Wind Energy Conversion System Matlab

Wind Energ. 2017; 00:1–15

This manuscript has been submitted to Wind Energy. Modeling space-time correlations of velocity fluctuations in

Wind Farms

Laura J. Lukassen1, Richard J. A. M. Stevens2, Charles Meneveau3 and Michael Wilczek1 1 Max Planck Institute for Dynamics and Self-Organization, Goettingen, Germany

Abstract

An analytical model for the streamwise velocity space-time correlations in turbulent flows is derived and applied to the special case of velocity fluctuations in large wind farms. The model is based on the Kraichnan-Tennekes random sweeping hypothesis, capturing the decorrelation in time while including a mean wind velocity in the streamwise direction. In the resulting model, the streamwise velocity space-time correlation is expressed as a convolution of the pure space correlation with an analytical temporal decorrelation kernel. Hence, the spatio-temporal structure of velocity fluctuations in wind farms can be derived from the spatial correlations only. We then explore the applicability of the model to predict spatio- temporal correlations in turbulent flows in wind farms. Comparisons of the model with data from a large eddy simulation of flow in a large, spatially periodic wind farm are performed, where needed model parameters such as spatial and temporal integral scales and spatial correlations are determined from the large eddy simulation. Good agreement is obtained between the model and large eddy simulation data showing that spatial data may be used to model the full temporal structure of fluctuations in wind farms. Copyright c⃝2017 John Wiley & Sons, Ltd.

Keywords

wind farms; turbulent flows; space-time correlations; large eddy simulation; power output fluctuations

Correspondence

Michael Wilczek, Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Goettingen, Germany. Received . .

1. Introduction

The wind energy market is strongly growing with a record in new installations in 2015 [1, 2] and increasing importance expected over the next years, see also . Wind energy is a fluctuating resource that is converted into electrical power in a nonlinear fashion (see e.g. ). These characteristics constitute challenges with respect to power quality and grid stability, cf. , which raise significant interest in predicting short-term power output fluctuations in wind farms. Understanding and predicting the spatio-temporal structure of velocity fluctuations in wind farms which is a main source of the power output fluctuations, is a crucial step towards this goal.

The atmospheric conditions of wind approaching a wind farm can be described by basic atmospheric boundary layer (ABL) theory. For a recent overview, we refer to . Wind speed fluctuations in the ABL occur on different time scales, ranging from annual or seasonal variations, down to minutes and seconds [4, 7]. The short-term fluctuations on scales smaller than around, say, ten minutes are usually referred to as turbulent fluctuations . An introduction to the characteristics of atmospheric wind including typical turbulence intensities and naturally occurring extreme wind gusts (cf. also ), as well as other factors such as geographical (onshore, offshore), landscape and climate influences is given in . The evolution of wind velocities in the wind farm itself is influenced by the aerodynamics of the turbines (see e.g. ), the design of the whole wind farm, e.g. in terms of turbine spacing , and by the influence of turbines onto each other, primarily mediated by turbine wakes . Capturing the full complexity of turbulent flows in wind farms at all scales simultaneously is currently computationally out of reach, emphasizing the need for simplified, physics-based models.

Copyright C

⃝2017 John Wiley & Sons, Ltd.

1

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Space-Time Correlations In Wind Farms

L. J. Lukassen et al. In the present work, we derive such a model for the space-time correlation of the turbulent short-term fluctuations based on concepts from classical turbulence theory. Specifically, it is based on the Kraichnan-Tennekes random sweeping hypothesis with additional mean flow. Focusing on its simplicity, it depends on a minimal set of parameters and is set up to be applied both in the ABL without turbines, or in the wind turbine array boundary layer. With respect to wind energy applications, it provides a starting point for further developments, such as predictive models of the temporal power output fluctuations, or in the area of wind turbine control .

Velocity space-time correlations along with modeling approaches have been widely investigated (see for a classical and for recent reviews) and have also found application to turbulent ABLs. Thus, applying similar approaches to the modeling of wind farms appears worth exploring, which is the scope of the present paper.

Considering turbulence with mean flow, velocity fluctuations are primarily advected with the mean velocity, which forms the basis of Taylor’s frozen eddy hypothesis . Due to the frozen-in assumption, however, these fluctuations do not decorrelate in time along the mean flow. In the random sweeping hypothesis, as introduced by Kraichnan and Tennekes , it is assumed that small-scale velocity perturbations are advected randomly by large-scale velocity fluctuations. This random advection is the primary source of temporal decorrelation. Both Taylor’s and the Kraichnan- Tennekes hypotheses have been thoroughly investigated with respect to their range of validity, e.g. in . Due to their simplicity, they have laid the foundation for a range of models for space-time correlations both in real space [28, 29] as well as in spectral space . In the spectral formulation of space-time correlations in terms of wavenumber-frequency spectra, the random sweeping hypothesis can be applied to derive the wavenumber-frequency spectrum as a product of the wavenumber spectrum and a frequency distribution, whose properties are fixed by the decorrelation hypothesis .

In a real space formulation, as adopted in this paper, this corresponds to a convolution of the spatial correlation function with a temporal decorrelation kernel, consistent with the classical Kovasznay-Corrsin conjecture [18, 32, 33]. Recently, the random sweeping model with additional mean flow advection has been extended to include the physics of ABLs to establish a spectral description of ABL space-time correlations .

In the context of ABLs and wind farms, a variant of space-time correlations, so-called coherence functions, are widely used which contain space-frequency information. Modeling approaches usually combine Taylor’s frozen eddy hypothesis with a model for the spectrum of wind speed fluctuations, such as the Kaimal spectrum or the von K´arm´an spectrum , see . As an example, in the Mann model of turbulence , two-point spatial information is obtained on the basis of one-point spectra with the help of Taylor’s frozen eddy hypothesis. Instead of using Taylor’s frozen eddy hypothesis, de Mar´e and Mann combine the Mann model mentioned above with a model by Kristensen to model space-time correlations in high-Reynolds number flow. Kristensen’s longitudinal coherence is determined by the temporal decay of eddies and their transverse motion. Other coherence models which also avoid the use of Taylor’s frozen eddy hypothesis can be found in literature, e.g. .

In the interest of a broader overview, we also mention stochastic modeling approaches which are used to predict velocity increment statistics in wind farms , and wind power output fluctuations, e.g. . A description of wind power output fluctuations influenced by atmospheric turbulence can be found in .

The model for space-time correlations in wind farms based on the random sweeping hypothesis derived in this paper essentially represents a real space formulation of the previous spectral model for ABL introduced in , generalized here to include finite time correlations of the random sweeping velocity. Given that the flow through a wind farm is strongly inhomogeneous due to the presence of the wind turbines, a real-space formulation appears favorable as it allows us to discern different flow regions such as along and between streamwise lines of turbines. The finite-time correlation of the random advection velocity is crucial for capturing the decorrelation trends at large times more accurately.

The remainder of this paper is structured as follows. In section 2, the mathematical derivation of the model in real space is presented. In a next step, our model is compared to data obtained from large eddy simulation (LES) of turbulent flow in a wind farm. In section 3, we briefly describe the LES tool and simulated flow conditions. The comparison of the model to the numerical data is given in section 4. The principal aim of the analysis and comparison is to establish whether the space-time correlation model in terms of a convolution of the space correlation and a temporal decorrelation kernel can predict observations made from the LES wind farm data. Further details are discussed in an outlook, section 5.

2. Space-Time Correlations From An Advection Model

In the following, we provide a derivation for the space-time correlations of streamwise velocity fluctuations u′

1 In A

plane at hub height in wind farms. It is based on the Kraichnan-Tennekes random sweeping hypothesis [22, 23]; we assume that small-scale velocity fluctuations are advected with a large-scale random velocity v = (v1, v2), which we, for simplicity, restrict to the horizontal plane. The indices 1 and 2 refer to the streamwise and spanwise direction, respectively.

Additionally, we take into account an advection with a mean velocity U = (U, 0) in streamwise direction, corresponding to the mean wind in an ABL. For now, we neglect possible differences between the mean velocity and effective convection velocity, a topic of considerable interest in its own right, see e.g. for turbulent shear flows. The simple dynamical

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Space-Time Correlations In Wind Farms

model discussed in the following serves to motivate the analytical structure of the space-time correlations. We then conjecture that, in essence, the same model holds for the arguably more complex flow in the wind farm and compare the results to LES.

