## Abstract

The generation of wind fields is of interest in the study of the structural performance of wind turbines in critical events, such as thunderstorm downbursts. Various methods ranging from the use of empirical data to employing computational simulations are typically adopted to study the response of wind turbines in downburst flow fields. While the former approach is limited in the ability to account for accurate and spatially resolved details of the flow field, the latter is expensive and, therefore, has limitations in its use. As an alternative, in this work, we propose a paused downburst model in which a snapshot of a time-dependent computational fluid dynamics (CFD) simulation is used to generate “mean” wind fields during thunderstorm downbursts. The developed model for the mean wind field is validated against recorded downburst data in the literature. The turbulent component of the wind field is generated using computationally inexpensive techniques based on Fourier-based power spectral density functions and coherence functions. In an illustrative example, the combined mean and turbulence wind fields are generated and applied on a utility-scale wind turbine to study structural load characteristics during a downburst event.

## 1 Introduction

Recent studies have found that the structural performance of wind turbines is significantly affected by extreme atmospheric events, such as thunderstorm downbursts [1–4]. A thunderstorm downburst is defined as a strong downdraft that includes an outflow of potentially damaging winds at or near the ground [5]. Simulation of such a wind flow field is essential to study the time-dependent structural behavior of wind turbines. Since the wind flow in downbursts is extremely turbulent, direct numerical simulations are computationally infeasible. While large eddy simulations (LES) [6–11] may yield more accurate flow field representations, they are still computationally very expensive.

An alternative, computationally inexpensive strategy to simulate the wind field in a thunderstorm downburst is to decompose it into a non-turbulent time-averaged component and a fluctuating turbulent component. In wind turbine applications, the turbulent component is generally simulated using power spectral density functions and coherence functions [12–16], whereas analytical or empirical models are used for the non-turbulent component [17,18]. The use of semi-empirical models, although may be simple, can be restrictive in terms of obtaining accurate and complex features of a realistic wind field in a downburst. Therefore, in this work, we propose to use a model based on computational fluid dynamics (CFD) for the non-turbulent component of the wind field that can better capture the spatial variations. Since the computational model is only for the time-averaged components of flow field variables, we employ statistical modeling using a Reynolds-averaged Navier–Stokes (RANS) approach. Simulation of non-turbulent wind fields of downbursts using CFD modeling was carried out by many researchers in the past [19–24]. However, unlike the current study, such flow fields simulated in the previous works were not aimed at the study of structural loads in wind turbines. In a very recent study, researchers conducted wind turbine load analysis [25] using CFD simulations. However, the study involves simulation of the full time-history of the downburst flow field using CFD, which is still computationally expensive. In contrast, we propose a simulation approach which does not require a complete CFD analysis and uses intermediate data to extrapolate the entire time-dependent flow field of the downburst. To capture the transient nature of flow fields in a downburst, we use a time-dependent intensity function and account for the movement of the position of radial peak velocity with time to scale the velocity profiles obtained from the proposed CFD-based model. This approach shares the simplicity of semi-empirical models that employ a similar strategy to generate time-dependent downburst flow fields [14].

Although a downburst is a buoyancy-driven flow, the wind velocity profile of a downburst after touchdown has been found to have significant similarities with that of a radial wall jet [26]. A radial wall jet is typically simulated as a jet impinging on a flat surface; this is essentially a momentum flux-driven flow. Downbursts are easier to simulate computationally and study experimentally if they are thought to be represented as momentum flux-driven flow fields. A considerable amount of work has been done on impinging jets—both experimentally [27–29] as well as computationally [19,30,31]. Computational resources in recent years have helped increase the use of CFD to simulate flow fields in complex problems. The earliest research studies on downbursts focused mainly on the steady-state solution of a jet impinging on a wall or plate. Later, features were added to represent actual characteristics of the downburst. A moving downburst that included transient features of an actual downburst was simulated by Sengupta et al. [32] and Li et al. [33]. Inspired from all these works, we simulate downburst as a radial jet impinging on a surface, with appropriate boundary conditions. We note that in the proposed computational approach, we use the actual length and time scales of the downburst and wind turbines, and therefore, no further scaling parameters typically used in experimental models [34] are required.

In the current paper,^{2} we propose an approach that we refer to as paused downburst model [35], where the non-turbulent component of the wind field is extracted from a particular time snapshot of the CFD simulation and further post-processed to obtain the complete wind field of a downburst with desired physical characteristics. The paper is organized as follows. In Sec. 2, we discuss the assumptions related to the computational model that we develop. In Sec. 3, we specify the boundary conditions and solver details. In Sec. 4, we present the paused downburst computational model. In Secs. 5 and 6, we apply the model for wind turbine loads analysis, and conclude in Sec. 7.

## 2 Model Development

As discussed, past research studies have simulated downburst-like flows using various methods—for example, impinging jets, buoyancy-driven flows, etc. Several characteristics of a downburst, such as its intensity, translational speed, the shape of the velocity profiles, etc., are important to be represented realistically in any simulation. Steady-state simulation of an impinging jet on a wall [31,36] has been shown to yield vertical velocity profiles that match observed data quite well. If one were interested in a more realistic representation of the actual physics of the flow, a buoyancy-driven flow simulation using LES [7,37] is preferred. For example, buoyancy effects which are not considered in impinging jet models play a crucial role in accurately simulating certain downburst features like ring vortices and flow fields close to the ground [37]. In the context of wind turbine loads analysis, realistic representation of the inflow wind velocity field on the turbine rotor is of critical importance. Insufficient good quality data of recorded downbursts have limited the validation of and hence the application of the various sophisticated downburst simulation models developed. This provides us the motivation to use the impinging jet approach despite its limitations, for the sake of simplicity and versatility in its ability to match downburst time series data as discussed in this paper.

