An Aerodynamic Method for the Analysis of Isolated Horizontal-Axis Wind Turbines

The aerodynamic analysis of a wind turbine represents a very complex task since it involves 
an unsteady three-dimensional viscous flow. In most existing performance-analysis methods, 
wind turbines are considered isolated so that interference effects caused by other rotors or by 
the site topology are neglected. Studying these effects in order to optimize the arrangement 
and the positioning of Horizontal-Axis Wind Turbines (HAWTs) on a wind farm is one of the 
research activities of the Bombardier Aeronautical Chair. As a preliminary step in the 
progress of this project, a method that includes some of the essential ingredients for the 
analysis of wind farms has been developed and is presented in the paper. In this proposed 
method, the flow field around isolated HAWTs is predicted by solving the steady-state, 
incompressible, two-dimensional axisymmetric Navier-Stokes equations. The turbine is represented 
by a distribution of momentum sources. The resulting governing equations are 
solved using a Control-Volume Finite Element Method (CVFEM). This axisymmetric implementation 
efficiently illustrates the applicability and viability of the proposed methodology, 
by using a formulation that necessitates a minimum of computer resources. The axisymmetric 
method produces performance predictions for isolated machines with the same 
level of accuracy than the well-known momentum-strip theory. It can therefore be considered 
to be a useful tool for the design of HAWTs. Its main advantage, however, is its capacity to 
predict the flow in the wake which constitutes one of the essential features needed for the 
performance predictions of wind farms of dense cluster arrangements.


INTRODUCTION
After some unsuccessful attempts at constructing and operating very-large-scale isolated wind turbines, the recent tendency is to construct wind farms of medi- um-size machines (500 kW).The strategy currently used during the conception of such wind farms con- sists in installing the turbines far from each other in order to minimize the interference effects.This practice results in very sparse wind farms where the wind energy potential of a site is inefficiently used.It is justified by the performance losses associated to the *Corresponding author.Tel.: (514) 340-4582.Fax: (514) 340-5917.E-mail: christian, masson@meca.polymtl.ca.wake effects, which are significant in dense arrangements.However, a relatively dense but staggered ar- rangement of the turbines is expected to produce an increase in the performance of the downstream tur- bines with respect to the isolated-turbine situation.This is due to the beneficial venturi effects that occur between two adjacent turbine wakes.The efficiency of the dense staggered cluster is not expected to be significantly higher than that of the sparse arrange- ment.However, for a given number of turbines, a dense staggered cluster can be easily conceived that would occupy up to 25% less land than the sparse arrangement.Notwithstanding the farm efficiency in- crease induced by the venturi effects, a reduction of 25% of the wind farm area can represent a significant economy in operating expenses.Furthermore, some of the construction expenses, such as the grading and electric infrastructure costs, will be reduced.
Various aerodynamic methods, appropriate for the conception of isolated turbines, are available to de-  signers  (Gohard [1978], Paraschivoiu [1981], Strick- land et al. [1980], Templin [1974], Wilson [1984]).However, efficient and accurate methods for the anal- ysis of a dense cluster where the effects of the three- dimensional turbulent turbine wakes are included are not available.The development of a method that in- cludes the essential ingredients for the successful performance predictions of wind turbines in a dense ar- rangement is the authors' main objective.In the pro- posed method, the flow field of the wind farm is predicted by solving the steady-state, incompressible, three-dimensional Navier-Stokes equations.The tur- bines are represented by distributions of momentum sources, a technique introduced by Rajagopalan  [1984]  and Rajagopalan and Fanucci [1985].This is a general formulation which can be applied, in princi- ple, to horizontal-axis and vertical-axis wind turbines and can include the effects of hubs, towers, and local topography.The Navier-Stokes equations are solved using the three-dimensional Control-Volume Finite Element Method (CVFEM) of Saabas and Baliga  [1994].
The development of this method is still in its early stage, and the paper is aimed at presenting the progress and at demonstrating the applicability of the proposed method along with its capacity to analyze the performances of wind farms.The mathematical model and numerical method described in the" paper are a two-dimensional axisymmetric formulation applicable to isolated HAWTs.This implementation is used in the paper to demonstrate the applicability and viability of the proposed methodology at much lower CPU costs than a fully three-dimensional implemen- tation.Several aspects related to the numerical solu- tion of the mathematical model, such as (i) the appropriate extent of the computational domain, (ii) the grid spacing needed to obtain accurate performance predictions, and (iii) the optimum choice of the vari- ous control parameters to obtain converged solutions, can also be studied more efficiently using the axisymmetric formulation.Comparisons between the performance predictions obtained with the proposed formu- lation and those of the momentum-strip theory are presented to illustrate the accuracy of the proposed methodology.Comparisons with experimental data are also included.

GOVERNING EQUATIONS AND ROTOR REPRESENTATION
The flow around an isolated HAWT is governed by the unsteady three-dimensional Navier-Stokes equa- tions which have to be solved in a.domain with mov- ing boundaries.Analytical solutions of such a problem are hardly possible while numerical solutions represent a formidable task on today's computers.However, a more tractable model producing mean- ingful results can be obtained by time-averaging the governing equations and by representing the turbine with momentum sources.
The derivation of the,gov.erningequations andmo- mentum sources is based on time-averaging tech- niques and the blade-element theory.The interested reader is referred to the works of Rajagopalan [1984] and Rajagopalan and Fanucci [1985] for a detailed derivation of the source terms of vertical-axis wind turbines.In the case of horizontal-axis wind turbines, of interest here, the derivation is very similar and the reader is referred to Fig. 1 for the definition of the various parameters involved in the evaluation of the momentum sources.
In steady-state, laminar, two-dimensional axisym- metric flow around an isolated HAWT, the mathemat- ical model consists of a set of four differential equations: a continuity equation and three momentum equations.The four dependent variables are P (hav- ing the components u, v, and w in the x, r, and 0 directions) and p.The fluid density is represented by p, and its dynamic viscosity by ta.The turbine is com- posed of B blades having a coning angle y and a chord c that can vary along the blade.The turbine rotational speed is f.
The flow around an HAWT can be represented by the following general formulation: V.(I3V+)--V.FV+ -ff (Sp)qb q--(ST) + (1) The appropriate governing equations can be obtained from Eq. ( 1) by defining the dependent variable, the diffusion coefficient, F, and the volumetric source terms, (Sp)+ and (ST)+, according to Table I, where (ST) K (UC D + WCL) cos (2) (ST) -K (UC D WCL) sin 3' (3) The source terms (ST)u, (ST)v, (ST) are the x-, r-, and 0-component of the mutual time-averaged force, ex- erted by the fluid and the blades on one another, per unit volume of the fluid.They will be referred to as the momentum-source terms.CL and CD are the lift and drag coefficients of the blade-defining airfoil and (ST)w are, in general, functions of the angle of attack o and the local Reynolds number Re In the proposed model, these coefficients can be taken from either ex- perimental data or numerical results obtained over the appropriate two-dimensional airfoil.
In the above equations, Vcv is the control volume to which the conservation principles have been applied, and Af corresponds to the blades length that crosses this control volume.It is to be noted that Af is zero in the region of the flow which is not crossed by the blades.Consequently, the momentum-source terms are non-zero only in the region of the flow crossed by the turbine blades.
The mathematical model described in this section has been obtained by considering laminar flow.This assumption is difficult to justify physically, since it is well known that the wake of a wind turbine is turbu- lent.Nevertheless, the proposed mathematical model still includes some of the essential elements for accu- rate performance predictions of wind farms.The ma- jor features of the proposed model are (i) its capacity to model the details of the wake behind the turbines and (ii) its implicit introduction of the interferences between the rotors.The interferences are due to nu- merous effects such as the pressure variation and the blockage phenomenon.
In this early stage of development, the laminar- flow assumption is used in order to emphasize the various aspects related to the wind turbine modelling without introducing uncertainties inherent to the tur- bulence models.It is demonstrated in the RESULTS Section that accurate performance predictions can be obtained using the laminar-flow assumption in the case of isolated HAWTs since their performances are not highly influenced by the details of their wakes.However, when a turbine is located behind another, the turbulent wake has to be considered in order to C. MASSON et at.U obtain accurate performance predictions of the down- wind turbine.

NUMERICAL METHOD
The proposed numerical method is a CVFEM based on the primitive-variables, co-located, equal-order formulation of Masson et al. [1994].Detailed de- scriptions of this CVFEM, pertaining to the simula- tion of particulate two-phase flows and internal three- dimensional turbulent flows, are available in the works of Masson [1993] and Saabas [1991].There- fore, for sake of conciseness, only the aspects rele- vant to the successful simulation of the flow around a wind turbine are presented in this section.

