AERODYNAMIC OF WIND TURBINES IN THERMALLY STRATIFIED TURBULENT ATMOSPHERIC BOUNDARY LAYER Cedric Alinot1 and Christian Masson2
[email protected],
[email protected] 1 PhD student, Ecole de Technologie Sup´erieure, Montr´eal, (Qu´ebec), Canada 2 Professor, Ecole de Technologie Sup´erieure, Montr´eal, (Qu´ebec), Canada ABSTRACT This paper presents a numerical method for performance predictions of wind turbines immersed into stable, neutral, or unstable atmospheric boundary layer. The flowfield around a turbine is described by the threedimensional Reynolds’ averaged Navier-Stokes equations complemented by the k- turbulence model. The density variations are introduced into the momentum equation using the Boussinesq approximation and appropriate buoyancy terms are included into the k and equations. An original expression for the closure coefficient related to the buoyancy production term is proposed in order to improve the accuracy of the simulations. The turbine is idealized as actuator disk surface, on which external surficial forces exerted by the turbine blade on the flow are prescribed according to the blade element theory. This theory uses the experimental twodimensional airfoil aerodynamics properties coefficients. Therefore, three-dimensional and rotational effects are not taken into account. That induce an underprediction of the power output at high speed. To take into account of these effects, the stall-delay model of Du & Selig is used. The resulting mathematical model has been implemented in FLUENT. The results presented here include comparisons between experimental measurements and the proposed method for the power output and wake development under various thermal stratifications of an isolated wind turbine. Good agreement is obtained in terms of performance predictions. Simulations indicate that the wake velocity defect is more important in stable conditions than unstable conditions in agreement with observations. However, it is underestimated compared with experimental results. The wake velocity defect evolution has been found in good correlation with experimental observation and the predicted logarithmic slope is 10% lower than the experimental value. INTRODUCTION Aerodynamic analysis is one of the most critical steps in designing wind turbines. Mainly, three formulations have been used to perform such analysis and are
classified as follows. (i) Integral blade element momentum method (BEM) [2, 3] : the turbine is represented as actuator surface, the flow field described by the integral momentum equation. (ii) Vortex wake method [4] : the turbine blade effects are represented by means of distributed bound vortices. (iii) Local differential equations [5, 6, 7] : the turbine is represented as an actuator disk and the flow field is described either by NavierStokes or Euler equations. The numerical method proposed in this paper belongs to the third class. The mathematical model is similar to that presented in previous works [6, 7]. The flowfield around a turbine is described by the Reynolds’ averaged incompressible three-dimensional Navier-Stokes equations. The modified k- model for shear flow under gravitational influence [8] has been chosen for the closure of the time-averaged, turbulent flow equations. The turbine is idealized as actuator disk surface, on which external surficial forces exerted by the turbine blade on the flow are prescribed according to the blade element theory. In Refs. [6] and [7], the resulting mathematical models, valid only for neutral atmospheric boundary layers, were solved using a Control Volume Finite Element Method (CVFEM) [9]. In this paper, a mathematical model having the capabilities of handling atmospheric boundary layers in various thermal stratifications has been proposed and implemented in FLUENT. The main objective of this paper is to propose modifications to the k- model in order to improve its accuracy for thermally stratified boundary layer and to study the aerodynamics effects of wind turbine on the thermally stratified atmospheric boundary layer. MATHEMATICAL MODEL Actuator Disk Concept The actuator-disk concept consists in modeling the rotor as a permeable surface, defined by the rotor swept area, on which a distribution of forces acts upon the incoming flow at a rate defined by the period-averaged mechanical work that the rotor extracts from the fluid. Its action can be modeled by a distribution of forces, per unit area, on the actuator disk surface AR [10]. These
