Quasigasdynamic Research Articles

Simulation of the Knudsen Pump by means of Quasi Gasdynamic Equation System

The use of miniature micropumps based on the effect of thermal gas sliding along the walls with an applied temperature gradient (Knudsen pump) is very relevant when creating modern miniature devices. Currently, algorithmically complex methods such as DSMC are usually used to simulate gas flows in such pumps. Progress in the development of algorithms based on a quasi-gasdynamic system of equations, pushed the authors to apply such algorithm to modeling the flow of a binary gas mixture gases in one variant of a micropump. The equations of the QGD system are conservation laws with additional dissipative terms. These terms contain a coefficient determined by the characteristic time between particles collisions. In this paper, the problem of flow in a rectangular Knudsen pump is solved based on this algorithm for a gas mixture in a single-fluid approximation. Different variants of the composition of the gas mixture are considered. The Knudsen numbers for which it is possible to describe gas flows by the QGD system correspond to flows of dense or moderately rarefied gases. Calculations for not too large Knudsen numbers demonstrated a good agreement with the results obtained by the DSMC method which is much more complex and expensive.

High-speed rarefied gas flow simulations using Quasi-Gas Dynamic equations with slip and jump boundary conditions

The high-speed rarefied gas flow simulation is essential for the aerothermodynamic design of high-altitude vehicles. Recently, the Quasi-Gas Dynamic (QGD) equations have been implemented in OpenFOAM as a CFD solver named QGDFoam. In computational fluid dynamics (CFD), the accuracy of the prediction of the surface quantities depends on the slip and jump boundary conditions applied to the surfaces. In the present work, various first-order and second-order slip and jump boundary conditions have been numerically implemented into the solver QGDFoam in OpenFOAM to obtain a completed solver for simulating the rarefied gas flows. It then captures the surface quantities of the gas flows, such as the surface gas pressure, the slip velocity, and surface gas temperature. This completed solver is validated for the sharp-leading-edge wedge, the compression ramp, and the NACA 0012 micro-airfoil cases. Using the QGD model with the slip and jump conditions extends its application and enhances it to simulate high-speed rarefied gas flows. The simulation results show that the slip and jump conditions have been successfully employed with the QGD equations and give good results for predicting the surface quantities compared with the DSMC data.

Computational multiscale method for parabolic wave approximations in heterogeneous media

In this paper, we develop a computational multiscale method to solve the parabolic wave approximation with heterogeneous and variable media. Parabolic wave approximation is a technique to approximate the full wave equation. One benefit of the method is that one wave propagation direction can be taken as an evolution direction, and one then can discretize it using a classical scheme like backward Euler method. Consequently, one obtains a set of quasi-gas-dynamic (QGD) models with possibly different heterogeneous permeability fields. For coarse discretization, we employ constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to perform spatial model reduction. The resulting system can be solved by combining the central difference in time evolution. Due to the variable media, we apply the technique of proper orthogonal decomposition (POD) to further the dimension of the problem and solve the corresponding model problem in the POD space instead of in the whole multiscale space spanned by all possible multiscale basis functions. We prove the stability of the full discretization scheme and give the convergence analysis of the proposed approximation scheme. Numerical results verify the effectiveness of the proposed method.

Contrast-Independent Partially Explicit Time Discretizations for Quasi Gas Dynamics

In the paper, we study a design and stability of contrast-independent partially explicit time discretizations for Quasi-Gas-Dynamics (QGD) Equations in multiscale high-contrast media. In our previous works, we have introduced contrast-independent partially explicit time discretizations. In this paper, we extend these ideas to multiscale QGD problems. Because of high contrast, explicit methods require a very small time stepping. By designing appropriate spatial splitting and temporal splitting, partially explicit methods remove this constraint. The proposed partially explicit time discretization consists of two steps. First, we split the space into contrast dependent (fast) and contrast independent (slow) components on a coarse grid that is much larger compared to spatial heterogeneities. Secondly, we design a temporal splitting algorithm in a such way that it is stable and the time step is independent of the contrast and only depends on the coarse mesh size. Using proposed method, a few degrees of freedom are treated implicitly and the approach is mostly explicit. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (slow). We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.

