Abstract
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.
Original language | English |
---|---|
Qualification | Doctor of Philosophy |
Awarding Institution |
|
Supervisors/Advisors |
|
Award date | 19 Mar 2013 |
Place of Publication | Eindhoven |
Publisher | |
Print ISBNs | 978-90-386-3338-1 |
DOIs | |
Publication status | Published - 2013 |