forces per unit area of the actuator disk will be referred as the surficial forces in this work. For HAWTs, the actuator-disk geometry is defined by the blades’ swept area, which consists in a circular cone having a base radius given by R cos γ , where R is the blade length and γ is the coning angle of the blades. It is assumed that the rotor does not have any spanwise action on the flow, therefore the surficial force exerted by an elementary actuator-disk surface dA may be decomposed into a normal and a tangential component denoted by fn and ft , respectively. Their expressions are defined from the blade element theory. Blade Element Theory Blade-element theory implies that the local forces exerted on the blades by the flow are dependent on the local geometrical characteristics, the airfoil aerodynamic properties and the relative fluid velocity, Vrel . Decomposing these forces onto the normal and tangential axes, and time-averaging the forces exerted by the blades on the flow during one period of rotation, yields the following expressions for the normal and tangential components of the surficial force exerted by the rotor on the flow:
fn =
B ρ Vrel c [Ut CL + Un CD ] 2π r 2 (1)
B ρ Vrel c ft = − [Un CL − Ut CD ] 2π r 2 where ρ is the air density, Un and Ut are the normal and tangential components of the relative fluid velocity, Vrel , and CL and CD are the dynamic lift and drag coefficients of the blade-defining airfoil, respectively, and are obtained using an appropriate stall-delay model [1]. This model modifies the 2-D airfoil aerodynamic properties through a simple correction formula to obtain the lift and drag coefficients which include the rotational and three-dimensional effects. Furthermore, to take into account the blade tip vortices effects, the lift of the twodimensional airfoil has to be corrected using the Prandtl correction factor(see for example Ref.[11]). Governing Equations For unsteady and incompressible flow conditions around an HAWT, the time-averaged Navier-Stokes equations in tensor form notation can be written as ui ni dA = 0 (2) A
ρ uj ui ni dA = − A
τij ni dA + A
p δij ni dA+ A
(fn )j + (ft )j dA + ρ gj dV (3) AI
V
ρ cp V
A
∂T dV + ∂t
cp µt uj τij + σT
ρ cp T ui ni dA =
A
gi ∂T − ∂xi cp
ρ ui gi dV + V
ni dA
(4)
where A denotes an outer surface of fixed volume V , ui is the ith flow velocity component, ni is the appropriate cosine director of the outward unit vector n perpendicular to the elementary control surface dA, δij is the Kronecker delta, τij is the viscous-stress tensor which includes the Reynolds stress tensor. AI refers to the surface defined by the intersection of V and the rotor actuator disk surface (i.e. AI = V ∩ AR ). The term (f ) dA + (ft )j dA refers to the j th component of AI n j the time averaged force exerted by the rotor on the fluid in the control-volume V . This contribution is non-zero only for control volumes intersecting the actuator disk. The Boussinesq approximation is used: the density ρ is assumed to vary linearly with temperature only in the gravity term. Finally, gi is the ith component of the gravitational acceleration. Turbulence modeling of the flow A turbulence model is needed in order to close the above system of time-averaged equations. Atmospheric, wake shear-generated and blade-generated (viscous and pressure effects due to the blades) turbulence are the three major contributions to turbulence in the wake of a wind turbine. Various experiments have demonstrated that in the first two diameters downstream of a rotor, strong turbulence is generated by the degradation of tip vortices and by shear production [12]. Further downstream, turbulence properties recover their upstream values and thus, atmospheric turbulence controls the growth of the wake. Therefore, a model taking both shear-generated and atmospheric turbulence into account is a minimum requirement. Two-equation models meet this requirement. In this work, the k- model has been selected mainly because of its wide use and because of the availability of k and properties of atmospheric boundary layers in meteorological data. For a control volume V delimited by an outer surface A, standard equations for the k- model [13, 18] are: µt ∂k ρkui ni dA = [Pt − ρ + Gb ] dV+ ni dA(5) A V A σk ∂xi µt ∂ ρui ni dA = ni dA+ A A σ ∂xi 2 C1 (Pt + (1 − C3 ) Gb ) − C2 ρ dV (6) k k V ∂ui ∂uj ∂ui + (7) Pt = µt ∂xj ∂xi ∂xj gi µt ∂T (8) − Gb = βgi σT ∂xi cp