On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

We deal with 2D and 3D barotropic gas dynamics system of equations with two viscous regularizations: so-called quasi-gas dynamics (QGD) and quasi-hydrodynamics (QHD) ones. The system is linearized on a constant solution with any velocity, and an explicit two-level in time and symmetric three-point in each spatial direction finite-difference scheme on the uniform rectangular mesh is considered for the linearized system. We study L2-dissipativity of solutions to the Cauchy problem for this scheme by the spectral method and present a criterion in the form of a matrix inequality containing symbols of symmetric matrices of convective and regularizing terms. Analyzing these inequality and matrices, we also derive explicit sufficient conditions and necessary conditions in the Courant-type form which are rather close to each other. For the QHD regularization, such conditions are derived for the first time in 2D and 3D cases, whereas, for the QGD regularization, they improve those that have recently been obtained. Explicit formulas for a scheme parameter that guarantee taking the maximal time step are given for these conditions. An important moment is a new choice of an “average” spatial mesh step ensuring the independence of the conditions from the ratios of the spatial mesh steps and, for the QGD regularization, from the Mach number as well.

Simulation of Multi-Gas Jet Flows by Use of Quasi Gas Dynamic Equation System

In the present paper, we use the quasi gas dynamic (QGD) model together with a finite volume method for the simulation of a gas jet inflowing region filled with another gas in the presence of gravity forces. A flow picture for such flow strongly depends on the gases density ratio. Numerical simulations are held for a region filled with air under atmospheric pressure. Three variants of inflowing gas are considered: methane (light gas), butane (heavy gas) and helium (extremely light gas). A difference between flow pictures for these test cases is demonstrated. Results obtained with the presence of wind in the air are also compared. Grid convergence is established by use of different spatial meshes. Here, the the QGD model demonstrated good efficiency in modeling multi-gas jet flows. The calculations were also used for the adjustment of the numerical method parameters.

Computational multiscale methods for quasi-gas dynamic equations

In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been focused on problems with constant coefficients. In this paper, we investigate the QGD model in multiscale media, which can be used in porous media applications. This multiscale problem is interesting from a multiscale methodology point of view as the model problem has a hyperbolic multiscale term, and designing multiscale methods for hyperbolic equations is challenging. In the paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) combined with the central difference scheme in time to solve this problem. The CEM-GMsFEM provides a flexible and systematical framework to construct crucial multiscale basis functions for approximating the solution to the problem with reduced computational cost. With this approach of spatial discretization, we establish the stability of the fully discretized scheme, based on the coarse grid, under a coarse-scale CFL condition. Complete convergence analysis of the proposed method is presented. Numerical results are provided to illustrate and verify the theoretical findings.

Численное исследование аэродинамических характеристик треугольного крыла при различных углах атаки и больших числах Маха

The results of numerical simulation of the external supersonic flow around a triangular wing with a blunted leading edge at Mach numbers 6 and 9, obtained in the framework of a quasi-gas dynamic (QGD) algorithm, are presented. The formulation of the problem at Mach number 9 corresponds to experiments in a wind tunnel. The values of the aerodynamic coefficients obtained at different angles of attack are in good agreement with the experimental data. It is shown that the modeling carried out using more detailed grids is more accurate.

Computational Experiment in the Problem of Supersonic Flow around a Blunt Body with Tail Expansion

The results of the direct numerical simulation of a supersonic flow around a cylindrical blunt body with tail expansion are presented. The data obtained are compared with the results of laboratory experiments for the Mach numbers 3 and 4 and the angles of attack of 10 and 20 degrees. The results are obtained by using the quasi-gas-dynamic (QGD) equations.

Моделирование инфразвукового пистонфона

There are presented the results of numerical simulation of an applied acoustic problem – modeling of gas processes occurring in the measuring chamber of the infrasound pistonphone 3202 at different frequencies of piston oscillation (0.1 – 1000 Hz) and characterized by extremely small Mach numbers (9.1·10-7÷9.1·10-3). The simulation was performed using quasi-gas-dynamic (QGD) and quasi-hydrodynamic (QHD) equations of a viscous compressible heat-conducting gas with the use of a time-explicit difference scheme, all spatial derivatives was approximated by central differences. It is shown that QGD and QHD models can be used for a simulation of applied acoustics and, in particular, to the simulation of infrasonic pistonphone: the stability limits of the QGD and QHD algorithms in this problem were determined, the dependence of sound pressure on the tuning parameter α is investigated and it is shown that this dependence is quite small. The spectra of sound pressure at the control point calculated by QGD and QHD are given, their dependence on the tuning parameter α is shown, both models equally predict the value of the sound pressure amplitude at the fundamental frequency oscillations. At the end of the article, the sound pressure at the control point at the fundamental frequency oscillations obtained by using QGD and QHD is compared with the values calculated by using semi-empirical formula of sound pressure at closed volume for a case of small oscillations using the polytropic index obtained by Henry Gerber instead of the adiabatic coefficient.

