Eects of wind veer on a yawed wind turbine wake in atmospheric boundary layer ow Ghanesh Narasimhan Dennice F. Gayme and Charles Meneveau

2025-05-03 0 0 3.99MB 23 页 10玖币

侵权投诉

Eﬀects of wind veer on a yawed wind turbine wake

in atmospheric boundary layer ﬂow

Ghanesh Narasimhan, Dennice F. Gayme and Charles Meneveau

Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA

Large Eddy Simulations (LES) are used to study the eﬀects of veer (the height-dependent lateral

deﬂection of wind velocity due to Coriolis acceleration) on the evolution of wind turbine wakes.

Speciﬁcally, this work focuses on turbines that are yawed with respect to the mean incoming wind

velocity, which produces laterally deﬂected wakes that have a curled (crescent-shaped) structure.

These eﬀects can be attributed to the introduction of streamwise mean vorticity and the formation

of a Counter-rotating Vortex Pair (CVP) on the top and bottom of the wake. In a Truly Neutral

Boundary Layer (TNBL) in which wind veer eﬀects are absent, these eﬀects can be captured well

with existing analytical wake models (Bastankhah et al. J. Fluid Mech. (2022), 933, A2). However,

in the more realistic case of atmospheric boundary layers subjected to Coriolis acceleration, existing

models need to be re-examined and generalized to include the eﬀects of wind veer. To this end, the

ﬂow in a Conventionally Neutral Atmospheric Boundary Layer (CNBL) interacting with a yawed

wind turbine is investigated in this study. Results indicate that in the presence of veer the CVP’s top

and bottom vortices exhibit considerable asymmetry. However, upon removing the veer component

of vorticity, the resulting distribution is much more symmetric and agrees well with that observed

in a TNBL. These results are used to develop a simple correction to predict the mean velocity

distribution in the wake of a yawing turbine in a CNBL using analytical models. The correction

includes the veer-induced sideways wake deformation, as proposed by Abkar et al. (Energies (2018),

11(7), 1838). The resulting model predictions are compared to mean velocity distributions from the

LES and good agreement is obtained.

I. INTRODUCTION

Yawing a turbine deﬂects its wake, decreasing wake interactions and potentially increasing the power

output of downstream turbines [1]. Coordinating such actions over a wind farm could improve its overall

eﬃciency [2]. Wake deﬂection due to yaw was studied experimentally [3–7], while Ref. [8] performed an

early Large Eddy Simulation (LES) study and proposed a simple analytical model for predicting the initial

wake skewing angle just behind the turbine. In a subsequent wind tunnel study [9], the formation of an

axial Counter-rotating Vortex Pair (CVP) was observed behind a yawed actuator disk, in the presence of

a uniform inﬂow. The deﬂection of the wake was attributed to the CVP because the vortices (one above

and the other below the actuator disk) induce a side wash velocity that deﬂects the wake from the center of

the turbine. The vortices also deform the wake shape into a curled (crescent-shaped) structure. Ref. [10]

performed further wind tunnel studies of a model wind turbine in a turbulent boundary layer where the

CVP formation was also observed.

Ref. [11] proposed considering the turbine as a lifting surface (applying a height-dependent sideways force

onto the ﬂuid), i.e., analogous to a vertically placed airfoil that sheds streamwise (tip) vortices in the presence

of an incoming mean ﬂow. Evaluation of the induced strength of the CVP near the turbine enabled predicting

the yaw-induced wake deﬂection quite accurately [11]. Other vortex-based models describe the vorticity at

the turbine as a distribution of multiple, discrete point-vortices [12–14] whose downstream transport and

diﬀusion are modeled numerically. Following these studies, Ref. [15] proposed a theory for the generation

and downstream evolution of the CVP. The analytical predictions for the decay of the maximum vorticity

and circulation strength of the vortices showed very good agreement with the LES data, while still assuming

a circular shape of the wake. In a more detailed recent study [16], it was shown that an analytical vortex

sheet-based model can successfully predict the curled wake shape behind yawed turbines. In this model, the

wake edge was treated as a vortex sheet and analytical solutions using truncated power series expansions

were obtained based on the decaying circulation strength estimate of the CVP from Ref. [15]. The Gaussian

wake model for the axial velocity deﬁcit in [10] was then modiﬁed to include the deformation caused by the

vortex sheet that predicted the curled shape and the deﬂection of the wake quite accurately.

