Кинетические методы решения нестационарных задач со струйными течениями

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.Изучению нестационарных течений разреженного газа в настоящее время уделяется большое внимание. Такой интерес к этим задачам  вызван созданием импульсных струй, используемых при нанесении тонких пленок и специальных покрытий на твердые поверхности. Однако проблемы, связанные с нестационарным течением разреженного газа недостаточно изучены из-за их большой вычислительной сложности. В этой статье рассматриваются вычислительные аспекты исследования нестационарного движения отраженного потока газа от стенки и вытекающего через внезапно образованную щель. Целью этого исследования является анализ возможных численных кинетических подходов для решения таких нестационарных задач и выявление трудностей, возникающих при их решении.При моделировании процессов, происходящих при сильном разрежении необходимо использовать кинетическое уравнение Больцмана, численная реализация которого, как правило, достаточно трудоемка. Чтобы использовать более простые подходы, основанные, например, на аппроксимирующих кинетических уравнениях (Эллипсоидально-статистической модели, модели Шахова), важно оценить отличие решений модельных уравнений от решения уравнения Больцмана. Для этого рассматриваются две вспомогательные задачи: отражение потока газа от стенки и истечение свободной струи в разреженный фоновый газ.Численное решение этих задач показывает слабую зависимость решения от типа оператора столкновения в разреженной области, но при этом сильную зависимость поведения макропараметров от шага скоростной сетки. Детальная скоростная сетка необходима, чтобы избежать немонотонного поведения макропараметров, вызванных так называемым эффектом луча. Для уменьшения вычислительных затрат решения на детальной скоростной сетке предлагается гибридный метод, основанный на синтезе модельных уравнений и уравнения Больцмана. Такой подход может быть перспективным, поскольку уменьшает область применения интеграла столкновений Больцмана.Результаты, представленные в этой статье, получены использованием двух различных программных комплексов Unified Flow Solver (UFS) [13] и Несветай-3Д [14-15]. Отметим, что в UFS реализован метод дискретных ординат для скоростного пространства на равномерной сетке и иерархическая адаптивная сетка в физическом пространстве как для уравнения Больцмана, так и модельных уравнений. Программный комплекс Несветай-3Д создан для решения модельного уравнения Шахова на неструктурированных неравномерных сетках, как в скоростном, так и в физическом пространствах

Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, there would be immediate value in an efficient simulation method. The collision integral component of the equation is the main contributor of the large complexity. A plethora of new mathematical and numerical approaches have been proposed in an effort to reduce the computational cost of solving the Boltzmann collision integral, yet it still remains prohibitively expensive for large problems. This paper aims to accelerate the computation of this integral via machine learning methods. In particular, we build a deep convolutional neural network to encode/decode the solution vector, and enforce conservation laws during post-processing of the collision integral before each time-step. Our preliminary results for the spatially homogeneous Boltzmann equation show a drastic reduction of computational cost. Specifically, our algorithm requires O(n3) operations, while asymptotically converging direct discretization algorithms require O(n6), where n is the number of discrete velocity points in one velocity dimension. Our method demonstrated a speed up of 270 times compared to these methods while still maintaining reasonable accuracy

Численное сравнение решений кинетических модельных уравнений

The collision integral approximation by different model equations has created a whole new trend in the theory of rarefied gas. One widely used model is the Shakhov model (S-model) obtained by expansion of inverse collisions integral in a series of Hermite polynomials up to the third order. Using the same expansion with another value of free parameters leads to a linearized ellipsoidal statistical model (ESL).Both model equations (S and ESL) have the same properties, as they give the correct relaxation of non-equilibrium stress tensor components and heat flux vector, the correct Prandtl number at the transition to the hydrodynamic regime and do not guarantee the positivity of the distribution function.The article presents numerical comparison of solutions of Shakhov equation, ESL- model and full Boltzmann equation in the four Riemann problems for molecules of hard spheres.We have considered the expansion of two gas flows, contact discontinuity, the problem of the gas counter-flows and the problem of the shock wave structure. For the numerical solution of the kinetic equations the method of discrete ordinates is used.The comparison shows that solution has a weak sensitivity to the form of collision operator in the problem of expansions of two gas flows and results obtained by the model and the kinetic Boltzmann equations coincide.In the problem of the contact discontinuity the solution of model equations differs from full kinetic solutions at the point of the initial discontinuity. The non-equilibrium stress tensor has the maximum errors, the error of the heat flux is much smaller, and the ESL - model gives the exact value of the extremum of heat flux.In the problems of gas counter-flows and shock wave structure the model equations give significant distortion profiles of heat flux and non-equilibrium stress tensor components in front of the shock waves. This behavior is due to fact that in the models under consideration there is no dependency of the collision frequency on the molecular velocity.As calculations show, the ESL-model describes more accurately the non-equilibrium flow regime, but gives a greater deviation from the Boltzmann equation, than the Shahov model in front of shock waves.DOI: 10.7463/mathm.0615.0823537Модель Шахова, дающая правильное число Прандтля при переходе к гидродинамическому режиму, является обобщением модели Бхатнагара - Гросса - Крука (БГК, BGK). Она получена разложением в ряд интеграла обратных столкновений по полиномам Эрмита до третьего порядка. Использование этого же разложения при другом выборе свободного параметра приводит к линеаризованной эллипсоидальной статистической модели (ESL), дающей также правильное число Прандтля. Точность ESL- модели при решении задач течения разреженного газа ранее не исследовалась. В работе проводится численное сравнение решений модельных уравнений с решением полного уравнения Больцмана для трех тестовых задач распада разрыва. При вычислениях используется консервативный метод дискретных ординат. Сравнение показывает, что модельные уравнения могут существенно искажать профили макропараметров перед фронтом ударной волны, при этом ESL-модель дает значительно большее отклонение от решения уравнения Больцмана, чем модель Шахова.DOI: 10.7463/mathm.0615.082353

