Soutenance mémoire
EVALUATION OF RANS MODELLING OF WIND TURBINE WAKE FLOW USING WIND TUNNEL MEASUREMENTS
Jonathon Sumner1 and Christian Masson1, 1Département de Génie Mécanique, École de Technologie Supérieure, 1100 Notre-Dame Ouest, Montreal, Quebec, Canada H3C 1K3 Article published online November 2011 by Wiley-Blackell in the International Journal for Numerical Methods in Fluids.
Foreword
In this chapter, the fundamental problem of simulating a steady, incompressible, shear-driven, boundary-layer flow using the RANS equations with standard k − ε closure is analyzed in depth. Although this topic is well-founded in theory and has been previously discussed in the literature, most notably by Richards and Hoxey (1993), is has nonetheless attracted new attention recently (by e.g. Blocken et al. (2007); Hargreaves and Wright (2007); Gorlé et al. (2009); Richards and Norris (2011); Parente et al. (2011)). This renewed interest in such a basic flow has been largely motivated by the inability of many commercial CFD solvers to exactly reproduce the known analytical solution, raising questions regarding their general validity for atmospheric flows. As it turns out, the errors in predicted velocity and turbulence properties can be significantly reduced with properly specified boundary conditions. The most stubborn case is the near-wall turbulent kinetic energy profile, which often contains a sharp anomalous peak in the first few cells. In the following, a systematic analysis of common finite-volume discretization schemes is presented to illustrate the source of this error and demonstrate how it may be remedied.
Abstract
The RANS/k −ε approach is the popular and practical choice for carrying out simulations involving the atmospheric boundary layer. However, despite its widespread use, implementation of this approach is not without its challenges – even when considering the simplest case of horizontally homogeneous conditions. Most notably, the distributions of turbulent kinetic energy and its dissipation rate have proved difficult to maintain near solid boundaries, particularly in wind engineering applications where the near-wall grid is relatively coarse. In this work, the origin of these errors is investigated and it is shown that by applying appropriate discretization schemes in conjunction with the Richards and Hoxey boundary conditions, truly invariant profiles of all flow properties can be obtained on such grids. Furthermore, based on this finding, a wall treatment for coarse grids is proposed that could be implemented for non-homogeneous conditions. All simulations are carried out using OpenFOAM-1.6.x.
Introduction
Simulation of atmospheric boundary layer (ABL) flow is a topic of increasing interest within the computational fluid dynamics community. An accurate description of the mean turbulent flow within the first few hundred metres of the atmosphere is especially pertinent in the analysis of pollutant dispersion, in the evaluation of wind-induced loading on structures, and in determining site-suitability for wind energy projects. Although large-eddy simulation is becoming increasingly popular, the RANS approach remains the practical tool of choice for such work. Within this context, by far the most popular closure scheme is the k − ε turbulence model of Jones and Launder (1972).
Numerically reproducing ABL flow using a RANS/k − ε approach can be divided into two main tasks: the derivation of appropriate boundary conditions and model constants, and their numerical implementation. The first task has been fully addressed by Richards and Hoxey (1993) (RH) and the widely accepted best practice for simulating neutral equilibrium surface layer flow using the k −ε model is laid out in their oft-cited paper (Franke et al., 2007). Their boundary conditions and prescription of model constants are mathematically consistent and ensure the inflow profiles are an exact solution of the model equations.
The second task has been more difficult to address. Maintaining turbulence properties under horizontally homogeneous conditions has proved problematic (Richards and Younis, 1990; Richards et al., 2002; Riddle et al., 2004; Blocken et al., 2008), largely due to challenges in implementing the full RH conditions in commercial software (Franke et al., 2007; Blocken et al., 2007; Hargreaves and Wright, 2007). The presence of streamwise gradients in flow properties can thus often be attributed to the use of an inconsistent set of boundary conditions or to limitations in ks-type wall functions for simulating ABL flow, or both, which initiates the development of an internal boundary layer at the domain inlet (see arguments by Blocken et al. (2007)). Recently, Hargreaves and Wright (2007) have addressed these problems by implementing the full RH boundary conditions, as well as a commonly used subset, in FluentTM, to highlight the importance of using the full set and to demonstrate how to better maintain inlet profiles over flat terrain using commercial software. They significantly reduced the presence of streamwise gradients everywhere except the near-wall region where a spike in the k distribution persists along with an overestimation of ε and an (albeit much smaller) underestimation of U. Even so, many of the original implementation difficulties with commercial software may be considered overcome and, at the same time, the recent availability of high-quality open-source CFD software obviates such challenges as users can modify the source code.
