polarization characteristics of radiation of different forms of polarization is considered. The results of model calculations have been confirmed by experimental data. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 67, No. 3, pp. 382–386, May–June, 2000. This work was carried out with partial support from the Belarusian Foundation for Fundamental Research (project F97-184).     

Turbulent gas-dispersed flow in a pipe with sudden expansion: numerical simulation

A mathematical model was developed to simulate two-phase gas-dispersed flow moving through a pipe with axisymmetric sudden expansion. In the model, the two-fluid Euler approach was used. The model is based on solving Reynolds-averaged Navier — Stokes equations for a two-phase stream. In calculating the fluctuating characteristics of the dispersed phase, equations borrowed from the models by Simonin (1991), Zaichik et al. (1994), and Derevich (2002) were used. Results of a comparative analysis with previously reported experimental and numerical data on two-phase flows with separation past sudden expansion in a plane channel and in a pipe are given. This work was supported by the President of the Russian Federation through the Foundation for Young Candidates of Sciences under Grant MK-186.2007.8 and by the Russian Foundation for Basic Research (Grants Nos. 05-08-33586 and 06-08-00967).     

Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations

We consider variational multiscale (VMS) methods with h-adaptive technique for the stationary incompressible Navier–Stokes equations. The natural combination of VMS with adaptive strategy retains the best features of both methods and overcomes many of their deficits. A reliable a posteriori projection error estimator is derived, which can be computed by two local Gauss integrations at the element level. Finally, some numerical tests are presented to illustrate the method’s efficiency.