An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet pr...An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the Archimedean, Stokes, and Navier problems. The exact solution is obtained with the help of the method of decomposition in invariant structures. Differential algebra is constructed for six families of random invariant structures: random scalar kinematic structures, time-complementary random scalar kinematic structures, random vector kinematic structures, time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector dynamic structures. Tedious computations are performed using the experimental and theoretical programming in Maple. The random scalar and vector kinematic structures and the time-complementary random scalar and vector kinematic structures are applied to solve the Stokes problem. The random scalar and vector dynamic structures are employed to expand scalar and vector variables of the Navier problem. Potentialization of the Navier field becomes available since vortex forces, which are expressed via the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, potential forces, which are described by the scalar potentials of the Helmholtz decomposition, superimpose to generate the gradient of a dynamic random pressure. Various constituents of the kinetic energy are ascribed to diverse interactions of random, three-dimensional, nonlinear, internal waves with a two-fold topology, which are termed random exponential oscillons and pulsons. Quantization of the kinetic energy of stochastic chaos is developed in terms of wave structures of random elementary oscillons, random elementary pulsons, random internal, diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons, random group pulsons, random internal, diagonal, and external group oscillons, a random energy pulson, random internal, diagonal, and external energy oscillons, and a random cumulative energy pulson.展开更多
In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic...In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic scalar dynamic structures, and the theoretical and experimental deterministic vector dynamic structures have been developed to compute the exact solution for deterministic chaos of the exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. To explore properties of the kinetic energy, rectangular, diagonal, and triangular summations of a matrix of the kinetic energy and general terms of various sums have been used in the current paper to develop quantization of the kinetic energy of deterministic chaos. Nested structures of a cumulative energy pulson, an energy pulson of propagation, an internal energy oscillon, a diagonal energy oscillon, and an external energy oscillon have been established. In turn, the energy pulsons and oscillons include group pulsons of propagation, internal group oscillons, diagonal group oscillons, and external group oscillons. Sequentially, the group pulsons and oscillons contain wave pulsons of propagation, internal wave oscillons, diagonal wave oscillons, and external wave oscillons. Consecutively, the wave pulsons and oscillons are composed of elementary pulsons of propagation, internal elementary oscillons, diagonal elementary oscillons, and external elementary oscillons. Topology, periodicity, and integral properties of the exponential pulsons and oscillons have been studied using the novel method of the inhomogeneous Fourier expansions via eigenfunctions in coordinates and time. Symbolic computations of the exact expansions have been performed using the experimental and theoretical programming in Maple. Results of the symbolic computations have been justified by probe visualizations.展开更多
In previous articles,the exact solutions for deterministic chaos,stochastic chaos,and wave turbulence have been developed in terms of exponential os-cillons and pulsons,which are governed by the nonstationary three-di...In previous articles,the exact solutions for deterministic chaos,stochastic chaos,and wave turbulence have been developed in terms of exponential os-cillons and pulsons,which are governed by the nonstationary three-dimen-sional Navier-Stokes equations.We have later considered theoretical quanti-zation of the deterministic chaos in invariant structures and experimental quantization in spatial and temporal eigenfunctions with the help of inhomo-geneous Fourier expansions.The study of exact wave turbulence was also con-tinued with the theoretical quantization of stochastic chaos and wave turbu-lence.The current paper proceeds with experimental quantization of the sto-chastic chaos and the wave turbulence in spatial x-eigenfunctions.The method of inhomogeneous Fourier expansions in the deterministic eigenfunctions has been extended to deterministic-random,random-deterministic,random,ex-ternal,and internal eigenfunctions.The previous results on theoretical quan-tization in invariant structures have been confirmed,analyzed,and visualized in this work using experimental quantization in the novel eigenfunctions.Ar-guments of exact solutions for quantized oscillons and pulsons are given by 1-,2-,3-,4-,5-,6-,8-,12-,15-,16,and 32-tuples of the spatial eigenfunctions.The exact solutions are grouped into the vector,deterministic-random,exter-nal oscillons,the vector,random-deterministic,external oscillons,the vector,deterministic-random,internal oscillons,the vector,turbulent,external oscil-lons,the vector,turbulent,diagonal oscillons,the vector,turbulent,internal,oscillons,and the vector,turbulent pulsons.We compute independent ran-dom parameters with the help of the random model of oscillatory cn-noise.Computation is performed by experimental and theoretical programming in Maple.The obtained results demonstrate a strong dependence of the quan-tized oscillons and pulsons on the Reynolds number.展开更多
