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wind farm simulations require intensive computational resources and advanced turbine models in order to capture rich amounts of information such as loads along the blades , aerodynamic forces , and power output accurately @xcite . the actuator line model ( alm )
is known to depend on a high grid resolution ( on the order of 50 grid points per rotor ) when using large - eddy simulations ( les ) to provide grid - converged results @xcite . however , in order to study large wind farms and the associated large - scale properties of coupling with the atmospheric boundary layer , wind farms consisting of many turbines must be considered @xcite .
when many turbines must be included in the computational domain , fine spatial resolution on each rotor is often not affordable , especially if one wishes to repeat the les varying flow conditions , rotor designs , etc .
it is thus of interest to study and better understand the performance of the alm approach when using very coarse discretization ( on the order of 10 les grid - points per rotor diameter ) .
it has been seen that power extracted from an alm in les is very dependent on the grid resolution and the width of the kernel used to smear the forces @xcite .
the wake structures are also affected by means of the spatial filter in the velocity field which is not able to resolve the structures smaller than the grid size .
the subgrid - scale closure of the les models the smaller scales of the flow , but because of the spatially filtered velocity may differ from the real unsmoothed velocity , differences of predicted forces and power at the rotor blade can exist
. the axial induction can be affected by the smearing of the velocity field .    here , we study the implementation of an alm in coarse simulations ( 14 les grid - points per rotor ) .
first , we examine the performance of the alm model for a single wind turbine with turbulent inflow conditions for which detailed wind tunnel experimental data are available .
second , a formulation is presented to address differences in power caused by the width of the kernel being used . a single turbine in uniform inflow
is studied to document the effect of the smearing kernel on blade loads and power production .
the current study uses lesgo , @xcite the johns hopkins turbulence research group s large - eddy simulations framework including a recent implementation of the alm . @xcite
lesgo solves the filtered navier - stokes equations for very large reynolds number @xmath0 where @xmath1 denotes spatial filtering , @xmath2 is the velocity , @xmath3 is pressure , @xmath4 is density , @xmath5 is the subgrid scale stress tensor , and @xmath6 is the body force .
the code uses pseudo - spectral representation in the two horizontal directions with the third ( vertical ) direction using second - order centered finite differencing .
the approach is kinetic - energy conserving .
time advancement is done using a second order adams - bashforth scheme .
subgrid - scale modeling is based on an eddy viscosity type scale - dependent lagrangian dynamic model @xcite .
the model represents the deviatoric part of the sgs stress tensor , @xmath7 , as @xmath8 where @xmath9 is the dynamically computed smagorinsky coefficient , @xmath10 is the grid resolution , and @xmath11 is the symmetric part of the resolved velocity gradient tensor @xcite .
the alm implementation in lesgo @xcite incorporates a wind turbine into the navier - stokes equation , see eq .
[ eq : ns ] , as a body force term @xcite .
this body force is calculated dynamically depending on the local velocity at the location of each actuator point .
the actuator points are used to calculate the blade forces with a high accuracy before they are projected on the computational grid used to calculate the flow field .
lift and drag are calculated based on tabulated airfoil data and then smeared onto the flow field as body forces by means of a gaussian kernel @xmath12 where @xmath13 is the distance from the actuator point , and @xmath14 establishes the width of the kernel .
the two turbines used for this study are the turbine used in the experiments from the group at cole polytechnique fdrale de lausanne ( epfl ) which has a 0.15 m diameter , @xcite and the nrel 5mw reference , which has a 126 m rotor diameter @xcite .
the turbine used by the group at epfl is a gws / ep-6030 .
@xcite . the chord and pitch angle along the blades are found in their previous work @xcite .
the lift and drag coefficient tables were obtained by using a cambered airfoil , i.e. we used airfoil ( 9 ) from the work by sunada et al .
