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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.
... 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 ...
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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]. ...
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... ., (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. ...
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