Direct Statistical Simulation Research Articles

Numerical Investigation of Radiation in a Shock-Heated CO2 and N2 Mixture by the Direct Statistical Simulation Method

Numerical Investigation of Radiation in a Shock-Heated CO2 and N2 Mixture by the Direct Statistical Simulation Method

Direct statistical simulation of the Busse annulus

We consider direct statistical simulation (DSS) of a paradigm system of convection interacting with mean flows. In the Busse annulus model, zonal jets are generated through the interaction of convectively driven turbulence and rotation; non-trivial dynamics including the emergence of multiple jets and bursting ‘predator–prey’ type dynamics can be found. We formulate the DSS by expanding around the mean flow in terms of equal-time cumulants and arrive at a closed set of equations of motion for the cumulants. Here, we present results using an expansion terminated at the second cumulant (CE2); it is fundamentally a quasilinear theory. We focus on particular cases including bursting and bistable multiple jets and demonstrate that CE2 can reproduce the results of direct numerical simulation if particular attention is given to symmetry considerations.

Direct statistical simulation of the Lorenz63 system.

We use direct statistical simulation to find the low-order statistics of the well-known dynamical system, the Lorenz63 model. Instead of accumulating statistics from numerical simulation of the dynamical system or solving the Fokker-Planck equation for the full probability distribution of the dynamical system, we directly solve the equations of motion for the low-order statistics after closing them by making several different choices for the truncation. Fixed points of the statistics are obtained either by time evolving or by iterative methods. The stability and statistical realizability of the fixed points of the statistics are analyzed, and the statistics so obtained are compared to those found by the traditional approach. Low-order statistics of the chaotic Lorenz63 system can be obtained from cumulant expansions more efficiently than by accumulation via direct numerical simulation or by solution of the Fokker-Planck equation.

Direct Statistical Simulation of Oxygen Radiation behind Shock Wave

Direct Statistical Simulation of Oxygen Radiation behind Shock Wave

Direct statistical simulation of low-order dynamosystems

In this paper, we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third order and has cubic nonlinearities while the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be used to solve for the statistics of these systems of equations both in the presence and the absence of stochastic terms, by truncating the cumulant hierarchy at either second or third order. We compare two different techniques for solving for the statistics: timestepping, which is able to locate only stable solutions of the equations for the statistics, and direct detection of the fixed points. We develop a complete methodology and symbolic package in Python for deriving the statistical equations governing the low-order dynamic systems in cumulant expansions. We demonstrate that although direct detection of the fixed points is efficient and accurate for DSS truncated at second order, the addition of higher order terms leads to the inclusion of many unstable fixed points that may be found by direct detection of the fixed point by iterative methods. In those cases, timestepping is a more robust protocol for finding meaningful solutions to DSS.

Вплив пропорцій орбітальних об’єктів простої форми на їхні аеродинамічні характеристики

When developing modern and promising aerospace technology models, the relevance of simulation of the flow around apparatuses (spacecraft) of various geometric shapes in a transitional mode corresponding to the flight in the region of the upper layers of the atmosphere and near space is growing. Solving the Boltzmann equation, which most adequately describes this process in the framework of kinetic theory, still remains a difficult task. To solve this equation, the Monte Carlo statistical methods are used quite successfully. An example of such a method is the direct statistical simulation, or, less common but also well-established in rarefied gas dynamics, the test particles method (TPM). The aim of this work is to study the effect of geometric proportions of simply-shaped orbiting objects during uncontrolled descent to dense layers of the atmosphere on their drag coefficients. Such objects may be elements of space debris or spacecraft of appropriate shapes and proportions. The studies were based on the results obtained by numerical simulation of TPMs on uniform rectangular grids. The shape of the orbital objects was set in the form of a circular cone, cylinder, rectangular parallelepiped of various elongations, and spheres. The calculations were carried out in a wide range of attack angles. The characteristic dimensions of the body class in question varied from 2 to 12 meters. According to the standard atmosphere for such characteristic dimensions, the transitional flow regime is realized at altitudes from 90 km to 180 km. It was found that the calculated values of the drag coefficients in the transition regime are in satisfactory agreement with the experimental data and calculations on the theory of local interaction, and at an altitude of 300 km, they correspond to the control free molecular values obtained by analytical formulas. The dependence of the drag coefficients of the bodies of the considered shapes on the angle of attack and flight altitude was studied. The influence of the choice of the characteristic area on the range of values of the calculation results is shown. The drag coefficient of the considered class of bodies at the entrance to the dense layers of the atmosphere using the TPM was calculated for the first time. Satisfactory agreement of the obtained results with the available experimental and calculated data confirms the effectiveness of the applied method in transition mode. This makes it possible to use it in practical calculations of the parameters of the external environment effect on the spacecraft in the most difficult to study altitude ranges corresponding to the transitional flow regime.