@xcite in order to match the wake profiles close to the center of the wake , a nacelle has to be included in the model .
power and thrust coefficient curves have been obtained using blade element momentum theory and are shown in figure [ fig : cpct ] .
it can be observed that this turbine provides a poor performance in terms of power , i.e. low @xmath15 value , compared to a large scale turbine which operates on a range closer to the betz limit .
the thrust coefficient , however , is closer to typical values observed for operational large scale turbines .
we now present simulation of a wind turbine with turbulent inflow .
we use the experiment from the epfl group @xcite , here focusing on the single - turbine case .
the inflow condition is matched by using a concurrent - precursor simulation matching the given roughness height and friction velocity of the experiment @xcite .
the tip speed ratio of the turbine is set to @xmath16 as in the experiment . as a first step
we compare the current implementation of the actuator line model ( alm ) with and without the effects of the nacelle . here
the les uses a coarse discretization , 14 points across the rotor .
the alm has been tested with and without a nacelle and tower modeled as drag objects with @xmath17 coefficients of 1.0 and 0.5 respectively .
is the mean velocity at hub - height at the inflow location ( -1d ) . ]
figures [ fig : multiplotu ] and [ fig : multiplotupup ] show streamwise velocity and turbulence intensity profiles across the wake for a single turbine in a turbulent boundary layer .
the non - dimensional scale used is the hub - height velocity @xmath18 .
it is to be noted that in order to obtain good agreement between the experiments and simulations a nacelle model needs to be included .
the current implementations provides excellent agreement for the velocity profiles of the wake .
the turbulence intensity computed in the les under - predicts the experimental measurements .
this is expected from les , since some of the scales are filtered , thus reducing the fluctuations .
the overall trends agree quite well , and it can be observed that at 2 - 3d downstream the wake of the nacelle and tower have a big influence in both the mean velocity and turbulence intensity .
the wake profiles obtained agree very well with the experiments .
however , quantities at the turbine , such as power and thrust , are very dependent on the grid resolution and the value of the kernel width @xmath14 .
@xcite the power coefficient obtained from the simulation for this turbine is @xmath19 .
this power coefficient is significantly higher than what is predicted by the turbine design using bem ( figure [ fig : cpct ] ) . while the overall thrust may be realistic ( since the wake profiles agree well ) , the differences between the predicted @xmath15 and the expected @xmath15 from bem points to possible problems due to the smearing of velocity inherent in alm with very coarse les resolutions . in order to better understand the discrepancy
, first we perform a simulation under uniform inflow , and then present some initial thoughts of how the model inaccuracies may be remedied .
we consider the simulation of the nrel wind turbine with laminar uniform inflow .
we use les with a grid resolution of @xmath20 4 m in each direction , an inflow velocity of 8 ( m / s ) , a prescribed tip - speed ratio of @xmath21 , and vary the gaussian kernel width @xmath14 .
it is known that wind turbine loads along the blades depend significantly on the filtering size of the gaussian kernel used to smear the forces of the turbine .
figure [ fig : bem ] shows angle of attack , axial velocity , lift and drag along the blade for an nrel 5mw reference turbine modeled as an actuator line with different kernel widths ( @xmath14 ) .            from this figure
we can see that the largest difference is in axial velocity .
this difference will cause differences in the lift and drag curves , that are more pronounced as we move further towards the tip .
these differences change the total power that the turbine is extracting from the flow .
next , we present some analytical considerations focusing on the effects of the spatial filtering inherent in les and its impacts on the alm computations of forces .
we take the lift component of the body force term per unit width in a 2d frame and write it ( simplifying ) as : @xmath22 where @xmath23 is the chord , @xmath4 is the density , @xmath24 is the lift coefficient , and @xmath25 is the magnitude of the local velocity on the reference frame of the moving airfoil section . the @xmath24 coefficient is a function of @xmath25 through its dependence on angle of attack . for small angle of attack , @xmath26 and thus the angle of attack
can be expressed as @xmath27 , where @xmath28 is the axial velocity .
the lift coefficient for small angles of attack is given by @xmath29 where @xmath30 is a constant and @xmath31 is an offset .
we will assume @xmath32 so that @xmath33 the filtered body force can be defined as @xmath34 now we can also write a similar equation for the unfiltered force : @xmath35 and from here : @xmath36 if we use the argument that the velocity magnitude @xmath25 is dominated by the rotational component of the velocity vector ( which is valid for high tsr s ) then we can take the velocity magnitude out of the filtering operation and we obtain @xmath37 the relationship in eq .
[ eq : ratio ] enables us to interpret the results shown in fig .
