Contents lists available at ScienceDirect

Applied Energy

journal homepage: www.elsevier.com/locate/apenergy

Assessing compressibility effects on the performance of large horizontal-axiswind turbines

Chi Yan, Cristina L. ArcherCollege of Earth, Ocean, and Environment, University of Delaware, Newark DE 19716, USA

H I G H L I G H T S

Assessments of compressibility effects around large wind turbines are conducted. Compressibility effects in flow around large wind turbines are not negligible. Compressibility effects start near the blade tips and impact the wake. Compressibility effects increase as upstream wind speed and tip speed ratio increase. Power generation of wind turbines is lower when compressibility is considered.

A R T I C L E I N F O

Keywords:Wind turbineComputational fluid dynamicsIncompressibleCompressibleBlade elementActuator line modelWind power

A B S T R A C T

The tips of large horizontal-axis wind turbines can easily reach high speeds, thus raising the concern thatcompressibility effects may influence turbine wakes and ultimately power production. All past studies haveassumed that these effects are negligible. Compressibility effects are assessed here in terms of blade aerodynamicproperties and variable density separately. Using the Blade Element Momentum (BEM) method, we find thatunder normal operating conditions (i.e., wind speed

systematically evaluate the limitations of incompressibility with respectto both aerodynamic coefficients of the blades and variable density.

The extraction of energy from the wind by a large wind turbineleaves a wake behind it, which propagates downstream and is char-acterized by lower wind speeds and higher turbulence than the ambientair. The behavior of the wind turbine wake and the possible interactionsbetween different wakes in a large wind farm have been extensivelystudied for more than three decades [5] via wind tunnel studies andcomputational simulations.

Experiments have been successfully conducted in wind tunnels tostudy wind turbine aerodynamics using scaled-down versions of small-size [68] and medium-size [911] rotors. Only two wind tunnel ex-periments were performed for full-size rotors, the National RenewableEnergy Laboratory (NREL) Phase VI rotor [12] and the Model Experi-ments In Controlled Conditions (MEXICO) rotor [13]. The main lim-itation of wind tunnel studies lies in the scale of the wind turbinemodels. Even the full-size rotors are much smaller than the turbinerotors used in the industry, which usually are O(100) m. When ex-tending the wind tunnel measurements to real applications, scalingeffects occur [14].

Studying individual and clustered real-size wind turbines has beenmade possible by computational fluid dynamics (CFD). With CFD, re-presenting large wind turbine rotors with high fidelity, i.e., fully re-solving the geometry, rotation, and effects of the turbine blades, ispossible in principle, but remains nearly impossible in practice becauseit is too computationally intensive, as reviewed in [15,5]. For highReynolds number flows, the length scale of the boundary layer thatforms around the turbine blades is O(103) m, while the length scale ofthe atmospheric boundary layer (ABL) domain is O(103) m. The numberof grid points required to properly simulate such a range of scales isenormous, although some parts of the domain can be resolved atcoarser resolution. To overcome this computational impediment,parameterizations of the aerodynamic forces on the turbine have beentherefore developed to reduce grid requirements. In general, the turbinerotor or blades can be represented by the Actuator Disk Model (ADM) orthe Actuator Line Model (ALM). For both, the aerodynamic forces areobtained with the Blade Element theory [16]. The original ADM uses acircular disk to simulate the rotor and the thrust force induced by thewind turbine is imposed to the flow [1719]; however, the rotationaleffects of the rotor are not taken into account. This limitation wasovercome by another version of the ADM, in which both thrust andtangential forces are imposed to the flow [2022]. The disadvantage ofthe ADM is that the aerodynamic forces imposed on the fluid areaveraged over the rotor area whereas the actual location of the bladeschanges with time. With the ALM, drag and lift forces are calculatedalong actuator lines that represent the rotating blades, therefore therotational effects and movements of the blades are taken into account[2325]. The ADM and ALM can be integrated with either the unsteadyReynolds Averaged Navier-Stokes (RANS) framework [21,26] or theLarge Eddy Simulation (LES) framework [19,27]. Finite ElementMethod (FEM) [28], Finite Difference Method (FDM) [27] and FiniteVolume Method (FVM) [29,30] have been used to solve the URANS andLES systems of equations, using the incompressible assumption.

Some efforts have been made to account for compressibility effectswhen modeling wind turbines and the flow around them, but either forsmall regions confined near the turbine blades or using certain sim-plifications or corrections. Wood [31] assumed that compressibilityeffects, being due primarily to the rotation of the blades, would beconfined to the region near the blades and performed calculations ofaerodynamic properties at various wind speeds using BEM theory. Hefound that, when the wind speeds were of the order of 30m s1, sig-nificant reductions in the wind turbine performance occurred due tocompressibility. Leishman and Beddoes [32] proposed a semi-empiricalstall model in which compressibility effects were simply representedwith a constant correction coefficient. Duque et al. [33] performedsuccessful simulations of compressible flow around a wind turbine

blade (the NREL phase II rotor) but using the so-called thin-layerNavier-Stokes equations. Later Duque et al. [34] simulated the flowaround blade of the NREL phase VI rotor using both CAMRAD II (alifting-line code with a free wake model) and OVERFLOW-D (a com-pressible solver with low Mach-number preconditioning capability); thepower prediction with OVERFLOW-D showed good agreement withmeasurements while CAMRAD II did not and modifications wereneeded. Xu and Sankar [35] solved the viscous compressible flowequations over a small region around the rotor and the other part of thedomain was modeled using an inviscid free-wake method. Pape andLecanu [36] performed 2D and 3D simulations of a two-bladed windturbine with a compressible solver, developed by ONERA [37], over adomain restricted to one 180 azimuthal sector by using periodicboundary conditions. Their 2D simulations showed good agreementwith experiments whereas the 3D computations did not, especially inthe high speed region. In summary, no information can be found in theliterature about assessments of the compressibility effects around largewind turbines in a realistically-sized domain.

The most widely used, averaged or filtered, governing momentumequation for wind turbine and wind farm simulations is the in-compressible, Boussinesq form of the Navier-Stokes equation as follows:

+

=

+

+ + +t

ux

u upx x

g f( ) ( ) ( ) ,ij

j ii j

ij t i i0 0 0ij(1)

where ui is the averaged or filtered velocity, ij is the mean or resolvedlaminar stress tensor, tij is the turbulent stress tensor, gi is the grav-itational acceleration, fi is the body force from the turbine blade model(ADM/ALM), and, from the Boussinesq approximation, air density isassumed constant everywhere (0) except in the gravity term ( ). Next,the buoyancy term can be linked to temperature to give the final formof the three governing equations (continuity, momentum, and tem-perature equations):

=ux

0,ii (2)

+

=

+

+ + +ut x

u u

px x

g f( ) 1 1 ( ) [1 ( )] ,ij

j ii j

ij t i i0 0

0ij

(3)

+

=

t x

u qx

q

x( ) ,

jj

j

j

t

j

j

(4)

where is the averaged or filtered potential temperature, 0 is the re-ference, constant, and uniform potential temperature, qj is the mean orresolved heat flux, qtj is the turbulent heat flux, is the coefficient ofvolume expansion.

Two problems arise when compressibility effects are taken intoaccount. First, the body force fi on the flow is equal and opposite to theforce exerted by the ADM/ALM, which is calculated using tabulatedairfoil lift and drag coefficients based on the incompressible assump-tion. Thus, these tabulated aerodynamic properties of each blade sec-tion can be safely used when the Mach number is small because theincompressible assumption is valid. However, the Mach number atblade sections near the tip of large wind turbines can easily reach up to0.20.3. Based on linearized, compressible, subsonic flow analysis, asthe Mach number increases, both lift and drag coefficients of the airfoilwill increase [38], thus compressibility corrections need to be appliedto these coefficients when modeling large wind turbines. This will beexplained in more detail in Section 2.

