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

Highlights

• Implementation of the Synthetic Eddy Method into the purely Lagrangian Vortex framework.

• To the author knowledge, one of the first attempt to model ambient turbulence in Lagrangian Vortex simulation.

• The use of variable size structures enable a better representation of the obtained Power Spectral Density.

• A linear relation is found between the mean structure size and the Taylor macro-scale.

• Applications to the computation of (Tidal or Wind) turbine wake are presented as an illustration of the method.

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.

Introduction

This paper deals with the simulation of ambient turbulence in the framework of the Lagrangian Vortex. Simulation of turbulence is among the most active current research topics as many problems remain unresolved, even with the increase of computational capacities. RANS (Reynolds Average Navier-Stokes), and its unsteady version U-RANS, LES (Large Eddy Simulation) and DNS (Direct Numerical Simulation) are the most frequent approaches to this problem in the Eulerian framework. DNS does not assume any model and therefore is the most reliable method, but its CPU time costs are incredibly high. For the present engineering applications, LES is one of the most popular implementations as it allows the computation of relatively complex and large configurations within a reasonable CPU time. Sagaut [1] presents a review of the different procedures commonly used for Large Eddy Simulation. This LES approach is also possible in the Lagrangian framework: some researchers [2], [3], [4] have already carried out LES computations using the Lagrangian Vortex Method. In their last configuration, Mansfield et al. [4] computed a 3D vortex ring collision. Although these computations assessed the influence of turbulence, no ambient turbulence could be taken into account with this approach.

However, in many industrial applications, the ambient turbulence intensity in the upstream flow plays a determining role. This is especially the case in the fields of wind or tidal energy, the latter of which is considered in the present document. Velocity fluctuations induced by ambient turbulence have an impact not only on the performances of an individual turbine, but also on the shape and length of its wake. This is of utmost importance in the design of turbine arrays, when considering the effect of a row of upstream turbines on the power output of any turbines positioned downstream. Experimental studies in potential tidal sites show that this turbulence intensity can range from approximately 3% to 20% [5], [6]. This percentage is calculated from the diagonal components of the Reynolds shear stress tensor and represents a characteristic percentage of the fluctuating velocity component with respect to the averaged incoming velocity field. Such non-negligible variations in inflow conditions must be taken into account when attempting to replicate numerically the true operating conditions of a tidal turbine.

Therefore, various methods have been developed to emulate the ambient turbulence in the context of Eulerian simulations, through the use of boundary conditions. The Synthetic Eddy Method (SEM) proposed by Jarrin et al. [7], [8] was initially formulated within this context: its original purpose is to generate inflow conditions for the Eulerian simulation of turbulent flows. Jarrin et al. [7] defined a set of turbulent structures to represent a desired fluctuating velocity field at the inlet of their computational domain. This approach was already used by several authors such as Afgan et al. [9], or Ahmed et al. [10]. As for marine current turbine simulations, Togneri et al. [11], [12] investigated a similar Synthetic Eddy Method in order to generate synthetic turbulent inflow conditions for their BEMT software. Others use the TurbSim generator from NREL [13] which generates the desired turbulent inflow conditions from a spectral representation of turbulence. For instance, Churchfield et al. [14] used TurbSim to generate inflow conditions for their tidal turbine farm computations. Togneri et al. [15], [16] compare the SEM and spectral based turbulent inflow generation methods in order to investigate fluctuations in loads on the turbine, using their BEMT code. Lastly, Mann’s algorithm [17] is another similar approach based on spectral representation of turbulence. Chatelain et al. [18], [19] used this algorithm to generate a turbulent inflows for wind turbines simulations. The numerical method used by Chatelain et al. [18], [19] is a particle-mesh method relying on an Eulerian mesh at some steps of the numerical scheme, which allows turbulent flow states to be used as inflow boundary conditions. All these cited approaches share the complication of their difficulty to maintain the chosen inflow ambient turbulence intensity level throughout the whole flow domain. As mentioned by Jarrin et al. [7], [8], the ambient turbulence intensity usually decreases as the flow progresses, and the desired input level is generally not recovered in the area of interest of the computational domain.

None of these above mentioned methods can be applied as such to a pure Lagrangian Vortex framework, and the adaptation of one of them is the topic of the present paper. Our numerical model represents the vortical flow field by means of a set of Lagrangian Vortex particles and the velocity field is obtained via the Biot & Savart equation. This paper presents an adaptation of the initial SEM method of Jarrin et al. which maintains the turbulence intensity level over the entire flow domain, ensured by the advection of turbulent structures with no stretching or diffusion. This can be of major importance for the simulation of wind or tidal turbine farms, to ensure that turbines perceive similar levels of turbulent intensity throughout the entire farm area and all the way downstream to the last row of turbines. Additionally the method presented here can function together with both of the most common treatments of diffusion in Lagrangian Vortex methods, namely the Particle Strength Exchange (PSE) [20], [21], [22] and the Diffusion Velocity Method (DVM) [23]. These methods can integrate turbulent diffusion models, such as Large Eddy Simulation [2], [3], [4], [24], to better represent all turbulent length scales.

First of all, an overview is given of the unsteady Lagrangian method, and its treatment of the diffusion is further detailed. The adapted SEM is then presented and analysed on a simple study case. Finally the combination of adapted SEM and Vortex methods is applied to the simulation of a simplified marine current turbine in varying turbulent conditions.