Numerical solution of the incompressible three-dimensional Navier-Stokes equations (original) (raw)
https://doi.org/10.1016/0045-7930(94)90009-4
Uploaded (2011) | Journal: Computers & fluids
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Related papers
2009
It is not known whether the three-dimensional (3D) incompressible Navier-Stokes equations possess unique smooth (continuously differentiable) solutions at high Reynolds numbers. This problem is quite important for basic science, practical applications, and numerical computations. This review presents a selective survey of the current state of the mathematical theory, focusing on the technical source of difficulties encountered with the construction of smooth solutions.
We present a robust numerical framework for solving the three-dimensional incompressible Navier-Stokes equations using the finite volume method. Our Python-based implementation employs explicit time integration, pressure correction via a Poisson solver, and advanced 3D visualization tools-including vortex identification and particle tracking. The simulations capture the formation, evolution, and dissipation of vortex structures, with a monotonic decay of kinetic energy consistent with the physics of viscous incompressible flows. While this work does not constitute a formal proof, our results provide new insights into the regularity and energy properties of solutions, directly addressing the Clay Mathematics Institute's Millennium Problem. All code and visualization tools are openly available to ensure full reproducibility and to foster further research on the existence and smoothness of Navier-Stokes solutions in three dimensions.While the numerical methods employed are well established, this work distinguishes itself by providing a fully open-source, Python-based 3D framework-complete with advanced visualization, detailed documentation, and explicit orientation towards the Millennium Problem. To our knowledge, no existing resource combines these features with such accessibility and pedagogical clarity.
Proteus Three-Dimensional Navier-Stokes Computer Code-Version 1.0, Volume 1Analysis Description
1993
A computer code called Proteus has been developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort has been to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation have been emphasized.
Navier Stokes Equations3Dsolutions
In this and next papers following shortly, we prove the existence of smooth solution for navier stokes equations. In this paper the equations solutions are proposed. And show the solutions fulfill all the conditions. We left behind how the solution is derived to the next papers. External force components of the navier stokes equations are set to zero for simplicity.
Applied Mathematics Letters, 2009
The three-dimensional incompressible Navier-Stokes equations with the continuity equation are solved analytically in this work. The spatial and temporal coordinates are transformed into a single coordinate ξ . The solution is proposed to be in the form V = ∇Φ + ∇ × Φ where Φ is a potential function that is defined as Φ = P(x, ξ )R(ξ ). The potential function is firstly substituted into the continuity equation to produce the solution for R and the resultant expression is used sequentially in the Navier-Stokes equations to reduce the problem to the class of nonlinear ordinary differential equations in P terms. Here, more general solutions are also obtained based on the particular solutions of P. Explicit analytical solutions are found to be mathematically similar for the cases of zero and constant pressure gradient. Two examples are given to illustrate the applicability of the method. It is also concluded that the selection of variables for the potential function can be interchanged from the beginning, resulting in similar explicit solutions.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.
Related papers
An integrated 2-D Navier–Stokes equation and its application to 3-D internal flows
International Journal of Computational Fluid Dynamics, 2006
The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden.
Navier-Stokes Three Dimensional Equations Solutions Volume Three
Journal of Mathematics Research; Vol. 10, No. 4, 2018
The existence of smooth solution for Navier-Stokes three-dimensional equations is proved by example. The equation is solved by writing initial velocity as the sum of sine and cosine series with proper coefficients.
Explicit solution of the incompressible Navier–Stokes equations with linear finite elements
Applied Mathematics Letters, 2007
A three-field finite element scheme for the explicit iterative solution of the stationary incompressible Navier-Stokes equations is studied. In linearized form the scheme is associated with a generalized time-dependent Stokes system discretized in time. The resulting system of equations allows for a stable approximation of velocity, pressure and stress deviator tensor, by means of continuous piecewise linear finite elements, in both two-and three-dimensional space. Convergence in an appropriate sense applying to this finite element discretization is demonstrated, for the stationary Stokes system.
Related topics
Cited by
Linear instability of the lid-driven flow in a cubic cavity
Theoretical and Computational Fluid Dynamics, 2019
Primary instability of the lid-driven flow in a cube is studied by a linear stability approach. Two cases, in which the lid moves parallel to the cube sidewall or parallel to the diagonal plane, are considered. It is shown that Krylov vectors required for application of the Newton and Arnoldi iteration methods can be evaluated by the SIMPLE procedure. The finite volume grid is gradually refined from 100 3 to 256 3 nodes. The computations result in grid converging values of the critical Reynolds number and oscillation frequency that allow for Richardson extrapolation to the zero grid size. Three-dimensional flow and most unstable perturbations are visualized by a recently proposed approach that allows for a better insight in the flow patterns and appearance of the instability. New arguments regarding the assumption that the centrifugal mechanism triggers the instability are given for both cases.