µt = ρCµ
k2
(9)
β is the air thermal volumetric expansion coefficient. In the original model of Jones and Launder [13], the values for the closure coefficients are: C1 = 1.44 C2 = 1.92 C3 = 1.0 (10) Cµ = 0.09 σk = 1.0 σ = 1.3 σT = 1.0 Description of the basic flow In this work, the flow upstream of the rotor is assumed to be the one observed in the first hundred meters of the atmospheric boundary layer, over uniform flat terrain in the case of various thermal stratification conditions (stable, neutral or unstable). The expressions describing this flow are based on the Monin-Obukhov similarity theory and have been taken from Panofsky and Dutton [16]. These expressions depend mainly of height, z, roughness length, z0 , turbulent friction coefficient, u∗ , and the Monin-Obukhov length, L (L > 0, L < 0 and L → ∞ indicate stable, unstable and neutral conditions respectively) and are: u∗ z 1+x2 1+x 2 ln − ln − z0 2 2 K π (11) L0 K
T0 (z) − Tw =
z0
L
T∗ ln zz0 − 2 ln 12 1 + x2 K g L0
(17)
An expression for C1 can be obtained from the transport equation (Eq.(6)) by introducing the MoninObukhov empirical expressions for the case of neutral stratification (i.e. L → ∞): K2 = 1.176 C1 = C2 − Cµ σ
(18)
These values of Cµ and C1 have been proposed by Crespo et al. [15] along with a value of 0.8 for C3 . In the cases of stable and unstable stratifications over a uniform flat terrain, the closure coefficients proposed by Crespo et al. [15] lead to a certain level of discrepancies with respect to the empirical solutions based on the Monin-Obukhov similarity theory (see [14]). In an effort to improve the accuracy of the predictions in the cases of stable and unstable stratifications, the following expression for C3 is proposed in this work: C3
1 where x is given by: x = 1 − 16 Lz 4 and K corresponds to the von Karman constant (K=0.42), Tw the surface temperature, cp the air specific heat at constant pressure, and g the gravitational acceleration module. Based on measurements of the turbulent kinetic energy budget terms in the surface layer of an atmospheric boundary layer over a flat terrain [16], one can find: u∗ 3 z φ (13) 0 (z) = Kz L where
z 1 − Lz φ = 1 + 4 Lz L
where the constant 5.48 has been experimentally determined for neutral atmospheric boundary layer [16]. And the non-dimensional wind shear, is given by the following relation: − 1
z 1 − 16 Lz 4 L0
z L
=
5 n=0
an
z n L
(19)
The coefficients of this series are given in Ref. [14]. These expressions represent the necessary conditions on C3 to ensure that the vertical distributions of k and given by Eqs.(13) and (15) are exact solutions of the k- model. NUMERICAL METHOD The solution of these equations is accomplished by employing FLUENT. FLUENT uses a control-volumebased technique to convert the governing equations to algebraic equations that can be solved numerically.
(14)
Finally, using of the k- model, a relation between the turbulent kinetic energy k and is obtained: 12 φ Lz νt0 0 2 = 5.48u∗ (15) k0 (z) = Cµ φm Lz
Computational Domain The flowfield in the vicinity of a turbine immersed in a non-uniform incoming flow parallel to the turbine axis of rotation is fully three-dimensional. Thus, the computational domain consists of a rectangular box that includes the wind turbine. This domain is first discretized
into quadrilateral elements defined by four nodes. Then, each element is considered as a control volume. All dependent variables are calculated at the centroide of the control volumes. To exactly account for the rotor presence, nodes are distributed conveniently over the swept area. A grid dependence study has been undertaken in Reference [19] in which the appropriate number of points and the location of the computational boundaries to achieve grid-independent solutions have been determined. Therefore, the inlet and the outlet boundaries should be at least 7.5 diameters upstream and 10 diameters downstream of the turbine respectively. In the present work, the three-dimensional rectangular domain used for the computations is 18 rotordiameters long in the windward direction, 5 rotordiameters long in the transversal direction, and 6 rotordiameters long in height. The wind turbine is located 7.5 rotor-diameters from the inlet boundary and a 211788 control-volume discretization is applied. The rotor swept areas is composed of 1232 control volumes.