The Andrews Air Force Base (AAFB) downburst (Fig. 1) is one of the best recorded downbursts in the literature. It shows two noticeable peaks in the wind speed record during the downburst. A second smaller peak follows the first larger one (when read from right to left in the figure). The low wind speed between the peaks corresponds to the eye of the downburst and is similar in effect to the stagnation region at the center of an impinging jet. The formation of a ring vortex is commonly observed in downbursts; the wind speed below the ring is usually the maximum wind speed experienced in a thunderstorm downburst [5]. The ring forms because of the Kelvin–Helmholtz instability that results at the interface of the downburst wind and the ambient wind [19]. The velocity gradient across the interface leads to the ring formation in impinging jet flows. In contrast, the ring vortex in a buoyancy-driven flow forms due to both shear as well as the thermodynamically driven buoyancy gradient [37]. The difference in magnitude between the two peaks in Fig. 1 is due to the effect of the downburst’s translational motion and the ambient wind on the front and rear ring vortices.

We aim to develop the CFD-based model for a downburst in such a way that it mimics the Andrews AFB wind speed time-history. Several CFD simulations using the commercial software, ansys fluent 12.0 [38], were carried out by modeling a stationary impinging jet, an impinging jet at steady state and a moving impinging jet. None of these models showed reasonable similarity with the velocity profiles from the recorded Andrews AFB downburst. Although observations similar to that of Kim and Hangan [19] were noted, such as in the development of quasi-periodic ring vortices at the interface of the ambient wind and the downburst, the wind speed time histories also did not resemble those recorded in the Andrews AFB event. This is not surprising for several reasons. First, an impinging jet is only an approximation to the physics associated with downburst flow fields. Second, it is unreasonable to expect to match the complex downburst flow phenomenon, kinetics, and movement using a jet simulation with selected simplified set of parameters (whether stationary or translating). Third, any single point’s recorded or simulated wind time series is an observer-dependent entity; it is entirely possible that the Andrews AFB downburst record did not capture the additional ring vortices and it could as well be the case that there were not multiple rings generated during the Andrews AFB downburst. There is not a wealth of good quality downburst data that can be used to validate computational simulations. Working only with the Andrews AFB downburst, other attempts were made to simulate wind speed time series that resembled that record alone.

Assuming that the Andrews AFB downburst record is representative of likely downburst wind speed time histories, a paused downburst model is developed to match this record. The idea behind developing a paused downburst model comes from the need to match the Andrews AFB data. This model uses a paused snapshot from the CFD simulation of a jet impinging on wall, which is post-processed to produce the complete non-turbulent wind field time series for wind turbine load calculations. This is explained in the following sections.

## 3 Computational Framework

### 3.1 Domain and Boundary Conditions.

The closed set of governing equations are to be solved using a turbulence model using RANS approach, within a physical domain with appropriately specified boundary conditions. Axisymmetric flow conditions may be assumed for a stationary impinging jet. The problem is, thus, solved in a two-dimensional computational domain. Only one-half of the two-dimensional plane including the axis of symmetry is employed. Figure 2 shows a sketch of the computational domain and associated boundary conditions. In the figure, *D* represents the diameter of the inlet jet while *H* represents the height of the origin of the inlet flow from the wall. Details related to the boundary conditions are given in Table 1. For the convergence studies discussed in this section, we choose *D* = 1000 m and *H* = 2000 m.

AB | Wall |

(no slip, stationary) | |

BC, CD | Pressure outlet |

Backflow turbulence intensity = 1% | |

Backflow hydraulic diameter = D | |

DE | Symmetry |

EF | Velocity inlet |

(V_{in} specified) | |

Turbulence intensity =1% | |

Hydraulic diameter = D | |

FA | Axis of symmetry |

AB | Wall |

(no slip, stationary) | |

BC, CD | Pressure outlet |

Backflow turbulence intensity = 1% | |

Backflow hydraulic diameter = D | |

DE | Symmetry |

EF | Velocity inlet |

(V_{in} specified) | |

Turbulence intensity =1% | |

Hydraulic diameter = D | |

FA | Axis of symmetry |

In the computational domain, the direction from A to B is the radial direction and that along the axis (i.e., from F to A) is the axial direction for the downburst. A discussion on convergence related to the computational grid is presented later in this section. The grid defined as mesh II is chosen for all the simulations. Mesh II has a grid spacing of 5 m in the direction perpendicular to the wall and a very fine spacing close to the wall to ensure that the dimensionless wall distance, *y*^{+}, is below 3. The grid spacing in the radial direction (AB) is 5 m for the entire mesh; hence, except in a region of about 20 m from the wall, the domain has a uniform grid of 5 m × 5 m over the entire computational domain.

### 3.2 Solution Methodology.

Table 2 provides some details on the models, schemes, and other information used with the commercial software, ansys fluent 12.0 [38], to carry out all the simulations. Chay et al. [31] discussed the use of a turbulence model specific to an impinging jet flow; there, the renormalization group (RNG) *k*–*ɛ* model was employed. The quadratic upstream interpolation for convective kinematics (QUICK) scheme has been employed as it is a preferred scheme for vortical flows.

General | Pressure-based solver |

2D axisymmetric space | |

Gravitational acceleration = 9.81 m/s^{2} | |

Models | Viscous |

k–ɛ (2-equation) RNG model | |

Enhanced wall treatment | |

Default model constants | |

Materials | Air |

Density = 1.225 kg/m^{3} | |

Dynamic viscosity = 1.7894 × 10^{−5} kg/m s | |

Solution methods | Pressure–velocity coupling: SIMPLE scheme |

Spatial discretization: | |

Gradient least squares cell-based | |

Pressure—standard | |

Momentum—QUICK | |

Turbulence kinetic energy—QUICK | |

Turbulence dissipation rate—QUICK | |

Transient formulation: second-order implicit | |

Solution controls | Default under-relaxation factors |

Monitors | Residual convergence criteria at 1 × 10^{−5} |

Run calculation | Time-step size = 0.5 s |

Max iterations/time-step = 80 |

General | Pressure-based solver |

2D axisymmetric space | |

Gravitational acceleration = 9.81 m/s^{2} | |

Models | Viscous |

k–ɛ (2-equation) RNG model | |

Enhanced wall treatment | |

Default model constants | |

Materials | Air |

Density = 1.225 kg/m^{3} | |

Dynamic viscosity = 1.7894 × 10^{−5} kg/m s | |

Solution methods | Pressure–velocity coupling: SIMPLE scheme |

Spatial discretization: | |

Gradient least squares cell-based | |

Pressure—standard | |

Momentum—QUICK | |

Turbulence kinetic energy—QUICK | |

Turbulence dissipation rate—QUICK | |

Transient formulation: second-order implicit | |

Solution controls | Default under-relaxation factors |

Monitors | Residual convergence criteria at 1 × 10^{−5} |

Run calculation | Time-step size = 0.5 s |

Max iterations/time-step = 80 |

Both time-dependent and steady-state simulations are carried out. The time-dependent simulations are more appropriate to describe the downburst event; hence, these simulations were used for the model development.

### 3.3 Solution Convergence and Choice of Inlet Velocity Profile.

Since the downburst flow has a very high Reynolds number, the computational domain requires a very fine grid size and a small time-step for accuracy. In addition, formation of the ring vortex at the interface of the downburst vertical jet and the ambient air further complicates the flow field. A resolution of the finer details of this ring in this high Reynolds number flow is impractical due to computational and other constraints. As the grid size and time-step are made smaller, the ring (vorticity) is better resolved but these show up small-scale effects that influence the final solution in a cumulative sense. The problem identified can be overcome if a weaker ring is generated.

The strength of the ring vortex that forms at the interface of the jet and the ambient air is a direct function of the difference in magnitude of the velocity (i.e., the vorticity) at the interface. If the inlet velocity has a profile with a higher magnitude at the interface, a stronger ring results. The shape of the profile of the inlet velocity influences the strength of the ring formed; this, in turn, controls the convergence of the solution with respect to grid and time-step. It can be assumed that if the magnitude of the velocity at the interface is zero, a very weak ring will result. At the interface, such a velocity profile can have different slopes. The effect of slope was studied by running simulations using different shapes for the inlet velocity. Changes in slope did not show any significant influence on the solution. It can be confirmed that the solution converges with respect to grid size. A parabolic inlet velocity profile was chosen for all other simulations used for development of the paused downburst model.

Generally, as in traditional approaches to CFD as well as with any computational numerical problems, a converged solution is expected to result as the grid size and/or time-step are reduced systematically. A series of simulations involving different meshes with different grid sizes and different time-steps were carried out.

Three meshes, mesh I, mesh II, and mesh III, each with uniform grid sizes of 10 m, 5 m, and 2 m, respectively, were created. Convergence was attained as we move from 5 m to 2 m mesh. The mesh with the 5 m grid spacing was chosen for simulations with different time-steps, Δ*t* = 0.1 s, 0.25 s, 0.5 s, 1.0 s, and 2.5 s. The time-step of 0.5 s proved to be a good choice based on the accuracy of solution obtained and computational efficiency.

## 4 Downburst Model

### 4.1 Paused Downburst Model.

The profiles (showing variation with elevation and radial distance) of the radial velocity from steady-state simulations are not in good agreement with profiles from actual downburst data or with validated semi-empirical models. The velocity profiles of the time-dependent simulations, at different time instants before the formation of the second ring vortex [19] near the ground, reveal better agreement with actual downburst data as well as with profiles suggested by semi-empirical models [12]. A snapshot profile (Fig. 3) of the time-dependent simulation, before the second ring vortex formation, can serve as an alternative (computational) model to a semi-empirical model. This profile used to describe the non-turbulent downburst wind field is referred to as a paused downburst model. We assume that this snapshot of a time-dependent CFD simulation captures the overall behavior of the downburst in an averaged sense.