Computable model on the collision integral of Boltzmann equation and application to rarefied aerodynamics

Due to its complexity in dealing with the collisional integral term of the Boltzmann equation and computational costs associated with multi-dimensional problems, deterministic methods are still restricted to simple flow such as one-dimensional linear flow. However, the recently emerged fast spectrum method has achieved breakthroughs in computational efficiency and accuracy, which can enable simulations for more realistic three-dimensional non-linear flows. In comparison with the dominant direct simulation Monte Carlo method, the deterministic method has advantages especially in simulating lowspeed flows where statistical variations prevail. Here, we review the development of fast spectrum method and discuss its applications for practical flow simulations. In particular, extended Boltzmann model is required for polyatomic and dense gases where the Boltzmann equation may not be valid. We present the applications of extended Boltzmann model for polyatomic gases in predicting spectra of both spontaneous and coherent Rayleigh-Brillouin Scattering, and in simulating space vehicle reentries with a broad range of Kn. Finally, we discuss the gas-kinetic unified algorithm (GKUA) of computable model Boltzmann equation and applications to the hypersonic aerodynamics of space reentry covering various flow regimes

Lattice Boltzmann methods for direct numerical simulation of turbulent fluid flows

We study the use of lattice Boltzmann (LB) methods for simulation of turbulent fluid flows motivated by their high computational throughput and amenability to highly parallel platforms such as graphics processing units (GPUs). Several algorithmic improvements are unearthed including work on non-unit Courant numbers, the force operator, use of alternative topologies based on face and body centered cubic lattices and a new formulation using a generalized eigendecomposition that allows a new freedom in tuning the eigenvectors of the linearised collision operator. Applications include a variable bulk viscosity and the use of a stretched grid, our implementation of which reduces errors present in previous efforts. We present details for numerous lattices including all required matrices, their moments the procedures and programs used to generate these and perform linear stability analysis. Small Mach number flows where density variations are negligible except in the buoyancy force term allow the use of a highly accurate finite volume solver to simulate the evolution of the buoyancy field which is coupled to the LB simulation as an external force. We use a multidimensional flux limited third order flux integral based advection scheme. The simplified algorithm we have devised is easier to implement, has higher performance and does not sacrifice any accuracy compared to the leading alternative. Our algorithm is particularly suited to an outflow based implementation which furthers the stated benefits. We present numerical experiments confirming the third order accuracy of our scheme when applied to multidimensional advection. The coupled solver is implemented in a new code that runs in parallel across multiple machines using GPUs. Our code achieves high computational throughput and accuracy and is used to simulate a range of turbulent flows. Details regarding turbulent channel flow and sheared convective boundary layer simulations are presented including some new insight into the scaling properties of the latter flow

Efficient simulation of internal multiscale gas flows

We develop, validate, and apply an efficient multiscale method for the simulation of a large class of low-speed internal rarefied gas flows, which are critical to a range of future technologies. The method is based on an existing multiscale approach for the simulation of small-scale dense-fluid flows of high-aspect ratio, but has been extended to support fluid compressibility, non-isothermal conditions, three dimensional domains, and transience. Furthermore, the method is able to treat a broader range of flows: periodic, non-periodic, body-force-driven, pressure-driven, thermally-driven, and shear-driven. It also incorporates pseudospectral methods, and so boasts excellent convergence characteristics and accuracy. All verification cases presented herein are designed to be amenable to solution by a full molecular treatment (where scale separation is not exploited). The computationally demanding simulation technique known as direct simulation Monte Carlo (DSMC) is employed to obtain reference solutions, allowing for comparison with those computed by the multiscale method: excellent agreement is observed throughout. The unsteady (time-marching) implementation of the method, which allows for the resolution of transient flows, is validated by comparison with time dependent experimental data. Again, agreement is excellent. The computational efficiency of the multiscale method is exceptional. It provides efficiency gains of multiple orders of magnitude, relative to full molecular simulations (by the DSMC method); in some cases, the multiscale method allows for the solution of otherwise computationally intractable problems. Note, highly scale-separated systems are simulated with even greater efficiency. Following the experimental validation of the method, it is applied to the study of thermal-transpiration compressors (and implicitly Knudsen compressors). We characterise the effectiveness of these devices by considering the maximum pressure difference attainable for various combinations of (realistic) thermodynamic and geometric conditions. The development time required to obtain this pressure difference, which is also considered as a performance indicator, is also computed

Applied Mathematics and Computational Physics

As faster and more efficient numerical algorithms become available, the understanding of the physics and the mathematical foundation behind these new methods will play an increasingly important role. This Special Issue provides a platform for researchers from both academia and industry to present their novel computational methods that have engineering and physics applications