Numerical Modelling of Stratified ABL Flow over Complex Terrain

Numerical Modelling of Stratified ABL Flow over Complex Terrain 

With the electronics and computer revolution experienced since the mid-twentieth century, the field of high-performance CFD has been increasingly transformed into a demanding scientific discipline, in which researchers are devoted to develop “ice-breaking” numerical methods, problem-specific codes and advanced post-processing techniques. The capabilities of each numerical solver for partial differential equations are diverse with respect to the dynamical and physical models, grid generation and post-processing of the results. However, none of these milestone novelties has fully succeeded the precise prediction or realization of turbulent flows, which is by far too complex to be completely characterized with any particular method.

Multiple spatial and time discretization, physical parameterization, numerical approximation and initialization schemes have been implemented, intensively tested, compared, enhanced and validated over the past 50 years. These efforts have paved the way to achieve a better understanding and prediction of turbulent flows, but a basic and stiff constraint persists. While the equations of motion can be applied directly to turbulent flows, the CFD models most appropriate for this application would need an exceedingly small grid spacing, and there would still be eddies that would not be resolved on the model grid. The effects of these subgrid eddies are yet to be accounted for in some way, which usually is based upon a statistical approach to eddy effects. Namely, it is clear that for a pseudo-complete description and prediction of turbulent flow, the grid spacing (implicit filter) or explicit filter size employed should not go beyond the length scale of the smallest turbulent eddy (η ). For a 3D simulation, the grid size needed to account for the smallest eddies should be on the order of 9 4 Re or in the range of 9 12 10 −10 grid points (Pope 2000, Zikanov 2010).

Thus, amongst all the numerical approaches developed up to this date, the Direct Numerical Simulation (DNS) and Large-eddy Simulation (LES) techniques replicate the closest realization of turbulent flow patterns, DNS being the most precise since it solves the NavierStokes equations without modelling modifications or assumptions. Nonetheless, the overwhelming requirement for accurate DNS approximations of flow features with very fine grid stepping leads to large computational grids when the Reynolds number exceeds 10⁴  . On the contrary, the LES approach is able to represent transient flow phenomena by applying spatial filtering on the prognostic variables, yielding an accurate realization of large and medium size flow features without an unrealistic large grid. Reynolds Averaged Navier-Stokes (RANS) technique is also widely used and accepted, although it recasts the turbulence transient evolution only based on mean flow quantities, hence, corresponding to the expectations of these characteristics that could be obtained after averaging over several realizations (Moin and Mahesh 1998, Stensrud 2007, Davidson 2015). Many RANS studies can be based on steady state and/or 2D realizations, while LES is inherently time-dependent and 3D.

Although the LES does not provide a description of the full spectrum of motions that DNS permits (due to its inherent modelling error of the small scale approximation), and may demand a slightly higher computational overhead than RANS depending on the desired accuracy, it occupies an intermediate position with balanced capabilities for predicting sufficient turbulent fluctuations for practical scientific computations. Additionally, the RANS-LES hybrid approach, proposed by Spalart and other researchers (Strelets 2001, Spalart 2009, Bechmann and Sørensen 2010, Cabezón 2013), takes advantage of the time-averaged modelling for mean statistics of the wall-bounded flow carried out with RANS and the filtered large-scale resolved flow’s interior obtained with LES. The hybrid RANS-LES modelling method, also known as detached eddy simulation (DES), merges both mean statistics and low-pass-filtered large-scale resolved structures to reproduce the turbulent flows at high resolution.

These methods are applicable both separately or combined for general-purpose engineeringtype flow analysis and atmospheric flow research. Particularly, for ABL turbulence parameterization, column models and RANS K − models have been historically preferred (Gasset, 2014). Different first and second-order closure schemes based on turbulent kinetic energy ( K ), dissipation (ε ), specific dissipation (ε / K ), turbulent length scale (l ) and time scale (τ ), fluctuating transverse velocity (v′), pressure-strain correlations and other Reynolds stress relations have been implemented successfully and appear to be well-suited for mesoscale simulations (e.g. K −ε SST, RNG K −ε , K −ω SST, 2 K −ω , K −ε −ω , K −l , RSM, ASM, etc.). Nonetheless, the LES models have gained significant attention and appreciation for microscale simulations, since they are more versatile, comprehensive and just slightly more computationally demanding with parallelized codes than RANS K − models (Wilcox 2006, Bechmann et al. 2011, Bengston 2015, Breton et al. 2017). Then, imbedded LES methods in mesoscale solvers seem a logical and bright combination to enable multiscale capabilities. In this scenario, a distinctive imprint of the coupled mesoscale-LES method is given by the interaction between the numerical solution method and subgrid scale (SGS) parameterization scheme (e.g. standard or dynamical eddy-viscosity models, standard or unsteady RANS models, scale-dependent or scale-invariant models, algebraic or spectral reconstruction models, etc.) (Germano et al. 1991, Lilly 1992, Mason and Thompson 1992, Porté Agel et al. 2000, Meneveau and Katz 2000, Ding et al. 2001, Chow et al. 2005, Sumner et al. 2010, Dellwik and Arnqvist 2014, Yu et al. 2017).