[ fig : bem ] ( top right ) in the sense that the ratio of unfiltered force equals the ratio of unfiltered to filtered _ axial _ velocity .
we wish to determine the unfiltered force but only know the filtered force .
if we can model ratio of unfiltered and filtered axial velocity , eq [ eq : ratio ] would enable us to determine the correct ( unfiltered ) force .    while the derivation of eq .
[ eq : ratio ] was based on spatial filtering , we now interpret variables filtered at a scale @xmath14 as the variables one obtains by running les and alm with a kernel width @xmath14 .
while this assumption can not be firmly established on theoretical grounds , in practice it is a useful assumption and is commonly made in les .
for example , if we denote the axial velocity and force obtained in a simulation with a kernel @xmath14 as @xmath38 and @xmath39 , and associate the scale @xmath14 with the filtering scale , then @xmath40 and @xmath41 . as a next step ,
we filter eq .
[ eq : ratio ] at another filter scale @xmath42 and assume that the @xmath14-variables are not affected by this smaller - scale filtering
. then one can rewrite eq . [
eq : ratio ] according to @xmath43 we can use simulations performed at various @xmath14 s to test this relationship , as a first step in establishing possible correction strategies for the alm approach .    returning to our simulations of the nrel case , we consider @xmath44 m , and attempt to use the results from the other 4 runs with @xmath45 to rescale the force @xmath46 . if eq .
[ eq : ua2 ] holds , the distribution of @xmath47 should coincide with the rhs of eq .
[ eq : ua2 ] , i.e. with @xmath48 , regardless of @xmath45 chosen .
the first region of the blade is made of cylindrical sections which produce no lift , thus the approach presented is not used in this section . as seen in figure [ fig : ua ] the curves collapse through most of the blade except close to the tip .
the reason for this is that near the tip , the 3d effects of the kernel become more noticeable
.     values ( left ) and re - scaled axial velocity ( right ) to @xmath49 8 m case as a function of blade length , where @xmath50 and @xmath51 m. the data points coincide with the actuator points used in the simulation.,title="fig : " ]   values ( left ) and re - scaled axial velocity ( right ) to @xmath49 8 m case as a function of blade length , where @xmath50 and @xmath51 m. the data points coincide with the actuator points used in the simulation.,title="fig : " ]
the alm has been validated on a relatively coarse resolution with experimental measurements from the group at epfl @xcite .
the les provides good agreement with the experiments , but a nacelle and tower model are required to obtain the proper wake deficit .
when analyzing power output computed by the alm , the values are higher than those predicted by blade element momentum theory . a theoretical analysis is presented that attempts to relate filtered forces to filtered velocities , highlighting the impact of filtering of the axial velocity on the force distribution .
the derived relationship is tested based on simulations done at various @xmath14 values and is seen to hold rather well , except when approaching the tip of the blade .
the results imply that one may be able to predict the fine - grained force corresponding to an @xmath14 smaller than that of the simulation , if one were able to deduce the corresponding axial velocity from the data at coarser @xmath14 .
such a model - based defiltering operation is the focus of ongoing investigations .
lamt was supported by nsf ( igert ) , while international travel and cm were supported by nsf ( iia 124382 , the windinspire project ) .
rjams is supported by the research program fellowships for young energy scientists ( yes ! ) of the foundation for fundamental research on matter ( fom ) supported by the netherlands organization for scientific research ( nwo ) .
in this work the accuracy of the actuator line model ( alm ) in large eddy simulations of wind turbine flow is studied under the specific conditions of very coarse spatial resolutions . for finely - resolved conditions 
 , it is known that alm provides better accuracy compared to the standard actuator disk model ( adm ) without rotation . 
 however , we show here that on very coarse resolutions , flow induction occurring at rotor scales can affect the predicted inflow angle and can adversely affect the alm predictions . 
 we first provide an illustration of coarse les to reproduce wind tunnel measurements . 
 the resulting flow predictions are good , but the challenges in predicting power outputs from the detailed alm motivate more detailed analysis on a case with uniform inflow . 
 we present a theoretical framework to compare the filtered quantities that enter the large - eddy simulation equations as body forces with a scaling relation between the filtered and unfiltered quantities . 
 the study aims to apply the theoretical derivation to the simulation framework and improve the current results for an alm , especially in the near wake where the largest differences are observed .