In a preceding paper, the theoretical and experimental, deterministic and random, scalar and vector, kinematic structures, the theoretical and experimental, deterministic-deterministic, deterministic-random, random-de...In a preceding paper, the theoretical and experimental, deterministic and random, scalar and vector, kinematic structures, the theoretical and experimental, deterministic-deterministic, deterministic-random, random-deterministic, random-random, scalar and vector, dynamic structures have been developed to compute the exact solution for wave turbulence of exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. The rectangular, diagonal, and triangular summations of matrices of the turbulent kinetic energy and general terms of numerous sums have been used in the current paper to develop theoretical quantization of the kinetic energy of exact wave turbulence. Nested structures of a cumulative energy pulson, a deterministic energy pulson, a deterministic internal energy oscillon, a deterministic-random internal energy oscillon, a random internal energy oscillon, a random energy pulson, a deterministic diagonal energy oscillon, a deterministic external energy oscillon, a deterministic-random external energy oscillon, a random external energy oscillon, and a random diagonal energy oscillon have been established. In turn, the energy pulsons and oscillons include deterministic group pulsons, deterministic internal group oscillons, deterministic-random internal group oscillons, random internal group oscillons, random group pulsons, deterministic diagonal group oscillons, deterministic external group oscillons, deterministic-random external group oscillons, random external group oscillons, and random diagonal group oscillons. Sequentially, the group pulsons and oscillons contain deterministic wave pulsons, deterministic internal wave oscillons, deterministic-random internal wave oscillons, random internal wave oscillons, random wave pulsons, deterministic diagonal wave oscillons, deterministic external wave oscillons, deterministic-random external wave oscillons, random external wave oscillons, random diagonal wave oscillons. Consecutively, the wave pulsons and oscillons are composed of deterministic elementary pulsons, deterministic internal elementary oscillons, deterministic-random internal elementary oscillons, random internal elementary oscillons, random elementary pulsons, deterministic diagonal elementary oscillons, deterministic external elementary oscillons, deterministic-random external elementary oscillons, random-deterministic external elementary oscillons, random external elementary oscillons, and random diagonal elementary oscillons. Symbolic computations of exact expansions have been performed using experimental and theoretical programming in Maple.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In a preceding paper, the theoretical and experimental, deterministic and random, scalar and vector, kinematic structures, the theoretical and experimental, deterministic-deterministic, deterministic-random, random-deterministic, random-random, scalar and vector, dynamic structures have been developed to compute the exact solution for wave turbulence of exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. The rectangular, diagonal, and triangular summations of matrices of the turbulent kinetic energy and general terms of numerous sums have been used in the current paper to develop theoretical quantization of the kinetic energy of exact wave turbulence. Nested structures of a cumulative energy pulson, a deterministic energy pulson, a deterministic internal energy oscillon, a deterministic-random internal energy oscillon, a random internal energy oscillon, a random energy pulson, a deterministic diagonal energy oscillon, a deterministic external energy oscillon, a deterministic-random external energy oscillon, a random external energy oscillon, and a random diagonal energy oscillon have been established. In turn, the energy pulsons and oscillons include deterministic group pulsons, deterministic internal group oscillons, deterministic-random internal group oscillons, random internal group oscillons, random group pulsons, deterministic diagonal group oscillons, deterministic external group oscillons, deterministic-random external group oscillons, random external group oscillons, and random diagonal group oscillons. Sequentially, the group pulsons and oscillons contain deterministic wave pulsons, deterministic internal wave oscillons, deterministic-random internal wave oscillons, random internal wave oscillons, random wave pulsons, deterministic diagonal wave oscillons, deterministic external wave oscillons, deterministic-random external wave oscillons, random external wave oscillons, random diagonal wave oscillons. Consecutively, the wave pulsons and oscillons are composed of deterministic elementary pulsons, deterministic internal elementary oscillons, deterministic-random internal elementary oscillons, random internal elementary oscillons, random elementary pulsons, deterministic diagonal elementary oscillons, deterministic external elementary oscillons, deterministic-random external elementary oscillons, random-deterministic external elementary oscillons, random external elementary oscillons, and random diagonal elementary oscillons. Symbolic computations of exact expansions have been performed using experimental and theoretical programming in Maple.
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.