The Use of a Supersonic Jet of Gas Activated in a Microwave Discharge in Diamond Deposition

The gas-jet method of the deposition of diamond-like structures using super-high-frequency (SHF) discharge for activating gases is considered. Direct statistical simulation is applied to analyze the gasdynamic processes occurring in the conditions of the gas-jet deposition of diamond structures. An activated gas flow in a nozzle is investigated and a considerable influence of the processes in the compressed layer formed near the substrate on the rate of synthesis of diamond structures is established. The regimes of flow of mixture components activated in the discharge chamber toward the substrate are determined and a considerable deceleration of heavy components in the compressed layer is found to exist. It is shown that with increase in the pressure in the deposition chamber from 10 to 80 Torr the crystal growth rate increases fourfold at the expense of an increase in the specific flux of the activated components toward the substrate.

Dimensional reduction of direct statistical simulation

Direct statistical simulation (DSS) solves the equations of motion for the statistics of turbulent flows in place of the traditional route of accumulating statistics by direct numerical simulation (DNS). That low-order statistics usually evolve slowly compared with instantaneous dynamics is one important advantage of DSS. Depending on the symmetry of the problem and the choice of averaging operation, however, DSS is usually more expensive computationally than DNS because even low-order statistics typically have higher dimension than the underlying fields. Here we show that it is in some cases possible to go much further by using a form of unsupervised learning, proper orthogonal decomposition, to address the ‘curse of dimensionality’. We apply proper orthogonal decomposition directly to DSS in the form of expansions in equal-time cumulants to second order. We explore two averaging operations (zonal and ensemble) and test the approach on two idealized barotropic models of fluid on a rotating sphere (a jet that relaxes deterministically towards an unstable profile and a stochastically driven flow that spontaneously organizes into jets). We show that the method offers the possibility of parameter continuation, in the reduced basis, for the lower-order statistics of the flow. Order-of-magnitude savings in computational cost are sometimes obtained in the reduced basis, potentially enabling access to parameter regimes beyond the reach of DNS.

Direct statistical simulation of Monte Carlo in the study of argon radiation behind the front of a strong shock wave

Direct statistical simulation of Monte Carlo in the study of argon radiation behind the front of a strong shock wave

Joint instability and abrupt nonlinear transitions in a differentially rotating plasma

Global magnetohydrodynamic (MHD) instabilities are investigated in a computationally tractable two-dimensional model of the solar tachocline. The model’s differential rotation yields stability in the absence of a magnetic field, but if a magnetic field is present, a joint instability is observed. We analyse the nonlinear development of the instability via fully nonlinear direct numerical simulation, the generalized quasi-linear approximation (GQL) and direct statistical simulation (DSS) based upon low-order expansion in equal-time cumulants. As the magnetic diffusivity is decreased, the nonlinear development of the instability becomes more complicated until eventually a set of parameters is identified that produces a previously unidentified long-term cycle in which energy is transformed from kinetic energy to magnetic energy and back. We find that the periodic transitions, which mimic some aspects of solar variability – for example, the quasiperiodic seasonal exchange of energy between toroidal field and waves or eddies – are unable to be reproduced when eddy-scattering processes are excluded from the model.

Application of the Monte Carlo Method for Simulation of Pattern Formation by Ion-Beam Sputtering of Amorphous Bodies

The formation of ordered structures by ion-beam sputtering of a surface of an amorphous body in the case when a strong nonlinearity has a significant effect on the morphology of the irradiated surface is studied. Three modifications of the Monte Carlo method are used for the numerical simulation of the process, and the first of them is a kind of imitational simulation. It is shown that the direct (imitating) statistical simulation of the ion bombardment of the surface of the target, which best matches the considered physical process and is widely used in other papers, has a serious disadvantage. In the case of imitational simulation, random fluctuations of the depth of the sputtering roughen the target surface to such an extent that none of the modes provided by the continuous model can be observed. It especially concerns the modes set after the long ion bombardment of the surface of the target. However, solutions of the continuous model can be investigated numerically by means of other modifications of the Monte Carlo method with decreased dispersion. Two of these modifications are developed in this paper. Applying these methods, under certain conditions, an ordered structure composed of hexagonally symmetrical hollows is obtained on the surface of the target after the target’s long-term exposure to irradiation by a normal ion beam.

Direct statistical simulation of jets and vortices in 2D flows

In this paper, we perform direct statistical simulations of a model of two-dimensional flow that exhibits a transition from jets to vortices. The model employs two-scale Kolmogorov forcing, with energy injected directly into the zonal mean of the flow. We compare these results with those from direct numerical simulations. For square domains, the solution takes the form of jets, but as the aspect ratio is increased, a transition to isolated coherent vortices is found. We find that a truncation at second order in the equal-time but nonlocal cumulants that employ zonal averaging (zonal CE2) is capable of capturing the form of the jets for a range of Reynolds numbers as well as the transition to the vortex state but, unsurprisingly, is unable to reproduce the correlations found for the fully nonlinear (non-zonally symmetric) vortex state. This result continues the program of promising advances in statistical theories of turbulence championed by Kraichnan.

