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All content in this area was uploaded by Alexandre Chorin on Sep 30, 2015

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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.

Content uploaded by Alexandre Chorin

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All content in this area was uploaded by Alexandre Chorin on Sep 30, 2015

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... In the present work the authors developed a computational code based on a lagrangian description; the numerical method used is called discrete vortex method (Chorin, 1973;Lewis, 1999;Alcântara Pereira et al., 2004;Bimbato et al., 2018). The vortex method discretizes a flow property (in this case, the vorticity field) using discrete particles and offers a number of advantages over the traditional eulerian schemes, such as: (i) it is not necessary the use of a mesh to discretize the fluid domain, avoiding instability problems; (ii) it is not necessary to explicitly treat convective derivates; (iii) only the rotational flow regions are solved; (iv) the boundary condition at the downstream end of the flow domain is automatically satisfied. ...

... In the discrete vortex method, the numerical simulation of the vorticity field evolution in time is realized following Lamb discrete particles. Using the viscous splitting algorithm, proposed by Chorin (1973), the solution is divided in two steps: advection and diffusion, which are solved separately but in the same time increment of the numerical simulation. This procedure makes the numerical implementation of the discrete vortex method simplest. ...

... Several methods have been proposed to solve the diffusion step and here two are highlighted: the random walk method, proposed by Chorin (1973) and modified by Lewis (1991), whereby the diffusion is simulated through the generation of random values used to compute radial and circumferential displacements of the discrete vortices; the core spreading method, proposed by Leonard (1980) and modified by Rossi (1996), in which the diffusion is simulated by the growth and subsequent partition of the discrete vortices in four new particles, according to previously prescribed parameters that govern the partition process. The first method is probabilistic and it has already been implemented in the computaional code developed by the authors (Vidille et al., 2021), but its accuracy is low in comparation with the other diffusion methods, especially when it is compared to the core spreading method. ...

... At first, a pair of discrete vortices was used to simulate the free vortices released by the wing tips; the time progress of the vortices was followed in a Lagrangian representation (the Vortex Method, e.g. references Chorin, 1973;Sarpakaya, 1989;Sethian, 1991;Lewis, 1999;Bimbato, et al., 2018) due they interact with the nascent vortices close to the ground (Hirata et al., 2002). As the pair of vortices move away and rebound, because of the ground effect, one can note the change in the mainly vortical structure along with the secondary structures that develop in the flow, close to the ground. ...

... Using the viscous splitting algorithm originally proposed by Chorin (1973). The vorticity transport equation is split into two steps: one to simulate de advection of the vortex cloud and other to simulate the diffusion of the vortex cloud. ...

... For example, for a fluid dynamics problem, a natural approach is to solve the inviscid and the viscous parts of the equations successively in substeps. This algorithm is known as viscous splitting, see [5]. The case when one of the operators is stiff (this commonly occurs in combustion, air pollution, and reactive flows) is considered in [23]. ...

... Splitting methods are one-step methods of the form (5). Other splitting methods we consider in this paper are the Strang-Marchuk-splitting [25], [19], for which ...

We compare the stability preserving properties of the Lie-Trotter, Strang-Marchuk, and symmetrically weighted sequential splitting schemes for a simple 2-dimensional linear system. We evaluate the trace and determinant of the split systems in terms the trace and determinant of the continuous system to establish stability criteria. We find that the stability region has a fingered structure, implying that stability is not a monotonic property of the splitting timestep. We provide estimates for the thickness of the stability fingers as well as the gaps between them. Counterintuitively, both the thickness and the size of gaps grow with decreasing splitting time step.

... According to the viscous splitting algorithm, proposed by Chorin (1973), in the same simulation time increment, ∆ * , the advective and diffusive terms of the vorticity transport equation (Eq. 7) can be considered separately: ...

... Where the ̅( ) is the velocity filtered field computed at position occupied by k−th vortex blob according to the Biot-Savart law (vortex-vortex interaction), panel method (vortex-panel interaction) and free stream velocity (vortex-mainstream interaction) contributions; for more details about these three contributions, see (1991). The diffusive displacement of each vortex blob governed by Eq. 10 is solved using to the random walk method in the following form (Chorin, 1973): ...

... We have refrained from doing that here because it would raise, in our context, the question how to incorporate diffusion in the particle method. One possibility would be a method similar to Chorin's random walk method for viscous fluid dynamics [12,16]. ...

We derive a differential-integral equation akin to the Hegselmann-Krause model of opinion dynamics, and propose a particle method for solving the equation. Numerical experiments demonstrate second-order convergence of the method in a weak sense. We also show that our differential-integral equation can equivalently be stated as a system of differential equations. An integration-by-parts argument that would typically yield an energy dissipation inequality in physical problems then yields a concentration inequality, showing that a natural measure of concentration increases monotonically.

... ., (1.10) with the initial iteration defined by u (0) (x,t) = u 0 (x) for all x and t ≥ 0. Let us point out that the computational schemes for simulations of turbulent flows based on different formulations of the vorticity equations have been very fruitful in the past, cf. [6,27] for an overview. Our analysis below shows that the iteration scheme developed in this paper does converge to the strong solution with inherent gradient estimates of the iteration solutions uniformly. ...

We develop a new approach for regularity estimates, especially vorticity estimates, of solutions of the three-dimensional Navier-Stokes equations with periodic initial data, by exploiting carefully formulated linearized vorticity equations. An appealing feature of the linearized vorticity equations is the inheritance of the divergence-free property of solutions, so that it can intrinsically be employed to construct and estimate solutions of the Navier-Stokes equations. New regularity estimates of strong solutions of the three-dimensional Navier-Stokes equations are obtained by deriving new explicit a priori estimates for the heat kernel (i.e., the fundamental solution) of the corresponding heterogeneous drift-diffusion operator. These new a priori estimates are derived by using various functional integral representations of the heat kernel in terms of the associated diffusion processes and their conditional laws, including a Bismut-type formula for the gradient of the heat kernel. Then the a priori estimates of solutions of the linearized vorticity equations are established by employing a Feynman-Kac-type formula. The existence of strong solutions and their regularity estimates up to a time proportional to the reciprocal of the square of the maximum initial vorticity are established. All the estimates established in this paper contain known constants that can be explicitly computed.

We present a novel Monte Carlo-based fluid simulation approach capable of pointwise and stochastic estimation of fluid motion. Drawing on the Feynman-Kac representation of the vorticity transport equation, we propose a recursive Monte Carlo estimator of the Biot-Savart law and extend it with a stream function formulation that allows us to treat free-slip boundary conditions using a Walk-on-Spheres algorithm. Inspired by the Monte Carlo literature in rendering, we design and compare variance reduction schemes suited to a fluid simulation context for the first time, show its applicability to complex boundary settings, and detail a simple and practical implementation with temporal grid caching. We validate the correctness of our approach via quantitative and qualitative evaluations - across a range of settings and domain geometries - and thoroughly explore its parameters' design space. Finally, we provide an in-depth discussion of several axes of future work building on this new numerical simulation modality.

The flexible cantilever riser, as a special form of the marine riser, can be encountered in a deep-sea mining system, where the bottom of the long vertical lifting pipeline is connected with the intermediate warehouse. The main objective of this paper is to investigate the effects of the bottom weight caused by the intermediate warehouse and the flow speed on the dynamic responses of the cantilever pipeline. A quasi-3D coupling algorithm based on the discrete vortex method and finite element method is employed to calculate the unsteady hydrodynamic forces and vortex-induced vibrations of this pipeline in the time domain, respectively. We first simulate the VIV of a long flexible riser with two fixed ends in a stepped flow to validate the feasibility of the present method. Then, systematic simulations of cross-flow VIV of the cantilever riser are carried out under a wide range of bottom weights and different current speeds. The number of the vibration mode shows the decreasing tendency with the increase of the bottom weight. In a certain range of the weight, the number of the dominant mode remains unchanged, while the vibration amplitude declines with increasing weight. An amplitude jump phenomenon can be observed when the transition of the dominant mode in two contiguous mode clusters occurs. Moreover, the higher-order modes are excited with the increase of the current speed.

Despite the impressive progress that has been made in numerical modelling of reactive transport in natural porous media, there are still technical issues that need to be further investigated. With significant impact in many applications are the numerical diffusion in coupled flow and transport problems, the solution feasibility for linear flow equations with highly heterogeneous coefficients, and the challenges posed by conducting code verification and convergence tests for degenerate Richards equation. These cumbersome issues are described and commented from the perspective of a generalized random walk solution approach.

The global random walk on grid method (GRWG) is developed for solving two-dimensional nonlinear systems of equations, the Navier–Stokes and Burgers equations. This study extends the GRWG which we have earlier developed for solving the nonlinear drift-diffusion-Poisson equation of semiconductors (Physica A 556 (2020), Article ID 124800). The Burgers equation is solved by a direct iteration of a system of linear drift-diffusion equations, while the Navier–Stokes equation is solved in the stream function-vorticity formulation.

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.