A stochastic method to account for the ambient turbulence in Lagrangian Vortex computations

Abstract

This paper describes a detailed implementation of the Synthetic Eddy Method (SEM) initially presented in Jarrin et al. (2006) applied to the Lagrangian Vortex simulation. While the treatment of turbulent diffusion is already extensively covered in scientific literature, this is one of the first attempts to represent ambient turbulence in a fully Lagrangian framework. This implementation is well suited to the integration of PSE (Particle Strength Exchange) or DVM (Diffusion Velocity Method), often used to account for molecular and turbulent diffusion in Lagrangian simulations. The adaptation and implementation of the SEM into a Lagrangian method using the PSE diffusion model is presented, and the turbulent velocity fields produced by this method are then analysed. In this adaptation, SEM turbulent structures are simply advected, without stretching or diffusion of their own, over the flow domain. This implementation proves its ability to produce turbulent velocity fields in accordance with any desired turbulent flow parameters. As the SEM is a purely mathematical and stochastic model, turbulent spectra and turbulent length scales are also investigated. With the addition of variation in the turbulent structures sizes, a satisfying representation of turbulent spectra is recovered, and a linear relation is obtained between the turbulent structures sizes and the Taylor macroscale. Lastly, the model is applied to the computation of a tidal turbine wake for different ambient turbulence levels, demonstrating the ability of this new implementation to emulate experimentally observed tendencies.

Lagrangian Vortex Computations of a Four Tidal Turbine Array: An Example Based on the NEPTHYD Layout in the Alderney Race

“…In order to take into account the ambient turbulence in the Lagrangian Vortex Blob formulation, the Synthetic Eddy Method (SEM) of Jarrin et al [26] was adapted and implemented (see [31,32]). After a brief presentation of this method, a few numerical parameters will be detailed.…”

“…More details about Jarrin's formulation can be found in [26,54]. A detailed presentation of the adaptation of the SEM to the Lagrangian Vortex Blob formalism is proposed in Choma Bex et al [32]. In particular, different shape functions f λ are considered.…”

“…It is a parameter which can be chosen based on the characteristics of the ambient turbulence which the user desires to reproduce. The results obtained by Choma Bex et al [32] show that λ is proportional to the integral length scale L of the generated flow (estimated from the autocorrelation function of the streamwise velocity). For a Gaussian shape function f λ , L ≈ 0.654λ.…”

“…If only one value is used for λ, all turbulent structures have the same size and this results in a poor reproduction of the velocity power spectral density [31,32]. Thus, in order to better represent the true physical phenomenon of the turbulence, a standard deviation of the size of turbulent structures, denoted σ(λ), was added as a parameter of the model by Choma Bex et al [32]. The size of all turbulent structures are calculated using a normal distribution law centred around an average value still noted λ.…”

“…Before any add-ons, Lagrangian approaches have the benefit of being much less dissipative than their Eulerian counterparts on the advection term of the Navier-Stokes equations. Based on the method proposed by Jarrin et al [26], Pinon et al [31] and more recently Choma Bex et al [32] applied the SEM ambient turbulence generation technique to the Lagrangian Vortex Method. This was not the first attempt with the Lagrangian approach because Chatelain et al [33] already performed computations with ambient turbulence in their Vortex-in-cell formalism.…”