Formation and energy relaxation of a fast electron beam in cathode regions of a glow discharge in helium

Results of a three-dimensional direct statistical simulation of formation and energy relaxation of a group of high-energy electrons in cathode regions of the low pressure (p < 1 Torr) glow discharge in helium are given. It is demonstrated that the electron distribution function at the exit from the cathode sheath contains a group (beam) of high-energy electrons which did not suffer inelastic collisions and had energy close to the cathode fall potential. The beam is shown to be a main source of charged particle production in the negative glow region. The calculated electron energy distribution function is compared with experimental data.

Numerical simulation of flows over vacuometers by means of the Monte-Carlo method of direct statistical simulation

An upper atmosphere probe is considered. For the atmosphere parameters, a reconstruction method using the density data by a vacuometer located behind a wire screen is constructed. In the context of the Monte-Carlo method of direct statistical simulation, we propose a simplified simulation algorithm for the interactions of a molecule with the wire screen treated as a semipermeable membrane. To obtain a relationship between the transmitter reading and the parameters of the undisturbed atmosphere, we numerically simulate the flow over the transmitter.

On the accuracy of the direct discrete simulation of the Landau collision integral by the Boltzmann integral

The Landau (Fokker–Planck) integral of Coulomb collisions is an integral component of the physical and mathematical models of both laboratory and space plasma, in which the intermediate collisionality regime is important. This paper discusses the method for the direct statistical simulation of the Monte Carlo type for a kinetic equation with a nonlinear operator of the Landau–Fokker–Planck (LFP) Coulomb collisions. This method is based on the approximation of the Landau collision integral by the Boltzmann collision integral. The paper has two main objectives: firstly, to obtain numerical estimates of the order of approximation of the Landau collision integral by the Boltzmann integral and, secondly, to explore the possibilities of optimizing the algorithm for multiply charged ions. The results are illustrated by the calculations of the problem on the relaxation of the initial distribution to equilibrium for one and two components.

Recent progress in understanding deconfinement and chiral restoration phase transitions

Paradigme shift in gauge topology, from instantons to their constituents -- instanton-dyons -- has recently lead to very significant advances. Like instantons, they have fermionic zero modes, and their collectivization at sufficiently high density explains the chiral symmetry breaking. Unlike instantons, these objects have electric and magnetic charges. Their back reaction on the mean value of the Polyakov line (holonomy) allows to explain the deconfinement transition. The talk summarizes recent works on the dyon ensemble, done in the mean field approximation (MFA), and also by direct numerical statistical simulation. Introduction of non-trivial quark periodicity conditions leads to drastic changes in both deconfinement and chiral transitions. In particulaly, in the so called Z(N_c)-QCD model the former gets much stronger, while the latter does not seem to occur at all.

Recent Progress in Understanding Deconfinement and Chiral Restoration Phase Transitions

Paradigme shift in gauge topology, from instantons to their constituents – instanton-dyons – has recently lead to very significant advances. Like instantons, they have fermionic zero modes, and their collectivization at sufficiently high density explains the chiral symmetry breaking. Unlike instantons, these objects have electric and magnetic charges. Their back reaction on the mean value of the Polyakov line (holonomy) allows to explain the deconfinement transition. The talk summarizes recent works on the dyon ensemble, done in the mean field approximation (MFA), and also by direct numerical statistical simulation. Introduction of non-trivial quark periodicity conditions leads to drastic changes in both deconfinement and chiral transitions. In particulaly, in the so called Z (Nc ) – QCD model the former gets much stronger, while the latter does not seem to occur at all.

Three-dimensional rotating Couette flow via the generalised quasilinear approximation

We examine the effectiveness of the generalised quasilinear (GQL) approximation introduced by Marston et al. (Phys. Rev. Lett., vol. 116 (21), 2016, 214501). This approximation splits the variables into large and small scales in directions where there is a translational symmetry and removes nonlinear interactions involving only small scales. We utilise as a paradigm problem three-dimensional, turbulent, rotating Couette flow. We compare the results obtained from direct numerical solution of the equations with those from quasilinear (QL) and GQL calculations. In this three-dimensional setting, there is a choice of cutoff wavenumber for the GQL approximation both in the streamwise and in the spanwise directions. We demonstrate that the GQL approximation significantly improves the accuracy of mean flows, spectra and two-point correlation functions over models that are quasilinear in any of the translationally invariant directions, even if only a few streamwise and spanwise modes are included. We argue that this provides significant support for a programme of direct statistical simulation utilising the GQL approximation.

Heat transfer in the cylindrical rarefied Couette flow

The efficiency of the self-similar interpolation method is demonstrated with reference to the solution of the problem of heat transfer in a rarefied gas between two coaxial cylinders rotating relative one another. The analytical solution of the problem is compared with the results obtained by direct statistical simulation. The most interesting result is the energy flux nonmonotonicity and the reversal of its sign with variation in the Knudsen number.

Generalized Quasilinear Approximation: Application to Zonal Jets.

Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. We present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through nonlocal spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta plane and show that it is accurate even for a small number of large-scale modes. As GQL is formally linear in the small zonal scales, it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems.