scholarly journals The Utilization of the Generating Function Technique in the Discovery of Solutions for the Three-Dimensional Navier-Stokes Equation System

2020 ◽  
Author(s):  
Robert Jackson
Keyword(s):  
Stokes Equation ◽  
Generating Function ◽  
Three Dimensional ◽  
Equation System ◽  
Navier Stokes ◽  
Function Technique ◽  
Navier Stokes Equation

scholarly journals About local fractional three-dimensional compressible Navier-Stokes equations in Cantor-type cylindrical co-ordinate system

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 847-851 ◽  
Author(s):  
Guo-Ping Gao ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Local Fractional Derivative

In this article, we investigate the local fractional 3-D compressible Navier-Stokes equation via local fractional derivative. We use the Cantor-type cylindrical co-ordinate method to transfer 3-D compressible Navier-Stokes equation from the Cantorian co-ordinate system to the Cantor-type cylindrical co-ordinate system.

scholarly journals ISWFoam: A numerical model for internal solitary wave simulation in continuously stratified fluids

2021 ◽  
Author(s):  
Jingyuan Li ◽  
Qinghe Zhang ◽  
Tongqing Chen
Keyword(s):  
Numerical Model ◽  
Stokes Equation ◽  
Nonlinear Models ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Discrete Equation ◽  
Stratified Fluids ◽  
Flat Bottom ◽  
Navier Stokes Equation ◽  
Laboratory Test Results

Abstract. A numerical model, ISWFoam, for simulating internal solitary waves (ISWs) in continuously stratified, incompressible, viscous fluids is developed based on a fully three-dimensional (3D) Navier-Stokes equation using the open source code OpenFOAM. This model combines the density transport equation with the Reynolds-averaged Navier-Stokes equation with the Coriolis force, and the model discrete equation adopts the finite volume method. The k-ω SST turbulence model has also been modified accordingly to the variable density field. ISWFoam provides two initial wave generation methods to generate an ISW in continuously stratified fluids, including solving the weakly nonlinear models of the extended Korteweg–de Vries (eKdV) equation and the fully nonlinear models of the Dubreil-Jacotin-Long (DJL) equation. Grid independence tests for ISWFoam are performed, considering the accuracy and computing efficiency, the appropriate grid size of the ISW simulation is recommended to be one-one hundred and fiftieth of the characteristic length and one-twenty fifth of the ISW amplitude. Model verifications are conducted through comparisons between the simulated and experimental data for ISW propagation examples over a flat bottom section, including laboratory scale and actual ocean scale, a submerged triangular ridge, a Gaussian ridge and slope. The laboratory test results, including the ISW profile, wave breaking location, ISW arrival time, and the spatial and temporal changes in the mixture region, are well reproduced by ISWFoam. The ISWFoam model with unstructured grids and local mesh refinement can accurately simulate the generation and evolution of ISWs, the ISW breaking phenomenon and the interaction between ISWs and complex structures and topography.

The ADC Method: A Rapid Numerical Technique for Solving Three-Dimensional Navier-Stokes Equation

1982 ◽  
Author(s):  
Gao Ge
Keyword(s):  
Stokes Equation ◽  
Numerical Technique ◽  
Linear Equations ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Navier Stokes ◽  
Damping Factor ◽  
Navier Stokes Equation ◽  
Artificial Damping ◽  
Order Of Magnitude

The artificial damping and compressibility method (ADC) is a rapid numerical technique for solving steady three-dimensional incompressible and compressible Navier-Stokes equation. The principle of this method lies in the introduction of an artificial damping factor into the group of linear equations, by which the elliptic equation is replaced, and the final results do not depend on the artificial terms. The consistency of this method for viscous flow is proved theoretically. The advantages of the SMAC method, the artificial compressibility method and the dynamic relaxation method are retained in the ADC method. Sample calculation of a bending jet is included. The results show that the convergence rate for calculating three-dimensional elliptic flow problems increases by 3–4 order of magnitude as compared with the SOR method.

scholarly journals Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows

2015 ◽  
Vol 783 ◽  
pp. 412-447 ◽  
Author(s):  
Basile Gallet
Keyword(s):  
Reynolds Number ◽  
Stokes Equation ◽  
Three Dimensional ◽  
Threshold Value ◽  
Navier Stokes ◽  
Rossby Number ◽  
Navier Stokes Equation ◽  
Rotating Turbulence ◽  
2D Flow ◽  
Long Time

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body force. This system admits vertically invariant solutions that satisfy the 2D Navier–Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations. We first consider the 3D rotating Navier–Stokes equation linearized around a statistically steady 2D flow solution. We show that this base flow is linearly stable to vertically dependent perturbations when the global rotation is fast enough: under a Reynolds-number-dependent threshold value$Ro_{c}(Re)$of the Rossby number, the flow becomes exactly 2D in the long-time limit, provided that the initial 3D perturbations are small. We call this property linear two-dimensionalization. We compute explicit lower bounds on$Ro_{c}(Re)$and therefore determine regions of the parameter space$(Re,Ro)$where such exact two-dimensionalization takes place. We present similar results in terms of the forcing strength instead of the root-mean-square velocity: the global attractor of the 2D Navier–Stokes equation is linearly stable to vertically dependent perturbations when the forcing-based Rossby number$Ro^{(f)}$is lower than a Grashof-number-dependent threshold value$Ro_{c}^{(f)}(Gr)$. We then consider the fully nonlinear 3D rotating Navier–Stokes equation and prove absolute two-dimensionalization: we show that, below some threshold value$Ro_{\mathit{abs}}^{(f)}(Gr)$of the forcing-based Rossby number, the flow becomes two-dimensional in the long-time limit, regardless of the initial condition (including initial 3D perturbations of arbitrarily large amplitude). These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number$Re$and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone–anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.

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