Wind turbine wake properties and the performance of wind farms also depend on the prevailing properties

arXiv:2210.09525v1 [physics.flu-dyn] 18 Oct 2022

of the atmospheric boundary layer (ABL). For instance, it is well known that the wake recovery rate (i.e., the

wake expansion coeﬃcient) is aﬀected by the ABL’s thermal stratiﬁcation conditions [17]. At the same time,

the Coriolis acceleration due to Earth’s rotation causes an Ekman spiral ﬂow in the surface layer of the ABL

[18]. This leads to a height-dependent lateral realignment of the incoming wind direction called wind veer,

which can signiﬁcantly aﬀect the wind farm power output [19]. Veer eﬀects have been previously considered

for an unyawed turbine in a stably stratiﬁed ABL [20]. That work modiﬁed the Gaussian wake model with

a veer correction term which successfully predicted the skewed/sheared wake structure arising from the

spanwise shear due to the wind veer. Ref. [21] considered the eﬀects of both yaw and veer on individual

turbine blade aerodynamics. However, the combined eﬀects of veer and yawing on wind turbine wakes and

their modeling via analytical approaches have not received signiﬁcant attention so far. The objective of the

present study is to examine the evolution of turbine wakes in the presence of both turbines yawing and veer,

as well as to include both of these eﬀects in analytical models of the turbine wake.

In order to generate the relevant data, an LES of a yawed wind turbine in the presence of an incoming

mean ﬂow typical of a Conventionally-Neutral ABL (CNBL) that includes veer is performed. The CNBL is

a type of ABL characterized by a neutrally stratiﬁed turbulent boundary layer region separated from the

Geostrophic and stably stratiﬁed free atmosphere by a capping inversion layer at the boundary layer height

[22]. As a reference, we also perform the LES of a yawed wind turbine in a truly neutral boundary layer

(TNBL). The details of the LES are described in §II. The results are analyzed in section III, which focuses

on the eﬀect of wind veer on the downstream evolution of mean vorticity. The results are used to introduce

modiﬁcations to existing analytical models so that both ABL veer and turbine yaw can be represented

eﬃciently and accurately as described in §IV. The main conclusions are summarized in §V.

II. LARGE EDDY SIMULATION OF A YAWED WIND TURBINE IN A CNBL

This section details the LES setup of CNBL and TNBL simulations used to generate data to study the

eﬀect of wind veer on a yawed turbine wake. We use the open-source code LESGO [23], an LES solver

primarily developed to simulate ABL ﬂows [24,25]. The code includes various dynamic sub-grid stress

parameterizations, wall models, wind turbine representations using actuator disk/line models, and inﬂow

generation using the concurrent-precursor approach [26]. The code has been validated by several previous

studies [11,15,25–31]. The governing equations and numerical method, initial conditions for the velocity,

and potential temperature are discussed in §II A. The simulation setup for the current LES study is described

in §II B. The characteristics of the CNBL and TNBL to be simulated are described in §II C, which documents

the main diﬀerences in mean ﬂow velocity proﬁles between both cases.

