Stochastic performance evaluation of horizontal axis wind turbine blades using non-deterministic CFD simulations

Energy 73 (2014) 126e136

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Stochastic performance evaluation of horizontal axis wind turbine blades using non-deterministic CFD simulations* ZhiYi Liu, XiaoDong Wang, Shun Kang* Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, Ministry of Education, North China Electric Power University, Beijing 102206, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 October 2013 Received in revised form 29 May 2014 Accepted 30 May 2014 Available online 4 July 2014

In this paper, non-deterministic CFD (computational fluid dynamics) simulations have been performed to investigate the uncertain effects of stochastic boundary conditions on the aerodynamic performance of wind turbines. A NIPRC (non-intrusive probabilistic collocation) method is employed, which is coupled with a commercial flow solver. A 2D (two-dimensional) airfoil case is used to validate the nondeterministic simulation, where the angle of attack is considered as an uncertain parameter in a Gaussian distribution. The simulation results are compared with Monte Carlo simulation results. Based on the validation, non-deterministic CFD simulations were performed on a 3D (three-dimensional) wind turbine blades case, where the wind speed is considered as an uncertain parameter. The discussions mainly focus on the total performance variations and the uncertainty propagation in the fluid field. The simulation results show that the input uncertainty of the inlet velocity results in a high variation zone in the pressure distribution near the blade root, and which decreases from the root to the tip. With the wind speed increases, flow separation is observed. The separation vortex regions correspond to the maximum variation area, and the maximum variation extends to the trailing edge even to the whole suction side. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Wind turbine Uncertainty Probabilistic collocation method Numerical simulation

1. Introduction CFD (computational fluid dynamics) methods have been developed rapidly in recent decades. The credibility of CFD simulations gains increasing attentions [1,2]. The AIAA Journal devoted a special issue to the topic of Verification, Validation and Uncertainties in CFD [1]. According to the reference [1], the verification is to ensure the accuracy of the numerical methodology, and the validation is to assess the approximation of the numerical model to the real physical model or the experiment data [3]. Ref. [4,5] summarized the possible sources of numerical error, such as grid density, computational domain scale, convergence tolerance et al. All these factors are included in the verification process. Regarding to the validation process, the comparison error (difference between experimental data and simulation results) and

* This research was supported by the National Natural Science Foundation of China (No. 51176046 & No. 51206052). * Corresponding author. E-mail address: [email protected] (S. Kang).

http://dx.doi.org/10.1016/j.energy.2014.05.107 0360-5442/© 2014 Elsevier Ltd. All rights reserved.

validation uncertainty (combination of uncertainties in data and portion of simulation uncertainties that can be estimated) are investigated. Many uncertainties may exist in the geometry, boundary conditions or working parameters, which may have substantial impact on the aerodynamic performance operation stability and system reliability of a design based on CFD. Obviously, it is inconsequential to compare a deterministic CFD result with the statistical experimental data containing uncertainty effects [6]. Moreover, there will be large potential failure risks for industrial designs when a deterministic CFD technology is used. Hence, the non-deterministic method should be introduced into CFD simulations to achieve highly confident and reliable numerical results. Several non-deterministic methods have been developed and used in structural mechanics fields and fatigue problems for many years. However, due to the complexity of fluid mechanics (nonlinear conservation laws, approximate turbulence models …), the application of these methods to CFD simulations has only been performed in recent years [3]. The most common used method is the sampling method, including Monte Carlo [7] and Latin hypercube sampling, etc. However, the number of samples needed to

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Fig. 1. A close-up view of the 2D grid.

