This article reviews the state-of-the-art numerical calculation of wind turbine wake aerodynamics. Different computational fluid dynamics techniques for modeling the rotor and the wake are discussed. Regarding rotor modeling, recent advances in the generalized actuator approach and the direct model are discussed, as far as it attributes to the wake description. For the wake, the focus is on the different turbulence models that are employed to study wake effects on downstream turbines.

This paper examines a class of explicit finite-difference advection schemes derived along the method of lines. An important application field is large-scale atmospheric transport. The paper therefore focuses on the demand of positivity. For the spatial discretization, attention is confined to conservative schemes using five points per direction. The fourth-order central scheme and the family of K-schemes, comprising the second-order central, the second-order upwind, and the third-order upwind biased, are studied. Positivity is enforced through flux limiting. It is concluded that the limited third-order upwind discretization is the best candidate from the four examined. For the time integration attention is confined to a number of explicit Runge-Kutta methods of orders two up to four. With regard to the demand of positivity, these integration methods turn out to behave almost equally and no best method could be identified. '~' 1995 Academic Press, Inc. l. INTRODUCTIONThe subject of this paper is the numerical solution of the partial differential equation for linear advection of a scalar quantity w in an arbitrary velocity field u, given byLinear advection is an important (classical) problem in computational fluid dynamics and has been the subject of numerous investigations. The central theme is how to approximate the advection term \7 · (uw), such that the resulting errors in both phase and amplitude are minimized and the computational cost is still affordable. An important application we have in mind concerns atmospheric transport of chemical species. Then w represents a concentration or density and u a wind field. In addition to the usual accuracy and efficiency requirements, here the main consideration is that the transported concentrations must remain positive, because in actual applications also chemical reactions are modeled for which positivity is a prerequisite for avoiding non-physical chemical instabilities. We emphasize *The research reported helongs to the projects EUSMOG and CIRK which are carried out in cooperation with the Air Laboratory of the RIVM-Tbe Dutch National Institute of Public Health and Environmental Protection. The RIVM is acknowledged for financial support. 35that the demand of positivity is important and that it severely restricts the choice of method, as it is essentially equivalent to the demand of avoiding numerical under-and overshoots in regions of strong variation.The research objective of this paper is to ex.amine a class of positive, finite-difference advection schemes which we consider promising for atmospheric transport applications and to select from this class the best possible candidate. We hereby follow the method-of-lines approach which means that the spatial discretization and temporal integration are considered separately.For the spatial discretization we confine ourselves to stencils using five points per (spatial) direction. We consider this a good starting point since a 5-point stencil is computationally attractive for the following reasons. First, a 5-point stenc...

a b s t r a c tA new formulation of Kapila's five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The new formulation uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which exchange terms are derived from physical laws. All equations are written as a single system of equations in integral form. No equation is used to describe the topology of the two-fluid flow. Relations for the Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the exchange terms is proposed. The exchange terms have distinct contributions in the cell interior and at the cell faces. For the exchange-term evaluation at the cell faces, the same Riemann solver as used for the flux evaluation is exploited. Numerical results are presented for two-fluid shock-tube and shock-bubble-interaction problems, the former also for a two-fluid mixture case. All results show good resemblance with reference results.

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a b s t r a c tThis paper investigates the temporal accuracy of the velocity and pressure when explicit Runge-Kutta methods are applied to the incompressible Navier-Stokes equations. It is shown that, at least up to and including fourth order, the velocity attains the classical order of accuracy without further constraints. However, in case of a time-dependent gradient operator, which can appear in case of time-varying meshes, additional order conditions need to be satisfied to ensure the correct order of accuracy. Furthermore, the pressure is only first-order accurate unless additional order conditions are satisfied. Two new methods that lead to a second-order accurate pressure are proposed, which are applicable to a certain class of three-and four-stage methods. A special case appears when the boundary conditions for the continuity equation are independent of time, since in that case the pressure can be computed to the same accuracy as the velocity field, without additional cost. Relevant computations of decaying vortices and of an actuator disk in a time-dependent inflow support the analysis and the proposed methods.

The simulation of viscous free-surface water flow is a subject that has reached a certain maturity and is nowadays used in industrial applications, like the simulation of the flow around ships. While almost all methods used are based on the Navier-Stokes equations, the discretisation methods for the water surface differ widely. Many of these highly different methods are being used with success.We review three of these methods, by describing in detail their implementation in one particular code that is being used in industrial practice. The descriptions concern the principle of the method, numerical details, and the method's strengths and limitations. For each code, examples are given of its use. Finally, the methods are compared to determine the best field of application for each.The following surface descretisation methods are reviewed. First, surface fitting/mesh deformation in PARNAS-SOS, developed by MARIN; the description focuses on the efficient steady-state solution method of this code. Then surface capturing with Volume-of-Fluid in ISIS-CFD, developed by CNRS/Ecole Centrale de Nantes; the main topic of this review are the compressive flux discretisation schemes for the volume fraction that are used in this code. And finally, the Level Set method in SURF, developed by NMRI;

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