Simulation of transonic low-Reynolds jets using quasi-gas dynamics equations

In this paper application of the OpenFOAM solver QGDFoam for numerical simulation of transonic viscous flows is considered. The developed solver, which implements regularized or quasi- gas dynamics (QGD) algorithms, is validated using the transonic low-Re jet flow case (Ma=0.9, Re=3600). The conducted numerical simulations allow the assessing applicability of the solver for modelling hydrodynamic instabilities and their interaction with transonic flow. Results of the numerical simulations are compared with experimental observations and Navier-Stokes-based code simulation. Results of the present study formulate a guideline for choosing (values of) QGD-algorithm tuning parameters.

Numerical simulation of gravity instabilities in gas flows by use of the quasi gas dynamic equation system

Paper presents two models for gravity instability simulations using compressible gas formulations, and especially, using regularised gas dynamic equations, or, in other words, the quasi-gas dynamic (QGD) equation system. The first part of the paper describes the layer flow for a single-component gas and the second one is one-fluid approximation for a gas mixture. Here a density equation for a mixture is split in two equations for each gas species. Momentum and energy equations are presented for a mixture with unified velocity and temperature and aggregated density and pressure. Examples of numerical calculations of Rayleigh-Taylor instability for different Atwood numbers are shown. The obtained results correspond to the well-known behaviour features of the Rayleigh-Taylor instability.

VERIFICATION OF AN ENTROPY DISSIPATIVE QGD-SCHEME

An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the previously constructed QGD-schemes and its merits are noticed. Practical convergence rates in the mesh L1-norm are computed. We also analyze the practical relevance in the nonlinear statement as the Mach number grows of recently derived necessary conditions for L2-dissipativity of the Cauchy problem for a linearized QGD-scheme.

High Performance Computing in Multiscale Problems of Gas Dynamics

The work is devoted to the organization of high performance computing in the solution of multiscale problems of gas dynamics relevant for the implementation of modern nanotechnologies. The base of the presented computing technology is a multiscale two-level approach that combines calculations at macro- and microlevels. The approach makes it possible to study micro- and nanoflows of a gaseous medium under conditions of complex geometry of technical systems used in the production cycle in order to obtain new nanomaterials and nanocoatings. Within the framework of the approach a system of the quasigasdynamic (QGD) equations and a system of the molecular dynamics (MD) equations are considered as two basic mathematical models. These models are aggregated using the method of splitting by physical processes and scales. The QGD system is solved by the finite volume method on grids of arbitrary type. The MD equations are solved according to the Verlet integration. In view of the complexity of the problem a high performance computing is used for realization of the approach. Parallel implementation of the approach is based on the methods of domain decomposition and functional parallelism and is oriented towards the use of computer systems with hybrid architecture. The implementation uses MPI, OpenMP and CUDA technologies. Testing of the developed approach and parallel tool was performed using the example of the problem of spraying the nanoparticles on a substrate. Numerical experiments confirm the effectiveness of the developed computing technology.

Development of a new OpenFOAM solver using regularized gas dynamic equations

The paper introduces the development of a new OpenFOAM solver QGDFoam for the numerical simulation of viscous compressible flows within a wide range of Mach numbers in the framework of the OpenFOAM formalism. The new solver is based on the implementation of regularized, or quasi-gas dynamic (QGD) equations. The mixed finite-volume and finite-difference approximation is constructed on unstructured space grids with co-located variables storage and explicit time scheme for convection approximation. The solver has been tested for a range of 1D Riemann problems and 2D cases, comparing results with analytic solutions and OpenFOAM’s implementation of the Kurganov–Tadmor scheme known as rhoCentralFoam.

Application of Quasi-Gas Dynamic Equations to Numerical Simulation of Near-Wall Turbulent Flows

We describe the possibilities of a numerical method based on quasi-gas dynamic (QGD) equations for the numerical simulation of a turbulent boundary layer. A subsonic Couette flow in nitrogen is used as an example, with dynamic Reynolds numbers of 153 and 198. The QGD system differs from the system of Navier---Stokes equations by additional nonlinear dissipative terms with a small parameter as a coefficient. In turbulent flow simulation, these terms describe small-scale effects that are not resolved on the grid. Comparison of our results (velocity profiles and mean-square velocity pulsations) with direct numerical simulation (DNS) results and benchmark experiments show that the QGD algorithm adequately describes the viscous and the logarithmic layers near the wall. Compared with high-accuracy DNS methods, the QGD algorithm permits using a relatively large spatial grid increment in the interior part of the viscous sublayer. Thus, the total number of grid points in the turbulent boundary layer may be relatively small. Unlike various versions of the large-eddy simulation (LES) method, the QGD algorithm does not require introduction of near-wall functions, because the additional terms vanish near the wall. For small Reynolds numbers, the QGD algorithm describes laminar Couette flow.