obtain a reasonable accuracy is usually extremely large, which takes a high computational cost. Thus, the sampling method is usually used as the validation method for other methods. In recent years, some analytical methods based on spectral expansion have been developed. The PC (polynomial chaos) method is one of them. PC methods can be divided into nonintrusive and intrusive methods according to the coupling ways with a CFD solver [3]. In general, an intrusive approach will calculate the unknown polynomial coefficients by projecting the resulting equations onto basis functions (orthogonal polynomials) for different modes. The IPC (intrusive polynomial chaos) method was proposed firstly by Wiener [8], which is named as homogenous chaos. Le Maître [7] used this method to simulate the twodimensional incompressible channel flow and convection flow in a cavity with a uniform distributed viscosity coefficient. Xiu and Karniadakis [9] proposed the GPC (generalized polynomial chaos) method for several different kinds of probability distribution function. They found that an optimal polynomial exists corresponding to the specific probability distribution function, which ensures an exponential convergence of the spectral expansion. GPC was applied successfully in the simulation of a stochastic incompressible channel flow and the flow around a cylinder [10]. Liu et al. [11] used the multidimensional IPC method to simulate the nondeterministic cavity flow under the influence of multiple uncertainties. However, the intrusive method has to modify the deterministic CFD solver, which is usually time and labor consuming, and it may introduce new numerical errors into wellverified codes [12]. To overcome the drawback of the intrusive methods, nonintrusive methods have been developed, which can reconstruct the statistical prosperities of uncertainties based on a number of deterministic simulations. Thus, the existing deterministic CFD solver can be used as a black box [12], which saves a lot of labor in modifying the code and prevents introducing new numerical error sources into CFD simulations. NIPC (non-intrusive polynomial chaos) is one of the nonintrusive uncertainty quantification methods. There are two

Fig. 2. Validation of deterministic CFD results.

different approaches to build NIPC: the point-collocation approach [12] and the Galerkin approach. The former was introduced by Hosder et al. [13]. The polynomial chaos is built using collocation points. The latter is based on sampling [14] or quadrature methods [15]. The projected polynomial coefficients can be obtained by solving a linear system. Parussini et al. [16] used the tonsorial-expanded chaos collocation method to solve fluid dynamic problems with geometric uncertainties. Mathelin and Hussaini [17] proposed the SC (stochastic

Fig. 3. Convergence curves of NIPRC and MC.

Table 1 Collocation points and corresponding weights. Angle of attack (AOA) Unit:
. 1st Order

2nd Order

3rd Order

4th Order

5th Order

AOA

Weights

AOA

Weights

AOA

Weights

AOA

Weights

AOA

Weights

4.75 5.25

0.5 0.5

4.5670 5.0000 5.4330

0.166667 0.666667 0.166667

4.4164 4.8145 5.1855 5.5836

0.045876 0.454124 0.454124 0.045876

4.2858 4.6611 5.0000 5.3389 5.7142

0.011257 0.222076 0.533333 0.222076 0.011257

4.1689 4.5277 4.8458 5.1542 5.4723 5.8311

0.002556 0.088616 0.408829 0.408829 0.088616 0.002556

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Fig. 4. CDFs of Cl and Cd.

collocation) method. The probability distribution of the uncertain parameter forms the basis of a transformation to a probabilistic space. Collocation points are chosen in this probabilistic space and, with Lagrange interpolation, the probability distribution of the solution is constructed [12]. It is applied to a quasi-one-dimensional supersonic flow by Mathelin et al. [18], and the numerical results indicate that the SC method is substantially more efficient than the full Galerkin PC method. However, the computational cost of SC is still high for high dimensional stochastic problems. The NIPRC (non-intrusive probabilistic collocation) method combines the idea of chaos transformation and the collocation approach from the SC method. The NIPRC method is applied to a steady flow around an NACA0012 airfoil with uncertain free stream velocity [12]. It is shown that, for the same amount of computational cost, the NIPRC method can reach a higher accuracy than the NIPC method. Dinescu et al. [6] compared the NIPRC method and the IPC in the test case of rotor 37. Onorato et al. [19] compared the IPC method and NIPRC methodology for a 2D turbulent NaviereStokes flow, based on an RAE2822 airfoil with uncertainties imposed on the inlet Mach number and AOA (angle of attack). Wang et al. [20] used

Fig. 5. Pdf of Cl.