Coupling Large-Eddy Simulation (LES) with Mesoscale Modelling 

Because the LES computational domain usually spans over a limited area, it must obtain its lateral boundary conditions from observations, analyses or larger-model grids with resolutions on the mesoscale. In this sense, a coupled mesoscale-LES model can operate with constant or variable boundary conditions, depending on large-scale flow evolution for the problem under examination. Also, initial conditions are typically prescribed from relatively smooth and horizontally uniform fields to allow microscale forcing to develop from local orography and vegetation features. In many cases distinct models simulate independently the meso- and microscale structures, thus, allowing a one-way coupling interaction. On the contrary, there is a two-way scale interaction within modern coupled mesoscale-LES methods, where the dynamical core can run as a regular mesoscale model with inner grids using LES closures and outer grids using standard mesoscale closures (Wyngaard 2004, Sumner and Masson 2010, Bechmann et al. 2011, Warner 2011).

A significant issue with the mesoscale-LES boundary conditions is that the inflow boundary is generally defined by an atmosphere for which the turbulence is parameterized, as the motions unresolved by the model are treated with a subfilter-scale (SFS) closure. If the buffer zone between nested grids is not large enough, the microscale processes will not spin-up sufficiently as the bulk airflow enters the central region of the grid (Cushman-Roisin and Beckers, 2011). Hence, no resolved microscale turbulent structures are considered to enter the LES grid, and because of the short residence time of the airflow within those grid cells there may not be sufficient time for realistic turbulence to develop before the air exits the outflow boundary. This discussion is intimately related to Wyngaard’s pioneering analysis (Wyngaard, 2004), which clearly states there is still no sufficient consensus on how to apply multiscale modelling that achieves spatial-filter scales within the part of the spectrum containing the turbulent energy (i.e. the terra incognita). There is always a trade-off between the mesoscale long wavelength grid size (for which turbulence is clearly unresolved) and the required LES short wavelength spatial filter size sufficiently small to capture the flow structures exchanging turbulent kinetic energy (TKE) within the inertial range.

Another important situation-dependent aspect to consider in coupled mesoscale-LES methods is the sensitivity of the LES model solution to errors carried by lateral boundary conditions from the mesoscale results. For example, overestimated wind speeds may not impact significantly on the forecast of sensible weather, but it affects adversely the initial and boundary conditions of the LES microscale model. Although the aim of this study is to benefit from the important advantages of mesoscale-LES coupling, it must be understood there could be negative influences on the results attributed mainly to (Durran 2010, Warner 2011, Dellwik and Arnqvist 2014, Bengtsson 2015):
• Low resolution of mesoscale data, that impacts the interpolated boundary conditions;
• Errors in the mesoscale data, arising from the quality of the data assimilation methods, poor numerical noise control, physical parameterization inconsistencies, etc.;
• Lack of interaction between small and large scales, which may occur when there is only one-way communication from mesoscale to LES nested grids, or due to limited energy backscatter;
• Subfilter scale modelling errors in partitioned models, in terms of space and time discretizations, coordinate systems or numerical solution schemes that may cause spurious gradients and feedbacks; and
• Noise generation due to nonphysical inertia-gravity modes that may mix with meteorological solutions, and grow exponentially quickly causing floating-point overflow conditions that halt the model computations.

