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Immersed Boundary Methods for High-Resolution Simulation of Atmospheric Boundary-Layer Flow Over Complex Terrain by Katherine Ann Lundquist A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering - Mechanical Engineering in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Fotini Katopodes Chow, Co-Chair Professor Philip S. Marcus, Co-Chair Professor Ronald W. Yeung Professor Mark T. Stacey Professor Julie K. Lundquist Spring 2010 Immersed Boundary Methods for High-Resolution Simulation of Atmospheric Boundary-Layer Flow Over Complex Terrain Copyright c 2010 by Katherine Ann Lundquist Abstract Immersed Boundary Methods for High-Resolution Simulation of Atmospheric Boundary-Layer Flow Over Complex Terrain by Katherine Ann Lundquist Doctor of Philosophy in Engineering - Mechanical Engineering University of California, Berkeley Professor Fotini Katopodes Chow, Co-Chair Professor Philip S. Marcus, Co-Chair Mesoscale models, such as the Weather Research and Forecasting (WRF) model, are in- creasingly used for high resolution simulations, particularly in complex terrain, but errors associated with terrain-following coordinates degrade the accuracy of the solution. Use of an alternative Cartesian gridding technique, known as an immersed boundary method (IBM), alleviates coordinate transformation errors and eliminates restrictions on terrain slope which currently limit mesoscale models to slowly varying terrain. In this dissertation, an immersed boundary method is developed for use in numerical weather prediction. Use of the method facilitates explicit resolution of complex terrain, even urban terrain, in the WRF mesoscale model. First, the errors that arise in the WRF model when complex terrain is present are pre- sented. This is accomplished using a scalar advection test case, and comparing the numerical solution to the analytical solution. Results are presented for different orders of advection schemes, grid resolutions and aspect ratios, as well as various degrees of terrain slope. For comparison, results from the same simulation are presented using the IBM. Both two-dimensional and three-dimensional immersed boundary methods are then de- scribed, along with details that are specific to the implementation of IBM in the WRF code. Our IBM is capable of imposing both Dirichlet and Neumann boundary conditions. Additionally, a method for coupling atmospheric physics parameterizations at the immersed boundary is presented, making IB methods much more functional in the context of numerical weather prediction models. The two-dimensional IB method is verified through comparisons of solutions for gentle terrain slopes when using IBM and terrain-following grids. The canon- ical case of flow over a Witch of Agnesi hill provides validation of the basic no-slip and zero gradient boundary conditions. Specified diurnal heating in a valley, producing anabatic winds, is used to validate the use of flux (non-zero) boundary conditions. This anabatic flow set-up is further coupled to atmospheric physics parameterizations, which calculate surface 1 fluxes, demonstrating that the IBM can be coupled to various land-surface parameterizations in atmospheric models. Additionally, the IB method is extended to three dimensions, using both trilinear and inverse distance weighted interpolations. Results are presented for geostrophic flow over a three-dimensional hill. It is found that while the IB method using trilinear interpolation works well for simple three-dimensional geometries, a more flexible and robust method is needed for extremely complex geometries, as found in three-dimensional urban environments. A second, more flexible, immersed boundary method is devised using inverse distance weight- ing, and results are compared to the first IBM approach. Additionally, the functionality to nest a domain with resolved complex geometry inside of a parent domain without resolved complex geometry is described. The new IBM approach is used to model urban terrain from Oklahoma City in a one-way nested configuration, where lateral boundary conditions are provided by the parent domain. Finally, the IB method is extended to include wall model parameterizations for rough surfaces. Two possible implementations are presented, one which uses the log law to recon- struct velocities exterior to the solid domain, and one which reconstructs shear stress at the immersed boundary, rather than velocity. These methods are tested on the three-dimensional canonical case of neutral atmospheric boundary layer flow over flat terrain. 2 To my parents. i Contents Contents ii List of Figures vi List of Tables xiii Acknowledgements xiv 1 Introduction and overview 1 1.1 Motivation ..................................... 