Fig. 6. Mean value and StD of Cp.

it in robust optimization of rotor 37. The mean aerodynamic efficiency and the variation of the efficiency due to the stochastic outlet static pressure are two optimization objectives. The robust optimization results show a wider stable working range than that of the deterministic optimization when the highest efficiency is the same. The working conditions of wind turbine are usually complex and unstable. The uncertainties exist widely in geometry and boundary conditions, for instance, errors in blade manufacturing and assembly, roughness on blade surface, and the stochastic incoming wind speed and direction. These uncertainties may have great influence on the aerodynamic performance of wind turbine, and lead to big deviation to the deterministic prediction. Therefore, it is necessary to investigate and reduce the sensibility of aerodynamic performance to uncertain factors. Non-deterministic CFD simulations have been used to solve this problem in quite recently [21,22]. In our previous studies, considering the angles of attack (AOA) is a random variable in Gaussian distribution, non-deterministic CFD simulations were

Fig. 7. Locations of point A and B.

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129

Fig. 9. Computational grids.

case in Section 4. The discussions mainly focus on the total performance analysis and the uncertainty propagation in the fluid field. Some conclusions are drawn in Section 5. 2. Method descriptions 2.1. Non-intrusive NIPRC A stochastic variable, f(x,t,q), which is a function of time t, space x and the random variable q,q can be represented using a spectral expansion [24]. Fig. 8. PDFs of the Cp at point A and B.

fz

NPC X

fk ðx; tÞJk ðxðqÞÞ;

k¼0

performed on the airfoil under two typical AOA [23]. The present paper investigates the effect of the stochastic wind velocity on the wind turbine aerodynamic performance using non-deterministic CFD simulations. The NIPRC method coupled with a commercial CFD software is used for uncertainty quantification of the stochastic incoming wind speed for the NREL Phase VI wind turbine. This paper is organized as follows. Section 2 is devoted to the introduction of the NIPRC method. In Section 3, the non-deterministic is validated by comparing with Monte Carlo simulation, based on a two-dimensional airfoil case. Then, the NIPRC method is used in nondeterministic simulations of a three-dimensional wind turbine blade

Table 2 Collocation points for NIPRC and corresponding weights. Collocation points (m/s)

Weights

Speed 1

Speed 2

Speed 3

5.80 7.02 8.23

10.81 13.07 15.33

16.64 20.13 23.62

0.166667 0.666667 0.166667

Fig. 10. Boundary conditions and computational domain.

(1)

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The Gaussian quadrature can be written in the following format:

Zb f ðxÞdxz

n X

wðxk Þf ðxk Þ;

(5)

k¼1

a

where xk, associated with the zeroes of orthogonal polynomials, are the integration points and w(x)is the weighting function related to the orthogonal polynomials. Exponential convergence can be reached when the weighting function has the same form as the probability density function of the input variable [9]. For a random variable in Gaussian distribution, the probability density function is given by x2 1 f ðxÞ ¼ pffiffiffiffiffiffiffie 2 : 2p

(6)

Thus, the optimal polynomial is Hermite polynomial Hn(x) with weighting function: x2 1 wðxÞ ¼ pffiffiffiffiffiffiffie 2 : 2p

Fig. 11. Power curve with uncertainty bars.

where,x ¼ {x1(q), x2(q), …,xn(q)} is a random variable vector; xi(q) (i ¼ 1, …,n) is a set of uncorrelated random variables with a random event q; fk is the deterministic coefficient, which is approximately estimated through a number of deterministic solutions for nonintrusive approaches; and Jk(x) is the stochastic polynomial chaos with the highest degree of p. The total number of expansion terms is NPCþ1,with:

NPC ¼

ðp þ nÞ!  1: p!n!