Second, the body force fi in the incompressible framework is adensity-normalized force. However, to calculate torque, thrust, orpower output of the turbine, the body force needs to be multiplied byair density, which in principle is different at each point and at eachtime. Because of the incompressible and Boussinesq assumptions, airdensity is treated as a constant and therefore the body force is simplymultiplied by a constant reference density 0 (Fig. 1a). Choosing the

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

34

value of this reference density is arbitrary and different reference airdensities will cause a direct change in power prediction. For example,using 1.23 kgm3 instead of 1.18 kgm3, a 4.2% change, will cause adirect increase in power of 4.2%, which is non-negligible in terms ofpower output. The compressibility effects due to variable density canonly be accounted for by using a compressible framework where vari-able density is resolved in space and time and the turbine force is cal-culated directly as fi instead of fi0 (Fig. 1b), as will be done in Section3.

This is the first study that systematically assesses the importance ofcompressibility effects of large horizontal-axis wind turbines. First, weevaluate compressibility effects on the tabulated airfoil properties using1D BEM to assess the performance of the NREL-5MW wind turbine(Section 2). Second, we evaluate the variable density effects using 3Dcompressible unsteady RANS simulations of single and two NREL-5MWwind turbines and compare the results with incompressible simulations(Section 3). Lastly, we conduct compressible simulations of the Lill-grund wind farm to validate the results with field observations (Section4).

2. Compressibility effects on blade aerodynamic properties

The BEM method is used to calculate the wind turbine performancewith and without the compressibility correction. A blade element withradius r (Fig. 2) experiences a local relative velocity Urel, which can becalculated as:

= + + U a U a r((1 ) ) ((1 ) ) ,rel2 0 22 (5)

where is the rotational speed of the turbine, a is the axial inductionfactor, and a is the rotational induction factor. Two corrections arenecessary in order to obtain satisfactory values for a and a . The first isPrandtls tip loss factor, F, to account for the finite number of blades of awind turbine, defined as =F a a/ b, where a is the average induction

factor and ab is the value at the blades. The second correction is calledthe Glauert correction, which is an empirical correction to the thrustcoefficient for high axial induction factor values. Different forms ofthese corrections have been developed [39,40]. Here we used the workof [41,42] to obtain the tip loss factor:

=

F

N r Rr

2 cos exp ( )2 sin

,1(6)

where N is the number of blades. The power coefficient and thrustcoefficient are calculated as:

= =C QAU

C TAU

, ,p t12 0

3 12 0

2(7)

where Q and T are the total torque and thrust from the blade elements,given by:

= +dT N L D dr( cos sin ) , (8)

= dQ N L D rdr( sin cos ) , (9)

where L and D are the lift and drag forces on the blade element dr(Fig. 2b). Note that an increase in both L and D would always cause anincrease in the thrust but not necessarily an increase in the torque, dueto the minus sign in Eq. (9). An iterative procedure is needed to solvethese equations, as explained in Manwell et al. [43].

Usually, blade section data are highly proprietary, thus in this study,we choose the well tested NREL-5MW research wind turbine, which hasa diameter of 126m. Details of the blade section data are listed inTable 1. The NREL-5MW turbine blades were developed based on theDOWEC 6MW wind turbine blades [44,45].

The default, tabulated airfoil data used for incompressible calcula-tions contain modifications that account for three-dimensional beha-viors, including: the correction for rotational stall delay [46,47], theextrapolation of the force coefficients using the Viternas method [48],and the incorporation of Beddoes-Leishman dynamic-stall model [49].

Although three types of corrections were included in the tabulatedaerodynamic properties, they do not account for the changes in lift anddrag coefficients when the blade Mach number increases. Such com-pressibility corrections have been developed in previous studies, themost widely used of which is the Prandtl-Glauert compressibility cor-rection [50,51], which is a function of the local blade Mach number MBas follows:

=

=

CC

MC

C

M1,

1,l

l

Bd

d

B

,02

,02 (10)

where Cl,0 and Cd,0 are the lift and drag coefficients for incompressibleflow. The local Mach number is:

=M U c/ ,B rel (11)

where a constant speed of sound c=340m s1 is generally used foratmospheric applications. More sophisticated corrections exist, such asthe Karman-Tsien correction [52,53] and the Laitone correction [54],but the Prandtl-Glauert is used here because of its simplicity and nu-merous applications.

Using the above compressibility corrections to the BEM equations,we assessed the performance of the NREL-5MW wind turbine for uni-form and constant flows with incoming wind speed U0 =5, 10, 15 and20m s1 and for Tip Speed Ratio (TSR) ranging from 0.25 to 16 with a0.25 interval. Results showed that, when the incoming wind speed islow (e.g., 5 m s1), compressibility losses, manifested as a decrease inthe power coefficient Cp, are barely noticeable even at the highest TSRof 16, as indicated by the almost perfect overlap of the blue1 and black-dashed lines in Fig. 3a. As the wind speed increases, the power losses

Fig. 1. Workflow for (a) incompressible, Boussinesq and (b) compressible wind turbinemodeling. Note that, in the Actuator Line Model calculation of blade-induced forces andin the turbine power calculation, density is a constant 0 in (a), but it is a three-dimen-sional, time-dependent, fully-resolved variable in (b).

1 For interpretation of color in Figs. 3, 5, and 9, the reader is referred to the webversion of this article.

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

35

remain negligible at low TSR, but become more and more significantstarting around a TSR of 8, as shown by the larger separation betweenthe purple and the black-dashed lines in Fig. 3a. Using a threshold of5% power losses (indicated by the green-dashed line in Fig. 3c), definedas the percent difference between compressible and incompressiblepower, we conclude that, for wind speed lower than approximately15m s1 and TSR smaller than approximately 12, no correction isneeded for the default incompressible tabulated data in terms of powerproduction. On the other hand, at high wind speeds (>15m s1) witha large TSR (>12), compressibility effects are not negligible andpower losses exceed 20% and can be as high as 50%. This is due to theincrease in both lift and drag coefficients associated with the increasedtip Mach number (Eq. (10) and Fig. 3d), which causes: an increase inthe thrust coefficient (Fig. 3d) and in the total thrust (Eq. (8)); a de-crease in the power coefficient (Fig. 3a) and in the total torque (Eq. (9));and ultimately a decrease in the power generated (Fig. 3c).

In real-world wind farms, the operating TSRs are always keptaround the optimum value, which is around 8 in general and rarelygreater than 12 for large horizontal-axis wind turbines. On the otherhand, the incoming wind speeds, even for the front row turbines, arerarely above 15m s1. When encountering severe operating conditions,such as hurricanes or tornadoes, the wind turbines will simply be shutdown. Thus we conclude that it is acceptable to use the provided in-compressible aerodynamic coefficients to model the power production

of large wind turbines and that compressibility effects on aerodynamicproperties are negligible.

3. Variable density effects

Other than on the aerodynamic properties of turbines, compressi-bility effects can manifest also as variable density, which can cause adirect change on turbine power production as well as on the flow field.In this section, we conduct assessments of the variable-density effectsby performing all simulations using both the incompressible and thecompressible framework. Differences between the simulation resultsusing both frameworks are presented in order to highlight the effectsclearly.

3.1. Numerical methods

For the incompressible Boussinesq framework (Incomp hereafter),we adopt the open-source package SOWFA (Simulator for On/OffshoreWind Farm Applications) [29,30], which was developed at the U.S.Department of Energys NREL based on OpenFOAM (Open source FieldOperation And Manipulation), a set of open-source C++ libraries forthe development of customized numerical solvers. SOWFA is well es-tablished and validated for wind farm applications [5557]. Turbines inSOWFA are modeled using ALM, but the nacelle and tower of the tur-bines are not modeled. The governing equations used in SOWFA werediscussed in Section 1.

For the compressible framework (Comp hereafter), in order tomaintain high consistency with the incompressible framework andmake the results between the two frameworks comparable, we followthe procedure of SOWFA and develop our solver based on OpenFOAM,thus turbines are modeled using ALM (Fig. 1b). While assessing thevariable-density effects, we want to exclude the aerodynamic effects,thus the tabulated lift and drag coefficients of the blade airfoils are keptidentical in both solvers. This is consistent with the findings in Section 2that compressibility effects are negligible on aerodynamic properties ofwind turbines operating under normal flow and TSR conditions. Thegoverning equations in the compressible framework are the Favre-averaged continuity, momentum, and enthalpy equations:

+

=t x

u( ) 0,i

i (12)

+

=

+

+ + + t

ux

u upx x

g f( ) ( ) ( ) ,ij

j ii j

ij t i iij(13)

Fig. 2. Velocities and forces on a blade element. The angle between the local relative velocity and the rotor plane is , the local twist angle of the blade element is , and the angle ofattack is given by = .

Table 1Aerodynamic properties of the NREL-5MW wind turbine based on incompressible bladeelement theory.

Blade element Radius (m) Twist () Length (m) Chord (m) Airfoil

1 2.8667 13.308 2.7333 3.542 Cylinder12 5.6000 13.308 2.7333 3.854 Cylinder13 8.3333 13.308 2.7333 4.167 Cylinder24 11.7500 13.308 4.1000 4.557 DU40_A175 15.8500 11.480 4.1000 4.652 DU35_A176 19.9500 10.162 4.1000 4.458 DU35_A177 24.0500 9.011 4.1000 4.249 DU30_A178 28.1500 7.795 4.1000 4.007 DU25_A179 32.2500 6.544 4.1000 3.748 DU25_A1710 36.3500 5.361 4.1000 3.502 DU21_A1711 40.4500 4.188 4.1000 3.256 DU21_A1712 44.5500 3.125 4.1000 3.010 NACA64_A1713 48.6500 2.319 4.1000 2.764 NACA64_A1714 52.7500 1.562 4.1000 2.518 NACA64_A1715 56.1667 0.863 2.7333 2.313 NACA64_A1716 58.9000 0.370 2.7333 2.086 NACA64_A1717 61.6333 0.106 2.7333 1.419 NACA64_A17

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

36

+

=

t h

x u h

pt

Qx

Qx

( ) ( ) ,j

jj

j

t

j

j

(14)

= p RT , (15)

where =u u /i i and =h h / are the Favre-averaged velocity and

enthalpy, the laminar stress tensor =

+

( ) ( ) ij ux ux ux ij23ij ji kk andheat flux =

Qj

Pr

hxj. The turbulent stress tensor, split into a deviatoric

and an isotropic part, by adopting the eddy-viscosity hypothesis for thedeviatoric part, can be calculated as:

=

+

u

xux

ux

23

13

,t ti

j

j

i

k

kij kk ijij

(16)

where t is the turbulent viscosity, calculated with the k equations[58]. The turbulent heat flux, with the eddy-viscosity hypothesis, iscalculated as:

=

Q

Pr

hx

,t tt j

j(17)

where Prt is the turbulent Prandtl number. Again, the purpose of thissection is to discover the variable-density compressibility effects alone,thus high consistency is required between the two frameworks. Becausethe standard k model [58] is highly consistent between the two, itwas chosen in this study as the turbulence closure for the governingequations. The compressible form has been used with success in vari-able-density thermal stratified flow [59] and free shear flows withMach number effects [60].

Both the incompressible and compressible governing equations arediscretized using the finite volume method on unstructured meshes. Allvariables are cell-centered and collocated on the grid. Linear inter-polation (equal to second-order central differencing) is used to

interpolate cell-centered variables to cell faces. The system of equationsis solved using the predictorcorrector Pressure Implicit SplittingOperation (PISO) method [61] and the implicit terms are integrated intime using Crank-Nicolson discretization; one predictor with two cor-rectors are used in this study. The discretized momentum and enthalpy/temperature equations are solved using an iterative diagonal in-complete-LU preconditioned bi-conjugate gradient matrix solver; thediscretized pressure equations are solved using an iterative precondi-tioned conjugate gradient solver with a diagonal incomplete Choleskysmoother. Both the incompressible and compressible codes are paral-lelized using the message-passing interface (MPI). All simulations ofthis study are conducted on a high-performance computing cluster with192 processors.

The differences between the two frameworks will be always dis-cussed as Comp minus Incomp, where Incomp is taken as the re-ference.

3.2. Single wind turbine cases

This section explores the compressibility effects associated withvariable density in the flow field around a single NREL-5MW windturbine operating in the ABL. The simulations were carried out in aCartesian computational domain with streamwise, spanwise, and ver-tical lengths of 3024, 756, and 756m, respectively. Using the diameterof the NREL 5MW wind turbine as reference (D=126m), the domainsize can be expressed in non-dimensional form as 24D6D6D(Fig. 4a). The computational domain is evenly divided in each directioninto = N N N 312 144 144x y z grid points of sizes

x y z =9.7m5.25m5.25m.A constant geostrophic wind speed Ug is imposed at the domain top

and periodic boundary conditions are used at the spanwise andstreamwise boundaries, so that the two frameworks simulate an in-finitely-large atmospheric boundary layer. The Reynolds number is

Tip speed ratio2 4 6 8 10 12 14 16

Cp

0

0.1

0.2

0.3

0.4

0.5

0.6

No correctionU

0=5

U0=10

U0=15

U0=20

Tip speed ratio2 4 6 8 10 12 14 16

Ct

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Tip speed ratio7 8 9 10 11 12 13 14

Pow

er lo

ss (

%)

0

20

40

60

80

100

Tip speed ratio2 4 6 8 10 12 14 16

Tip

Mac

h nu

mbe

r

0

0.2

0.4

0.6

0.8

1

Fig. 3. Effects of compressibility correction: (a) power coefficient Cp, (b) thrust coefficient Ct, (c) power loss (%) and (d) tip Mach number.

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

37

sufficiently high to neglect molecular viscosity, except at the first gridpoint off the ground, where the Schumanns wall model is imposed [62]and the roughness length is set to be z0 =0.016m. The simulations arecarried out first for the ABL without wind turbines for 14,400 s (phy-sical time), which is long enough for turbulence to become fully de-veloped to capture the log-law of the ABL (precursor run). Then wecollect the flow information at the inflow boundaries from 7200 to14,400 s and start the new simulations with the addition of the windturbine and the inflow boundary conditions from the precursor run. Theturbine is located 3D downstream from the inlet section and at thecenter in the spanwise direction. The height of the turbine hub is 87.6m(or 0.7D). The actuator lines rotate counter-clockwise in the x-plane.For the simulations with wind turbine, periodic conditions are used atthe spanwise boundaries and the free-slip condition is used at the top.Since at the inlet section in the streamwise direction of the domain theinflow information comes from the precursor simulation, the inflow iseffectively non-periodic and unaffected by the wake of the wind tur-bine. At the outlet, a zero-gradient condition is imposed. Details of thesimulations are provided in Table 2.

The first variable of interest is wind speed, shown in Fig. 5 for thecontrol case (Case 2 in Table 2). In both compressible and in-compressible simulations, the flow decelerates in front of the turbinewhile accelerating in the region enveloping the wake. The acceleration

in the outer wake, also found in previous studies [56,63], vanishesquickly around 2D downstream of the wind turbine. This acceleration isslightly lower in Comp than in Incomp, thus Fig. 5c shows wings ofnegative wind speed differences outside of the rotor circle (light blue in5c). Similarly, the deceleration of the flow right in front of the turbine isstronger in Comp and therefore the wind speed right in front of therotor is slightly weaker with Comp than with Incomp (0.2 m s1 inFig. 5c).

In the wake, the wind speed simulated by Comp is slightly higherthan that in Incomp, or the wind speed deficit is slightly weaker in theComp than Incomp results, thus the positive wind speed difference inFig. 5c. The difference caused by compressibility effects has a typicalbowl shape, reaches its maximum in the near-wake region(0.3m s1) by 3D, and becomes negligible in the far-wake region.

The differences can be explained by comparing the two frameworksphenomenologically. In incompressible flow, when a volume of air hitsthe blades, no energy is used to compress the air volume and all of it isused to push the blades and deflect the air. In compressible flow, a smallfraction of the kinetic energy is used to slightly compress the air, thusthe wind speed is lower in front of the rotor, the power extracted is less,and the speed in the wake is higher.

Vertical and horizontal profiles of wind speed (Fig. 6) for bothframeworks confirm the previous findings. In general, the wind speeddeficit is slightly weaker in the Comp wake starting at 1D, but it isalmost recovered to the Incomp value by 5D (Fig. 6b). The shape of thehorizontal TKE distribution is similar in the two frameworks, with twopeaks at the left and right tips of the rotor, which merge into one at 3D(Fig. 7b). The Comp wake has always smaller values of TKE than theIncomp wake, by up to 15%, which is counter-intuitive in a wake withhigher wind speed [64]. In incompressible flow, fluctuations in velocitythat contribute to TKE are stronger because no energy is used to com-press the air, whereas in compressible flow the perturbations areslightly damped because some energy is used towards density changes.In the vertical, again, the wakes simulated by the two frameworks havesimilar TKE shapes with a strong peak at the top tip of the rotor and aweaker peak at the bottom tip (Fig. 7a), as found also in previousstudies [27], but the Comp TKE is lower than the Incomp TKE by up to10%.

Going back to the rule-of-thumb for incompressible flow, any in-crease in the flow velocity would increase the Mach number andtherefore increase the compressibility effects. Two properties thatwould increase the flow velocity are the hub-height wind speed (Uhub)and the rate of rotation of the turbine , both of which have been foundto influence the wind speed deficit and the turbulence properties ofwind turbine wakes [5,15,65,66]. The two properties are combined inthe TSR:

=TSR RU .

hub (18)

The sensitivity of Comp and Incomp results to both is analyzed here, forincreasing values of the TSR (corresponding to Cases 1, 2, 3 and 4 inTable 2).

Compressibility effects begin near the tip region, but are not con-fined there, as wind speed differences are found in the wake with thebowl shape in all cases (Fig. 8, left). An increase of the magnitude ofthe wind speed differences is observed as TSR increases. For Case 1 withthe lowest TSR, the largest difference is around 1%; for Case 2 and 3with the same TSR, the largest difference is around 2%; for Case 4 withthe highest TSR, the difference is up to 3%. Another finding is that theshape of the affected region is also related to TSR. For low TSR, whichmeans relatively high free-stream wind speed, the affected region ex-tends all the way to the far wake without a significant decay in themagnitude of the wind speed difference (Fig. 8, top); for intermediateTSR, the affected region still extends to the far-wake region, but with adecay in the magnitude (Fig. 8, middle); for high TSR, which meansrelatively low free-stream wind speed, the affected region is confined to

Fig. 4. Schematic of the computational domain (not to scale) for the single-turbine andtwo-turbine cases. Domain sizes are expressed as multiples of the diameter of the re-ference NREL 5-MW wind turbine (D=126m). The circles represent the turbine rotors.

Table 2Setup of the five cases considered, with different geostrophic and hub-height wind speeds(Ug and Uhub), rotational speed , and tip speed ratio (TSR). The TSR value for Case 5 isfor the front-row turbine only.

Case N. turbines Ug (m s1) Uhub(m s1)

1 (rpm) 2 (rpm) TSR

1 1 15 11.83 10 5.582 (Control) 1 15 11.83 15 8.373 1 10 7.88 10 8.374 1 10 7.88 15 12.56

5 2 15 11.83 15 10 8.37

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

38

the near-wake and becomes negligible in the far wake (Fig. 8, bottom).Compressibility also changes the turbulence properties of the wake

(Fig. 8, right). Bowl-shaped TKE differences are again observed in thenear-wake region, since the TKE differences also origin from the highMach number zone, i.e., the turbine tip circle region, and then the re-gion further downstream is affected. The Comp wake always exhibitsless TKE than the Incomp, thus the negative value of TKE differences.Both the magnitude and the extent of the TKE differences increase withTSR; however, for all TSRs, the TKE differences disappear past 10D.

Fig. 9 shows the cross-sections in the rotor plane of Comp minusIncomp wind speed differences for different TSR. The blue rings are

caused by the different accelerations outside of the rotor circle, as ob-served in Fig. 5c. The red rings are the origin of the bowl-shaped windspeed differences near the tips of the blades and the change of theirmagnitude is consistent with previous findings, with higher TSR cor-responding to larger differences between the two frameworks. Thesefigures show that the compressibility effects originate at the rotor tips.

Last, the effects on wind power production is discussed.Compressibility effects related to variable density show up as a slightlydegraded turbine performance (Fig. 10a) and they increase with TSR(Fig. 10b, circles). For Case 1, with a TSR of 5.58, the power coefficientdecreases by about 3% from Incomp to Comp; for Case 2 and 3

Fig. 5. Case 2: Horizontal cross-sections at hubheight of: (a) Comp wind speed, (b) Incomp windspeed, and (c) Comp-Incomp wind speed differ-ence, all in m s1. The 12m s1 contour line isshown in black in (a) and (b).

Fig. 6. Case 2: Vertical (a) and lateral (b) profilesof wind speed (m s1) along the wake centerlineat various distances downstream of the turbine(expressed as multiples of the turbine diameterD). The black dashed lines represent positions ofthe turbine hub and tips.

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

39

(TSR=8.37) by about 8% (note that the two cases are indistinguish-able in the figure); and for Case 4 (TSR=12.56) by 20%. Althoughthese values from compressible and incompressible simulations are notdirectly comparable with the theoretical curves shown in Fig. 3, whichwere obtained using blade element theory for a uniform incoming windspeed, we put them together in Fig. 10b. The two compressibility ef-fects: on aerodynamic properties (solid and dashed curves) and on airdensity (circles and filled circles), have a similar pattern, meaning thatboth cause power to be reduced switching from incompressible tocompressible and more so for high TSRs, although the variable-densityeffect is larger.

Several conclusions can be drawn already from the single-turbinecase. First, compressibility effects are not negligible and are not limited

to the region near the blade tips, but impact the entire wake. Themagnitude and extent of the effects in the wake depend on the tip speedratio. A wind turbine wake has a weaker wind speed deficit, higherwind speed and lower turbulence in the wake, and produces less powerwhen compressibility effects are taken into account than when they areignored. Since most wind farms include more than a single turbine, it isimportant to verify if and how these effects come to play when multipleturbine wakes interact with each other, as they do in a wind farm. Thisissue will be addressed in the next section.

3.3. Two aligned wind turbines

This section explores the compressibility effects in the flow field of

Fig. 7. Case 2: (a) vertical and (b) lateral profilesof TKE (m2 s2) along the wake centerline atvarious distances downstream of the turbine (ex-pressed as multiples of the turbine diameter D).The black dashed lines represent positions of theturbine hub and tips.

Fig. 8. Horizontal cross-sections of Comp-Incomp wind speed difference (left) and Comp-Incomp TKE difference (right) at hub height,normalized byUhub andUhub2 respectively, for thefour single-turbine cases in Table 2.

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

40

two NREL 5-MW wind turbines operating in the same ABL flow dis-cussed in the previous section. The numerical setup for two turbinesimulations is kept the same as the single turbine simulations, the onlydifference is that now we have two wind turbines aligned, one locatedat x= 3D and the other one at x= 7D, which are 4D apart (Fig. 4b).The rotational speed of the first and second turbine is 15 and 10 rpm,respectively (Table 2).

The general behavior of the wind speed difference for the front-rowturbine is similar to the single-turbine case, i.e., a weaker wind speeddeficit (indicated by the positive wind speed differences within therotor area in Fig. 11a), weaker acceleration outside of the rotor area,and lower TKE (Fig. 11b) with Comp than with Incomp. In overlappingwakes, some compressibility effects are enhanced and some aredamped. TKE differences are enhanced, meaning that the Comp flowfield becomes less and less turbulent than the Incomp flow field pasttwo or more turbines. On the other hand, wind speed differences aredamped, with a similar bowl-shaped pattern as in the control case, butgenerally weaker and less extended downstream past the second tur-bine. This suggests that, without considering TKE, wind power pro-duction is most different at the front row, with Incomp predictinghigher power than Comp, but the two converge eventually as thenumber of turbines downwind increases.

Horizontal profiles of wind speed and TKE for both frameworks(Fig. 12) confirm that wind speed and TKE respond differently inoverlapping wakes when compressibility effects are considered. Thewind speed deficit difference becomes insignificant at the second tur-bine and remains as such downstream. TKE is higher in the second wakewith both frameworks, but the TKE difference becomes even larger pastthe second turbine than past the first.

4. Validation

To validate the results, we acquired power production data at anexisting wind farm, the Lillgrund offshore wind farm [55], located offthe coast of Sweden in the Baltic Sea, for a period of approximatelythree years at a temporal resolution of 1min. Lillgrund contains 48Siemens SWT-2.3MW wind turbines [67] with rotor diameter D= 93mand hub height H=63.4 m. The prevailing wind direction at Lillgrundis from the southwest (225), for which the turbine spacing is 4.3Dalong the wind and 3.2D across the wind, similar to the spacing used inthe previous sections.