Interpolation of in Convective Term
In the derivation of algebraic approximations to sur- face integrals of the convective fluxes, two different interpolation schemes for + were presented by Saabas [1991]: the FLow Oriented upwind scheme (FLO); and a MAss Weighted upwind scheme (MAW).
The FLO scheme is based on the earlier work of Baliga andPatankar [1980,1988].The interpolation function used in this scheme responds appropriately to an element-based Peclet number and to the direc- tion of the element-averaged velocity vector.In planar two-dimensional problems that involve acute-an- gled triangular elements and relatively low element Peclet numbers, Saabas and Baliga [1994] have proven that the FLO scheme can be quite successful.If high values of the element Peclet number are en- countered, however, the FLO scheme can lead to neg- ative coefficients in the algebraic discretised equa- tions (Saabas and Baliga [1994]) and this difficulty is compounded when obtuse-angled triangular elements are used.The donor-cell scheme of Prakash [1987] is one way of ensuring positive coefficients: in this approach, the value of a scalar convected out of a con- trol volume, across its surface, is set equal to the value of the scalar at the node within the control volume.This approach guarantees positive coefficients, but takes little account of the influence of the direc- tion of the flow.Thus it is prone to considerable false diffusion (Prakash 1987]).
The MAW scheme is based on the positive-coeffi- cient schemes of Schneider and Raw [1986].It en- sures, at the element level, that the extent to which the dependent variable at a node exterior to a control volume contributes to the convective outflow is less than or equal to its contribution to the inflow by con- vection.Thus, it is a sufficient condition to ensure that the algebraic approximations to the convective terms add positively to the discretised equation.It should be noted that the MAW scheme takes better account of the influence of the direction of the flow than the donor-cell scheme of Prakash [1987], so it is less prone to false diffusion.Details of the formula- tion of the MAW scheme are presented in the work of Saabas 1991 ].
In problems with acute-angled triangular elements and relatively low element Peclet numbers, the FLO scheme is more accurate than the MAW scheme.As was mentioned earlier, however, when high element Peclet numbers are involved, especially in conjunction with obtuse-angled elements, the FLO scheme produces negative coefficients in the discretised equations.Negative coefficients in the' discretized equations can lead to convergence difficulties when itera- tive solution algorithms, such as SIMPLE or its vari- ants (Patankar [1980]) and CELS (Galpin et al.  [1985]), that use segregated or coupled equation line- by-line iterative algorithms to solve the linearized sets of discretised equations, are used.
In the numerical solution of the flow around a wind turbine, a fine grid has to be used in the immediate vicinity of the rotor in order to capture the large vari- ations of the flow properties.The grid size is then increased as the distance to the turbine increases.
This coarsening is applied in order to keep a reason- able number of grid points while ensuring the appli- cation of the boundary conditions to be far enough from the rotor.The use of such coarse grids results in large Peclet numbers since these are directly proportional to the grid size.Convergence difficulties were encountered during the simulations presented in this paper when the FLO scheme was used.Therefore, the MAW scheme has been used to produce all the results presented in this paper and its application is recom- mended for the successful solution of the flow around a wind turbine.related velocities (um, vm), as a function of a pseudo- velocity and a pressure gradient term: (9) where t/, and 9 are the pseudo-velocities, and d", and d are called the pressure-gradient coefficients.These expressions come directly from the discretised mo- mentum equations.However, instead of using the control-volume-averaged pressure gradient, the ele- mental pressure gradient is used to compute the mass-flow related velocity in each element.This prevents the occurrence of spurious pressure oscillations in the proposed CVFEM.

Momentum-Source Term Linearization
The momentum-source term, (ST) + is expressed in the following general form (Patankar [1980]): (ST) + (ST) C q-(ST)p+ (lO) In each triangular element, the values of (ST)c and (ST)p are stored at the vertices, and are assumed to prevail over the corresponding portions of the control volumes within that element.While the volume inte- gration of the momentum-source term is straightforward, its proper linearization is crucial to ensure con- vergence of the overall algorithm, especially in the context of the segregated iterative solution algorithm used in this work.The linearization consists in the specification of appropriate expressions for (ST)c and (ST)p" The momentum-source term can be linearized ex- plicitly in each iteration:

Mass Flow Rate Interpolation
The mass flow rate is calculated using a special treat- ment borrowed from the works of Prakash and Patankar [1985] and Saabas and Baliga [1994].This spe- cial treatment consists in expressing the mass-flow (ST) C (ST) + (ST)p 0 (11) where the superscript * means that the source term has been evaluated using the flow properties obtained at the previous iteration.Implementation of this lin- earization in the segregated iterative algorithm used in this work resulted in severe convergence problems for highly loaded wind turbines.In such conditions, the value of the momentum-source terms become large, and the resulting momentum equations are source-dominated.This results in very slow conver- gence rates, and, sometimes the segregated iterative algorithm even diverges.In an effort to improve the robustness of the iterative solution algorithm, the fol- lowing treatment is proposed: (ST); (ST) C 0 (ST)p +, This linearization has proven to be much less prone to convergence problems than the explicit lineariza- tion expressed by Eq. ( 11).