2.1. Advection Model

Assuming that advection with U and v gives the dominant contribution to capture the temporal decorrelation of velocity fluctuations, we arrive at the simple advection equation for the streamwise small-scale velocity component:

1 + (U + V) · ∇U′

1 = 0 .

(1)

To include the random character of the large-scale advection, we assume that v is spatially constant with a Gaussian ensemble distribution and decorrelates exponentially in time:

(2)

for the streamwise random-sweeping velocity v1, and analogously for the spanwise random-sweeping velocity v2. Here, ⟨. . ⟩denotes ensemble-averaging. For practical purposes in the application to wind farm LES data, the ensemble averages are replaced by averages over space and time which is further discussed in the corresponding sections. The assumption of an exponentially decaying correlation function allows us to introduce decorrelation time scales T1 and T2, respectively.

This assumption is tested below with our LES data. In the following, we explicitly compute the space-time correlations of

(3)

in the framework of the advection model, where r = (r1, r2) is the distance vector in the plane at hub height and τ is the time lag. The solution of equation (1) corresponds to a shifted initial condition such that



.

(4)

Under the assumption of homogeneity and statistical stationarity, there is no explicit x- and t-dependence on the left hand side of equation (3). Using equation (4), the space-time correlations can be obtained according to



.

(7)

The average here is taken over both, small-scale and large-scale fluctuations. To factorize the averages in the last step, we have assumed statistical independence of the large-scale random sweeping velocity v at all times in the interval [t, t + τ]

And The Streamwise Velocity Fluctuations U′

1. A similar assumption has been made by Corrsin in the context of the relation between Eulerian and Lagrangian velocity correlations [32, 33]. As a result of this factorization, the first average in the integrand is simply the instantaneous correlation function evaluated at r −a in the plane, i.e. R11(r −a). The averaged delta function represents the decorrelation kernel which determines the temporal decorrelation of the velocity fluctuations.

Under the Gaussian assumption for the large-scale velocity, it can be explicitly evaluated yielding



.

(10)

The effect of the Gaussian decorrelation kernel on the space-time correlations can be interpreted straightforwardly: the mean velocity advection induces a shift of the spatial correlation function, whereas the random advection blurs the spatial

Space-Time Correlations In Wind Farms

L. J. Lukassen et al. correlation by mixing a range of spatial scales. The functional forms of σ1 and σ2 are implied by the exponential decay of the large-scale velocities. While these formulas are already quite involved, the essential point here is that σ1(τ)2 ≈⟨v2

For |Τ| ≪T1 And Σ1(Τ)2 ≈2⟨V2

1⟩T1|τ| for |τ| ≫T1, i.e. the random displacement exhibits a ballistic behavior for short times and then transitions to a diffusive long-time behavior. This behavior, in fact, is not limited to exponentially decaying correlation functions but can be realized with a much larger class of correlation functions.

In the limit of |τ| →0, the decorrelation kernel becomes a delta function; as expected, the space-time correlation function reduces to the spatial correlation only, R11(r, 0) = R11(r). For large time increments, |τ| →∞, the decorrelation kernel becomes very broad and shallow, which leads to a vanishing correlation in time.

2.2. Reduction To A One-Dimensional Model

Because we aim for an even simpler, one-dimensional model for the streamwise velocity correlation in the streamwise direction r1, we make the assumption that the spatial correlation function in the plane can be factorized according



, i.e. the transverse decorrelation of the streamwise velocity is modeled as a Gaussian with a transverse integral length scale L2. Throughout the paper, we use the convention that functions such as R11(...) only contain those parameters as arguments which are not a priori set to zero. The factoring assumption, which is robust with respect to the functional form of the transverse decorrelation, allows to explicitly evaluate the transverse contribution of the convolution integral. As a result, we obtain

2

 .

(11)

The space-time correlation is given by a convolution of the instantaneous correlation function with a Gaussian temporal decorrelation kernel, which is parameterized by the mean velocity and the mean square random sweeping velocities. Interestingly, the width of the decorrelation kernel is given by the effect of streamwise random sweeping σ1, whereas its amplitude depends on both, streamwise and spanwise random sweeping. Furthermore, σ2(τ) and L2 only appear in

, and not independently of each other. The derivation of our model resembles parts of the derivation of the wavenumber-frequency model for ABLs , however, it is further extended by incorporating the temporal decorrelation of the random sweeping. Our resulting model also shows similarities to the Kovasznay-Corrsin convolution as presented in for homogeneous turbulence advecting in streamwise direction.

