The primary application of wind turbines is to generate energy using the wind. Hence, the aerodynamics is a very important aspect of wind turbines. Like most machines, wind turbines come in many different types, all of them based on different energy extraction concepts.
Though the details of the aerodynamics depend very much on the topology, some fundamental concepts apply to all turbines. Every topology has a maximum power for a given flow, and some topologies are better than others. The method used to extract power has a strong influence on this. In general, all turbines may be classified as either lift-based or drag-based, the former being more efficient. The difference between these groups is the aerodynamic force that is used to extract the energy.
The most common topology is the horizontal-axis wind turbine. It is a lift-based wind turbine with very good performance. Accordingly, it is a popular choice for commercial applications and much research has been applied to this turbine. Despite being a popular lift-based alternative in the latter part of the 20th century, the Darrieus wind turbine is rarely used today. The Savonius wind turbine is the most common drag type turbine. Despite its low efficiency, it remains in use because of its robustness and simplicity to build and maintain.
General aerodynamic considerations
The governing equation for power extraction is:
where P is the power, F is the force vector, and u is the velocity of the moving wind turbine part.
The force F is generated by the wind's interaction with the blade. The magnitude and distribution of this force is the primary focus of wind-turbine aerodynamics. The most familiar type of aerodynamic force is drag. The direction of the drag force is parallel to the relative wind. Typically, the wind turbine parts are moving, altering the flow around the part. An example of relative wind is the wind one would feel cycling on a calm day.
To extract power, the turbine part must move in the direction of the net force. In the drag force case, the relative wind speed decreases subsequently, and so does the drag force. The relative wind aspect dramatically limits the maximum power that can be extracted by a drag-based wind turbine. Lift-based wind turbines typically have lifting surfaces moving perpendicular to the flow. Here, the relative wind does not decrease; rather, it increases with rotor speed. Thus, the maximum power limits of these machines are much higher than those of drag-based machines.
Characteristic parameters
Wind turbines come in a variety of sizes. Once in operation, a wind turbine experiences a wide range of conditions. This variability complicates the comparison of different types of turbines. To deal with this, nondimensionalization is applied to various qualities. Nondimensionalization allows one to make comparisons between different turbines, without having to consider the effect of things like size and wind conditions from the comparison. One of the qualities of nondimensionalization is that though geometrically similar turbines will produce the same non-dimensional results, other factors (difference in scale, wind properties) cause them to produce very different dimensional properties.
Power Coefficient The coefficient of power is the most important variable in wind-turbine aerodynamics. The Buckingham π theorem can be applied to show that the non-dimensional variable for power is given by the equation below. This equation is similar to efficiency, so values between 0 and less than 1 are typical. However, this is not exactly the same as efficiency and thus in practice, some turbines can exhibit greater than unity power coefficients. In these circumstances, one cannot conclude the first law of thermodynamics is violated because this is not an efficiency term by the strict definition of efficiency.
where is the coefficient of power, is the air density, A is the area of the wind turbine, and V is the wind speed.
Thrust coefficient The thrust coefficient is another important dimensionless number in wind turbine aerodynamics.
Speed ratio Equation () shows two important dependents. The first is the speed (U) of the machine. The speed at the tip of the blade is usually used for this purpose, and is written as the product of the blade radius r and the rotational speed of the wind: , where is the rotational velocity in radians/second).[please clarify] This variable is nondimensionalized by the wind speed, to obtain the speed ratio:
Lift and drag The force vector is not straightforward, as stated earlier there are two types of aerodynamic forces, lift and drag. Accordingly, there are two non-dimensional parameters. However, both variables are non-dimensionalized in a similar way. The formula for lift is given below, the formula for drag is given after:
where is the lift coefficient, is the drag coefficient, is the relative wind as experienced by the wind turbine blade, and A is the area. Note that A may not be the same area used in the power non-dimensionalization of power.
Relative speed The aerodynamic forces have a dependency on W, this speed is the relative speed and it is given by the equation below. Note that this is vector subtraction.
Drag- versus lift-based machines
All wind turbines extract energy from the wind through aerodynamic forces. There are two important aerodynamic forces: drag and lift. Drag applies a force on the body in the direction of the relative flow, while lift applies a force perpendicular to the relative flow. Many machine topologies could be classified by the primary force used to extract the energy. For example, a Savonious wind turbine is a drag-based machine, while a Darrieus wind turbine and conventional horizontal-axis wind turbines are lift-based machines. Drag-based machines are conceptually simple, yet suffer from poor efficiency. Efficiency in this analysis is based on the power extracted vs. the plan-form area. Considering that the wind is free, but the blade materials are not, a plan-form-based definition of efficiency is more appropriate.