(2)

One can choose NPCþ1 points randomly to build the linear system, where each collocation point corresponds to one deterministic calculation. Solving the linear system gives the value of fk. However, the efficiency of this non-intrusive method is low, and the value of fk depends on the location of the chosen points, which is not unique. Instead of choosing NPCþ1 points randomly, the NIPRC method uses the Lagrange interpolation and Gaussian collocation points to approximate the polynomial. For the NIPRC method, each uncertain variable f can be expanded as follows:

f¼

NP X

fk ðx; tÞhk ðxðqÞÞ;

(3)

k¼1

where, fk (x,t) is the deterministic simulation solution f(x,t,q) at the collocation point qk, computed with the deterministic CFD software, and NP is the number of collocation points. Thus, the elaborate choice of collocation points in the stochastic space corresponding to the deterministic simulations is important. For polynomial approximation, n Gaussian quadrature points give the integrated polynomial accuracy of (2ne1) degree. Therefore, the collocation point qi is chosen corresponding to the Gaussian quadrature points. hk denotes the Lagrange interpolation polynomial chaos corresponding to the same point. The Lagrange interpolating polynomial chaos is the polynomial chaos hk(x(q)) of order NP1 that passes through the NP collocation points, which is given by

hk ðxðqÞÞ ¼

Np Y i¼1;isk

xðqÞ  xðqi Þ ; xðqk Þ  xðqi Þ

(4)

where hk(x(qi)) ¼ dki. Once the coefficients are available from deterministic solution, which means fk (x,t) and hk can be obtained easily. Then the f can be obtained as well.

(7)

The Gaussian quadrature points xk can be obtained by taking the kth zero of Hn(x). When the coefficients fk in Eq. (3) are available, the expression of the stochastic variable f(x,t,q) is obtained. Then, the PDF (probabilistic density function) and the CDF (cumulative distribution function) of f can be reconstructed easily. Moreover, the statistical characteristics can also be obtained. For instance, the mean and variance can be rebuilt using the following formula:

f¼

NP X

wk fk ðx; tÞ;

(8)

k¼1

s2 ¼

NP X

2

wk f2k ðx; tÞ  f ;

(9)

k¼1

where wk are the quadrature weights corresponding to the collocation points qk:

wðxk Þ ¼

pffiffiffi 2n1 n! p pffiffiffi i2 : n2 Hn1 2xk h

(10)

A brief NIPRC procedure can be described as follows: a)Generate N collocation points according to Gaussian quadrature points. b)Perform N deterministic calculations according to the points generated in Step a). c)Reconstruct the PDF or CDF of the response variables or calculate the statistical characteristics using Eq. (8) and Eq. (9). 2.2. CFD method By virtue of the non-intrusive property of the NIPRC method, an existing well-verified CFD code, the Fine™/Turbo software package from NUMECA Inc. is employed. Fine™/Turbo solves the RANS (Reynolds averaged NaviereStokes) equations using a finite volume method and a time-marching method. A standard central scheme with the Jameson type dissipation is used for spatial discretization, and an explicit fourth-order RungeeKutta scheme for the time integration. A full multi-grid strategy is used to accelerate the convergence. The one-equation SpalarteAllmaras turbulence model is selected for turbulence modeling in the present paper.

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Fig. 12. Distributions of the normal and tangential load coefficients with uncertainty bars at different wind speeds.

3. Validation of NIPRC method 3.1. Case description In this section, the NIPRC method is validated by being compared with the Monte Carlo simulation in the case of NREL S809 airfoil. The deterministic computational domain is extended 60 times of the airfoil chord. The total number of grid nodes is about 140,000 with 275 nodes around the airfoil and a certain clustering towards both the leading and trailing edges. Yþ (dimensionless wall distance) is less than 1 over most of the range from the leading edge to the trailing edge. The choice of the computational mesh and turbulence model is based on our previous studies [25,26],where the influence of the grid density and turbulence model have been

investigated in detail. A close view of the 2D grid nearby the airfoil is shown in Fig. 1. All outer boundaries of the computational domain are defined as external boundaries with a given velocity, atmospheric pressure and temperature. The airfoil surface is set to non-slip solid wall with full turbulence assumption. The AOA is considered as a normally distributed variable, with a mean of 5
and a StD (standard deviation) of 0.25
, 5% of the mean value. The collocation points and the corresponding weights are given in Table 1. Each collocation point (AOA value) corresponds to one deterministic CFD calculation. Note that, using n collocation points gives the (n1)th order PC expansion. MC (Monte Carlo) simulations using up to 129 samples are also performed, and each sample corresponds to one deterministic CFD calculation.