To sum up, the model depends upon the following parameters: the advection velocity U, the second moment of the streamwise and spanwise random sweeping velocities ⟨v2

1⟩, ⟨V2

2⟩, the correlation time scales of the random sweeping velocities T1, T2, the transverse integral length scale of the streamwise velocity fluctuations L2 and the space correlation of the streamwise velocity fluctuations R11(r1). By a comparison to LES, we explore the potential of the model presented here to describe space-time correlations of the streamwise velocity fluctuations in the hub height plane in wind farms. In the following, we discuss how the free parameters can be determined from LES. As an alternative, these parameters could also be prescribed by further modeling assumptions, which is beyond the scope of the present paper.

3. Large Eddy Simulations And Parameters

The applicability of the model in equation (11) for describing space-time velocity correlations in wind farms is investigated by LES data. The goal is to compare the velocity space-time correlations predicted by the model to velocity space-time correlation data from the LES.

3.1. Description Of Les

In the wind farm LES, we solve the filtered incompressible Navier-Stokes equations without buoyancy, system rotation or other effects to model a neutral pressure-driven ABL flow over a rough surface [51, 52]. In horizontal directions, a pseudo-spectral discretization with periodic boundary conditions is adopted and second-order centered finite differences are used in the vertical direction. The subgrid stresses are modeled using a scale-dependent Lagrangian dynamic approach in conjunction with the Smagorinsky model and a sharp spectral cutoff test-filter . The wall stress at the ground is captured by a standard rough-wall model using velocities test-filtered at twice the grid scale . For the top boundary we use a zero-vertical-velocity and zero-shear-stress boundary condition. Therefore, the modeled flow effectively corresponds to a ‘half-channel flow’ with an impermeable centerline boundary. Time-stepping is performed with a second- order Adams-Bashforth scheme. Here, we consider the idealized case of a fully developed wind turbine array boundary layer, modeled by a periodic array of wind turbines. The individual wind turbines are modeled as actuator disks .

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⟨U1⟩T(X)/U∗

Figure 1. Left: Streamwise velocity component normalized by the friction velocity u∗at one instant in time. 16 × 16 turbines are located in the domain indicated by the black lines. Right: Temporally averaged streamwise velocity component normalized by u∗. The individual turbines clearly introduce inhomogeneities in the flow field, especially along the lines of the turbines. One can also observe high-speed streaks between the turbines when averaging over time. All plots in this paper have been created with matplotlib .

Ti

Figure 2. Left: Temporally averaged second moment of the streamwise velocity fluctuations as given in equation (13), normalized by u∗2. The field is highly inhomogeneous. Right: Turbulence intensity of streamwise components, TI =

1 ⟩T(X)/⟨U1⟩T(X). The

highest intensity can be found at the edges of the turbine wakes. Their back-reaction on the flow is modeled by a thrust force added to the momentum equation. For a more complete account on wind farm modeling by means of LES, we refer the reader to .

Throughout the paper, x1 and x2 denote the streamwise and spanwise directions, respectively, and x3 denotes the vertical direction. The simulation is performed on a 512 × 256 × 128 grid discretizing a domain of size Lx1/H × Lx2/H × Lx3/H = 4π × 2π × 1, where H is the height of the ABL, i.e. the domain height. In this domain, 16 × 16 turbines are placed with a turbine diameter D = 0.1 H in an aligned configuration. The streamwise spacing of the turbines Sx1 is 7.85 D = 0.785 H, the spanwise spacing Sx2 is 3.93 D = 0.393 H. The simulation is conducted with a time step of 6 × 10−5 H/u∗, where u∗is the friction velocity associated with the driving pressure gradient. The data are collected after the statistically stationary state is reached. The velocity field at hub height (x3 = 0.1 H) is stored every ten time steps at 8192 snapshots, thus, over a total duration of about 4.9 H/u∗. This corresponds to about 3.5 flow through times at hub height, or about 55 inter-turbine travel times (i.e. the time for traveling from one turbine to the next). To give an impression of the flow, the left panel in figure 1 shows a visualization of the streamwise velocity. One can see that higher- velocity flow streaks meander between wind turbine columns. In the present paper, columns refer to turbines aligned in the streamwise direction, whereas rows refer to neighboring turbines in spanwise direction. The right panel of figure 1 shows the temporal mean velocity. Even after time averaging there remain spatial large scale inhomogeneous streaks, visible as elongated regions. It is well known that such features are very slow to converge in this type of flow . Here, we average over these in space as appropriate, i.e. when we investigate the applicability of the model between lines of turbines, we only average over all lines in the middle between the turbines in order to minimize the effect of these streaks. Furthermore, pronounced wakes behind the wind turbines are clearly visible.