The computational domain shown in Fig. 2 is used for all the simulations to develop the generalized model discussed later. In the following, *Z* refers to the height in meters from the wall, *Z*_{Max} is the height at which the overall maximum radial velocity occurs, *U*_{r} is the radial velocity, *U*_{r,max} is the peak radial velocity or the maximum radial velocity at each radial position, *r*, from the axis, *U*_{r,Max} is the overall maximum radial velocity considering all radial distances from the axis, and *R*_{Max} is the radial position where the overall maximum radial velocity occurs.

Consider a domain with inlet diameter, *D* = 1000 m, *H* = 2000 m, and with a parabolic inlet velocity profile with maximum velocity, *V*_{in, max} = 40 m/s. The values chosen for these parameters represent real observations discussed by Hjelmfelt [26]. These three entities—*D*, *H*, and *V*_{in,max}—are the input parameters for the CFD simulations carried out using ansys fluent. The paused downburst model results depend on the time at which the simulation is paused. Figure 3 shows a snapshot of a simulation paused at 107 s with the above parameters. This snapshot is equivalent to a computational time-independent model of a downburst. This can also be understood as an outcome of the paused downburst model for a given set of input parameters.

As stated earlier, the model represents the set of velocity component data in space. The velocity at any point in space has two components—radial (*U*_{r}) and axial (*U*_{a}). Figures 4 and 5 show the variation of the radial velocity in both directions. The radial velocity starts from zero at the axis, reaches a peak and then decreases (at a higher rate than the radial wall jet at steady state). In this example, *U*_{r,Max} occurs at a radial distance, *R*_{Max}, of approximately 450 m or 0.45*D* from the axis; also, *Z*_{Max} is approximately 60 m or 0.06*D* from the wall. The axial velocity profiles do not have a significant effect on wind turbine response during a downburst as they are of significantly smaller magnitude than the radial velocity profile; hence, a discussion on axial velocity profiles in not included here.

The paused time instant, which determines the mean flow profile of the downburst, is chosen based on best agreement of the radial velocity profiles with the mean representative real data from the literature [26]. This typically involves systematically comparing a set of time instants, from before the primary vortex touches the ground until just after, with the recorded downburst data. If the paused time instant were chosen as something other than 107 s, then the radial velocity profiles would be slightly different. The input parameters could be changed to yield a different set of velocity profiles. The dependence of the model on these input parameters and the time of pause is used to develop a generalized model later.

The time-independent model discussed is a static representation of a stationary downburst. In reality, a ring vortex forms in a downburst and moves radially outward from its axis. Also, the intensity of the downburst flow changes with time. This latter aspect can be accounted for by scaling the model output velocity field data from the simulation using a time-dependent amplitude modulation function. Consideration of the radial evolution of the downburst, however, requires the development of a spatial velocity field at each time-step created by moving *R*_{Max} away from the axis; this requires significantly greater computational effort. It can also not be achieved by an unsteady CFD simulation because the velocity profiles after the formation of a second ring vortex near the ground do not match actual downburst data.

The evolution of *R*_{Max} with time and development of the velocity field at every time-step can be achieved using normalized profiles from the time-independent paused downburst model. What is needed is that the normalized shape of the velocity profiles at any time-step must be similar to that of the time-independent model. As time elapses, the flow that is impinging on the ground rushes radially outward. With time, *R*_{Max} and, hence, the spatial extent of the storm’s influence will increase. In all the simulations, the velocity data have been collected at a 10 m spacing in the radial direction. Hence, when *R*_{Max} increases, the number of grid points at which velocity values need to be defined also increases. For example, if *R*_{Max} = 450 m, about 90 data points representing 900 m of radial extent are sufficient, but when *R*_{Max} = 800 m, about 160 data points are required. Developing velocity profiles at each time-step requires interpolation of the normalized velocity profiles at intermediate radial positions. This is discussed next.

*u*(

*r*

_{n}) and

*r*

_{n}represent the normalized peak radial velocity,

*U*

_{r,max}/

*U*

_{r,Max}, and the normalized radial distance,

*r*/

*R*

_{Max}, respectively.

In the current example, *R*_{Max} is assumed to increase at a constant rate of 10 m/s. It is evident that the normalized profiles (Fig. 7(a)) are the same at all times, but the non-normalized profiles (Fig. 7(b)) are expectedly different since *R*_{Max} increases with time.

A general procedure or algorithm to simulate a thunderstorm downburst-like event using the paused downburst model is described below. The final output parameters that define the characteristics of the simulated downburst are *U*_{r,Max}, *Z*_{Max}, and *R*_{Max}, which were defined earlier. These model output parameters depend on (i) *V*_{in, max} and *D*/*H*, which are input parameters for the CFD simulation and (ii) an intermediate parameter, the time of the downburst pause, *t*_{p} (s). The output and intermediate parameters are written in non-dimensional form as follows: *t*_{p}* = *t*_{p}*V*_{in,max}/*H*, *U*_{r,Max}* = *U*_{r,Max}/*V*_{in,max}, *Z*_{Max}* = *Z*_{Max}/*D*, and *R*_{Max}* = *R*_{Max}/*D*.

