Abstract
In this thesis numerical methods are developed for the simulation of turbulent
flows governed by the incompressible NavierStokes equations. This is inspired
by the need for accurate and efficient computations of the flow of air in windturbine
wakes. The stateoftheart in computing such flows is to use Large Eddy
Simulation (LES) as a turbulence model. In LES the NavierStokes equations are
filtered such that only the large, energycontaining 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 ‘underresolved’: 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 highorder energyconserving discretization
methods. Such methods mimic an important property of the continuous incompressible
NavierStokes 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
energyconserving 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 energyconserving 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 energyconserving finite volume discretizations
are limited to second order global accuracy in the presence of boundaries.
On properly chosen nonuniform 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
NavierStokes equations with RungeKutta methods. RungeKutta methods
are often applied to the spatially discretized incompressible NavierStokes equations,
but order of accuracy proofs that address both velocity and pressure are
missing. By viewing the spatially discretized NavierStokes equations as a system
of differentialalgebraic equations the order conditions for velocity and pressure
are derived. Based on these conditions new explicit RungeKutta methods
are derived, that have highorder accuracy for both velocity and pressure. These
explicit methods are not strictly energyconserving but can be efficient, depending
when the time step is determined by accuracy instead of stability. However,
for truly energyconserving RungeKutta methods implicit methods need to be
considered. Highorder RungeKutta methods based on Gauss quadrature are
proposed. In particular, the twostage fourth order Gauss method is investigated
and combined with the fourth order spatial discretization, resulting in a fourth
order energyconserving 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 Lstability. Therefore,
new additive RungeKutta methods are investigated: the diffusive terms are integrated
with an Lstable RungeKutta method, and the convective terms with
an energyconserving RungeKutta 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, nondiffusive, 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 NavierStokes solver: ECNS (EnergyConserving NavierStokes
solver). The resulting method combines stability, no numerical viscosity and
highorder accuracy. This makes it a valuable tool for simulating turbulent flow
problems governed by the incompressible NavierStokes equations and suitable
for the development and comparison of LES models. For the particular case
of windturbine 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  9789038633381 
DOIs  
Publication status  Published  2013 
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Cite this
Sanderse, B. (2013). Energyconserving discretization methods for the incompressible NavierStokes equations : application to the simulation of windturbine wakes. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR750543