A. Governing equations and numerical method

The code LESGO solves the ﬁltered Navier-Stokes equations (with the Boussinesq approximation for

buoyancy eﬀects) and the scalar potential temperature transport equation:

∂˜ui

∂xi

= 0,(1)

∂˜ui

∂t + ˜uj∂˜ui

∂xj−∂˜uj

∂xi=−1

ρ0

∂p∞

∂xi−∂˜p

∂xi

θ0

(˜

θ−˜

θ0)δi3−∂τij

∂xj

ρ0

fxδi1+1

ρ0

fyδi2−fc˜u δi2+fc˜v δi1,(2)

∂˜

∂t + ˜uj

∂˜

∂xj

=−∂Πj

∂xj

,(3)

where the tilde (˜

·) represents spatial ﬁltering operation such that ˜ui= (˜u, ˜v, ˜w) are the ﬁltered velocity

components in the streamwise, lateral and vertical directions, respectively, and ˜

θis the ﬁltered potential

temperature. The term τij =σij −(1/3)σkkδij is the deviatoric part of the Sub-Grid Scale (SGS) stress

tensor σij =guiuj−˜ui˜uj. The quantity ˜p= ˜p∗/ρ0+ (1/3)σkk + (1/2)˜uj˜ujis the modiﬁed pressure, where the

actual pressure ˜p∗divided by the ambient density ρ0is augmented with the trace of the SGS stress tensor

and the kinematic pressure arising from writing the non-linear terms in rotational form. The quantities

fi= ( ˜

fx,˜

fy,0) are the streamwise and spanwise component of the turbine’s force imparted on the ﬂuid. The

term −(1/ρ0)∂p∞/∂xiis the mean external pressure gradient applied to drive the ﬂow. The δij in equation

(2) is the Kronecker delta function determining the direction of buoyancy, turbine thrust and Coriolis forces.

In the buoyancy term, g= 9.8 m/s2is the gravitational acceleration, ˜

θ0is the reference potential temperature

scale taken to be 288 K for the CNBL case. In the Coriolis force terms, fc= 2Ω sin φ= 10−4s−1is the

Coriolis parameter at latitude φ= 45◦. In equation (3), the term Πj=g

ujθ−˜uj˜

θis the SGS heat ﬂux.

In the momentum equation (2), a constant mean pressure gradient ∇p∞= (∂p∞/∂x, ∂p∞/∂y, 0) is applied

to drive the ﬂow. For the CNBL case, it is written in terms of the Geostrophic velocity components Ug, Vg

using the Geostrophic balance equation

ρ0

∂p∞

∂x =fcVg,1

ρ0

∂p∞

∂y =−fcUg.(4)

The Geostrophic velocity components are speciﬁed as Ug=Gcos α, Vg=Gsin α, where G= (U2

g+V2

g)1/2is

the magnitude of the Geostrophic wind which is set to 8 m/s and α, is the angle made by the resultant wind

vector with respect to the streamwise xdirection. For the TNBL case, the dimensionless streamwise mean

pressure gradient is set to a constant value. This constant streamwise pressure gradient ensures the mean

ﬂow is streamwise aligned throughout the domain without any wind veer (V(z)≡0). In addition, since the

TNBL ﬂow is isothermal and neutrally buoyant throughout the domain, equation (3) is not solved for this

case, and also the buoyancy term in the momentum equation (2) vanishes.

In the CNBL simulation, we maintain a mean ﬂow direction such that it is streamwise aligned at the hub

height. This is achieved by choosing a value of αfor the Geostrophic wind such that the wind veer at hub

height is zero (V(z= 0) = 0). To compute the appropriate value of α, we use the Proportional-Integral (PI)

control approach introduced in Ref. [32] with a proportional gain KP= 10, and an integral gain KI= 0.5.

We also impose the constraint V(z= 0) = 0.