文摘An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the Archimedean, Stokes, and Navier problems. The exact solution is obtained with the help of the method of decomposition in invariant structures. Differential algebra is constructed for six families of random invariant structures: random scalar kinematic structures, time-complementary random scalar kinematic structures, random vector kinematic structures, time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector dynamic structures. Tedious computations are performed using the experimental and theoretical programming in Maple. The random scalar and vector kinematic structures and the time-complementary random scalar and vector kinematic structures are applied to solve the Stokes problem. The random scalar and vector dynamic structures are employed to expand scalar and vector variables of the Navier problem. Potentialization of the Navier field becomes available since vortex forces, which are expressed via the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, potential forces, which are described by the scalar potentials of the Helmholtz decomposition, superimpose to generate the gradient of a dynamic random pressure. Various constituents of the kinetic energy are ascribed to diverse interactions of random, three-dimensional, nonlinear, internal waves with a two-fold topology, which are termed random exponential oscillons and pulsons. Quantization of the kinetic energy of stochastic chaos is developed in terms of wave structures of random elementary oscillons, random elementary pulsons, random internal, diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons, random group pulsons, random internal, diagonal, and external group oscillons, a random energy pulson, random internal, diagonal, and external energy oscillons, and a random cumulative energy pulson.
文摘In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic scalar dynamic structures, and the theoretical and experimental deterministic vector dynamic structures have been developed to compute the exact solution for deterministic chaos of the exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. To explore properties of the kinetic energy, rectangular, diagonal, and triangular summations of a matrix of the kinetic energy and general terms of various sums have been used in the current paper to develop quantization of the kinetic energy of deterministic chaos. Nested structures of a cumulative energy pulson, an energy pulson of propagation, an internal energy oscillon, a diagonal energy oscillon, and an external energy oscillon have been established. In turn, the energy pulsons and oscillons include group pulsons of propagation, internal group oscillons, diagonal group oscillons, and external group oscillons. Sequentially, the group pulsons and oscillons contain wave pulsons of propagation, internal wave oscillons, diagonal wave oscillons, and external wave oscillons. Consecutively, the wave pulsons and oscillons are composed of elementary pulsons of propagation, internal elementary oscillons, diagonal elementary oscillons, and external elementary oscillons. Topology, periodicity, and integral properties of the exponential pulsons and oscillons have been studied using the novel method of the inhomogeneous Fourier expansions via eigenfunctions in coordinates and time. Symbolic computations of the exact expansions have been performed using the experimental and theoretical programming in Maple. Results of the symbolic computations have been justified by probe visualizations.
文摘In previous articles,the exact solutions for deterministic chaos,stochastic chaos,and wave turbulence have been developed in terms of exponential os-cillons and pulsons,which are governed by the nonstationary three-dimen-sional Navier-Stokes equations.We have later considered theoretical quanti-zation of the deterministic chaos in invariant structures and experimental quantization in spatial and temporal eigenfunctions with the help of inhomo-geneous Fourier expansions.The study of exact wave turbulence was also con-tinued with the theoretical quantization of stochastic chaos and wave turbu-lence.The current paper proceeds with experimental quantization of the sto-chastic chaos and the wave turbulence in spatial x-eigenfunctions.The method of inhomogeneous Fourier expansions in the deterministic eigenfunctions has been extended to deterministic-random,random-deterministic,random,ex-ternal,and internal eigenfunctions.The previous results on theoretical quan-tization in invariant structures have been confirmed,analyzed,and visualized in this work using experimental quantization in the novel eigenfunctions.Ar-guments of exact solutions for quantized oscillons and pulsons are given by 1-,2-,3-,4-,5-,6-,8-,12-,15-,16,and 32-tuples of the spatial eigenfunctions.The exact solutions are grouped into the vector,deterministic-random,exter-nal oscillons,the vector,random-deterministic,external oscillons,the vector,deterministic-random,internal oscillons,the vector,turbulent,external oscil-lons,the vector,turbulent,diagonal oscillons,the vector,turbulent,internal,oscillons,and the vector,turbulent pulsons.We compute independent ran-dom parameters with the help of the random model of oscillatory cn-noise.Computation is performed by experimental and theoretical programming in Maple.The obtained results demonstrate a strong dependence of the quan-tized oscillons and pulsons on the Reynolds number.
