Study of the higher-order asymptotic behavior of quantum field expansions in the theory of two-dimensional fully developed turbulence

We use a simplified (0+1)-dimensional theory to develop approaches for studying the higher-order asymptotic behavior of quantum field expansions in the two-dimensional theory of fully developed turbulence. We consider the asymptotic behavior of the correlation function in the small-time limit in the theory of fully developed turbulence and derive and investigate the stationarity equation. We show that the perturbation series in this limit has a finite convergence radius.     

Landau-Hopf scenario of passage to turbulence in some problems of elastic stability theory

We consider a boundary value problem for a nonlinear integro-differential equation modeling the vibrations of a pipe transporting a fluid and the vibrations of a thin cylinder in an axisymmetric flow. For the boundary conditions we take the hinge support conditions. We show that the Landau-Hopf scenario of passage to turbulence can be realized in such a problem, whereby stable invariant tori of increasing dimensions are generated as the main bifurcation parameter increases.     

Calculated spectra of decaying turbulence developed in the energy-containing and inertial ranges

We construct an approximate solution to the equation of energy spectrum balance for developed homogeneous isotropic turbulence in the dissipative range. This equation is closed in the framework of the statistical model we use. The model is based on the principle of maximum randomness, renormalization group technique, and -expansion. We calculate the energy spectra and the transfer functions in the cases where the wave numbers relate to the energy-containing and inertial ranges. The Kolmogorov constant is found to be C _K = 1.55. It is shown that the one-dimensional longitudinal energy spectrum calculated without any free fitting parameters and normalized in von Karman-Hovart energy units, is in good agreement with experimental data on decaying turbulence.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 416–424, March, 1996.Translated by M. V. Chekhova.     

On the Turbulence Modelling of 3D-Atmospheric Boundary Layer Flow

The paper presents a mathematical and numerical investigation of the 3D–flow in the atmospheric boundary layer (ABL) over complex relief. The two–equation k - ε model is applied to account for the turbulence. The flow is also supposed to be viscous, incompressible and stationary. The boundary conditions are realized through the wall-functions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonstationary wave turbulence

Summary Nonstationary regimes of the wave turbulence evolution are considered in the framework of isotropic kinetic equation. It is predicted analytically and confirmed by numerical experiment that there is a class of wave systems in which any initial distribution of the turbulence energy ink-space comes into a universal, Kolmogorovtype spectrum in a finite time. Before and after the formation of the Kolmogorov spectrum, two different self-similar regimes of evolution occur: the first one is responsible for explosively forming the universal spectrum and the second one determines energy dissipation.     

Effective Dynamics of the Nonlinear Schrödinger Equation on Large Domains

We consider the nonlinear Schrödinger (NLS) equation posed on the box [0,L]^d with periodic boundary conditions. The aim is to describe the long‐time dynamics by deriving effective equations for it when L is large and the characteristic size ɛ of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in the theory of statistical physics of dispersive waves, which goes by the name of “wave turbulence.” Our main result is deriving a new equation, the continuous resonant (CR) equation, which describes the effective dynamics for large L and small ɛ over very large timescales. Such timescales are well beyond the (a) nonlinear timescale of the equation, and (b) the euclidean timescale at which the effective dynamics are given by (NLS) on ℝ^d. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy‐Littlewood circle method, which are modified and extended to be applicable in a PDE setting.© 2018 Wiley Periodicals, Inc.     

On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients

We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. Furthermore, it is shown that the Galerkin approximation of the state solution can be calculated by means of the Kachanov method as the limit of a sequence of solutions to linearized problems.     

Elementary divisors of induced transformations on tensors via the representation theory of sl 2

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DA^t positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.     