We solve the equations for the high Reynolds number limit such that the molecular viscous and heat

diﬀusion terms are neglected in the equations (2) and (3). The necessary diﬀusion for the problem is

provided by modeling the deviatoric part of the SGS stress tensor (τij ) and the SGS heat ﬂux (Πj) as

τij =−2νSGS

T˜

Sij ,Πj=−κSGS

∂˜

∂xj

,(5)

where νSGS

Tis the SGS momentum diﬀusivity, κSGS

Tis the SGS heat diﬀusivity, ˜

Sij = (1/2)(∂˜ui/∂xj+

∂˜uj/∂xi) is the symmetric part of the velocity gradient tensor. The diﬀusivities are related by the SGS

Prandtl number P rSGS =νSGS

T/κSGS

Twhich is taken to be 0.4 in the current study [33]. The diﬀusivities

νSGS

Tand κSGS

Tare modeled as

νSGS

T= (Cs˜

∆)2q˜

Sij ˜

Sij , κSGS

T=P r−1

SGSνSGS

T=P r−1

SGS(Cs˜

∆)2q˜

Sij ˜

Sij .(6)

where Csis the Smagorinsky model coeﬃcient and ˜

∆ = (∆x∆y∆z)1/3is the ﬁlter width. The model

coeﬃcient Csis evaluated using the Lagrangian dynamic scale dependent model [25].

The code uses the pseudo-spectral method along the streamwise and spanwise directions. The wall-normal

direction is discretized using a second-order central ﬁnite diﬀerence method. The second-order accurate

Adams-Bashforth scheme is used for time advancement. A shifted periodic boundary condition is used in

the streamwise direction of the precursor domain to prevent artiﬁcially long ﬂow structures from developing.

This approach also enables the development of statistical homogeneity with a shorter precursor domain size

and less computational cost [34]. A stress-free boundary condition is imposed on the top boundaries of the

domains. The wall stress boundary condition from the equilibrium wall model is applied at the bottom wall

of both the precursor and wind turbine domains. Assuming the grid points near the surface is within the

logarithmic layer (Monin-Obukhov similarity in the absence of stratiﬁcation), the wall stress τwmagnitude

is evaluated as

τw=−˜urκ

ln(z1/z0)2

,(7)

where ˜ur=√˜u2+ ˜v2is the resultant horizontal velocity at the ﬁrst grid point z=z1= ∆z/2, κ= 0.41

is the Von Karman constant, and z0= 0.1 m is the assumed surface roughness height in the present study.

Using equation (7), the wall stress components are evaluated as

τi,3|w=τw

˜ui

˜ur

, i = 1,2,(8)

which are applied as the stress boundary conditions at the bottom boundary. For the CNBL simulation, we

apply a zero buoyancy ﬂux boundary condition q∗= 0 to maintain the neutral stratiﬁcation within the ABL.

The velocity ﬁelds of both the CNBL and TNBL LES are initialized with the log-law velocity proﬁle with

zero-mean white noise superimposed. The noise is initialized for the entire domain in the TNBL while only

for the ﬁrst 900 m from the bottom surface for the CNBL. To simulate the CNBL conditions in the LES,

an initial potential temperature proﬁle with a capping inversion layer is set up such that the boundary layer

is neutral below the layer and is stably stratiﬁed above it (see ﬁgure 1). The capping inversion height is

set to 1 km from the ground. The initial potential temperature (˜

θ) magnitude below the capping inversion

region is 288 K (the same as the reference temperature scale (˜

θ0)). The thickness of the capping inversion

layer where the potential temperature increases linearly from 288 K to 290.5 K is 100 m. Above this capping

inversion layer, the potential temperature increases with a lapse rate of 0.001 K/m.

To represent the wind turbine, we use the local thrust coeﬃcient-based actuator disk model (ADM)

[15,27,31,35]. The ADM treats the turbine as a drag disk of diameter Dand radius R=D/2 imparting

a total force (T) on the ﬂuid directed along the unit normal direction n= cos βi+ sin βjperpendicular to

the disk (see ﬁgure 2) given by

T=−1

2ρ0πR2C0

Tu2

d,(9)

where C0

Tis the local thrust coeﬃcient and udis the disk averaged velocity deﬁned as

ud=Z˜

u·nR(x)d3x=Z(˜ucos β+ ˜vsin β)R(x)d3x.(10)