Boundary Conditions
The computational domain consists of a simple cylinder that includes the wind turbine.Boundary con- ditions have to be prescribed on the three faces of this cylindrical domain.
Inlet Boundary: The inlet boundary is a r-0 plane located upstream of the wind turbine.In this plane, the three velocity components are given by the known freestream wind speed while the pressure is calculated from the discretised continuity equations.

Overall Solution Algorithm
The discretised equations form a set of coupled non- linear algebraic equations.In this work, the iterative variable adjustment procedure proposed by Saabas  and Baliga [1994] is used to solve the proposed math- ematical model.This procedure is akin to SIMPLER (Patankar [1980]) without the pressure correction equation.In order to facilitate implementation and testing of the proposed method, structured grids are used.Thus, a line Gauss-Seidel algorithm based on the tridiagonal matrix algorithm are used to solve the discretised equations for p, u, v, and w.
The momentum-source linearization proposed in this work can lead to large differences between the values of the two pressure-gradient coefficients.In the case of an horizontal-axis wind turbine com- pletely contained within a r-0 plane, for example, d" is typically much smaller than d.This large differ- ence in the values of the pressure-gradient coeffi- cients leads to difficulties in ensuring the overall mass conservation since the streamwise pressure gradient has a negligible effect in the discretised pressure equations.This difficulty is alleviated by prescribing a relatively large number of iterations in the line Gauss-Seidel algorithm used for the solution of the discretised pressure equations.

RESULTS
Outlet Boundary: The outlet boundary is a r-0 plane located downstream of the wind turbine.In this plane, the pressure is assumed to be uniform and given while the three velocity components are com- puted from the discretised momentum equations ob- tained using the outflow treatment of Patankar  [980].
Top Boundary: This is a curved surface located at a radial distance far from the wind turbine blade tip.In this plane, the velocity is set to its freestream value.Pressure is calculated from the discretised continuity equations.
The results presented in this section are aimed at demonstrating the capacity of the proposed method- ology to accurately predict the performances of iso- lated HAWTs, and to analyse wind farms.To this effect, the details of the computed flow field in the vicinity of an isolated HAWT are shown, and com- parisons between the predictions of the proposed method, those of the well-known momentum-strip theory, and experimental data when available are presented.Furthermore, the following aspects related to the successful numerical solution of the mathematical model are studied: (i) the determination of the mini- mum extent of the various boundaries necessary to obtain a computational-domain independent solution, and (ii) the sensitivity of the performance predictions with respect to the grid size.
The simulations presented in this section have been realized for two HAWTs: (i) the NASA/DOE Mod-0 100-kW Experimental HAWT (Puthoff and Sirocky  [1974]) operating at a rotational speed of 40 rpm and a blade pitch angle of 3 , and (ii) the INTA Experi- mental rotor (Hernandez and Crespo [1987]) operat- ing at a rotational speed of 1500 rpm and a blade pitch angle of 9.5 4.1 Extent of the Computational Domain A detailed study of the behaviour of power predic- tions with respect to the size of the computational domain has been undertaken in order to determine the minimum extent of the computational domain needed to produce relevant performance predictions for HAWTs.In the case of the simulation of the axisym- metric/swirling model, the size of the computational domain is characterized by three length parameters, namely Axue, AXDN, and RCD.These length parame- ters are presented in Fig. 2 which illustrates the grid topology.It is to be noted that the grids used in the axisymmetric/swirling simulations were much finer than the one shown in Fig. 2.
Figs. 3-5 show the results of the behaviour of the performance prediction with respect to the extent of the computational domain.These graphs present the variation of the difference between the performance prediction at a finite value of one of the length pa- rameters (denoted by either P(Axup), P(AXDN), P(RcD)) and the power predicted for a very large value of the corresponding length parameter (denoted by p(c)), as a function of a specific length parameter.The length parameters have been nondimensionalized with respect to the rotor diameter D. The grids used were of uniform type.with equal grid spacing in the r and x directions.This analysis has been undertaken at the most critical tip speed ratio TSR which is believed to be the TSR corresponding to the maximum power coefficient Cp.The power coefficient is defined by P ce 3) 2 where P is the mechanical power, U is the freestream wind velocity, and R is the rotor radius.The operational condition at which the maximum power coefficient occurs is the most critical situation since it corresponds to the regime where a greater portion of the energy available in the flow is ex- tracted by the rotor.For a blade pitch angle of 3 and a rotational speed of 40 rpm, the freestream wind speed at which the maximum Cp is reached for the NASA/DOE Mod-0 rotor is near 8 rrds.This study has revealed that the power predictions of an isolated rotor are significantly influenced by the length of the computational domain upstream of the rotor (i.e. AXup) and the position of the constant-radius bound- ary (i.e.RcD).Based on the results presented in Figs.
3 and 5, AXup/D 7.5 and RcD/D 4.0 seems to be large enough to produce performance predictions in- dependent of the extent of the domain.For the down- stream extent of the computational domain, a much smaller value can be used.Fig. 4 suggests that AXDN/D 4.5 is more than sufficient.These values of the extent of the computational domain have been used to produce the results presented in the remainder of this section.