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Fig. 13. Contours of StD of Cp and streamline patterns at the speed of 7 m/s.

3.2. Results Firstly, Fig. 2 shows the comparison of deterministic CFD results of Cl and Cd with the experimental data. The comparison shows a good agreement at the low angle of attack from about 5
to about 10
. When the deterministic computation results at each collocation point are available, the statistical properties of the response random variables, such as the Cl (lift coefficients), Cd (drag coefficients), and the Cp (pressure coefficient) on the blade surface, can be computed with the deterministic solutions and weights. Since the mean values of Cl and Cd converged quickly with the number of samples increase. For instance, the results calculated from the NIPRC method have only small difference (<3%) to the results of MC simulation with 65 and 129 samples, the convergence curves are not shown here. More attentions are paid to the convergence studies of standard deviation (StD). Fig. 3 shows the convergence study of the NIPRC and MC results. The convergence in terms of the relative error is defined as [14]:

Relative Error ¼

    StDref  StDi  StDref

(11)

where StDref stands for the results of the MC simulation using 129 samples, and StDi is the value for comparison. As can be seen in Fig. 3, when the number of MC samples is 65, the relative error of StD reduces to 4.5%. The same accuracy level can be achieved using only the second order NIPRC method. In other words, the accuracy level of the second order NIPRC method may be comparative with the MC simulation using at least 65 samples for the current case. Fig. 4 shows the Cl and Cd reconstructed CDFs from NIPRC results using three collocation points and the MC results using 129 samples. The curves show good agreement between the results of the two methods. PDF can also be rebuilt by the two methods. Since a large number of samples are needed for MC method to rebuild a smooth PDF curve, the number of 129 is far from enough. However, NIPRC can rebuild the PDF easily. 10,000 samples for AOA are generated randomly respecting a Gaussian distribution, with a mean of 5
and a standard deviation (StD) of 0.25. Then 10,000 corresponding Cl can be obtained using Eq. (3), and the PDF can be built according to its definition. Fig. 5 shows the PDF curve, from which it is clear to see that the Cl is in a Gaussian distribution. The reason is that the Cl is a linear function of AOA when AOA is from 7
to 8
, which can be seen from Fig. 2. The AOA analyzed in the current case is exactly within this interval.

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Fig. 14. Contours of StD of Cp and streamline patterns at the speed of 13 m/s.

Fig. 6 shows the mean and StD curves of Cp along the airfoil surface. Both the mean and MC StD calculated using two methods have good agreement. As can be seen, for a low AOA, the variation of Cp due to the AOA uncertainty is mainly located at the leading edge of the airfoil. However, it will change significantly for a high AOA according to previous studies [23]. The separated flow at a high AOA has great influence on the uncertainty propagation in flow field. The PDFs of Cp for two points marked as A and B, respectively, are rebuilt and shown in Fig. 8. The locations of both two points are close to the airfoil surface, which is illustrated in Fig. 7. Point A is close to the leading edge, while point B is located at about 50% chord length. It is seen from Fig. 8a) that the Cp at point A is approximately in a Gaussian distribution. The reason is that the AOA has approximately a linear influence on the flow at point A since it is quite close to the leading edge. However, the Cp at location B as shown Fig. 8b), is in a skewed distribution. The comparison results indicate that the probabilistic collocation method has a high capability to quantify the uncertainty propagation in the flow field. The results of the second-order NIPRC method can be competitive with the MC results in this case. Moreover, the NIPRC method has much lower computational cost than the MC method.

4. Three dimensional wind turbine blade analysis 4.1. Models and methods NREL Phase VI is a two-bladed and stall regulated wind turbine. The blade is twisted and tapered, which is designed based on the S809 airfoil. The rotational speed is 71.63 rpm. More geometry information can refer to [27]. The Unsteady Aerodynamics Experiment (UAE) is conducted in a wind tunnel. The test section is 24.4 m  36.6 m, which is sizeable enough to accommodate this 10m-diameter wind turbine and operated by NASA. The tests were performed in more than 1700 different operating conditions, corresponding to 35 different turbine configurations, such as downwind or upwind configurations, rigid or teetering mode hub, etc. This paper will refer to the UAE-S series, the upwind baseline configuration without flow probes to avoid unnecessary disturbances. In this paper, the wind speed imposed on the inlet boundary is supposed to be an uncertain parameter in a Gaussian distribution. The CFD simulations are performed at four mean wind speeds, 7, 13, 15 and 20 m/s, respectively, with a StD of 10% of each mean value. The second order NIPRC is used in this section. The collocation points and the corresponding weights are given in Table 2.

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Fig. 15. Contours of StD of Cp and streamline patterns at the speed of 20 m/s.

The deterministic computations are performed by solving the RANS equations with the SpalarteAllmaras turbulence model [28]. The slip Euler condition is applied at the hub. Wind speed and temperature are imposed at the inlet and atmospheric pressure is imposed at the outlet. The three-dimensional (3D) grid is generated using AutoGrid™ from NUMECA Inc. The O4H grid topology around the blade is shown in Fig. 9. The total number of grid nodes is about 2.9 million, with 161 grid nodes around in circumference, and 97 grid nodes across the span. Fig. 10 illustrates the boundary conditions and computational domain, where R stands for the radius of the rotor. Yþ of the first grid away from the blade surface is less than 2 over most of the blade surface. The computational mesh, turbulence model and numerical scheme have been validated in previous studies [28].

stochastic wind speed. The results show that the most sensitive speed condition is 13 m/s in four wind speeds, followed by 7 m/s and 15 m/s, and the variation at 20 m/s is very small. The experimental results [27] are also presented in this figure. It is known that the

4.2. Results Comparisons among the mean values from non-deterministic simulation, the deterministic results and the experiment results are presented in Figs. 11 and 12. Fig. 11 shows the curve of the power with uncertainty bars (in blue (in web version)) containing the value within 90% confidence intervals. Note that, the uncertainty bars shown here only indicate the sensitivity of power with respect to the

Fig. 16. Streamline pattern near suction surface at the speed of 20 m/s. Red line: 2D streamline on sections of 95% and 80% span. Blue line: 3D streamline across the blade. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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experimental results are the mean values of many repeated experiments. So the mean value of the non-deterministic simulation leads to a fundamental change in the way of CFD validation. Fig. 12 shows the normal(Cn) and tangential(Ct) load coefficients with the uncertainty bars, including the values within 90% confidence intervals at different speeds of 7 m/s, 13 m/s and 20 m/s as the flow phenomenon at 13 m/s is quite similar to that at 15 m/s. At a speed of 7 m/s, the uncertainty bar of Cn and Ct is high near the root of the blade. At a speed of 13 m/s, the variation in Cn is also high near the root, but it is very low in the middle of the blade. However, the Ct has inverse variation. When the wind speed reaches 20 m/s, both Cn and Ct have very small variations. Moreover, the mean values of the NIPRC, the deterministic results and the experiment results have a good agreement at low wind speeds, but have differences under stall conditions. This is because the load is more sensitive to the uncertain conditions in the stall areas: the mean values of NIPRC only display the results from the uncertain factor of the inlet wind speed; deterministic results do not contain the uncertain factors; however, all uncertain factors are included in the experiment results. Figs. 13e15 present StD contours of Cp and the streamline patterns at different sections (30%, 47%, 63%, 80% and 95% span). It is observed clearly that the input uncertainty of the inlet velocity results in a high uncertainty zone in the pressure distribution near the root, and decreases from root to tip. At the low wind speed of 7 m/s as shown in Fig. 13, the maximum variation of Cp occurs at the leading edge of the suction side. No separation is observed. In Fig. 14, the maximum variation of Cp also appears at the leading edge. Flow separations are observed near the suction side at most span sections. From the streamline patterns, it is seen that the detached vortex corresponds to the maximum variation area of Cp. Both the area and the value of the variations are larger than that in Fig. 13. Separation occurs near the suction side and the vortexes correspond to the areas of Cp maximum variation in the sections. At the 30% span, there is one vortex attached to the leading edge of the suction side, and in the same area in the contour, there is an area of high maximum variation. The same phenomenon also appears in the other spans. From the 47%e95% span, maximum variations appear near the trailing edge in the suction side, which is different from Fig. 13. At the 47% span, the maximum values near the trailing edge have left the surface, which is due to the separated flow, and could explain the minimum values of the uncertainty bar appearing in the middle of the blade of Cn in Fig. 12. The flow patterns for 20 m/s (Fig. 15) are same to that for 13 m/s in general. The separation zone is further enlarged, and the variation area extends to the suction side. The streamline patters at all span sections show that the vortices move away from the airfoil surface, which corresponds to the maximum variation area of Cp .Thus, the variation area and the value of the variations on the surface are getting smaller, which explains the uncertainty bars in Fig. 11. It also can be concluded that the most sensitive condition is 13 m/s in all four conditions presented in this paper. Moreover, since the normal and tangential load coefficients are the integral of the pressure coefficients Cp around the section, the contours of Cp also explain the uncertainty bar of normal and tangential load coefficients as shown in Fig. 12. Note that, a pair of vortexes is observed at the 30%, 47%, 63% and 80% spans, respectively. But at the 95% span, the vortex near the trailing edge is not visible. From the 3D streamline pattern in Fig. 16, we can see that the streamlines across the vortex near the trailing edge on the 80% span have not reattached to the tip of the blade, but issued forth into the wake flow, resulting in a single vortex structure at the 95% span.

135

It is seen from the results shown above that the maximum variation occurs at the leading edge of the suction side at low wind speeds. With the wind speed increases and flow separation appears, the maximum variation extends to the trailing edge even to the whole suction side, and the flow separation regions correspond to the maximum variation area. Therefore, the flow variables in the separated flow region are more sensitive to the uncertain inlet velocity than other flow regions. 5. Conclusions This paper applied the NIPRC method to the non-deterministic CFD simulations of the NREL S809 airfoil and NREL Phase VI turbine blades. The AOA and the wind speed imposed on the inlet boundary are supposed to be uncertain parameters in Gaussian distribution, respectively. The discussions mainly focus on the total performance and the uncertainty propagation in the fluid field. The following conclusions can be summarized: 1) The NIPRC is proven to be an efficient method for uncertainty quantification, being compared with the MC method. The results of the second-order NIPRC method can be competitive with the MC results. The PDF of response variable is easy to get from NIPRC method, but too large number samples are need with MC. The flow around the leading edge of the airfoil is more sensitive to the uncertainty of the AOA. 2) The simulation results suggest that the input uncertainty of inlet velocity has big influence on the aerodynamic performance of a wind turbine. The uncertain responses of the flow variables are bigger near the root of the blade, and decreases from root to tip. 3) The maximum variation of Cp occurs at the leading edge of the suction side at low wind speeds. With the wind speed increases and flow separation appears, the maximum variation extends to the trailing edge, even to the whole suction side. The maximum variation of Cp corresponds to the separated flow region. 4) The non-deterministic CFD method can provide abundant information from stochastic responses with respect to uncertain factors. In this sense, the non-deterministic simulation leads to a fundamental change in the way of CFD validation. Nomenclature 2D 3D AOA CDF CFD GPC IPC MC NIPC NIPRC PC PDF RANS SC StD TE Yþ Cd Cl Cn Cp Ct

two-dimensional three-dimensional angle of attack cumulative distribution function computational fluid dynamics generalized polynomial chaos intrusive polynomial chaos Monte Carlo non-intrusive polynomial chaos non-intrusive probabilistic collocation polynomial chaos probabilistic density function Reynolds averaged NaviereStokes stochastic collocation standard deviation trailing edge dimensionless wall distance drag coefficient lift coefficient normal load coefficient pressure coefficient tangential coefficient

136

t x

f s

Z. Liu et al. / Energy 73 (2014) 126e136

time space stochastic variable standard deviation

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