For the subsequent analysis, we decompose the total streamwise velocity into its temporal mean and the remaining

U1(X, T) = ⟨U1⟩T(X) + U′

1(x, t) .

(12)

The ⟨...⟩t-brackets denote an average over time. As a result, the temporal mean velocity is a function of the position in the plane. The mean square velocity fluctuations are obtained analogously:

1⟩T(X) −⟨U1⟩2

t(x) .

Space-Time Correlations In Wind Farms

L. J. Lukassen et al. The mean square velocity fluctuations are shown in the left panel of figure 2. As can be seen, the inhomogeneities introduced by the wind farm also carry over to the statistical quantities and have to be taken into account when computing statistical averages. As expected, the temporal mean velocity is higher between the lines of wind turbines than along the lines, cf. right panel in figure 1. There, the velocity deficit behind the wind turbines is clearly visible.

As for the fluctuations, they appear lower between the turbine columns. The highest velocity fluctuations are generated at the edges of the wind turbine wakes. The turbulence intensity, defined as the relation of the root mean square fluctuations and the temporal mean velocity, is shown in the right panel of figure 2. The highest turbulence intensity can be found at the edges of the turbine wakes accompanying the regions of the highest velocity fluctuations. Furthermore, it is visible that along the lines of the turbines, the highest intensity is located in the region behind the turbines within half the distance to the following turbines. This is followed by a less intense region in front of the following turbine.

3.2. Determination of parameters and spatial correlations from LES In the following, we compare the model space-time correlation to the space-time correlation obtained directly from the LES data. To this end, we compute the wavenumber-frequency spectrum from the LES data and determine the velocity space-time correlation by an inverse Fourier transform. To evaluate the model (11), we also compute the spatial correlation function. Based on the LES data, we determine values for the mean velocity U, the streamwise random sweeping velocity

Fluctuations ⟨V2

1⟩, the corresponding integral time scale T1, as well as values for the spanwise random sweeping fluctuations

⟨V2

2⟩and T2, and the transverse integral length scale L2. In the following, we discuss how to obtain these parameters from the LES data. At this stage it is worth recalling that the derivation of the advection model contains a number of simplifying assumptions. To derive the analytical model, scale separation between the spatially constant random sweeping advection velocity and the advected velocity fluctuations has been assumed. It has been discussed by Kraichnan , that even with spatially slowly varying large-scale random sweeping fluctuations, the random sweeping hypothesis as discussed here remains valid. To make it more realistic, we have additionally included a large-scale flow with a finite correlation time in our model. In the application of the model to LES data, however, we do not distinguish anymore between small- and large- scale fluctuations, but rather apply the model to the overall velocity fluctuations. For the model, we furthermore restrict ourselves to advection in the plane at hub height; any vertical transport of streamwise velocity fluctuations is therefore neglected. Also, effects like large-scale shear which have been discussed, e.g. in the context of the elliptic and the Mann models, are currently not taken into account. Additionally, we have assumed a statistically homogeneous flow throughout the derivation in section 2, which is not met in the wind farm application: due to the equidistantly spaced wind turbines, the continuous shift invariance of statistical quantities implied by homogeneity reduces to a discrete one.

For averages in spanwise direction, we take these spatial inhomogeneities into account and distinguish two cases in the following: Firstly, we investigate the space-time correlations along the wind turbine columns. The description of velocity correlations along these lines is the primary purpose of the application of the model. In a second case, the lines between the wind turbine columns are investigated for a comparison. As a consequence, averages to compute the reference space-time correlations from the LES and the space correlation to feed the model space-time correlation are taken over all streamwise positions and times, whereas the spanwise average is restricted to the corresponding subset of lines. Also the correlation times T1 and T2 are determined for the two cases separately. The distinction in two cases is not done for U, ⟨v2

2⟩, And

L2 as discussed below. In the following, averaging brackets such as ⟨...⟩xt denote an average over all times and the whole x1, x2-plane. A subscript ˜x at the ⟨...⟩˜xt-brackets denotes an average over all x1 in streamwise direction and a spanwise average over ˜x2 referring to the reduced subset of lines.

Figure 3 shows the longitudinal and transverse spatial velocity correlations. The longitudinal correlations for the two cases are shown in the left plot. For a comparison, the right panel of figure 3 shows the transverse space correlation which is used for the computation of the integral length scale L2 below.

As for the mean velocity U, we use the overall mean velocity, i.e. the temporally and spatially averaged streamwise velocity ⟨u1⟩xt = 7.88 u∗for both cases. However, as a comparison, the mean velocity along the lines of the turbines is ⟨u1⟩˜xt = 6.51 u∗, while it is ⟨u1⟩˜xt = 8.59 u∗over the subset of lines between the turbines.

The second moment of the streamwise and spanwise random sweeping fluctuations are obtained from the LES straightforwardly by computing the variance of the respective velocity components, i.e. ⟨v2

2⟩= ⟨U2

2⟩xt = 1.48 u∗2 (note that there is no imposed mean flow in the spanwise direction). The way

⟨V2

1⟩is computed does not yield the same result as a spatial average of equation (13), i.e. ⟨u′2

1 ⟩Xt = 2.68 U∗2. For ⟨V1⟩We

do not distinguish the two cases because the highest turbulence intensity is at the edges of the wakes behind the turbine, cf. figure 2 which is neither exactly on the lines of the turbines nor exactly between the turbines. For the spanwise random sweeping fluctuations we also use the overall mean square spanwise fluctuations from LES.

We distinguish the two cases for the determination of the correlation time scales of the random sweeping velocities. The model time scales T1 and T2 are determined from the temporal correlation of the LES streamwise and spanwise velocity

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With L2 From Les

Figure 3. Left: Comparison of the longitudinal normalized streamwise velocity space correlations obtained from LES data for the

1(X1 + R1, X2, T)U′

1(x1, x2, t)⟩˜xt. The decay of the correlation is higher on the lines of the turbines. Right: transverse normalized spatial correlation of the streamwise velocity fluctuation, R11(r2) = ⟨u′

1(X1, X2, T)⟩Xt. The

average here is taken over the whole x2-range. The integral over the positively correlated range of the blue dashed line (LES) is used to determine L2, cf. equation 15. The transverse velocity correlation is modeled as a Gaussian (red solid line) in the modeling process in section 2.2.

With T2 From Les

Figure 4. Comparison of the normalized temporal velocity correlations on the turbine columns from LES data (blue dashed line) to the exponential decorrelation in equation (2), (red solid line). Left: normalized streamwise temporal velocity correlation

1(X, T + Τ)U′

1(x, t)⟩˜xt from LES data and exponential decorrelation with the same streamwise integral time scale as given through the LES. Right: normalized spanwise temporal velocity correlation R22(τ) = ⟨u2(x, t + τ)u2(x, t)⟩˜xt from LES data and exponential decorrelation with the same spanwise integral time scale as given through the LES.

fluctuations with respect to the subset of lines in the respective cases. The integration stops at the first zero crossing:

0

dτ ⟨u2(x, t + τ)u2(x, t)⟩˜xt .

(14)

The ˜t indicates the first zero-crossing of the correlation so that for the integral time scales only the positively correlated range is counted. As a result, the model integral time scales T1 and T2 on the lines of turbines (case 1) are T1 = 0.033 H/u∗and T2 = 0.016 H/u∗. The time scales between the lines of the turbines (case 2) result in T1 = 0.054 H/u∗ and T2 = 0.021 H/u∗. The dashed blue lines in figures 4 and 5 are the normalized streamwise and the spanwise temporal correlation from simulation data. The solid red lines are exponentials with the above given integral time scales which compare favorably to the LES data. As a reminder, we also use the LES data for obtaining the streamwise space correlation R11(r1) for the model. The transverse correlation R11(r2) is modeled as a Gaussian in section 2.2 which requires the determination of the transverse integral length scale L2. The transverse integral length scale L2 of the streamwise velocity

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

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In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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