The udis an average of the velocity in the direction normal to the disk, ˜

u·n= ˜ucos β+ ˜vsin βwith the

integration performed over the actuator disk using the indicator function R(x) (deﬁned below). The local

thrust coeﬃcient is set to C0

T= 1.33, the same value as used in previous studies [15,27,31,35] to represent

standard wind turbine operating conditions. The force is spatially distributed using the smoothed indicator

function R(x) such that the ﬁltered force vector is

f=TR(x)n=TR(x) cos βi+T R(x) sin βj.(11)

The smoothed indicator function is deﬁned according to [23,31]

R(x) = ZG(x−x0)I(x0)d3x0,(12)

where, I(x) and G(x) are the normalized indicator function and Gaussian ﬁltering kernel, respectively, given

I(x) = 1

sπR2[H(x+s/2) −H(x−s/2)] H(R−r),(13)

G(x) = 6

π∆23/2

exp −6||x||2

∆2.(14)

In equation (13), sis the x-direction thickness of the forcing region which is set to 10 m, H(x) is the Heaviside

function which is used to localize the disk within the region −s/2< x < s/2 and r < R, where r=py2+z2.

In the ﬁltering kernel (14), ∆ is the ﬁlter width deﬁned as ∆ = 1.5h, where h=p∆x2+ ∆y2+ ∆z2is the

eﬀective grid size.

We note that the Actuator Line Model (ALM) is a high-ﬁdelity representation of the turbine as it can

capture the eﬀects of root and tip vortices behind the turbine which are not resolved by the ADM. However,

owing to the high computational costs of running LES with ALM and that there are not many diﬀerences

in the far-wake behavior between the ADM and ALM [36], we choose ADM over ALM in this study.

In the following sections, simulation setup and results from the precursor simulation of the CNBL and

TNBL LES cases are discussed.

B. Simulation setup

As shown in the schematic ﬁgure 1, the simulation is performed using two computational domains, the

precursor and wind turbine domains. The turbine is placed in the wind turbine domain while the turbulent

inﬂow simulating the CNBL or TNBL conditions is generated in the precursor domain. The details of the

domain size and number of grid points are summarized in table Iand relevant dimensions are also shown in

ﬁgure 1.

FIG. 1. Schematic of the CNBL simulation setup with the turbine in the concurrent wind-turbine domain, inﬂow

mean velocity proﬁle with streamwise U(z) and wind veer V(z) components, and initial potential temperature proﬁle

θ(t= 0) in the precursor domain.

Case Precursor & wind turbine Number of grid points Grid resolution

domain size (Nx×Ny×Nz) (∆x(m) ×∆y(m) ×∆z(m))

(Lx(km) ×Ly(km) ×Lz(km))

CNBL 3.75 ×1.5×2 360 ×144 ×432

10.4×10.4×4.6

TNBL 3.75 ×1.5×1 360 ×144 ×216

TABLE I: Computational domain size and grid points for LES of yawed wind turbine in the CNBL and

TNBL cases. Note that the grid resolution is the same for both LES domains.

Figure 1includes a sketch of the initial potential temperature (˜

θ) proﬁle (blue line) used to simulate the

CNBL atmospheric condition in the precursor domain. A sponge (or Rayleigh damping) layer at the top

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

EectsofwindveeronayawedwindturbinewakeinatmosphericboundarylayerowGhaneshNarasimhan,DenniceF.GaymeandCharlesMeneveauDepartmentofMechanicalEngineering,JohnsHopkinsUniversity,Baltimore,Maryland21218,USALargeEddySimulations(LES)areusedtostudytheeectsofveer(theheight-dependentlateraldeectionofwindvelo...

展开>> 收起<<

Eects of wind veer on a yawed wind turbine wake in atmospheric boundary layer ow Ghanesh Narasimhan Dennice F. Gayme and Charles Meneveau.pdf

共23页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Eects of wind veer on a yawed wind turbine wake in atmospheric boundary layer ow Ghanesh Narasimhan Dennice F. Gayme and Charles Meneveau

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: