Energy-conserving discretization methods for the incompressible Navier-Stokes equations

In this thesis numerical methods are developed for the simulation of turbulent flows governed by the incompressible Navier-Stokes equations. This is inspired by the need for accurate and efficient computations of the flow of air in windturbine wakes. The state-of-the-art in computing such flows is to use Large Eddy Simulation (LES) as a turbulence model. In LES the Navier-Stokes equations are filtered such that only the large, energy-containing scales of motion are simulated - the smaller scales are modeled. However, even with such a model, LES simulations remain expensive (not only in wind energy applications) and are typically ‘under-resolved’: the mesh is too coarse to resolve all important scales. Thus, an ongoing challenge is to construct numerical methods that are stable and accurate even on coarse meshes, and do not introduce false (‘artificial’) diffusion that can destroy the delicate features of turbulent flows. The approach taken is to construct high-order energy-conserving discretization methods. Such methods mimic an important property of the continuous incompressible Navier-Stokes equations, namely the conservation of kinetic energy in the limit of vanishing viscosity. The energy equation is, for incompressible flows, derived from the equations for conservation of mass and momentum. An energy-conserving discretization method is nonlinearly stable, independent of mesh, time step, or viscosity, and does not introduce artificial diffusion. The first part of this thesis addresses spatially energy-conserving discretization methods, in particular second and fourth order finite volume methods on staggered cartesian grids. Special attention is paid to the proper treatment of boundary conditions for high order methods. New boundary conditions are derived such that the boundary contributions to the discrete energy equation mimic the boundary contributions of the continuous equations. An important theoretical result is obtained: higher order energy-conserving finite volume discretizations are limited to second order global accuracy in the presence of boundaries. On properly chosen non-uniform grids, designed such that the maximum error is not at the boundary, fourth order accuracy can be recovered. The second part of this thesis addresses time integration of the incompressible Navier-Stokes equations with Runge-Kutta methods. Runge-Kutta methods are often applied to the spatially discretized incompressible Navier-Stokes equations, but order of accuracy proofs that address both velocity and pressure are missing. By viewing the spatially discretized Navier-Stokes equations as a system of differential-algebraic equations the order conditions for velocity and pressure are derived. Based on these conditions new explicit Runge-Kutta methods are derived, that have high-order accuracy for both velocity and pressure. These explicit methods are not strictly energy-conserving but can be efficient, depending when the time step is determined by accuracy instead of stability. However, for truly energy-conserving Runge-Kutta methods implicit methods need to be considered. High-order Runge-Kutta methods based on Gauss quadrature are proposed. In particular, the two-stage fourth order Gauss method is investigated and combined with the fourth order spatial discretization, resulting in a fourth order energy-conserving method in space and time, which is stable for any mesh and any time step. A disadvantage of the Gauss methods is that they are less suitable for integrating the diffusive terms, since they lack L-stability. Therefore, new additive Runge-Kutta methods are investigated: the diffusive terms are integrated with an L-stable Runge-Kutta method, and the convective terms with an energy-conserving Runge-Kutta method, both based on the same quadrature points. Unfortunately, their low stage order does not make them more efficient than the original Gauss methods. In practice, the second order Gauss method (implicit midpoint) is therefore the preferred time integration method. The third part of this thesis addresses actuator methods. Actuator methods are simplified models to represent the effect of a body (such as a wind turbine) on a flow field, without requiring the actual geometry of the body to be taken into account. Actuator forces introduce discontinuities in flow variables and should therefore be treated carefully. A new immersed interface method in finite volume formulation is proposed, which leads to a sharp, non-diffusive, representation of the actuator. This does not require the choice for a discrete Dirac function and regularization parameter. The ideas put forth in this thesis have been implemented in a new parallel 3D incompressible Navier-Stokes solver: ECNS (Energy-Conserving Navier-Stokes solver). The resulting method combines stability, no numerical viscosity and high-order accuracy. This makes it a valuable tool for simulating turbulent flow problems governed by the incompressible Navier-Stokes equations and suitable for the development and comparison of LES models. For the particular case of wind-turbine wake aerodynamics a number of simulations have been performed: flow over a wing as a model for a wind turbine blade, and flow through an array of actuator disks representing a wind farm.

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