power. The rotor diameter, rotational speed and hubheight are 23m, 42 rpm and 35m respectively. Simulations have been undertaken for a hub-height wind velocity and turbulence intensity of 10 m/s and 10%, respectively in three thermal stratifications (L = −231.5m, L = 514.5m, and L → ∞) corresponding to unstable, slightly stable and neutral conditions. The turbulence intensity has been maintained to a constant level for the three thermal stratifications, knowing that this results in simulating different ground roughnesses for each stratification. Since power output and wake development are strongly dependent on the turbulence intensity [21], it was important to remove this effect in order to evaluate solely the effects of the thermal stratification.
Boundary Conditions Inlet boundary : The inlet boundary is y −z plane located upstream of the wind. The velocity field and the turbulent properties are set to the undisturbed flow conditions given by the Monin-Obukhov similarity theory. The pressure is calculated from the continuity equation. Outlet boundary : The outlet boundary is a y − z plane located downstream of the wind turbine. Here, the velocity field is calculated using the outflow treatment of Patankar [20]. Lateral surfaces : These surfaces are x − y plane (top boundary) and x − z planes (side boundaries). On these boundaries, undisturbed flow conditions are prescribed while pressure is calculated from the continuity equations. Wall boundary : The classical turbulent law of the wall is applied on the wall neighboring control volumes.
Figure 1: Stable atmospheric boundary layer with the original k- turbulence model and L = 514.5m
Wind Turbine Modelisation Essentially, the originality of the proposed method consists in the implementation of the rotor modeling in FLUENT. For this purpose, FLUENT allows the introduction of ”user-defined functions” which provide the capability to customize FLUENT to solve the appropriate mathematical model. Therefore, a customized FLUENT code that enables the simulation of HAWTs has been developed. RESULTS In order to investigate the validity of the proposed method, computations have been carried out over a stall-regulated, upwind, three-bladed, horizontal axis Danwin wind turbine rated at 180kW of electrical
Figure 2: Stable atmospheric boundary layer with the modified k- turbulence model and L = 514.5m The first part of the validation process consisted in assessing the capabilities of the proposed method to reproduce accurately the properties of the atmospheric boundary layer over a uniform flat terrain without the presence of the turbine. This work has already been carried out in a previous paper [14]. Therefore, results presented in this report will not include all the validations, but just one case for stable conditions. For more details, the interested reader can consult Ref. [14]. Figs. 1 and 2 present comparisons between the results produced
Figure 4: Unsdisturbed velocity profiles Figure 3: Comparison between experimental results and proposed method with stall-delay model by the original and modified k- turbulence model and the empirical expressions based on the Monin-Obukhov similarity theory [16] for a stable atmospheric boundary layer with a velocity of 10 m/s and a turbulence intensity of 10% at 35m. The original k- turbulence model leads to significant differences between simulations and the empirical expressions especially for the turbulent kinetic energy (TKE), k, and . The proposed modifications improve the accuracy of the simulations. It is clear from these comparisons that the mathematical model used is appropriate. Fig. 3 presents comparisons between the predicted power curve and the experimental measurements [17]. Results show the accuracy of the proposed method to reproduce wind turbine performances in the stall region. One can see on this curve that for a same wind velocity there is an important variation in the power output. This difference is mainly owed to the orientation of the wind that varied during recordings, this one was not always aligned with the turbine axis rotation [17]. It is interesting to look at the undisturbed velocity and TKE profiles in order to better understand the aerodynamics effects of a wind turbine on the atmospheric boundary layer. Fig. 4 and Fig. 5 present the undisturbed velocity and TKE profiles for three types of thermal stratification and a same hub-height velocity and turbulence intensity. Depending on the atmospheric stability, significant differences are observed. This is particularly important for the TKE in unstable conditions. These variations are expected to have an influence on the flowfield characteristics such as the wind-turbine wake velocity defect. This is confirmed by experimental measurements [22] on Figs. 6 and 7. The normalized velocity defect, Ud , is defined by: Ud =
u0 (z) − uw (z) u0 (z)
(20)
where the indice 0 indicates the undisturbed velocity profile and the indice w the velocity profile in the wake.