In order to simulate a downburst event with specified output parameters, *U*_{r,Max}, *Z*_{Max}, and *R*_{Max}, a wide range of combinations of the input parameters, *V*_{in, max} and *D*/*H*, and of the intermediate parameter, *t*_{p}*, may be used. A large number of simulations were carried out for different input and intermediate parameters and a generalized model has been developed and is presented in the form of three charts that can help in selecting input parameters to obtain the target output parameters. These charts are described in detail in the study by Pratapa [35]. One must make iterative use of the charts to obtain the input parameters needed for the CFD simulation if the paused downburst model is employed.

Results from the paused downburst should be subjected to the post-processing using the normalized radial profiles, discussed earlier, to yield the full time-dependent downburst model output that could be used for any engineering application. A non-dimensional time of pause, *t*_{p}*, in the range from 2.1 to 2.5 is seen to best match actual downburst data. Values of *t*_{p}* below 2.1 are not advised; also, values of *t*_{p}* above 2.4 tend to cause maximum velocities very close to the ground (around 5–10 m above ground) and these values are again not supported by actual observed data on downbursts.

Other storm parameters—such as the storm intensity, the storm translation velocity, and the direction of the storm path—can be added to the model in order to simulate the evolution of a downburst. This would be required in order to carry out wind turbine loads analysis. Such a procedure was used by Nguyen et al. [12] and Pratapa et al. [13].

### 4.2 Validation of the Model.

Downburst data available in the literature are not sufficient to fully describe time-dependent characteristics of the storm. Hence, the goal was to develop a time-independent model (from CFD simulations) that could be validated against actual data; time-dependent characteristics could be added later as described. This has led to the development of the paused downburst model with time-dependent characteristics incorporated into that model as described. Before using the velocity profiles from the paused downburst model for any engineering application, it is important to validate the velocity data against any available downburst data. Hjelmfelt [26] presented observations and characteristics from several recorded downbursts; a resulting “mean” radial velocity profile showing variation with height and radial distance was also presented. This mean profile is thought to be representative of all the recorded downbursts of the study.

Figures 4 and 5 that show the paused downburst model’s radial velocity distributions also include Hjelmfelt’s mean profiles. It is noted here that the flow field based on the model paused at 107 s matched the mean from observations better than that at any other time instants. The adequacy of validation of the paused downburst model against data depends on the time of the downburst pause. It is important to note that Hjelmfelt’s mean profiles represent only a few downbursts and are applicable to a specific location and period. Hence, other pause times of the model developed here, although not in good agreement with the Hjelmfelt’s data, might still be realistic models for use in downburst simulations. We note that a pause time beyond the formation of the second ring vortex near the wall is not a good choice because a second ring significantly distorts the radial distribution and, hence, cannot be validated.

An important feature of the paused downburst model is that the output time series can be simulated at a specified location. The ability to generate such time series led to the development of the model in the manner described rather than from a steady-state simulation of a wall jet. Figure 8 shows a comparison of the time series created by the paused downburst model at a specified location and the time series recorded during the Andrews AFB downburst (Fig. 1). Here, the ambient wind speed was superimposed onto the paused downburst wind field. Specifically, we have chosen a radial velocity profile with peak magnitude of 48 m/s and an ambient wind speed of 12 m/s so that the two peaks of the AAFB downburst can be closely matched. This demonstrates the versatility of the model to match certain types of observed downburst time series data. We see that the model-generated time series matches characteristics of the observed data quite well; such comparisons serve to support the use of the paused downburst model for engineering applications where wind speed time series are needed as in wind turbine loads analyses.

### 4.3 Limitations of the Model.

While the paused downburst model has been seen to offer comparable wind speed time series characteristics to those in recorded data such as from the Andrews AFB microburst, it does not necessarily represent the entire time-dependent downburst phenomenon. Because the paused downburst model was developed with the intent of representing the Andrews AFB data, it may not match, as effectively, downbursts that display other characteristics such as the occurrence of multiple ring vortices at or near the ground, the effect of storm translation, etc. There are not a sufficient number of real downburst records available to establish the effects of these various characteristics on model-generated wind speed time series. Another limitation of the model arises from the basic physics assumptions used to simulate the flow; since the model is based on a momentum flux-driven flow (jet impingement), it does not provide any information related to temperature and precipitation. Finally, an assumption made in the development of this model is that the flow field is axisymmetric; downbursts that do not touch down normal to the ground and those that translate or move do no generate axisymmetric flow fields.

### 4.4 Turbulent Wind Velocity Field.

The turbulent part of the wind velocity field in a thunderstorm downburst may be simulated as the product of a desired turbulence intensity *ɛ*, a zero-mean unit-variance Gaussian process, *κ*(*x*, *y*, *z*, *t*), and the non-turbulent wind velocity, *U*_{m}(*x*, *y*, *z*, *t*) [39]; the latter is derived from the CFD-based paused downburst model or some semi-empirical representation.

There are two common approaches for simulating spatial components of correlated wind velocity time series at specified locations, given target turbulence power spectral density functions and coherence functions: the auto-regressive moving average method employed in the time domain or a Fourier-based method applied in the frequency domain. In this study, the frequency domain approach [40] is utilized to generate the turbulent wind field; this method is summarized in the study by Nguyen et al. [12].