Grid Dependence Study
Fig. 6 presents the difference between the performance prediction obtained at a given number of grid points N, P(N), and the grid-independent power pre- diction, p(c).This grid dependence study has been HORIZONTAL-AXIS WIND TURBINES 29 realized on uniform grids with equal grid spacing in the r and x directions in order to facilitate the deter- mination of the grid-independent power prediction.Fig. 6 shows that the grid-independent solution is reached near N 25 000.This very large number of grid points needed to obtain the grid-independent power prediction is due to the use of uniform grids.
However, for practical calculations, nonuniform grids should be used in order to minimize the number of grid points needed.Using a uniform and fine grid in the vicinity of the rotor along with an expanding grid in the rest of the computational domain, it has been possible to reduce the number of points needed to obtain a solution close to the grid-independent one to 3 000 (see Fig. 6).and 8 show comparisons between the performance predictions of the NASA/DOE Mod-0 100-kW Experimental HAWT produced by the pro- posed methodology and the results of the momentum- strip theory.The agreement between the two methods is good.This was to be expected since the factor limiting the accuracy of the results is the use of static two-dimensional lift and drag coefficients, on which both methods are based.As stated before, the main motivation in developing the proposed method re- sides in its inherent modelling of the rotor/rotor, ro- tor/ground, and rotor/tower interactions and its capacity to produce the details of the flow field around the turbines.
Figs. 9 and 10 present similar comparisons for the case of the INTA Experimental HAWT.In this case, the agreement is very good between the two methods.Fig. 10 also shows some experimental data.The per- formance predictions produced by the two methods are in relatively good agreement with the measured performances.

Predicted Flow Field
Using the nonuniform 3 000-point grid, the flow field around the NASA/DOE Mod-0 HAWT has been computed.The resulting pressure and velocity fields in the vicinity of the rotor are presented in Figs. 11and 12.These results illustrate the capacity of the proposed methodology to produce the details of the flow behind a wind turbine, which constitutes one of the essential features for the successful analysis of wind farms.

CONCLUSION
The idea of using the venturi effects in wind farms is a new concept.More fundamental research is re- quired to determine its applicability.For instance, the sensitivity of the cluster efficiency to the wind orien- tation is a crucial aspect related to the viability of this concept.The development of a method appropriate for the analysis of dense cluster arrangements corre- sponds to the first step in this feasibility study.Such a method can allow the determination of the optimum cluster arrangement, which is expected to be highly related to a specific site (wind speed and orientation variations, topography, etc.).Furthermore, the digital version of such a method represents a powerful tool for the designers of wind farm projects.
The method presented in this paper includes some of the important ingredients needed for the successful analysis of dense cluster arrangements.The implementation of this method has been realized under the assumption of axisymmetric swirling flow.This im- plementation has allowed to efficiently illustrate the applicability and viability of the proposed methodol- ogy by using a formulation that.necessitates a mini- mum of computer resources.Additional features have to be included in this method before its application to the performance predictions of wind farms.The fully three-dimensional formulation has to be implemented.Furthermore, an appropriate turbulence model should be selected and implemented in order to obtain accurate performance predictions for tur- bines located behind others.Nevertheless, the axi- symmetric formulation produces performance predic- tions for isolated HAWTs with the same level of ac- curacy than the well-known momentum-strip theory.It can be considered to be a useful tool for the design of horizontal-axis wind turbines.

E EN NE ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy Systems
Economic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions. FIGURE

[FIGURE 3
FIGURE 2 Computational Domain and Grid Topology.

FIGURE 5
FIGURE 4 Variation of the Performance Prediction with Respect to AXDN.

FIGURE 6
FIGURE 6 Variation of the Performance Prediction with Respect to N.
FIGURE 8 Power Coefficient Predictions for the NASA/DOE Mod-0 HAWT.

FIGURE 7
FIGURE 7 Power Predictions for the NASA/DOE Mod-0 HAWT.

FIGURE
FIGUREComputed Pressure Field for the NASA/DOE Mod-0 HAWT.

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