**Numerical Experiments with Energy-Conserving ****schemes**

**To predict the power produced by a wind farm, one must at least know the velocity of the air flowing through the farm. The wind velocity can be measured using meteorological masts with devices that can assess wind speed and direction a given point or even over a limited area. However, this data is not enough to study complex flow phenomena like gusts, kinetic energy transport, and the effect of the wind farm on local weather. This article explains the challenges involved in complementing experimental data with that from simulations.**

A wind farm consists of a cluster of wind turbines placed over a piece of land or at sea, that work together to produce electricity from the wind. The wind turbines are generally placed in a manner that leads to efficient generation of electricity. For example, one programs the turbines’ control systems to make the turbine face the wind as the latter changes direction, because the amount of power a turbine can extract from the wind is proportional to the cube of the wind speed. Further, the control system must also be able to switch the turbine off when the wind speed is too high to allow safe operation. Such tuning is required to reduce the higher cost of maintenance, while ensuring longevity. Thus, prior to a wind farm’s establishment, one must perform a thorough study of the farm’s design and placement, while aiming not only to maximize power output but also the operational life of a turbine or the farm as a whole.

To perform such a study, one must have good knowledge of the flow of air through a wind farm, a subject that is known as wind farm aerodynamics. The reason being that whilst a turbine is in operation, it extracts energy from the air and produces a wake that bears a velocity lower than that of the undisturbed atmospheric flow. In most cases, as a turbine’s wake travels along the length of the wind farm, it interacts with downstream turbines, which results in the latter being exposed to slower air with a lower potential for generating power. A wind farm can have over fifty turbines, providing ample chances for turbine-wake interactions and also wake-wake interactions, all of which can affect the local flow velocity, leading to reduction in power.

Additionally, the lower wake velocity or available energy is accompanied by a concomitant increase in turbulence. Turbulent flows show rapid fluctuation in velocity and pressure, which leads to an increased transport of momentum and energy, especially if the flow is dominated by inertia or its viscosity is too low to dissipate any increase in momentum transport through turbulence. Upon interaction with turbines, these fluctuations increase the loading on the turbine and reduce its longevity. Further, fluctuations could also lead to resonance that reinforces the turbine’s vibrations, which could lead to structural failure.

To avoid a detailed aerodynamic analysis, one could simply place the turbines far apart to avert any turbine’s wake from interacting with another turbine. However, this demands more area, which adds to the costs in terms of land, connecting cables that would relay the electricity produced and also increases the costs of maintenance. Alternatively, turbines of different height can be used. But the earth’s surface creates the atmospheric boundary layer that has a lower velocity close to the surface. Placing turbines of different heights would mean that the shorter ones generate less power due to the slower oncoming air, while the taller ones may be exposed to wind speeds greater than what they could bear. Therefore, an aerodynamic analysis on the entire wind farm is very desirable.

Aerodynamic data can be gathered by placing suitable apparatus on masts within a wind farm. But due to measurement equipment interfering with the flow, on can only place a small number of masts that are not enough to provide detailed information at a resolution that would allow tracing every wake and determining the velocity field at most relevant locations on a farm. Further, the erratic nature of the atmosphere only hampers the analysis of data as one can never be certain if what is being measured downstream of a turbine is solely due to the turbine’s activity alone and is free from gusts that occur frequently, a wake interaction from another turbine, etc. Thus, scientists now resort to the simulations to determine how wakes develop and interact. Simulations seek to model the wind turbines and the atmospheric boundary layer mathematically with an accuracy that is enough to resolve the wakes and their interactions. However, simulations rely greatly on the numerical schemes used for modeling the flow, i.e., solving the *Navier-Stokes* equations and the availability of computational resources that determines the level of accuracy that one can achieve. Of the above, the current project deals with the development of numerical schemes and flow models that are not only accurate but demand relatively lesser computational effort.

A turbulent flow, like that through a wind farm, comprises various regions of swirling flow known as eddies. These eddies vary in size across the flow field and their velocity fluctuations are generally correlated in both, space and time. In short, the presence of obstacles at one part in a flow alters the velocity field, which by virtue of eddies, is felt in other regions of the flow, depending on the size of the eddies and strength of the disturbance. Further, the higher the flow’s Reynolds number, or the molecular viscosity’s inability to control the effect of disturbances, the greater is the range of eddies. On a wind farm, these eddies range from nearly a kilometer in size to those of a few millimeters created by the grass over the surface, for instance. A thorough analysis would obviously involve modeling all possible eddies in a flow, a technique known as *Direct Numerical Simulation.* DNS is computationally expensive and nearly impossible for a wind farm given the range of eddies.

As an alternative, one models only the larger eddies that carry most of the flow’s mass, momentum and energy, and which mainly interact with the turbines and are carried by the wakes. In short, one must analyze the flow as discrete volumes, which should be small enough to provide an accurate representation of the flow’s energy but large enough to not lead to a high number of parcels such that their analysis is made impossible. This technique is known as *Large Ed**dy Simulation*.

Like every numerical code, the discretization of the flow into small volumes needs mathematical approximations; similar to how one uses the trapezoidal or the Simpson’s rule to approximate the area under a curve. Discretizing a flow and hence the *Navier-Stokes* equations is rather complicated. At times, numerical schemes go awry to the extent that they themselves end up influencing the development of the flow, rather than just predicting it. With regards to a wind farm simulation, poor numerical schemes can lead to wakes with abnormally high or low velocities, and the same for turbulence, leading to a faulty power and load prediction. This can mislead designers trying to optimize how a wind farm must be established.

This project aims at developing numerical schemes that while solving the *Navier-Stokes *equations do not spuriously affect the flow’s evolution. Of the various schemes that can do this, this project uses *Energy-Conserving *schemes. An energy-conserving scheme ensures that the flow’s energy is only altered by physical processes like dissipation through viscosity, forcing through wind turbines etc., and not spuriously through numerical approximations that lead to faulty velocity fields. Apart from their ability to resolve the flow accurately, these schemes are numerically stable even if the discretization is very coarse. This enables faster yet reliable calculations. These energy-conserving schemes are being developed as part of the *Energy-Conserving Navier-Stokes *solver at the *Energy Research Centre of the Netherlands. *Once ready, these schemes will help simulate wind farms with an accuracy that is sufficient to determine the power that a wind farm can produce and also the loads that the turbines encounter while doing so, thus enabling the design of more efficient and long-lasting wind farms.

*by Dhruv Mehta, PhD Candidate, AWEP*

References:

- D Mehta, AH van Zuijlen and H Bijl, “Energy-Conserving schemes for Wind Farm Aerodynamic
*s*”,*Journal of Physics Conference Series,* - D Mehta, AH van Zuijlen, B Koren, JG Holierhoek and H Bijl, “Large Eddy Simulation of Wind Farm Aerodynamics: A Review”,
*Journal of Wind**Engineering and Industrial Aerodynamics,*