1 1.2 Background .................................... 2 1.2.1 Differences between computational fluid dynamics and numerical weather prediction ............................ 2 1.2.2 Complex terrain in numerical weather prediction ............ 4 1.2.3 Improvements for numerical simulations of flow over complex terrain 9 1.3 Overview ...................................... 10 1.4 Summary of contributions ............................ 11 2 Background of LES boundary conditions and the immersed boundary method 13 2.1 Large-eddy simulation of wall bounded turbulent flows ............ 14 2.1.1 Structure of the turbulent boundary layer ............... 14 2.1.2 Resolution requirements for DNS and LES ............... 17 2.1.3 Classes of wall models .......................... 18 2.1.4 Wall models for atmospheric applications ................ 22 2.2 Background of the immersed boundary method ................ 23 ii 2.2.1 Formulation of the forcing term ..................... 24 2.2.2 Interpolation methods for boundary reconstruction .......... 26 2.2.3 Combining the immersed boundary method with wall models ..... 31 3 Details of the Weather Research and Forecasting model 33 3.1 General description of WRF ........................... 34 3.2 The governing equations of WRF ........................ 34 3.2.1 The compressible Euler equations .................... 36 3.2.2 Transformation to pressure coordinates ................. 37 3.2.3 Transformation to terrain-following pressure coordinates ....... 40 3.2.4 Perturbation form of the governing equations ............. 42 3.3 Time integration ................................. 43 3.4 Boundary conditions ............................... 43 4 Analysis of numerical errors arising from complex terrain 47 4.1 Introduction .................................... 47 4.2 Numerical method ................................ 48 4.2.1 Coordinate definitions .......................... 48 4.2.2 Metric terms in the governing equations ................ 50 4.3 Idealized advection test .............................. 51 4.3.1 Model set-up and initialization ..................... 52 4.3.2 Results using the default WRF settings ................. 53 4.4 Analysis of truncation errors ........................... 58 4.4.1 Effect of advection scheme ........................ 58 4.4.2 Effect of spatial resolution and grid aspect ratio ............ 60 4.4.3 Effect of coordinate transformation ................... 61 4.5 Conclusions .................................... 63 5 Implementation of the immersed boundary method 64 5.1 Introduction .................................... 64 5.2 Formulation of the numerical solver ....................... 66 5.2.1 Coordinate definition ........................... 66 iii 5.2.2 Governing equations ........................... 67 5.2.3 Discretization schemes .......................... 68 5.3 Treatment at the immersed boundary ...................... 69 5.3.1 Flow reconstruction at the immersed boundary ............ 69 5.3.2 IBM in WRF ............................... 72 5.4 Boundary conditions ............................... 74 5.4.1 Comparison of native and immersed boundary conditions ....... 75 5.4.2 Implementation of the no-slip boundary condition ........... 76 5.5 Inclusion of atmospheric parameterizations ................... 77 5.5.1 Radiation models ............................. 78 5.5.2 Surface physics .............................. 78 5.6 Conclusions .................................... 79 6 Validation of the immersed boundary method 81 6.1 Introduction .................................... 81 6.2 Witch of Agnesi hill simulations ......................... 82 6.3 Idealized valley simulations ............................ 87 6.3.1 Specified surface heating ......................... 89 6.3.2 Coupled surface heating ......................... 93 6.4 Two-dimensional urban terrain simulations ................... 99 6.5 Conclusions .................................... 101 7 Extended implementations of the immersed boundary method 102 7.1 Introduction .................................... 102 7.2 Interpolation in three dimensions ........................ 103 7.2.1 Trilinear interpolation .......................... 103 7.2.2 Inverse distance weighted interpolation ................. 105 7.3 Verification case .................................. 108 7.3.1 Model set-up and initialization ..................... 108 7.3.2 Results ................................... 109 7.4 Flow in urban environments ........................... 116 7.4.1 Model set-up and initialization ..................... 117 iv 7.4.2 Results for flow through urban terrain ................. 121 7.5 Conclusions .................................... 122 8 Wall modeling at the immersed boundary 125 8.1 Introduction

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