Figure 5: Unsdisturbed turbulent kinetic energy profiles Figs. 6 to 9 present the vertical distributions of the experimental and predicted normalized velocity defect, Ud , at two downstream locations, 4.2D and 6.1D respectively for stable and unstable stratifications. Results show that the velocity defect is lower in unstable conditions than stable conditions. Indeed, the ambiant level of TKE is more important in unstable conditions (see Fig. 5) producing a faster velocity recovery. These results show qualitative agreement between simulations and measurements, but the values of the velocity defect are significantly underestimated. In order to quantify the effects of stratification a wake velocity defect deviation, δ, has been defined: δ=
Ud (stable) − Ud (unstable) Ud (stable)
(21)
Table 1 presents comparisons of predicted and experimental wake velocity defect deviation at the hub height. The predicted deviation is in good agreement with experimental results. One can see that it increases slightly with downstream location. These differences are important and prove that stratification effects will be important in the prediction of wind park performances since they induce a significant variation of the characteristics of the flow downstream of a wind turbine. Figs. 10 show the velocity defect at the hub height along the wake of the Danwin 180kW wind turbine. It has been found that the velocity defect varies as a x α , where α represent the logarithmic function of D
Figure 6: Experimental Velocity defect [22] in stable and unstable conditions at 4.2D
Figure 9: Predicted Velocity defect in stable and unstable conditions at 6.1D Downstream location Experimental measurements Proposed method
4.2D
6.1D
22% 23%
25% 23.6%
Table 1: Velocity defect differences between stable and unstable conditions Figure 7: Experimental Velocity defect [22] in stable and unstable conditions at 6.1D slope, in agreement with observations. Table 2 presents a comparison between predicted and experimental logarithmic slopes for three types of thermal stratification. Simulations show that the predicted logarithmic slope is 10% lower than the slope obtained experimentaly [22]. CONCLUSION The main purpose of this paper was to investigate the effects of thermal stratification of the atmospheric boundary layer on the flowfield downstream of a wind turbine. The flowfield around the turbine is described by the three-dimensionnal Reynolds’ averaged NavierStokes equations complemented by the k- turbulence model. An original expression for the closure coefficient related to the buoyancy production term is proposed in an effort to improve the accuracy of the simulations.
Figure 8: Predicted Velocity defect in stable and unstable conditions at 4.2D
This new expression has improved significantly the accuracy of the simulations. A stall-delay model has been implemented with success for predicted wind turbine performances in the stall region. In agreement with observations, it has been found that the wind turbine wake velocity recovery is faster in unstable than stable conditions . However, the velocity defect is underestimated compared with experimental results. The wake velocity defect evolution has been found in good correlation with experimental observation and the predicted logarithmic slope is 10% lower than the experimental value. ACKNOWLEDGMENTS This work is supported by the Minist`ere des ressources naturelles du Qu´ebec through the programme d’aide au d´eveloppement des technologies de l’´energie. The support of the Natural Sciences and Engineering Research
Figure 10: Decay of the wake velocity defect at hub height downstream of the Danwin 180kW
type of stratification slightly stable neutral unstable
α Experimental -0.8 x -0.8
α Predicted -0.72 -0.72 -0.73
Table 2: Logarithmic slope of the velocity defect curve under various thermal stratifications Council of Canada (NSERC) in the form of research grant to C. Masson is gratefully acknowledged.