## 5 Five-Megawatt Turbine Model and Parameters of a Downburst Example

Table 3 shows a schematic diagram of the 5-MW wind turbine and includes a summary of important turbine properties and dimensions. Table 4 shows a summary of the parameters of the thunderstorm downburst used in this simulation study. For the turbine response simulations, assumptions on turbine control and performance are made as follows: (i) When yaw control is required, the yaw rate is first set based on a time-varying moving average of the changing inflow wind direction as long as this rate of change in wind direction is small; if/when this rate of change in wind direction becomes significant, this yaw rate is set to a maximum of 0.3 deg/s. Up to 45 deg of yaw error/misalignment is allowed (i.e., the turbine is assumed to continue to operate as long as the yaw misalignment stays below 45 deg) and (ii) wind speeds above the turbine’s cut-out wind speed are permitted; the pitch control logic for wind speeds above cut-out is assumed to be the same as that for wind speeds below cut-out. Figure 9 shows the pitch control, turbine rotor speed, and power curve as a function of the wind speed at hub height (90 m above ground level (AGL)).

Properties/dimensions | Values | |

Power rating | 5 mW | |

Rotor type | Upwind/3 blades | |

Rotor diameter | 126 (m) | |

Hub height | 90 (m) | |

Cut-in, rated, cut-out | 3, 11.4, 25 (m/s) | |

Rated rotor speeds | 12.1 (rpm) | |

Rotor mass | 110,000 (kg) | |

Nacelle mass | 240,000 (kg) | |

Tower mass | 347,460 (kg) |

Properties/dimensions | Values | |

Power rating | 5 mW | |

Rotor type | Upwind/3 blades | |

Rotor diameter | 126 (m) | |

Hub height | 90 (m) | |

Cut-in, rated, cut-out | 3, 11.4, 25 (m/s) | |

Rated rotor speeds | 12.1 (rpm) | |

Rotor mass | 110,000 (kg) | |

Nacelle mass | 240,000 (kg) | |

Tower mass | 347,460 (kg) |

Downburst | Numerical | |
---|---|---|

Parameter | value | |

U_{r,Max} | (m/s) | 26.4 |

Z_{Max} | (m) | 60 |

R_{Max} | (m) | 430 |

k_{rm} | (m/s) | 1.0 |

t_{d} | (min) | 16 |

U_{trans} | (m/s) | 8 |

U_{amb} | (m/s) | 6 |

Downburst | Numerical | |
---|---|---|

Parameter | value | |

U_{r,Max} | (m/s) | 26.4 |

Z_{Max} | (m) | 60 |

R_{Max} | (m) | 430 |

k_{rm} | (m/s) | 1.0 |

t_{d} | (min) | 16 |

U_{trans} | (m/s) | 8 |

U_{amb} | (m/s) | 6 |

*r*

_{m}(

*t*) =

*R*

_{Max}+

*k*

_{rm}

*t*is the time-dependent radial distance to maximum radial velocity, where

*R*

_{Max}is the initial value of

*r*

_{m}(

*t*) and

*k*

_{rm}is the rate of change of

*r*

_{m}(

*t*) with time; and Π(

*t*) = sin (

*πt*/

*t*

_{d}) is an assumed time-dependent storm intensity function. The velocity field of the storm is the product of the field developed either by the paused downburst or the semi-empirical model and the intensity function, Π(

*t*). The storm touchdown point (i.e., the storm center at

*t*= 0) is assumed to be defined by polar coordinates, (

*R*

_{0},

*θ*

_{0}), relative to a coordinate system defined at the turbine. The storm is assumed to move in a straight line (its track) at a constant translation speed,

*U*

_{trans}. The angle,

*ϕ*, which defines the direction of the storm translation relative to the ambient wind,

*U*

_{amb}, is assumed to be constant. The storm’s outflow is assumed to be perturbed by the boundary layer environmental winds,

*U*

_{amb}(

*z*) [26]; this ambient wind profile,

*U*

_{amb}(

*z*) (directed along the

*x*direction), is modeled by a power law, with a shear exponent of 0.2 in this study. The wind velocity field that the turbine experiences at any time,

*t*, during the downburst is related to the distance,

*d*(

*t*), from the storm center to the turbine at that time:

For the turbine response simulations, the non-turbulent wind velocity field is generated using a semi-empirical model (discussed by Nguyen et al. [12]) as well as the CFD-based paused downburst model. Together with the downburst parameters given in Table 4, the storm touchdown point and direction of translation are chosen such that *R*_{0} = 4 km, *θ*_{0} = 180 deg, and *ϕ* = 15 deg (see Fig. 10). An ambient wind speed of 6 m/s at 90 m is considered. The turbine response is simulated assuming yaw control is available. Input parameters used for the CFD simulation of the paused downburst model are *H* = 2000 m, *D* = 1000 m, and *V*_{in,max} = 40 m/s, along with a selected time of pause, *t*_{p} = 107 s (i.e., *t*_{p}* = 2.14). The output from the CFD simulation is subjected to the required post-processing to yield the non-turbulent wind field.

