CABARET scheme on moving grids for two-dimensional equations of gas dynamics and dynamic elasticity

Схема КАБАРЕ, являющаяся представителем семейства балансно-характеристических методов, широко используется при решении многих задач для систем дифференциальных уравнений гиперболического типа в эйлеровых переменных. Возрастающая актуальность задач взаимодействия деформируемых тел с потоками жидкости и газа требует адаптации этого метода на лагранжевы и смешанные эйлерово-лагранжевы переменные. Ранее схема КАБАРЕ была построена для одномерных уравнений газовой динамики в массовых лагранжевых переменных, а также для трехмерных уравнений динамической упругости. В первом случае построенную схему не удалось обобщить на многомерные задачи, а во втором — использовался необратимый по времени алгоритм передвижения сетки. В данной работе представлено обобщение метода КАБАРЕ на двумерные уравнения газовой динамики и динамической упругости в смешанных эйлерово-лагранжевых и лагранжевых переменных. Построенный метод является явным, легко масштабируемым и обладает свойством временн´ой обратимости. Метод тестируется на различных одномерных и двумерных задачах для обеих систем уравнений (соударение упругих тел, поперечные колебания упругой балки, движение свободной границы идеального газа). The conservative-characteristic CABARET scheme is widely used in solving many problems for systems of differential equations of hyperbolic type in Euler variables. The increasing urgency of the problems of interaction of deformable bodies with liquid and gas flows requires the adaptation of this method to Lagrangian and arbitrary Lagrangian-Eulerian variables. Earlier, the CABARET scheme was constructed for one-dimensional equations of gas dynamics in mass Lagrangian variables, as well as for three-dimensional equations of dynamic elasticity. In the first case, the constructed scheme could not be generalized to multidimensional problems, and in the second, a time-irreversible grid movement algorithm was used. This paper presents a generalization of the CABARET method to two-dimensional equations of gas dynamics and dynamic elasticity in arbitrary Lagrangian-Eulerian and Lagrangian variables. The constructed method is explicit, easily scalable, and has the property of temporal reversibility. The method is tested on various one-dimensional and two-dimensional problems for both systems of equations (collision of elastic bodies, transverse vibrations of an elastic beam, motion of the free boundary of an ideal gas).

An iterative method for solving difference problems of gas dynamics in the mixed Euler-Lagrangian variables

The method proposed is intended to solve implicit conservative operator difference schemes for a grid initial-boundary value problems on a simplex grid for a system of equations of gas dynamics in the mixed Euler-Lagrangian variables. To find a solution to such a scheme at a time step, it is represented as a single equation for a nonlinear function of two arguments from space – the direct product of the grid spaces of gas-dynamic quantities. To solve such an equation, a combination of the generalized Gauss-Seidel iterative method (external iterations) and an implicit two-layer iteration scheme (internal iterations at each external iteration) is used. The feature of the method is that, the equation, which is solved by internal iterations, is obtained from the equation of the difference scheme using symmetrization – such a non-degenerate linear transformation that the function in this equation has a self-adjoint positive Frechet derivative.

High precision methods for solving a system of cold plasma equations taking into account electron–ion collisions

High precision simulation algorithms are proposed and justified for modelling cold plasma oscillations taking into account electron–ion collisions in the non-relativistic case. The specific feature of the approach is the use of Lagrangian variables for approximate solution of the problem formulated initially in Eulerian variables. High accuracy is achieved both through the use of analytical solutions on trajectories of particles and due to sufficient smoothness of the solution in numerical integration of Cauchy problems. Numerical experiments clearly illustrate the obtained theoretical results. As a practical application, a simulation of the well-known breaking effect of multi-period relativistic oscillations is carried out. It is shown that with an increase in the collision coefficient one can observe that the breaking process slows down until it is completely eliminated.

A numerical study of variational discretizations of the Camassa–Holm equation

We present two semidiscretizations of the Camassa–Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa–Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.

On the existence of global solution of the system of equations of liquid movement in porous medium

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.