The principle of maximum randomness in the theory of fully developed turbulence. II. Isotropic decaying turbulence

A statistical model for describing the decay of developed isotropic turbulence of an incompressible fluid is proposed. The model uses the distribution function of the velocity pulsations introduced earlier by the authors on the basis of the principle of maximum randomness of the velocity field for a given spectral energy flux. The renormalization-group technique and expansion are used to calculate the correlation functions of the velocity that occur in the equation of spectral energy balance. This leads to a closed equation for the dependence of the energy spectrum on the integral turbulence scaler _c(t). In the inertial interval, this equation gives the Kolmogorov asymptotic spectrum, while for the time dependence ofr _c(t) and the pulsation energye(t) it predicts the power lawsr _c(t)t^2/5 andr(t)t ^–6/5.Physics Research Institute of the St Petersburg University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 150–159, July, 1993.     

关于湍流理论中的不封闭性的讨论*

有一种观点认为湍流的不封闭性是由于N-S方程的非线性,这是一似是而非的错误说法。因为工程上要求的量并不仅仅限于湍流的平均速度和平均压力,即使方程式是线性的,也无法把工程上要求的这些量逐个求出来。本文论证了湍流的不封闭性来源于目前的湍流理论中缺少了一个与实际情况符合的有关分布函数,并且进一步说明了湍流模式理论的局限性和用N-S方程直接计算机模拟亦是相当困难的原因。     

Some experimental results on flow in a diverging 2D channel

The effect of the free stream turbulence (FST) on the essential flow characteristics was investigated in the diverging 2D channel. The diverging channel was modelled in the closed type working section by fastening a displacement body above the flat plate that is parallel with the wind-tunnel axis. The angle of the channel expansion 11 degree induced the pressure gradient parameter with the flow velocity U₀ 30 m/s at the start of expansion, x = 0. The height of the channel is 0.15 m at x = 0. FST was either natural 0.3 percent or amplified by turbulence generating grids/screens with turbulence levels up to 5 percent. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact soliton solution for the fourth‐order nonlinear Schrödinger equation with generalized cubic‐quintic nonlinearity

In this paper, we investigate the fourth‐order nonlinear Schrödinger equation with parameterized nonlinearity that is generalized from regular cubic‐quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth‐order generalized cubic‐quintic nonlinear Schrödinger equation through modified F‐expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.     

On the nature of turbulence in Rayleigh-Benard convection

In the present paper, we suggest a numerical method for the analysis of the motion of a viscous incompressible fluid under the passage to the turbulent mode in Rayleigh-Benard convection. We show that one possible scenario of the onset of turbulence in the problem is given by a subharmonic cascade of bifurcations of stable two-dimensional tori.     

Dynamical systems and simulation of turbulence

We propose an approach to the analysis of turbulent oscillations described by nonlinear boundary-value problems for partial differential equations. This approach is based on passing to a dynamical system of shifts along solutions and uses the notion of ideal turbulence (a mathematical phenomenon in which an attractor of an infinite-dimensional dynamical system is contained not in the phase space of the system but in a wider functional space and there are fractal or random functions among the attractor “points”). A scenario for ideal turbulence in systems with regular dynamics on an attractor is described; in this case, the space-time chaotization of a system (in particular, intermixing, self-stochasticity, and the cascade process of formation of structures) is due to the very complicated internal organization of attractor “points” (elements of a certain wider functional space). Such a scenario is realized in some idealized models of distributed systems of electrodynamics, acoustics, and radiophysics. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 217–230, February, 2007.     

Incompressible liquid in thermodynamics, new entropy, and the scenario for the occurrence of turbulence for the Navier-Stokes equation

In this paper, we propose a new scenario for the occurrence of turbulence on the basis of the new notion of entropy of an ideal liquid and the theory of coherent vortices.     

Turbulent flow computations on 3D unstructured grids

An edge-based finite element method is presented for the simulation of compressible turbulent flows on unstructured tetrahedral grids. A two equation k–ω turbulence model is employed and the standard Galerkin approach is used for spatial discretisation. Stabilisation of the resulting procedure is achieved by the addition of an appropriate diffusion. An explicit multistage time-stepping scheme is used to advance the solution in time to steady state. The performance of the algorithm is demonstrated for the simulation of a high Reynolds number transonic separated flow over a wing.     