The unmitigated spike in k is often attributed to an overestimation of the turbulence production term in the first few cells nearest the wall. More precisely, it is likely due to an imbalance between calculated production and dissipation terms stemming from the fact that both are dependent on quantities that vary rapidly as z → 0 and are thus poorly approximated using standard finite volume method (FVM) discretization schemes unless cell heights are exceedingly small. This leads to a third task: treatment of the inevitable discretization errors that arise due to the use of coarse grids recurrent in wind engineering applications. This last topic has received less attention and is the focus of the present study. This work thus aims to clearly identify the source of and provide corrections for numerical errors near solid boundaries that arise when using the k − ε turbulence model for ABL flow. Strategies are suggested herein which aim to yield truly horizontally homogeneous distributions of all turbulent flow properties on practical grids; grids which, for whatever reason (e.g. computational time, memory requirements, etc.), do not have sufficient fineness to properly resolve strong near-wall gradients in flow properties. Specifically, wall-damping-style functions are proposed, in the spirit of low-Re k − ε models (Lam and Bremhorst, 1981; Chien, 1982), to adjust the source terms in the k and ε transport equations to correct for discretization errors. Furthermore, it is proposed to correct diffusion terms in the ε and momentum equations by replacing piecewise linear approximations with other weighting schemes inspired by analytically derived near-wall distributions of these quantities. Since it is postulated that these errors are entirely numerical in nature, all corrections are formulated purely in terms of grid geometry. Although the open-source CFD software OpenFOAM (OpenCFD, 2009b) has been used to analyze and develop improved discretization schemes, by interpreting them in terms of corrections to standard discretization methods (at least partial) implementation in commercial software should be possible.
The following section summarizes the RANS/k − ε model as applied to surface layer flow. Section 1.3 outlines the case study used for the analysis of discretization error and a review of boundary conditions is given in section 1.4. Section 1.5 presents, term-by-term, the discretization of the governing equations and derivation of the required corrections. Simulations with corrected discretization schemes and a newly proposed wall treatment are also included.
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Table des matières
INTRODUCTION
0.1 Context
0.1.1 Technical challenges
0.1.2 The need for advanced models
0.2 Scope and methodology
0.2.1 Conservation laws and mathematical modelling
0.2.2 Justification of approach
0.3 Thesis organization LITERATURE REVIEW
0.4 RANS modelling of atmospheric flows
0.4.1 Homogeneous conditions
0.4.2 Heterogeneous conditions
0.5 RANS modelling of wind turbine wakes
0.5.1 Single wake
0.5.2 Multiple wakes
0.6 Contributions
CHAPTER 1 k −ε SIMULATIONS OF THE NEUTRAL ATMOSPHERIC BOUNDARY LAYER: ANALYSIS AND CORRECTION OF DISCRETIZATION ERRORS ON PRACTICAL GRIDS
1.1 Introduction
1.2 Governing equations
1.2.1 RANS equations with k −ε closure
1.2.2 Two-dimensional surface layer flow
1.3 Case study
1.4 Boundary conditions
1.5 Analysis and correction of discretization errors
1.5.1 Solution using standard FVM schemes
1.5.2 Grid sensitivity analysis
1.5.3 Derivation of corrections to standard FVM schemes
1.5.3.1 k −ε equations
1.5.3.2 The momentum equation
1.5.4 Solution using corrected FVM schemes
1.5.5 Implementation as part of wall treatment
1.6 Conclusions
CHAPTER 2 THE APSLEY AND CASTRO LIMITED-LENGTH-SCALE k−ε MODEL REVISITED FOR IMPROVED PERFORMANCE IN THE ATMOSPHERIC SURFACE LAYER
2.1 Introduction
2.2 The equilibrium surface layer
2.2.1 Governing equations
2.2.2 A comment on k −ε closure for stably-stratified surface-layer flow
2.2.3 The mixing length
2.2.4 Imposing a mixing-length limit
2.3 Revised limited-length-scale model
2.3.1 Definition of a weighting function
2.3.2 Derivation of an exact weighting function
2.3.2.1 Stable conditions
2.3.2.2 Neutral conditions
2.3.2.3 Exact expression in the Apsley and Castro form
2.4 One-dimensional simulations
2.4.1 Grid, boundary conditions, and numerics
2.4.2 Neutral length-limited surface layer
2.4.3 Stable surface layer simulation
2.5 Regarding Cε3
2.6 Conclusions
CHAPTER 3 EVALUATION OF RANS MODELLING OF WIND TURBINE WAKE FLOW USING WIND TUNNEL MEASUREMENTS
3.1 Introduction
3.2 Wind tunnel experiments
3.2.1 Reduction of scale
3.2.2 Porous disk and actuator disk theory
3.2.3 Flow conditions
3.2.4 Deducing disk thrust from wake data
3.3 Mathematical models
3.3.1 RANS equations
3.3.2 Turbulence
3.3.3 Computational domain and grid generation
3.3.4 Boundary conditions
3.3.5 Computational considerations
3.4 Results
3.4.1 Velocity defect
3.4.2 Turbulent kinetic energy
3.5 Discussion
3.5.1 The ε equation
3.5.2 Similitude
3.6 Closing remarks
CONCLUSION
ANNEX I RANS SIMULATIONS OF BOLUND
ANNEX II A CLOSER LOOK AT SECOND-ORDER CLOSURE FOR WIND FARM ANALYSIS
APPENDIX 2 DERIVATION OF EXACT WEIGHTING FUNCTION FOR STABLE
CONDITIONS
REFERENCES .
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