Two-scale computation of N2–H2 jet flow based on QGD and MMD on heterogeneous multi-core hardware

Gas mixture flows take place in a wide range of scientific and technical problems, including space applications, microchannel streams, nanotechnology and others. This paper deals with numerical simulation of gas mixture flows in microchannels. For such flows speeds can have supersonic, subsonic and transonic ranges according to parameters of gas stream and properties. The supersonic expansion of gas mixtures is accompanied by several simultaneous nonequilibrium processes: interfusion of layers, shock waves, heat exchange with walls etc. To investigate these complex problems, it is essential to develop adequate numerical technique that contains stable and robust numerical algorithms and calculations based on fine grids. These calculations are supposed to be carried out by means of high performance computer systems.This paper presents a new method for simulating gas mixture flows that enables us to solve similar issues and reduce computing time (Garzo et al., 1989; Elizarova, 2009). Particularity of our approach is combination of continuum mechanics and Newton's dynamics models. For computer realization of the approach we use grid methods for solution of macroscopic quasigasdynamic (QGD) equations and molecular dynamics methods (MMD). Some results of numerical modelling are discussed.

Smoothed MHD equations for numerical simulations of ideal quasi-neutral gas dynamic flows

We introduce a mathematical model and related numerical method for numerical modeling of ideal magnetohydrodynamic (MHD) gas flows as an extension of previously known quasi-gasdynamic (QGD) equations. This approach is based on smoothing, or averaging of the original MHD equation system over a small time interval that leads to a new equation system, named quasi-MHD, or QMHD system. The QMHD equations are closely related to the original MHD system except for additional strongly non-linear dissipative τ-terms with a small parameter τ as a factor. The τ-terms depend on the solution itself and decrease in regions with the small space gradients of the solution. In this sense the QMHD system could be regarded as an approach with adaptive artificial dissipation. The QMHD is a generalization of regularized (or quasi-) gas dynamic equation system suggested in last three decades. In the QMHD numerical method the evolution of all physical variables is presented in a non-split divergence form. Divergence-free evolution of the magnetic field provides by using a constrained transport method based on Faraday’s law of induction. Accuracy and convergence of the QMHD method is verified on a wide set of standard MHD tests including the 3D Orszag–Tang vortex flow.

On quasi-gasdynamic and quasi-hydrodynamic equations for binary gas mixtures

The quasigasdynamic (QGD) approach makes it possible to construct convenient and reliable differ� ence schemes for the numerical solution of various gasdynamic problems. Its description can be found in (1-3). More specifically, in (2, Chapter 9) (see also (4)), the Boltzmann kinetic equation for a mixture of monatomic gases (5) is used to derive and test QGD equations for binary mixtures of nonreactive ideal polytropic gases. In this paper, we analyze and expand the capabili� ties of the QGD approach in this area. The original equations from (2) are rewritten as conservation laws, which are more conventional in viscous gas dynamics and convenient for discretization. Additionally, an external force and a heat source are taken into account. We briefly discuss the parabolicity of the sys� tem in the sense of Petrovskii, which ensures that the system is well defined. An entropy balance equation is derived, and the entropy production for a gas mixture is shown to be nonnegative, which ensures that the sys� tem is physically consistent (but does not hold in all available descriptions of gas mixtures). Importantly, to achieve the latter property, the expressions for the exchange terms in the total energy balance equation (initially derived only for monatomic gas mixtures) are properly generalized. Additionally, we introduce a simplification of the QGD system for binary mixtures, which is referred to as a quasihydrodynamic system and is used for the numerical simulation of weakly compressible suband transonic flows. At the end of this paper, we present simplified barotropic versions of both systems and derive a corresponding energy bal� ance equation with nonpositive energy production. The QGD system for binary gas mixtures a and b (see (2)) consists of the following mass balance, momentum, and total energy equations for the gas α:

Simulation of laminar–turbulent transition in compressible Taylor–Green flow basing on quasi-gas dynamic equations

We report the results of numerical simulation of laminar–turbulent transition in the Taylor–Green vortex for viscous compressible gas flow basing on quasi-gas-dynamic (QGD) equations. Here the QGD system is obtained by a temporal averaging of the Navier–Stokes equations. The additional dissipative terms in QGD system serve to model the effects of the unresolved subgrid scales. Comparison with direct numerical simulation and large eddy simulation reference data demonstrates that QGD numerical algorithm provides a uniform and adequate simulation of both laminar and turbulent evolution of the vortex for Reynolds numbers from 100 up to 5000, including transition.