Direct and Large Eddy Simulation of Environmental FlowsSutanu Sarkar1& Vincenzo Armenio2To appear in Handbook of Environmental Fluid Dynamics (2010).1 IntroductionNumerical simulation is a powerful tool to both understand and predict environmental flows.These flows are multiscale e.g. a mesoscale eddy in the ocean has scales of motion that rangefrom the eddy’s horizontal extent of say 100 km to the 1 cm scale at which molecular mixingis accomplished. Different spatial scales are governed by qualitatively different physicalprocesses that include geostrophic flow, quasi-geostrophic turbulence, linear and nonlineargravity waves, shear instabilities, convective instabilities and three-dimensional turbulence.Time scales are similarly disparate even if we exclude the interest in local va r iability ofclimate and could rang e from months to hours to seconds.In response to the multiscale and multiphysics nature of environmental flows and todifferences in user-set objectives, different types of numerical models have been developed.These include two-dimensional (in space) or quasi two-dimensional models, hydrostatic three-dimensional models and non-hydrostatic three-dimensional models. Turbulent t r ansport inthe environment is parameterized or resolved in numerical models. Parameterization maybe as simple (and correspondingly inaccurate) as a constant eddy viscosity/diffusivity orconsiderably more complicated such as additional differential equations that govern turbu-lence quantities, e.g. turbulent kinetic energy (TKE) or turbulence length scales. Turbulenceresolving simulations can be o f t he direct numerical simulation (DNS) type or large eddysimulation (LES) type. When a simulation resolves all the dynamically important scales ofmotion, it is t ermed as DNS. Simulations with the order of 500 million grid points have beenperformed by several research gr oups in Environmental Fluid Dynamics (EFD) includinga recent simulation of a stratified wake by [7] that employed a grid with approximately 2billion grid points. The largest DNS to date is the simulation of f orced isotropic turbulence1Department of Mechanical and Aerospace Engineering, University of C alifornia at San Diego2Dipartimento di Ingegneria Civile e Ambientale, Universit`a di Trieste, Trieste Italy1on a 40963grid by [23]. Although DNS with current computing resources can be at Reynoldsnumbers lower tha n in application and is customarily performed in non-complex geometry,it does result in detailed and high-accuracy description of complex nonlinear dynamics andturbulence in environmental flows. Also, the Reynolds number in several simulations hasbeen larger than in corresponding laboratory experiments. LES, when employed with suf-ficient resolution a nd an appropr iate subgrid model, resolves most of the scales responsiblefor turbulent transport but does not resolve the fine scales at which molecular dissipationoccurs.The present chapter reviews DNS and LES of three-dimensional environmental flows.The emphasis is o n canonical flows which provide high-fidelity descriptions of one or moreof the following complications peculiar to t he natural environment: density stratification,rotation, oscillation (as in an ocean tide) or topography. References t o related experimentaland analytical studies of importance will be kept sparse since they will likely be coveredelsewhere in this ha ndbook. Numerical simulation of atmospheric flows is discussed in othercha pters and, therefore, will not be considered here. Subgrid models for LES are discussedin another chapter and will also not be covered here.2 Computational methodsThe three-dimensional, unsteady Navier-Stokes equations (NSE) under the Boussinesq ap-proximation fo r density variation are solved without any additional model in the DNS ap-proach while t he LES approach requires an additional model for the unknown subfilter (orsubgrid) fluxes. There are many texts on computational fluid dynamics which can be con-sulted f or numerical alg orithms for the unsteady, multidimensional Navier-Sokes equations.Most fundamental DNS and LES studies of turbulent mixing in flows of interest to EFDhave been carried out in simple geometrical conditions. In these cases, the Cartesian gridformulation of the NSE has been used and the equations have been numerically integratedusing highly efficient algo rithms. In this section, we will briefly discuss methods fo r flows insimple geometry f ollowed by methods for complex geometry.2DNS of density-stratified turbulent flows started with the study of turbulence tha t isstatistically homogeneous, i.e., statistics of velocity and density fluctuations do not varyin space but may vary in time. Although statistically homogeneous, the fluctuations arestrongly anisotro pic owing to the direct effect of buoyancy in the vertical direction. The firstDNS was perfo rmed by [37] who simulated unforced turbulence, also called box turbulence,subject to uniform strat ification. Periodic boundary conditions were applied in all directionsand a pseudo-spectral algorithm was a dopted. This first DNS at a micro-scale Reynoldsnumber, Reλ= 27.2 utilized only 323points (!) but clearly showed suppression of certaintypes of nonlinear transfers as well as the existence of wave-like behavior. Both observa-tions were consistent with the evolution of stratified turbulent flows in the laboratory. Thenext problem with statistically homogeneous turbulence that was simulated correspondedto uniform mean shear and mean stratification. This enabled study of the competing rolesof mean shear and stratification on turbulent fluctuations. A pseudo-spectral algorit hm andresultant spectral accuracy was possible by transforming to a coordinate system advectedwith the mean flow.In shear flows, turbulence statistics often have one direction of strong variation, e.g., thecross-stream direction in flows such as a finite-thickness shear layer, a plane jet or a planefar wake. Periodic boundary conditions in the horizontal directions are natural for suchflows and, therefore, Fourier collocation can be used in the horizontal. Finite difference(possibly with high accuracy such as Pad´e derivatives) is used in the vertical. Solution ofthe pressure equation, equivalently the enforcement of the divergence-free constraint o n thevelocity, is necessary. A projection method or fractional step method is customarily utilizedfor this purpose. Time advancement can be accomplished by a Runge-Kutta (RK)