The Effect of Turbulence Modelling on the Assessment of Platelet Activation

Pathological platelet activation induced by abnormal shear stresses is regarded as a main clinical complication in recipients of cardiovascular biomedical implantable devices and prostheses. In order to improve their performance computational fluid dynamics (CFD) has been used to evaluate flow fields and related shear stresses. More recently CFD models have been equipped with mathematical models that describe the relation between fluid dynamics variables, and in particular shear stresses, and the platelet activation state (PAS). These mathematical models typically use a Lagrangian approach to extract the shear stresses along possible platelet trajectories. However, in the case of turbulent flow, the choice of the proper turbulence closure model is still debated for both concerning its effect on Lagrangian statistics and shear stress calculation. In our study five numerical simulations of the flow through a mechanical heart valve were performed and then compared in terms of Eulerian and Lagrangian quantities: a direct numerical simulation (DNS), a large eddy simulation (LES), two Reynolds-averaged Navier-Stokes (RANS) simulations (SST k-ω and RSM) and a “Laminar” (no turbulence modelling on a Taylor microscale-based grid) simulation. Results exhibit a large variability in the PAS assessment depending on the turbulence model adopted. “Laminar” and RSM estimates of platelet activation are about 60% below DNS, while LES is 16% less. Surprisingly, PAS estimated from the SST k-ω velocity field is only 8% less than from DNS data. This appears more artificial than physical as can be inferred after comparing frequency distributions of PAS and of the different Lagrangian variables of the mechano-biological model of platelet activation. Our study indicates that turbulence closures can lead to a severe underestimation of platelet activation and suggests that turbulence should be fully resolved by DNS when assessing blood damage in blood contacting devices.

Immersed Boundary Method for Simulating Interfacial Problems

We review the immersed boundary (IB) method in order to investigate the fluid-structure interaction problems governed by the Navier–Stokes equation. The configuration is described by the Lagrangian variables, and the velocity and pressure of the fluid are defined in Cartesian coordinates. The interaction between two different coordinates is involved in a discrete Dirac-delta function. We describe the IB method and its numerical implementation. Standard numerical simulations are performed in order to show the effect of the parameters and discrete Dirac-delta functions. Simulations of flow around a cylinder and movement of Caenorhabditis elegans are introduced as rigid and flexible boundary problems, respectively. Furthermore, we provide the MATLAB codes for our simulation.

LAGRANGIAN FORMALISM IN PROBLEMS OF SMALL OSCILLATIONS OF VORTEX FLOWS AND ITS CONNECTION WITH THE VARIATIONAL PRINCIPLE FOR IDEAL INCOMPRESSIBLE HYDRODYNAMICS OF VORTEX LINES

The paper discusses the description of vortex flows of an ideal incompressible fluid based on the formalism of Lagrangian mechanics. Using the displacement field of liquid particles as a generalized coordinate, we write out the Lagrangian describing the dynamics of small perturbations (Kopiev, Chernyshev, 2018). The corresponding Lagrange equations are the equation for the displacement field (Drazim, Reid, 1981): This equation is equivalent to the Helmholtz equation for vorticity perturbations. The displacement field is defined as the difference in the positions of liquid particles on trajectories in disturbed and undisturbed flows. Although this definition is given in terms of Lagrangian variables associated with liquid particles, the displacement field itself is an Euler variable, expressed through velocity and vorticity perturbations. An example of using Lagrangian to solve the problem of conservation of the quadrupole moment of a vortex flow is considered. Using the Noether theorem, conditions on a stationary flow are obtained, under which the quadrupole moment of small perturbations of this flow is an integral of motion (Kopiev, Chernyshev, 2018). It is shown that these conditions are satisfied for the jet flows uniform along the longitudinal coordinate. The result obtained is important in aeroacoustics due to the fact that the quadrupole moment of the vortex flow represents the main term of the decomposition of a compact acoustic source in Machnumber (Lighthill, 1952; Crow, 1970; Kopiev, Chernyshev, 1995). The generalization of these results to the nonlinear case is considered. The Lagrangian is obtained for an arbitrary nonlinear displacement field: nowhere Gis Green’s function of the Laplace equation. The corresponding Lagrange equations coincide with the differential equations describing the nonlinear dynamics of the displacement field (Drazin, Reid, 1981). Expansion of the Lagrangian in small perturbations to quadratic terms gives the Lagrangian of the linear system. The question of the relationship of the proposed approach to the description of the dynamics of an incompressible fluid and known approaches based on the formalism of Lagrangian mechanics with the coordinates of liquid particles as generalized coordinates (Chapman, 1978; Goncharov, Pavlov, 2008; Kuznetsov, Ruban, 1998) is considered. It is shown that the transformation of the Lagrangian obtained in (Kuznetsov, Ruban, 1998) to the Lagrangian can be carried out by transforming Lagrangian variables (coordinates of liquid particles) to Eulerian variables (displacement field). This study was supported by the Russian Science Foundation, project No. 17-11-01271.

Influence of electron temperature on breaking of plasma oscillations

The influence of thermal motion of electrons on the processes of relativistic plasma oscillations is studied analytically and numerically. It is shown that if the temperature of electrons grows and exceeds a certain critical level, then the breaking effect vanishes due to transformation of plasma oscillations into travelling waves. Analytical conclusions are made in the framework of the theory of small perturbations based on Lagrangian variables. Numerical simulation of the transformation is performed using three different algorithms constructed on the basis of the method of finite differences in Eulerian variables. The analytical results are in good agreement with numerical experiments.