## 6 An Illustrative Turbine Response Simulation

Figure 11 shows a plan view of the moving downburst and the turbine for the selected downburst simulation. The centers of the circles describe the downburst locations at the beginning of the event and after 8 min. Since the duration of the downburst, *t*_{d}, is 16 min, the storm is at its peak intensity at 8 min, when the maximum radial velocity is 26.4 m/s. The points on the circumference of the circles are where the maximum radial velocity occurs; note that the radius to maximum winds, *r*_{m}(*t*), increases with time (at *t* = 8 min, the radius to maximum winds is 910 m). The storm touchdown point is initially 4 km from the turbine; in this simulation, the moving downburst is within 1.0 km of the turbine at *t* = 8 min, and the radial velocity experienced by the turbine then is close to 26.4 m/s. Figure 12 provides a summary of the procedure adopted to generate the paused downburst model data and subsequent steps involved in generating the mean velocity data at the turbine location at any time instant during the downburst event.

Figure 13 compares the wind speed time series at the hub height simulated using the paused downburst model and the semi-empirical model. The wind speed plotted is the sum of the non-turbulent component simulated using any of the above models and the turbulent component simulated using a frequency domain approach. Figure 14 corresponds to the scenario where the ambient wind speed at 90 m is 6 m/s and yaw control is applied. The upper panel of the figure shows the variation with time of the distance, *d*(*t*), separating the center of the storm from the turbine; the radial distance from the storm center, *r*_{m}(*t*), where the maximum velocity occurs; and the radial velocity at the turbine, shown with color variation, on the *d*(*t*) plot. The storm-to-turbine distance, *d*(*t*), indicates whether the storm is moving toward or away from the turbine. The radial distance to the maximum velocity is an indication of the changing size (i.e., spatial extent) of the storm. The plots of *d*(*t*) and *r*_{m}(*t*), taken together, give an indication of the changing strength of the wind velocity that the turbine experiences through the storm’s evolution. At points in time where the values of *d*(*t*) and *r*_{m}(*t*) are close, the turbine is experiencing the largest instantaneous winds—for instance, from *t* ≈ 400 s to *t* ≈ 600 s, the turbine is at points during the storm where radial velocities are largest. Note that it is not only at the peak intensity of the storm that the turbine experiences maximum wind speeds. Also note that the wind field created using the paused downburst model is similar to that based on the semi-empirical model used by Nguyen et al. [12].

The next panels in Fig. 14 show the yaw misalignment angle (or yaw error) time series; the simulated wind velocity normal to the rotor plane at hub height, *U*_{norm}; the time-varying blade pitch angle; the flapwise bending moment at a blade root due to non-turbulent wind field only; and the flapwise bending moment at a blade root due to both non-turbulent and turbulent wind fields. Between *t* = 0 s and *t* = 200 s, i.e., before the storm’s intensity is felt at the turbine, the mean wind speed at hub height (*z* = 90 m) is approximately 6 m/s, which is below the rated wind speed (11.4 m/s) for this turbine. At these low winds, the blade pitch angle is zero and the power generated is less than the rated 5 mW level (see Fig. 9). As the storm picks up in intensity, the wind speed increases and the flapwise bending moment increases as well, while the blade pitch angle remains at zero until the rated wind speed is reached. When the wind speed reaches rated, i.e., at *t* ≈ 350 s, a first peak (or kink) in the flapwise bending moment time series is seen. Since the winds are continuing to increase, the blade pitch angle must increase in order to maintain the rated power of 5 mW and to limit structural loads (see Fig. 9). Higher wind speeds would be expected to increase loads on the blades; however, the increase in blade pitch angle decreases the blade loads at a faster rate and leads to a decreasing flapwise bending moment starting at *t* ≈ 350 s. It is the change in blade pitch angle from a zero value that produces the kink in the response time series. As the winds get larger, the blade pitch angle continues to increase and the flapwise bending moment continues to decrease. As the winds begin to decrease after *t* = 480 s, the blade pitch angle decreases and the flapwise bending moment starts to pick up. At *t* ≈ 650 s, the blade pitch angle goes to zero while the winds keep decreasing; then, flapwise loads also begin to decrease (this explains the second peak or kink in the flapwise bending moment time series). From this discussion, one can see that pitch control clearly influences the turbine response; whenever the pitch angle increases from zero or decreases to zero, there is a noticeable abrupt change or kink in the response. The kinks are not as clear in *FlapBM*_{2} as they are in *FlapBM*_{1} due to the effect of turbulent fluctuations.

In Fig. 14, note that for the semi-empirical model, the applied yaw control is adequate until *t* = 704 s when the yaw error exceeds 45 deg and the simulation is stopped. With the same downburst parameters, the yaw error does not exceed 45 deg with the paused downburst model. This can be confirmed from Fig. 15 and is also noted in the yaw error panel of Fig. 14. Yaw error represents the difference between the horizontal direction of the resultant wind speed at hub height and the nacelle yaw angle. Although with both the models, the nacelle yaw rate is almost the same, the rate of change of the horizontal wind direction for the semi-empirical model is greater than that for the paused downburst model while the storm is receding. This causes a larger yaw error for the semi-empirical model that eventually exceeds 45 deg. This difference between the two models for the non-turbulent wind field may be explained by their contrasting radial velocity profiles. The radial velocity profile for the semi-empirical model decays faster than it does for the paused downburst model (see Fig. 5). The horizontal direction of the resultant wind speed at hub height is directly dependent on the radial velocity relative to the storm center. If the radial velocity decreases faster (radially), then the horizontal wind direction will also change more rapidly. This, then, leads to larger yaw error for the semi-empirical model than in the case for the CFD-based paused downburst model.

From comparison of the turbine response using the two non-turbulent wind field models, a few general conclusions may be made. The response of the turbine is quite similar with both models, at least with regard to the magnitude of the wind velocities and the loads experienced by the wind turbine. Because the radial velocity profiles for the two models are slightly different, the yaw error experienced by the turbine is also slightly different for the two cases. With the CFD-based paused downburst model, the jet impingement type flow causes radial velocity decays with radial distance that are not as fast as for the semi-empirical model. Since the radial velocity profile used for the paused downburst model was validated against the Hjelmfelt [26] average wind profile in downbursts, we conclude that the model is acceptable for use in turbine loads studies despite its small differences relative to the semi-empirical model [12].

Based on the preceding discussions, it is clear that large loads are not always associated with high winds. Excessive loads may result from fast-changing wind fields during which pitch and yaw controls may not respond sufficiently to limit loads. For example, rapidly changing winds from below-rated to above-rated levels, especially if accompanied by rapid wind direction changes can cause large loads. It is the storm translation speed and the storm duration that determine how fast a wind field will change. Fast-moving storms generally create unfavorable conditions for a wind turbine when both wind speed and wind direction can change rapidly. The ambient wind also has some effect on a turbine’s response during a downburst event; if the ambient wind (at hub height) is below the rated wind speed (11.4 m/s for the 5-MW turbine), loads will likely get higher during the storm. If the ambient wind is already above the rated wind speed, loads will likely decrease during the storm as the winds increase, assuming available pitch/yaw control; such scenarios were discussed in a previous study [12].

## 7 Concluding Remarks

A model for the simulation of the non-turbulent wind field in thunderstorm downbursts was presented. In turbine response studies, non-turbulent and turbulent components of the wind field are generated separately and combined. A detailed procedure for simulating the non-turbulent wind field using a CFD-based model was discussed. This model referred to as a paused downburst model was developed for use in wind turbine loads analysis; the model was validated against limited recorded downburst data available in the literature.

The paused downburst model is compared to a semi-empirical model through the wind turbine response simulation. The following observations were noted. The results shown in this work using both paused downburst model and the semi-empirical model are similar except for the yaw misalignment. This was attributed to the faster decay of radial velocity profiles in the semi-empirical model. But overall, the CFD-based model proves to be a valid alternative model for the semi-empirical model.

During thunderstorm downbursts, high wind speeds and rapid direction changes can occur. Blade pitch control effected by changes to blade pitch angles influence the rotor aerodynamics and can lead to a reduction in turbine loads during high winds that accompany downbursts. Yaw control limits yaw misalignment and can, as well, reduce turbine loads especially in yawed flow conditions as when wind direction changes are rapid.

We note that surface roughness and its corresponding effects on the vertical wind shear were not considered within the scope of the current work. The idea of the paused downburst model is pursued at a qualitative level to mimic highly transient downburst time series data at turbine hub height levels, where surface roughness is expected to have minimal influence. For application of the model to other structures and general engineering purposes, the role of surface roughness can be investigated to update the model in future studies.

## Footnote

This paper is based on the proceedings of the 2012 ASME Wind Energy Symposium.

## Acknowledgment

The authors are pleased to acknowledge the financial support received from Sandia National Laboratories by way of Contract No. 743358 (Manager: Mr. Joshua Paquette). They are grateful too for the technical oversight from Dr. Paul S. Veers, formerly of Sandia National Laboratories, and express their gratitude to Dr. Jason Jonkman of the National Renewable Energy Laboratory for continued assistance with Fatigue, Aerodynamics, Structures, and Turbulence (FAST) and with the turbine model.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*r*=radial distance (m or km) from the center of the downburst

*t*=time variable (s or min)

*D*=diameter (m or km) of inlet jet

*H*=height (m or km) of the origin of the inlet flow from the wall

*Z*=height (m or km) from the wall

*k*_{rm}=rate of change of

*r*_{m}(*t*) (m/s)*r*_{n}=normalized radial distance

*t*_{d}=duration of the storm (s or min)

*t*_{p}=time of pause (s)

*R*_{0}=initial distance (m or km) between downburst and wind turbine

*R*_{Max}=radial position (m or km) where the overall maximum radial velocity occurs

*U*_{amb}=ambient wind speed (m/s)

*U*_{r}=radial velocity (m/s)

*U*_{r,max}=maximum radial velocity (m/s) at distance

*r**U*_{r,Max}=overall maximum radial velocity

*U*_{trans}=translational velocity of the downburst (m/s)

*V*_{in, max}=maximum inlet velocity (m/s)

*Z*_{Max}=height (m or km) at which the overall maximum radial velocity occurs

*t*_{p}* =non-dimensional time of pause

*R*_{Max}* =non-dimensional radial position where the overall maximum radial velocity occurs

*U*_{r,Max}* =non-dimensional overall maximum radial velocity

*Z*_{Max}* =non-dimensional height at which the overall maximum radial velocity occurs

*d*(*t*) =distance (m) between downburst and wind turbine at time

*t**r*_{m}(*t*) =radial distance (m or km) to maximum radial velocity from the center of the downburst

*u*(*r*_{n}) =normalized peak radial velocity

*θ*_{0}=initial orientation between downburst and wind turbine (deg)

- Π(
*t*) = storm intensity parameter

*ϕ*=direction of storm translation (deg)