We conducted a simulation of the entire Lillgrund wind farm usingthe compressible framework with the aim of comparing the simulatedpower output against the observed to validate the results. The compu-tational domain has streamwise, spanwise, and vertical lengths of 4000,4000, and 756m, respectively (Fig. 13a). The computational domain isdivided in each direction into =N N Nx y z

=512 512 144 37,748,736 grid points of sizes xy z =7.8m7.8m5.25m. Again, precursor simulations were

performed using periodic lateral conditions (north to south, west toeast) with a constant geostrophic wind speed of 15m s1 coming fromthe southwest (225). The precursor simulations were run for 14,400 sto develop a fully turbulent, neutrally stratified boundary layer. Thevalues of wind speed and temperature at the south and west boundarieswere saved from 7200 s to 14,400 s and used as the inlet boundaryvalues to start the wind farm simulations with the 48 wind turbines; topand bottom boundary conditions remained the same as the precursor.The numerical discretization method and the algorithms to solve thesystem of governing equations are kept the same as in the single- and

Fig. 9. Vertical cross-sections in the rotor plane of Comp-Incomp wind speed difference, normalized by Uhub, for the four single-turbine cases in Table 2.

Fig. 10. (a) Power production of the four incompressible and four compressible single-turbine cases described in Table 2, normalized by the power of the incompressible cases. (b) Effectsof compressibility on the power coefficient Cp from BEM (lines) and Comp/Incomp simulations (circles).

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

41

two-turbine cases (Section 3.1). Time-averaged horizontal velocitynormalized by hub height wind speed is shown in Fig. 13b, where thewakes of the 48 turbines are clearly visible. As far as we know, this isthe largest domain and the finest-resolution simulation of a wind farmconducted to date using a compressible framework.

To validate the compressible simulation results, we extracted asubset of the observed power data at each turbine for wind directionsbetween 215 and 235 (using the yaw angle as a proxy for wind di-rection, as in [56]) and with wind speeds between 11m s1 and12m s1 (corresponding to a geostrophic wind speed of 15m s1), fora total of approximately three months of data. Relative power wascomputed next as the power of each wind turbine normalized by thepower of the corresponding front turbine of each column and theaverage was calculated over the three months of interest. Qualitychecks were performed to ensure that cases with turbine shut-downs formaintenance, or with pitch controls not reproducible in the simulations,or with excessive yaw biases would not be retained. After cleaning thedata from all such scenarios, observed relative power for selected

columns (E-H) was obtained and compared against the simulated re-lative power.

In general, the compressible simulation results are in great agree-ment with the observations Fig. 14, as the simulated relative power isalways within one standard deviation of the observed and often thematch is very good. For example, the drop in relative power betweenthe first and second turbine in each column was correctly reproduced,especially in Column H, and so was the flattening of the curves after thesecond turbine. Note that, in Column E, the third turbine (34) shows arecovery of the relative power because there is a hole in the middle ofthe farm with no turbines (Fig. 13a), a feature well reproduced by thesimulation. The root mean square error for the four columns is 0.053,0.038, 0.037 and 0.01, respectively, thus we conclude that the com-pressible framework proposed here is successful and should be used forfuture simulations of wind turbine/flow interactions.

Fig. 11. Case 5: horizontal cross-sections of: (a)Comp-Incomp wind speed difference and (b)Comp-Incomp TKE difference at hub height, nor-malized byUhub andUhub2 respectively, for the two-turbine case in Table 2.

Fig. 12. Case 5: horizontal profiles of: (a) windspeed (m s1) and (b) TKE (m2 s2) at hub heightfor the two turbine case, starting at the position ofthe second turbine (4D after the first turbine).

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

42

5. Conclusions and future work

The booming development of wind energy in the past decade, withturbines becoming increasingly taller and with longer blades, thuslarger diameters and rotor areas, brings new challenges to numericalmodeling of wind turbines/wind farms and power prediction. For ex-ample, the tip Mach number of large wind turbines operating in normalconditions can easily reach 0.20.3, which is usually treated as theupper threshold of incompressible flow. Compressibility effects mayarise and alter the flow field as well as the turbine performance. Here,

an assessment of such compressibility effects is performed for the firsttime from two points of view, aerodynamic properties of the turbineblades and variable-density effects.

The Prandtl-Glauert rules are applied to calculate the compressi-bility corrections to the lift and drag coefficients of the blade airfoils forvarious combinations of incoming wind speeds and wind turbine rota-tion speeds. The power coefficient in incompressible flow (no correc-tions) only depends on TSR and does not change with different in-coming wind speeds. However, when compressibility effects are takeninto account only to correct the aerodynamic coefficients, the

Fig. 13. (a) Computational domain and location of the 48 turbines in the Lillgrund wind farm; and (b) time-averaged horizontal velocity normalized by hub height wind speed with windcoming from 225 using the compressible framework.

Column E

313233343536

Rel

ativ

e po

wer

0.2

0.4

0.6

0.8

1

1.2

Column F

3738394041

Rel

ativ

e po

wer

0.2

0.4

0.6

0.8

1

1.2CompObservation

Column G

42434445

Rel

ativ

e po

wer

0.2

0.4

0.6

0.8

1

1.2

Column H

464748

Rel

ativ

e po

wer

0.2

0.4

0.6

0.8

1

1.2

Fig. 14. Time-averaged relative power simulated with the compressible framework versus observed for columns E-H at the Lillgrund wind farm.

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

43

performance of the wind turbine is degraded, but negligibly for lowwind speeds and low TSRs. The degradation starts becoming significantwhen the incoming wind speeds are larger than 15m s1 and TSR islarger than 12, both of which are high and rarely occur in real windfarm applications. Therefore, our first conclusion is that compressibilitycorrections to the aerodynamic coefficients of wind turbine blades (Cdand Cl) are not necessary for simulations of wind farms under normaloperating conditions.

Variable-density effects are assessed by performing simulationsusing both the compressible and the incompressible frameworks andcomputing their differences. Consistent with our previous conclusion,the aerodynamic coefficients of lift and drag are kept the same in bothComp and Incomp simulations in order to isolate the variable-densityeffects. Differences between the Comp and Incomp frameworks start toshow up when wind turbines are present in the ABL. For a single turbinein the ABL, compressibility effects are not negligible already for Machnumber of 0.1 and, although they originate near the tips, they are notlimited to the turbine tip region, but are found also in the wake. As thewake propagates, the wind speed and TKE differences between Compand Incomp also propagate, which leads to lower TKE (up to 15%) anda slightly weaker wind speed deficit in the compressible wake. Theexact distribution of the wind speed and TKE differences between Compand Incomp simulations, in terms of magnitude and horizontal extent,depends strongly on the tip Mach number, thus on TSR. A higher TSRleads to larger compressibility effects and a more confined affectedarea, while with a lower TSR, the compressibility effects are smaller inmagnitude but the affected area extends farther.

In terms of power output, TSR again was the most important factor.For low TSR, smaller than the optimum value of 8, power losses due tovariable-density effects are very small and so are the effects on theaerodynamic coefficients; thus, for a turbine operating at low TSR,compressibility effects can be safely neglected. When the operating TSRis around the optimum value, the correction in the aerodynamic coef-ficients still does not affect the power production significantly whilepower losses due to the variable-density effects start to grow and can nolonger be neglected; thus, for a wind turbine operating around theoptimum TSR, we expect power losses of about 58% due to com-pressibility. When the operating TSR reaches a high value (>12), thepower losses due to variable-density effects and the corrections of theaerodynamic coefficients are both large, thus numerical modeling ofintensively operating wind turbines using the incompressible frame-work might cause large errors; however, wind turbines rarely keepoperating in such severe conditions.