Zeros of Racah coefficients and the pell equation

The interface between Racah coefficients and mathematics is reviewed and several unsolved problems pointed out. The specific goal of this investigation is to determine zeros of these coefficients. The general polynomial is given whose set of zeros contains all nontrivial zeros of Racah (6j) coefficients [this polynomial is also given for the Wigner-Clebsch-Gordan (3j) coefficients]. Zeros of weight 1 3j- and 6j-coefficients are known to be related to the solutions of classic Diophantine equations. Here it is shown how solutions of the quadratic Diophantine equation known as Pell's equation are related to weight 2 zeros of 3j- and 6j-coefficients. This relation involves transformations of quadratic forms over the integers, the orbit classification of zeros of Pell's equation, and an algorithm for determining numerically the fundamental solutions of Pell's equation. The symbol manipulation program MACSYMA was used extensively to effect various factorings and transformations and to give a proof.The results of this paper were presented in an invited talk by one of us (JDL) at the NSF-CBMS Regional Conference on Special Functions, Physics and Computer Algebra, May 20–24, 1985, Arizona State University, Tempe, AZ.     

Influence of turbulence on the performance of a laboratory scale HAWT

The oncoming wind to horizontal axis wind turbines (HAWT) may change its speed and direction stochastically in time. Hence, turbine blades are exposed to flows both with fluctuating angle of attack and fluctuating yaw angles. The modern wind turbines are reacting to those changes by pitch angle and torque control not only to exploit as much power as possible but also stabilize energy production and prevent any damage of the turbine. However, time scales of wind fluctuations and sudden changes of wind properties can be very short and with very high in amplitude. In the present study we focus on the influence of turbulence on the performance of a HAWT. Main motivation of the investigations is to figure out best strategies for the aerodynamic design of the blades operating under turbulent conditions. A laboratory scale HAWT and a performance measurement set-up are employed to measure the influence of the oncoming wind. The tests are conducted in the closed loop wind tunnel of our institute. The test section of the tunnel is 1.87 m in width, 1.4 m in height and 2 m in length. The rotor blades are specially designed and optimized for this wind tunnel and the generator used. The turbulence is generated by two static squared mesh grids; fine and coarse one. Hence, two mainly different turbulence scales are obtained. In addition, the distance between the wind-turbine and the grid is adjusted to have additional sub-turbulence scales for each grid. The turbulence is nearly isotropic and decays in the flow direction. The developments of Taylor's micro scale (λ_g) and integral scale (L_g) of the turbulence in the flow direction at various incoming wind velocities (8−16 m/s) are measured. Hence, the facility allows to expose the wind-turbine to turbulence with various energy and length scale content. Those measurements are conducted with hot-wire anemometry in the absence of the wind-turbine. Upstream and downstream turbulence intensities (TI) distributions are measured to give insight on the surrounding free stream and turbine wake interaction and how can different turbulence eddies scales contribute in the influence of the performance of the turbine. Performance measurements are conducted with and without turbulence and the results are compared. The study shows that the higher the turbulence, the more the power extracted by the turbine. This is due to the higher interaction of large eddies with the turbine wake and with the boundary layer, which helps to keeping it attached. Furthermore, higher TI's help in suppressing the tip vortex, thus, reduce turbine tip losses. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Turbulence in Nonlinear Schrödinger Equations

We consider the small-dispersion and small-diffusion nonlinear Schr?dinger equation , , where the space-variable x belongs to the unit n-cube () and u satisfies Dirichlet boundary conditions. Assuming that the force is a zero-meanvalue random field, smooth in x and stationary in t with decaying correlations, we prove that the C ^m -norms in x with of solutions u, averaged in ensemble and locally averaged in time, are larger than , . This means that the length-scale of a solution u decays with as its positive degree (at least, as and - in a sense - proves existence of turbulence for this equation. Submitted: August 1996, revised version: May 1997