The analysis is focused on comparing the maximum power extraction modes and nothing else. Accordingly, several idealizations are made to simplify the analysis, further considerations are required to apply this analysis to real turbines. For example, in this comparison the effects of axial momentum theory are ignored. Axial momentum theory demonstrates how the wind turbine imparts an influence on the wind which in-turn decelerates the flow and limits the maximum power. For more details see Betz's law. Since this effect is the same for both lift and drag-based machines it can be ignored for comparison purposes. The topology of the machine can introduce additional losses, for example trailing vorticity in horizontal axis machines degrades the performance at the tip. Typically these losses are minor and can be ignored in this analysis (for example tip loss effects can be reduced by using high aspect-ratio blades).
Maximum power of a drag-based wind turbine
Equation () will be the starting point in this derivation. Equation () is used to define the force, and equation () is used for the relative speed. These substitutions give the following formula for power.
The formulas () and () are applied to express () in nondimensional form:
It can be shown through calculus that equation () achieves a maximum at . By inspection one can see that equation () will achieve larger values for . In these circumstances, the scalar product in equation () makes the result negative. Thus, one can conclude that the maximum power is given by:
Experimentally it has been determined that a large is 1.2, thus the maximum is approximately 0.1778.
Maximum power of a lift-based wind turbine The derivation for the maximum power of a lift-based machine is similar, with some modifications. First we must recognize that drag is always present, and thus cannot be ignored. It will be shown that neglecting drag leads to a final solution of infinite power. This result is clearly invalid, hence we will proceed with drag. As before, equations (), () and () will be used along with () to define the power below expression.
Similarly, this is non-dimensionalized with equations () and (). However, in this derivation the parameter is also used:
Solving the optimal speed ratio is complicated by the dependency on and the fact that the optimal speed ratio is a solution to a cubic polynomial. Numerical methods can then be applied to determine this solution and the corresponding solution for a range of results. Some sample solutions are given in the table below.
Optimal Optimal 0.5 1.23 0.75 0.2 3.29 3.87 0.1 6.64 14.98 0.05 13.32 59.43 0.04 16.66 92.76 0.03 22.2 164.78 0.02 33.3 370.54 0.01 66.7 1481.65 0.007 95.23 3023.6
Experiments have shown that it is not unreasonable to achieve a drag ratio () of about 0.01 at a lift coefficient of 0.6. This would give a of about 889. This is substantially better than the best drag-based machine, and explains why lift-based machines are superior.
In the analysis given here, there is an inconsistency compared to typical wind turbine non-dimensionalization. As stated in the preceding section, the A (area) in the non-dimensionalization is not always the same as the A in the force equations () and (). Typically for the A is the area swept by the rotor blade in its motion. For and A is the area of the turbine wing section. For drag based machines, these two areas are almost identical so there is little difference. To make the lift based results comparable to the drag results, the area of the wing section was used to non-dimensionalize power. The results here could be interpreted as power per unit of material. Given that the material represents the cost (wind is free), this is a better variable for comparison.
If one were to apply conventional non-dimensionalization, more information on the motion of the blade would be required. However the discussion on horizontal-axis wind turbines will show that the maximum there is 16/27. Thus, even by conventional non-dimensional analysis lift based machines are superior to drag based machines.
There are several idealizations to the analysis. In any lift-based machine (aircraft included) with finite wings, there is a wake that affects the incoming flow and creates induced drag. This phenomenon exists in wind turbines and was neglected in this analysis. Including induced drag requires information specific to the topology. In these cases it is expected that both the optimal speed-ratio and the optimal would be less. The analysis focused on the aerodynamic potential but neglected structural aspects. In reality most optimal wind-turbine design becomes a compromise between optimal aerodynamic design and optimal structural design.
Horizontal-axis wind turbine
The aerodynamics of a horizontal-axis wind turbine are not straightforward. The air flow at the blades is not the same as the airflow further away from the turbine. The very nature of the way in which energy is extracted from the air also causes air to be deflected by the turbine. In addition, the aerodynamics of a wind turbine at the rotor surface exhibit phenomena rarely seen in other aerodynamic fields.
Axial momentum and the Lanchester–Betz–Joukowsky limit