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Numerical Study of Slightly Viscous Flow
Abstract
A numerical method for solving the time-dependent Navier–Stokes equations in two space dimensions at high Reynolds number is presented. The crux of the method lies in the numerical simulation of the process of vorticity generation and dispersal, using computer-generated pseudo-random numbers. An application to flow past a circular cylinder is presented.
... There is that Laplacian term! One of the earliest applications of probabilistic ideas in connection with incompressible Navier-Stokes was introduced by Chorin [13], based on the associated vorticity equation. The vector cross product ω(x, t) = ∇ ∧ u(x, t), i.e. the curl of the velocity field, defines the vorticity. ...
... For numerical computations Chorin [13] viewed (19) by an operator splitting, i.e. along the lines of a Trotter-Kato semi-group product, into successive incremental advective and diffusive flow processes using distinct numerical schemes. ...
... This is the starting point for an approach which has been exploited heavily as a numerical Monte-Carlo method, and has been studied closely by analysts and probabilists alike; e.g. Chorin [13], Marchioro and Pulverenti [39] Goodman [24], Long [38], Szumbarski and Wald [51], Meleard [41]. In this regard, Busnello, Flandoli, and Romito [8] also obtain a natural probabilistic version of the Biot-Savart law in three-dimensions. ...
This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman [Probab. Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al. [Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140, 2004, in press]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.
... It can be treated by the well-known vortex method introduced by Chorin in 1973 [14]. The convergence of the vortex method for two and three dimensional inviscid (σ = 0) incompressible fluid flows was first proved by Hald et al. [25,26], Beale and Majda [2,3]. ...
We rigorously justify the mean-field limit of a N-particle system subject to the Brownian motion and interacting through a Newtonian potential in . Our result leads to a derivation of the Vlasov-Poisson-Fokkker-Planck (VPFP) equation from the microscopic N-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and trajectories following the the mean-field is bounded by () for a system with blob size () up to a probability for any . Moreover, we prove the convergence rate between the empirical measure associated to the particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates crucially rely on the randomness coming from the initial data and from the Brownian motion.
... Connections between stochastic evolution and the deterministic Navier-Stokes equations have been established in seminal work of Chorin [6]. In two dimensions, the nonlinear equation obeyed by the vorticity has the form a Fokker-Planck (forward Kolmogorov) equation. ...
In this paper we derive a representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self-contained proof of local existence for the nonlinear stochastic system, and can be extended to formulate stochastic representations of related hydrodynamic-type equations, including viscous Burgers equations and LANS-alpha models.
... One of our ultimate objectives is to further develop, using the multisymplectic approach, some methods and techniques which were derived in the infinite-dimensional framework and which proved to be very useful. One of them is the vortex blob method developed by Chorin [1973], which recently has been linked to the so-called averaged Euler equations of ideal fluid (see Oliver and Shkoller [1999]). ...
This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper.
In this paper, we introduce a novel neural networks (NN)-based approach for approximating solutions to the two-dimensional (2D) incompressible Navier–Stokes equations, which is an extension of so-called Deep Random Vortex Methods (DRVM), that does not require the knowledge of the Biot–Savart kernel associated with the computational domain. Our algorithm uses a neural network (NN), which approximates the vorticity based on a loss function that uses a computationally efficient formulation of the random vortex dynamics. The neural vorticity estimator is then combined with traditional numerical solvers for partial differential equations, which can be considered as a final implicit linear layer of the network for the Poisson equation to compute the velocity field. The main advantage of our method compared to the standard DRVM and other NN-based numerical algorithms is that it strictly enforces physical properties, such as incompressibility or (no slip) boundary conditions, which might be hard to guarantee otherwise. The approximation abilities of our algorithm, and its capability for incorporating measurement data, are validated by several numerical experiments.
Numerical simulation of fluid plays an important role in the research of engineering, weather, and climate; the classical methods solve incompressible Navier–Stokes equations, providing the most detailed flow information in many cases. However, an increase in the number of grids causes the computing cost to increase significantly. In this paper, we propose a deep learning-based numerical solver for incompressible flow to improve the accuracy of numerical simulation on coarse-resolution grids. The solver uses the Swin Transformer—a hierarchical vision transformer using shifted windows—to build independent subnetworks and learn the interpolating coefficients for the variable values on the cell edge and, thus, to obtain fluxes in different directions. Numerical experiments show that our proposed solver can perform better than the traditional numerical scheme, predicting the solution well and maintaining long-term computational stability.
We consider a new vortex approximation for solving the initial-value problem for the Euler equations in two dimensions. We assume there exists a smooth solution to these equations and that the vorticity has compact support. Then we show that our approximation to the velocity field converges uniformly in space and time for a short time interval.