References [1] Du, Z. and Selig, M. S., A 3-D Stall-Delay Model for Horizontal Axis Wind Turbine Performance Prediction, Proc. of the 1998 ASME Wind Energy Symposium, Reno, pp. 9-19, January, 1998. [2] Glauert, H., Airplane propellers, Aerodynamic theory, (Ed. W.F. Durant), 1935, Vol. 4, Division L (Springer, Berlin) [3] Wilson, R. E. and Lissaman, P.B.S., Applied Aerodynamics of Wind Power Machines, NTIS PB 238594, Oregon State University, 1974. [4] Wilson, R. E., Lissaman, P.B.S. and Walker, S. N. Aerodynamic performance of wind turbines, Oregon State University, 1976. [5] Rajagopalan, R.G., Fanucci, J.B., Finite Difference Model for the Vertical Axis Wind Turbines, J. of Propulsion and Power, Vol. 1, pp. 432-436, (1985). [6] Masson, C., Ammara, I., Paraschivoiu, I., An Aerodynamic Method for the Analysis of Isolated Horizontal-Axis Wind Turbines, Int. J. of Rotating Machinery, Vol. 3, No. 1, 21-32,(1997). [7] Ammara, I., Leclerc, C., Masson, C., A Fully Three-Dimensional Turbulent Method for the Analysis of Horizontal-Axis Wind Turbines in Wind Farms, Proc. of the Seventh Int. Symposium on Transport Phenomena and Dynamics of Rotating Machinery, 1782-1793, Hawaii (1998). [8] Gibson, M.M., Launder, B.E., On the Calculation of Horizontal, Turbulent, Free Shear Flows Under Gravitational Influence, ASME of Heat Transfer, February, 81-87, (1976). [9] Masson, C., Saabas, H.J., Baliga, B.R., Co-Located Equal-Order Control-Volume FiniteElement Method for Two-Dimensional Axisymmetric Incompressible Flow, Int. J. for Numerical Methods in Fluids, Vol. 18, 1-26, (1994).
[10] Madsen, H. A., The Actuator Cylinder a Flow Model for Vertical Axis Wind Turbines, Aalborg University Centre, Denmark, (1992). [11] Durand, W.F., Aerodynamic Theory Vol. IV, Dover Publications, (1963). [12] H¨ ogstr¨om, U., Asimakopoulos, D. N., Kambezidis, H., Helmis, C. G., Smedman, A., A Field Study of the Wake behind a 2 MW Wind Turbine, Atmospheric Environment, Vol. 22, No. 4, pp. 803820, (1988). [13] Jones, W.P., Launder, B.E., The Prediction of Laminarization with a Two-Equation Model of Turbulence, Int. J. of Heat and Mass Transfer, Vol. 15, pp. 301-314, (1972). [14] Alinot, C., Masson, C., Aerodynamic Simulations of Wind Turbines Operating in Atmospheric Boundary Layer with Various Thermal Stratifications, Proc. of the 2002 ASME Wind Energy Symposium, Reno, pp. 206-215, january, 2002. [15] Crespo, A., Manuel, F., Moreno, D., Fraga, E., Hernandez, J., Numerical Analysis of Wind Turbine Wakes, Workshop on Wind Energy Applications, Delphi, Greece, (1985). [16] Panofsky, H., Dutton, J., Atmospheric Turbulence, John Wiley&Sons, (1984). [17] Magnusson, M., Smedman, A.-S., Air Flow behind Wind Turbines, J. of Wind Engineering Industrial Aerodynynamics, Vol. 80, n 1-2, pp. 169189, (1999). [18] Rodi, W., Turbulent Buoyant Jets and Plumes, Pergamon Press, New York, 184 p., (1982). [19] Masson, C., Ammara, I., Paraschivoiu, I., An aerodynamic method for the analysis of isolated horizontal-axis wind turbines, Int. J. of Rotating Machinery, Vol. 3, pp. 21-32, (1997). [20] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, (1980). [21] Masson, C., Leclerc, C., Dabin, N. Abnormal High Power Output of Wind Turbine in Cold Weather : A Quantitative Analysis of Temperature Influence on Power Output, Proc. of the 2001 ASME Wind Energy Symposium, Reno, pp. 123-133, January, 2001. [22] Magnusson, M., Smedman, A.-S., Influence of Atmospheric Stability on Wind Turbine Wakes, Wind Engineering, Vol. 18, No. 3, pp. 139-152, (1994).