The interactions between turbine wakes are then studied via simu-lations of two aligned turbines. The general behavior of the first turbinein the two-turbine case is the same as in the single-turbine case.However, after the second turbine, the effects of compressibility on theoverlapping wakes behave differently. Looking at wind speed first, thedifference between Comp and Incomp becomes less noticeable than inthe single-turbine case, thus the power output of the second turbine isapproximately the same with the two frameworks. This suggests thatthe power losses due to compressibility might be limited to the frontrow of a wind farm, even with multiple rows. However, when con-sidering TKE, compressibility effects appear to be enhanced downwindof the second turbine, with increasingly lower TKE in Comp thanIncomp. This suggests that the air flow in a large wind farm withmultiple rows of turbines may be significantly less turbulent than pre-viously thought, due to compressibility effects.

The most important implications of these findings are related towind energy generation and are relevant in many real-world applica-tions. First, since the front row of a wind farm always generates themost power, using the Incomp framework, which is the commonpractice today, may introduce overestimates of the total wind farmpower. Second, since the power generated by the front turbines is usedto calculate relative power, which is the power generated by eachturbine divided by that of the front turbine, the relative performance of

the inner turbines may be better than previously estimated with theIncomp approach. Third, understanding turbulent wakes, reducingwake losses, and optimizing wind turbine layout including optimizationof the turbine hub heights are becoming urgent and important issues inthe wind industry [55,68]. This study finds that compressibility effectsshould not be neglected for such applications because predicting thewake development correctly with the Comp framework could lead tomore efficient layout designs, which could lead to benefits of the orderof millions of dollars over the lifetime of a wind project.

Since this work is the first to assess compressibility effects in windfarm simulations, more research is obviously recommended. An accu-rate compressible simulation would require the use of both the cor-rected aerodynamic coefficients and the variable-density compressibleframework together, as opposed to separately as was done here.Without the need for consistency between Incomp and Comp that wascrucial in this paper to compute meaningful differences, more advancednumerical methods can be chosen for the compressible simulations,such as a more sophisticated turbulence closure or large eddy simula-tion with a dynamic turbulence model. Third, it is recommended thatthe combined effects of buoyancy and compressibility via simulations ofthe stable and unstable atmospheric boundary layer in the presence ofturbines be assessed using the compressible framework, since this paperhas focused on neutral stability with effectively no buoyancy. Lastly,aeroelastic coupling between the compressible air flow and the turbineblade structure, which requires algorithms linking structural dynamicsand aerodynamics [66], is needed to assess the effects of compressi-bility on the flexibility of the blades.

In conclusion, compressibility effects associated with large hor-izontal-axis wind turbines cannot be neglected because ignoring themwould cause an overly optimistic prediction of a wind farm powerproduction. It is therefore recommended that future numerical studiesof the flow around wind turbines be based on a compressible frame-work, as opposed to the commonly-used incompressible and Boussinesqframework.

Acknowledgments

This research was funded by the U.S. National Science Foundationunder the EAGER award n. 1357649, with partial support by theDelaware Municipal Electric Corporation. The authors would like tothank Dr. Niranjan Ghaisas for helpful comments. The SOWFA code wasdeveloped by the U.S. National Renewable Energy Laboratory and thesimulations were conducted on the Farber high-performance computercluster of the University of Delaware.

References

[1] MHI Vestas Offshore Wind. Worlds most powerful wind turbine selected forBelgiums largest offshore wind park; 2016. .

[2] Siemens AG. SWT-8.0-154 product brochure; 2016.< https://www.siemens.com/content/dam/internet/siemens-com/global/market-specific-solutions/wind/data_sheets/swt-8.0-154-data-sheet-wind-turbine.pdf > .

[3] Offshore Wind. Ming Yang installs SCD 6.5MW turbine; 2014.< http://www.offshorewind.biz/2014/11/03/ming-yang-installs-scd-6-5mw-turbine/ > .

[4] Adwen. 8 MW platform; 2016.< http://www.adwenoffshore.com/products-services/products/8-mw-turbines/> .

[5] Vermeer LJ, Srensen JN, Crespo A. Wind turbine wake aerodynamics. Prog AerospSci 2003;39:467510.

[6] Ainslie JF, Hassan U, Parkinson HG, Taylor GJ. A wind tunnel investigation of thewake structure within small wind turbine farms. Wind Eng 1990;14:2431.

[7] Chamorro LP, Port-Agel F. Effects of thermal stability and incoming boundary-layer flow characteristics on wind-turbine wakes: a wind-tunnel study. Bound-LayerMeteorol 2010;136:51533.

[8] Chen TY, Liou LR. Blockage corrections in wind tunnel tests of small horizontal-axiswind turbines. Exp Therm Fluid Sci 2011;35:5659.

[9] Haans W, Sant T, van Kuik G, van Bussel G. Measurement of tip vortex paths in thewake of a HAWT under yawed flow conditions. J Sol Energy Eng 2005;127:45663.

[10] Krogstad P-, Eriksen PE. Blind test calculations of the performance and wakedevelopment for a model wind turbine. Renew Energy 2013;50:32533.

[11] Cho T, Kim C. Wind tunnel test results for a 2/4.5 scale MEXICO rotor. Renew

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

44

http://www.mhivestasoffshore.com/norther-foi/http://www.mhivestasoffshore.com/norther-foi/https://www.siemens.com/content/dam/internet/siemens-com/global/market-specific-solutions/wind/data_sheets/swt-8.0-154-data-sheet-wind-turbine.pdfhttps://www.siemens.com/content/dam/internet/siemens-com/global/market-specific-solutions/wind/data_sheets/swt-8.0-154-data-sheet-wind-turbine.pdfhttps://www.siemens.com/content/dam/internet/siemens-com/global/market-specific-solutions/wind/data_sheets/swt-8.0-154-data-sheet-wind-turbine.pdfhttp://www.offshorewind.biz/2014/11/03/ming-yang-installs-scd-6-5mw-turbine/http://www.offshorewind.biz/2014/11/03/ming-yang-installs-scd-6-5mw-turbine/http://www.adwenoffshore.com/products-services/products/8-mw-turbines/http://www.adwenoffshore.com/products-services/products/8-mw-turbines/http://refhub.elsevier.com/S0306-2619(17)31734-8/h0025http://refhub.elsevier.com/S0306-2619(17)31734-8/h0025http://refhub.elsevier.com/S0306-2619(17)31734-8/h0030http://refhub.elsevier.com/S0306-2619(17)31734-8/h0030http://refhub.elsevier.com/S0306-2619(17)31734-8/h0035http://refhub.elsevier.com/S0306-2619(17)31734-8/h0035http://refhub.elsevier.com/S0306-2619(17)31734-8/h0035http://refhub.elsevier.com/S0306-2619(17)31734-8/h0040http://refhub.elsevier.com/S0306-2619(17)31734-8/h0040http://refhub.elsevier.com/S0306-2619(17)31734-8/h0045http://refhub.elsevier.com/S0306-2619(17)31734-8/h0045http://refhub.elsevier.com/S0306-2619(17)31734-8/h0050http://refhub.elsevier.com/S0306-2619(17)31734-8/h0050http://refhub.elsevier.com/S0306-2619(17)31734-8/h0055

Energy 2012;42:1526.[12] Simms D, Schreck Scott J, Hand M, Fingersh Lee J. NREL unsteady aerodynamics

experiment in the NASA-Ames wind tunnel: a comparison of predictions to mea-surements. NREL/TP-500-29494. 2001.

[13] Snel H, Schepers JG, Montgomerie B. The MEXICO project (Model Experiments inControlled Conditions): the database and first results of data processing and in-terpretation. J Phys: Conf Ser 2007;75:012014.

[14] McTavish S, Feszty D, Nitzsche F. Evaluating Reynolds number effects in small-scalewind turbine experiments. J Wind Eng Ind Aerodyn 2013;120:8190.

[15] Sanderse B, van der Pijl SP, Koren B. Review of computational fluid dynamics forwind turbine wake aerodynamics. Wind Energy 2011;14:799819.

[16] Glauert H. Airplane propellers. Springer Berlin Heidelberg; 1935. p. 169360.[17] Gmez-Elvira R, Crespo A, Migoya E, Manuel F, Hernndez J. Anisotropy of tur-

bulence in wind turbine wakes. J Wind Eng Ind Aerodyn 2005;93:797814.[18] Jimenez A, Crespo A, Migoya E, Garcia J. Large-eddy simulation of spectral co-

herence in a wind turbine wake. Environ Res Lett 2008;3:15004.[19] Calaf M, Meneveau C, Meyers J. Large eddy simulation study of fully developed

wind-turbine array boundary layers. Phys Fluids 2010;22:116.[20] Srensen JN, Kock CW. A model for unsteady rotor aerodynamics. J Wind Eng Ind

Aerodyn 1995;58:25975.[21] Masson C, Smali A, Leclerc C. Aerodynamic analysis of HAWTs operating in un-

steady conditions. Wind Energy 2001;4:122.[22] Alinot C, Masson C. Aerodynamic simulations of wind turbines operating in at-

mospheric boundary layer with various thermal stratifications. ASME 2002 windenergy symposium; 2002. p. 20615.

[23] Srensen JN, Shen WZ. Numerical modeling of wind turbine wakes. J Fluids Eng2002;124:3939.

[24] Troldborg N, Sorensen JN, Mikkelsen R. Numerical simulations of wake char-acteristics of a wind turbine in uniform inflow. Wind Energy 2010;13:8699.

[25] Lu H, Port-Agel F. Large-eddy simulation of a very large wind farm in a stableatmospheric boundary layer. Phys Fluids 2011;23.

[26] Schluntz J, Willden RH. An actuator line method with novel blade flow field cou-pling based on potential flow equivalence. Wind Energy 2015;18:146985.

[27] Xie S, Archer C. Self-similarity and turbulence characteristics of wind turbine wakesvia large-eddy simulation. Wind Energy 2015;18:18151838.

[28] Hsu M-C, Akkerman I, Bazilevs Y. Finite element simulation of wind turbine aero-dynamics: validation study using NREL Phase VI experiment. Wind Energy2014;17:46181.

[29] Churchfield MJ, Lee S, Michalakes J, Moriarty PJ. A numerical study of the effectsof atmospheric and wake turbulence on wind turbine dynamics. J Turbul2012;13:N14.

[30] Churchfield MJ, Lee S, Moriarty PJ, Martinez L, Leonardi S, Vijayakumar G, et al. Alarge-eddy simulation of wind-plant aerodynamics. In: 50th AIAA aerospace sci-ences meeting including the new horizons forum and aerospace exposition, aero-space sciences meetings; 2012b.

[31] Wood DH. Some effects of compressibility on small horizontal-axis wind turbines.Renew Energy 1997;10:117.

[32] Leishman JG, Beddoes TS. A semi-empirical model for dynamic stall. J AmHelicopter Soc 1989;34:317.

[33] Duque E, van Dam C, Hughes S. Navier-Stokes simulations of the NREL combinedexperiment phase II rotor. In: 37th Aerospace sciences meeting and exhibit, Reno(NV); 1999.

[34] Duque E, Burklund MD, Johnson W. Navier-Stokes and comprehensive analysisperformance predictions of the NREL phase VI experiment. J Sol Energy Eng2003;125:45767.

[35] Xu G, Sankar L. Effects of transition, turbulence and yaw on the performance ofhorizontal axis wind turbines. In: 2000 ASME wind energy symposium, Reno (NV);2000.

[36] Pape AL, Lecanu J. 3D Navier-Stokes computations of a stall-regulated wind tur-bine. Wind Energy 2004;7:30924.

[37] Cambier L, Gazaix M. elsA an efficient object-oriented solution to CFD complexity.In: 40th AIAA aerospace sciences meeting & exhibit, aerospace sciences meetings;2002.

[38] Leishman GJ. Principles of helicopter aerodynamics with CD extra. CambridgeUniversity Press; 2006.

[39] Wilson RE, Lissaman PBS. Applied aerodynamics of wind power machines. Corvallis(USA): Oregon State Univ.; 1974.

[40] de Vries O. Fluid dynamic aspects of wind energy conversion. Technical report.DTIC Document; 1979.

[41] Shen WZ, Mikkelsen R, Srensen JN, Bak C. Tip loss corrections for wind turbinecomputations. Wind Energy 2005;8:45775.

[42] Wang X, Shen WZ, Zhu WJ, Srensen JN, Chen J. Shape optimization of windturbine blades. Wind Energy 2009;12:781803.

[43] Manwell JF, McGowan JG, Rogers AL. Wind energy explained. John Wiley & Sons,Ltd; 2010.

[44] Kooijman HJT, Lindenburg C, Winkelaar D, van der Hooft EL. DOWEC 6 MW pre-design: aero-elastic modeling of the DOWEC 6 MW pre-design in PHATAS.Technical report DOWEC 10046_009. Energy Research Center of the Netherlands;2003.

[45] Lindenburg C. Aeroelastic modeling of the LMH64-5 blade. Technical reportDOWEC 10083_001. Energy Research Center of the Netherlands; 2002.

[46] Du Z, Selig M. A 3-D stall-delay model for horizontal axis wind turbine performanceprediction. In: 1998 ASME wind energy symposium, aerospace sciences meetings;1998.

[47] Eggers AJ, Chaney K, Digumarthi R. An Assessment of approximate modeling ofaerodynamic loads on the UAE Rotor. In: ASME 2003 wind energy symposium,aerospace sciences meetings; 2003.

[48] Viterna LA, Janetzke DC. Theoretical and experimental power from large hor-izontal-axis wind turbines. Technical teport NASA TM-82944. U.S. Department ofEnergy; 1982.

[49] Jonkman J, Butterfield S, Musial W, Scott G. Definition of a 5-MW reference windturbine for offshore system development. Technical report NREL/TP-500-38060.National Renewable Energy Laboratory; 2009.

[50] Glauert H. The effect of compressibility on the lift of an aerofoil. Proc Roy Soc LondA: Math, Phys Eng Sci 1928;118:1139.

[51] Glauert H. The elements of aerofoil and airscrew theory. Cambridge UniversityPress; 1983.

[52] Tsien HS. Two-dimensional subsonic flow of compressible fluids. J Aeronaut Sci1939;6:399407.

[53] von Karman T. Compressibility effects in aerodynamics. J Aeronaut Sci1941;8:33756.

[54] Laitone EV. New compressibility correction for two-dimensional subsonic flow. JAeronaut Sci 1951;18:350.

[55] Archer CL, Mirzaeisefat S, Lee S. Quantifying the sensitivity of wind farm perfor-mance to array layout options using large-eddy simulation. Geophys Res Lett2013;40:496370.

[56] Ghaisas NS, Archer CL, Xie S, Wu S, Maguire E. Evaluation of layout and atmo-spheric stability effects in wind farms using large-eddy simulation. Wind Energy2017;20:122740.

[57] Bhaganagar K, Debnath M. The effects of mean atmospheric forcings of the stableatmospheric boundary layer on wind turbine wake. J Renew Sustain Energy2015;7:013124.

[58] Launder BE, Spalding DB. The numerical computation of turbulent flows. ComputMethods Appl Mech Eng 1974;3:26989.

[59] Khalil EE, Spalding DB, Whitelaw JH. The calculation of local flow properties intwo-dimensional furnaces. Int J Heat Mass Transfer 1975;18:77591.

[60] Launder BE, Morse A, Rodi W, Spalding DB. Prediction of free shear flows: acomparison of the performance of six turbulence models. Technical report NASASP-311. NASA; 1973.

[61] Issa RI. Solution of the implicitly discretised fluid flow equations by operator-splitting. J Comput Phys 1986;62:4065.

[62] Schumann U. Subgrid scale model for finite difference simulations of turbulentflows in plane channels and annuli. J Comput Phys 1975;18:376404.

[63] Ghaisas NS, Archer CL. Geometry-based models for studying the effects of windfarm layout. J Atmosph Ocean Technol 2016;33:481501.

[64] Archer CL, Colle BA, Veron DL, Veron F, Sienkiewicz MJ. On the predominance ofunstable atmospheric conditions in the marine boundary layer offshore of the U.S.Northeastern Coast. J Geophys Res: Atmosph 2016;121:886985.

[65] Hansen MOL, Srensen JN, Voutsinas S, Srensen N, Madsen HA. State of the art inwind turbine aerodynamics and aeroelasticity. Prog Aerosp Sci 2006;42:285330.

[66] Srensen JN. Aerodynamic aspects of wind energy conversion. Annu Rev FluidMech 2011;43:42748.

[67] Siemens AG. SWT-2.3-93 product brochure; 2009.< https://www.energy.siemens.com/mx/pool/hq/power-generation/wind-power/E50001-W310-A102-V6-4A00_WS_SWT-2.3-93_US.pdf > .

[68] Vasel-Be-Hagh A, Archer CL. Wind farm hub height optimization. Appl Energy2017;195:90521.

C. Yan, C.L. Archer Applied Energy 212 (2018) 3345

45

http://refhub.elsevier.com/S0306-2619(17)31734-8/h0055http://refhub.elsevier.com/S0306-2619(17)31734-8/h0060http://refhub.elsevier.com/S0306-2619(17)31734-8/h0060http://refhub.elsevier.com/S0306-2619(17)31734-8/h0060http://refhub.elsevier.com/S0306-2619(17)31734-8/h0065http://refhub.elsevier.com/S0306-2619(17)31734-8/h0065http://refhub.elsevier.com/S0306-2619(17)31734-8/h0065http://refhub.elsevier.com/S0306-2619(17)31734-8/h0070http://refhub.elsevier.com/S0306-2619(17)31734-8/h0070http://refhub.elsevier.com/S0306-2619(17)31734-8/h0075http://refhub.elsevier.com/S0306-2619(17)31734-8/h0075http://refhub.elsevier.com/S0306-2619(17)31734-8/h0080http://refhub.elsevier.com/S0306-2619(17)31734-8/h0085http://refhub.elsevier.com/S0306-2619(17)31734-8/h0085http://refhub.elsevier.com/S0306-2619(17)31734-8/h0090http://refhub.elsevier.com/S0306-2619(17)31734-8/h0090http://refhub.elsevier.com/S0306-2619(17)31734-8/h0095http://refhub.elsevier.com/S0306-2619(17)31734-8/h0095http://refhub.elsevier.com/S0306-2619(17)31734-8/h0100http://refhub.elsevier.com/S0306-2619(17)31734-8/h0100http://refhub.elsevier.com/S0306-2619(17)31734-8/h0105http://refhub.elsevier.com/S0306-2619(17)31734-8/h0105http://refhub.elsevier.com/S0306-2619(17)31734-8/h0115http://refhub.elsevier.com/S0306-2619(17)31734-8/h0115http://refhub.elsevier.com/S0306-2619(17)31734-8/h0120http://refhub.elsevier.com/S0306-2619(17)31734-8/h0120http://refhub.elsevier.com/S0306-2619(17)31734-8/h0125http://refhub.elsevier.com/S0306-2619(17)31734-8/h0125http://refhub.elsevier.com/S0306-2619(17)31734-8/h0130http://refhub.elsevier.com/S0306-2619(17)31734-8/h0130http://refhub.elsevier.com/S0306-2619(17)31734-8/h0135http://refhub.elsevier.com/S0306-2619(17)31734-8/h0135http://refhub.elsevier.com/S0306-2619(17)31734-8/h0140http://refhub.elsevier.com/S0306-2619(17)31734-8/h0140http://refhub.elsevier.com/S0306-2619(17)31734-8/h0140http://refhub.elsevier.com/S0306-2619(17)31734-8/h0145http://refhub.elsevier.com/S0306-2619(17)31734-8/h0145http://refhub.elsevier.com/S0306-2619(17)31734-8/h0145http://refhub.elsevier.com/S0306-2619(17)31734-8/h0155http://refhub.elsevier.com/S0306-2619(17)31734-8/h0155http://refhub.elsevier.com/S0306-2619(17)31734-8/h0160http://refhub.elsevier.com/S0306-2619(17)31734-8/h0160http://refhub.elsevier.com/S0306-2619(17)31734-8/h0170http://refhub.elsevier.com/S0306-2619(17)31734-8/h0170http://refhub.elsevier.com/S0306-2619(17)31734-8/h0170http://refhub.elsevier.com/S0306-2619(17)31734-8/h0180http://refhub.elsevier.com/S0306-2619(17)31734-8/h0180http://refhub.elsevier.com/S0306-2619(17)31734-8/h0190http://refhub.elsevier.com/S0306-2619(17)31734-8/h0190http://refhub.elsevier.com/S0306-2619(17)31734-8/h0195http://refhub.elsevier.com/S0306-2619(17)31734-8/h0195http://refhub.elsevier.com/S0306-2619(17)31734-8/h0205http://refhub.elsevier.com/S0306-2619(17)31734-8/h0205http://refhub.elsevier.com/S0306-2619(17)31734-8/h0210http://refhub.elsevier.com/S0306-2619(17)31734-8/h0210http://refhub.elsevier.com/S0306-2619(17)31734-8/h0215http://refhub.elsevier.com/S0306-2619(17)31734-8/h0215http://refhub.elsevier.com/S0306-2619(17)31734-8/h0250http://refhub.elsevier.com/S0306-2619(17)31734-8/h0250http://refhub.elsevier.com/S0306-2619(17)31734-8/h0255http://refhub.elsevier.com/S0306-2619(17)31734-8/h0255http://refhub.elsevier.com/S0306-2619(17)31734-8/h0260http://refhub.elsevier.com/S0306-2619(17)31734-8/h0260http://refhub.elsevier.com/S0306-2619(17)31734-8/h0265http://refhub.elsevier.com/S0306-2619(17)31734-8/h0265http://refhub.elsevier.com/S0306-2619(17)31734-8/h0270http://refhub.elsevier.com/S0306-2619(17)31734-8/h0270http://refhub.elsevier.com/S0306-2619(17)31734-8/h0275http://refhub.elsevier.com/S0306-2619(17)31734-8/h0275http://refhub.elsevier.com/S0306-2619(17)31734-8/h0275http://refhub.elsevier.com/S0306-2619(17)31734-8/h0280http://refhub.elsevier.com/S0306-2619(17)31734-8/h0280http://refhub.elsevier.com/S0306-2619(17)31734-8/h0280http://refhub.elsevier.com/S0306-2619(17)31734-8/h0285http://refhub.elsevier.com/S0306-2619(17)31734-8/h0285http://refhub.elsevier.com/S0306-2619(17)31734-8/h0285http://refhub.elsevier.com/S0306-2619(17)31734-8/h0290http://refhub.elsevier.com/S0306-2619(17)31734-8/h0290http://refhub.elsevier.com/S0306-2619(17)31734-8/h0295http://refhub.elsevier.com/S0306-2619(17)31734-8/h0295http://refhub.elsevier.com/S0306-2619(17)31734-8/h0305http://refhub.elsevier.com/S0306-2619(17)31734-8/h0305http://refhub.elsevier.com/S0306-2619(17)31734-8/h0310http://refhub.elsevier.com/S0306-2619(17)31734-8/h0310http://refhub.elsevier.com/S0306-2619(17)31734-8/h0315http://refhub.elsevier.com/S0306-2619(17)31734-8/h0315http://refhub.elsevier.com/S0306-2619(17)31734-8/h0320http://refhub.elsevier.com/S0306-2619(17)31734-8/h0320http://refhub.elsevier.com/S0306-2619(17)31734-8/h0320http://refhub.elsevier.com/S0306-2619(17)31734-8/h0325http://refhub.elsevier.com/S0306-2619(17)31734-8/h0325http://refhub.elsevier.com/S0306-2619(17)31734-8/h0330http://refhub.elsevier.com/S0306-2619(17)31734-8/h0330https://www.energy.siemens.com/mx/pool/hq/power-generation/wind-power/E50001-W310-A102-V6-4A00_WS_SWT-2.3-93_US.pdfhttps://www.energy.siemens.com/mx/pool/hq/power-generation/wind-power/E50001-W310-A102-V6-4A00_WS_SWT-2.3-93_US.pdfhttps://www.energy.siemens.com/mx/pool/hq/power-generation/wind-power/E50001-W310-A102-V6-4A00_WS_SWT-2.3-93_US.pdfhttp://refhub.elsevier.com/S0306-2619(17)31734-8/h0340http://refhub.elsevier.com/S0306-2619(17)31734-8/h0340

Assessing compressibility effects on the performance of large horizontal-axis wind turbinesIntroductionCompressibility effects on blade aerodynamic propertiesVariable density effectsNumerical methodsSingle wind turbine casesTwo aligned wind turbines

ValidationConclusions and future workAcknowledgmentsReferences