Solutions for each of the limitations stated above, except for the last one, have been adopted to some extent in previous studies for the mesoscale-LES method used and exanimated within the present work (Girard et al. 2005, Gasset et al. 2014). Our effort will concentrate, not entirely though, on the implementation of a new semi-implicit time discretization for mesoscale modelling to remove the inherent spurious computational mode and terrain-induced noise in presence of steep slopes, ensuring the best possible quality of the imbedded large-eddy simulation method. This is performed considering the important advantages of mesoscale-LES coupled capabilities in engineering applications, such as (Warner, 2011):
• Better understanding of microscale atmospheric turbulence, that can lead to improved numerical parameterizations of the ABL physical phenomena;
• More insightful study of wind turbine siting over complex terrain to maximize the wind power harvest, as well as to minimize the turbulent loading on the generator;
• Wake turbulence analysis obtained for specific types of structures (e.g. buildings or wind turbines) that allow safe spacing to reduce turbulent loading or shadowing, as well as wind farm array interactions;
• More precise modelling of transport phenomena within urban sites, to prevent hazardous gases, aerosols or light pollutants to diffuse from a transportation or industrial accident.

In the context of turbulent ABL flow modelling over topography, the conventional requirement for a successful mesoscale-LES implementation is the appropriate realization of the neutrally stratified flow over a homogenous flat surface, achieving the constant-flux similarity within the surface layer and the equilibrium of all scales across the domain. Once this is achieved, it is necessary to replicate the expected flow features accounting for rotational, thermal stratification and terrain-induced effects over heterogeneous surfaces, which comprises complex orography as well as variable land use, surface cover, local weather, etc. This implies the use of suitable discretization and solution schemes, turbulence parameterization, boundary conditions and computational grid. Numerically reproducing the equilibrium of the TKE production and its dissipation rate for neutrally stratified flow over flat or complex terrain has proved to be a difficult task mostly near solid boundaries (Maurizi 2000, Dalpé and Masson 2008, Sumner and Masson 2010, Brasseur and Wei 2010).

Table des matières

INTRODUCTION
CHAPTER 1 ATMOSPHERIC BOUNDARY LAYER FUNDAMENTALS AND MODELLING – CONCEPTUAL FRAMEWORK AND LITERATURE REVIEW
1.1 Dynamics of the Atmospheric Boundary Layer (ABL)
1.1.1 Atmospheric Thermal Stratification and the ABL Structure
1.1.2 Rotational and Topographic Effects on Stratified ABL Flow
1.1.3 Turbulence and Diffusion in Stratified ABL Flow
1.2 Numerical Modelling of Stratified ABL Flow over Complex Terrain
1.2.1 Coupling Large-Eddy Simulation (LES) with Mesoscale Modelling
1.2.2 The Mesoscale Compressible Community (MC2) Model
1.2.3 Numerical Stability of the SISL Leapfrog Method
1.2.4 Model Equations Filtering and Turbulence Parameterization
1.2.5 Boundary Conditions and Computational Grid
CHAPTER 2 NUMERICAL STABILITY AND NOISE CONTROL OF A NEW SEMIIMPLICIT SCHEME FOR MESOSCALE MODELLING OVER STEEP TERRAIN
2.1 Background and Context
2.2 Basic Semi-discrete Model Equations
2.3 Stability Analysis of the Original SI (O-SI) Scheme
2.4 The New Semi-implicit (N-SI) Scheme
2.5 Validation and Discussion
2.5.1 Atmosphere-at-rest Simulations with Different Terrain Slopes
2.5.2 Atmosphere-at-rest Simulations with Different Parametric Combinations
2.5.3 Atmosphere-at-rest Multi-layer Simulations over Steep Parallel Ridges87
2.6 Summary
CHAPTER 3 ENHANCED MESOSCALE MODELLING OF THE STRATIFIED SURFACE LAYER OVER STEEP TERRAIN FOR WIND RESOURCE ASSESSMENT
3.1 Background and Context
3.2 Model Description and Main Issues
3.3 Model Enhancement
3.3.1 The New Wind Climate-state Classification for the SDD Method
3.3.2 The New Semi-Implicit (N-SI) Scheme for Numerical Noise Reduction
3.4 Validation and Discussion
3.4.1 Validation of the Model Enhancements
3.4.2 Numerical Simulation of Strongly Stratified Wind over the Whitehorse Valley
3.5 Summary
CHAPTER 4 WIND MODELLING OVER STEEP TERRAIN WITH LARGE EDDY SIMULATION EMBEDDED IN A MESOSCALE ATMOSPHERIC MODEL
4.1 Background and Context
4.2 Model Equations and Numerical Enhancements
4.3 Validation and Discussion
4.3.1 Neutral ABL Simulations over Flat Terrain for Model Calibration
4.3.2 Neutral ABL Simulations over an Isolated Gaussian Ridge
4.3.3 Neutral ABL Simulations over the RUSHIL H3 Ridge
4.4 Summary
CONCLUSION

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