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Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage
CJA 748 26 December 2016 Chinese Journal of Aeronautics, (2016), xxx(xx): xxx–xxx
No. of Pages 11
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage
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Hossein Sheikhi, Abas Saghaie *
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Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
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Received 2 February 2016; revised 6 May 2016; accepted 4 July 2016
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KEYWORDS
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Aerodynamic coefficients; Computational fluid dynamics (CFD); Engineering-statistical model; Helicopter fuselage; Wind tunnel test
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Abstract The design of the geometric shape of a helicopter fuselage poses a serious challenge for designers. The most important parameter in determining the shape of the helicopter fuselage is its aerodynamic coefficients. These coefficients are determined using two methods: wind tunnel test and computational fluid dynamics (CFD) simulation. The first method is expensive, time-consuming and limited. In addition, estimates in regions away from data can be poor. The second method, due to the limitations of numerical solution, the number of nodes and the used solution, is often inaccurate. In this paper, with the aim of accelerating the design process and achieving results with reasonable engineering accuracy, an engineering-statistical model which is useful for estimating the aerodynamic coefficients was developed, which mitigated the drawbacks of these two methods. First, by combining CFD simulation and regression techniques, an engineering model was presented for the estimation of aerodynamic coefficients. Then, by using the data from a wind tunnel test and implementation of statistical adjustment, the engineering model was modified and an engineering-statistical model was obtained. By spending less time and cost, the final model provided the aerodynamic coefficients of a helicopter fuselage at the desired angles of attack with reasonable accuracy. Finally, three numerical examples were provided to illustrate the application of the proposed model. Comparative results demonstrate the effectiveness of the engineering-statistical model in estimating the aerodynamic coefficients of a helicopter fuselage. Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author. E-mail addresses: [email protected] (H. Sheikhi), [email protected] (A. Saghaie). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
1. Introduction
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Models which describe the performance of a physical process are essential for prediction, process control and optimization.1–3 In general, two approaches exist for the development of these models. The first approach involves the development of models based on engineering/physical laws governing the process, which includes analytical models and numerical
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http://dx.doi.org/10.1016/j.cja.2016.12.015 1000-9361 Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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simulation.4–7 These kinds of models are called engineering models.1,8 The other development approach involves the postulation of statistical models and their estimation on the basis of generated data from physical experiments.9 These models are known as statistical models.1,2 Both modeling approaches possess some shortcomings and limitations.1,2,8 Predictions derived from the engineering models are often not accurate due to some assumptions in modeling,1,8 limitations in numerical solution, number of nodes and used solution method.10,11 Statistical models can provide good predictions at points close to the observed data, however, in an attempt to carry out predictions in regions away from the data, the predictions are usually poor. Moreover, the preparation of experimental data required to estimate the statistical models is costly and time consuming.1,8,10–12 Some researchers began to study various methods in order to discover models by combining engineering models with statistical models, which are useful in adjusting the shortcomings and limitations of the aforementioned models.1,2,8,13–16 These models are called engineering-statistical models.1 Engineering-statistical models are expected to provide more realistic predictions than the engineering models, and they are less expensive to estimate than the statistical models.1,2 In helicopters, one important characteristic of quality is the shape of the fuselage. The shape of the fuselage is impressive on flight endurance, cruise speed, stability and controllability, altitude, fuel consumption, maneuverability, etc. One of the technical characteristics which determines the suitability of the shape of the fuselage is its aerodynamic coefficients, which is achieved based on macroscopic fluid flow analysis around the helicopter, using Newton’s laws of motion and the basic principles of the laws of conservation of mass, momentum, energy and chemical species.17 Based on the nature of the problem and the desired parameters, these basic concepts can be described as algebraic, differential or integral equations.17,18 With regards to a body with complex geometric shape, such as helicopter fuselage, there is no exact solution to these equations, and physical experiments (wind tunnel tests) or numerical simulation techniques (computational fluid dynamics) are used to obtain these coefficients.12,19–22 Both wind tunnel test and CFD methods used in the analysis of fluid flow around helicopters have shortcomings and limitations expressed in engineering and statistical models. The main objective of this study is to construct an engineering-statistical model in order to accelerate the design process, thereby achieving results with appropriate engineering precision and cost reduction in computing the aerodynamic coefficients of helicopter fuselage which are: the drag coefficient CD, lift coefficient CL and pitching moment coefficient Cm. The engineering-statistical model is far more superior to the statistical model and engineering model because it behaves like the engineering model and has values those are close to the data.1 Therefore, by firstly using the CFD technique to model different well known helicopter fuselage in medium and intermediate weights, including the Bell 412, Bell 212, Bell 214, Bell 205, Agusta A109, Dauphin SA365N and UH-60 (here, for the ease of reference, they are called F1, F2, F3, F4, F5, F6, and F7, respectively), with the aid of a Fluent software, and running it, the aerodynamic coefficients of a helicopter fuselage are obtained for a number of angles of attack. Therefore, the trend of changes in these coefficients can be understood in
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H. Sheikhi, A. Saghaie terms of angle of attack of a helicopter. Since the observed trends are nonlinear, by using nonlinear regression, a function obtains fitting on the data. At this stage, an engineering model is achieved which estimates the aerodynamic coefficients of helicopters of medium and intermediate weights at the desired angles of attack. To achieve the considered engineeringstatistical model, a modified version of the sequential model building strategy of Joseph and Melkote1 was applied (the JM method). The JM method provides a sequential model building strategy, which helps to identify a prediction model that introduces minimal changes to the engineering model. This prevents additional adjustments which could cause the formation of a model that differs from the concept of the physical phenomena.1 Lack of a validation process is a shortcoming of the JM method. Hence, the modified version of the JM method is proposed. Therefore, the resultant model will be one which overcomes the shortcomings and limitations of wind tunnel test and CFD methods, and which can also estimate the aerodynamic coefficients with reasonable engineering accuracy. The rest of this paper is organized as follows. Section 2 presents the data of various helicopter fuselage modeling in Fluent software and engineering modeling method for estimation of the aerodynamic coefficients. In Section 3, the strategy of sequential engineering-statistical model building of Joseph and Melkote1 is introduced and a validation method of the model is proposed. Section 4 presents three numerical examples to demonstrate the application of the proposed models and the effectiveness of engineering-statistical model in estimating the aerodynamic coefficients of a helicopter fuselage. The conclusions and directions of future work are provided in Section 5.
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2. Engineering model develop by combining CFD simulation and regression techniques
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The modern helicopter industry requires design tools which are able to accurately and efficiently predict aerodynamic flow.10,18 In recent years, CFD methods have been increasingly used in the design and analysis of helicopters. This tendency was made by advances in CFD algorithms and the access to more powerful affordable computers. CFD is a virtual simulation technique. A flow can be fully simulated using CFD.12 In this paper, at the first step, by CFD technique, helicopters fuselage of F1, F2, F3, F4, F5, F6 and F7 (Fig. 1), were modeled by Fluent 6.3 software. Simulations were implemented in 3D and steady state, using implicit second order method, density based turbulent flow ((with aduption model) K-x shear stress transport (SST)), the ideal gas model, the Sutherland viscous model, at Mach number Ma = 0.2 and the number of meshes for different angles of attack a = 3.70 106 to 5.43 106. The software output values for aerodynamic coefficients of CD, CL and Cm are presented in Fig. 2. As shown in the figure, aerodynamic coefficients in terms of angle of attack, show nonlinear behavior and this trend is similar in almost all helicopters. Fitting a function to a particular data requires a parametric model that relates the response data to the predictor data with one or more coefficients. The result of the fitting process is an estimate of the model coefficients.23 Through trial and error, and also through the survey of various nonlinear
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Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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Fig. 1 Shape of Bell 412 (F1), Bell 212 (F2), Bell 214 (F3), Bell 205 (F4), Agusta A109 (F5), Dauphin SA365N (F6) and UH-60 (F7) helicopters. 157 149 150 151 152 153 154
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regression models, the best fit to the data is achieved for the Fourier model (Model 1) and fifth-order polynomial model (Model 2), respectively. Hence, each of the Models 1 and 2 can be fitted on the CFD data in order to obtain the engineering model of aerodynamic coefficients CD, CL and Cm. fðaÞ ¼ a0 þ a1 cosðwaÞ þ b1 sinðwaÞ þ a2 cosð2waÞ þ b2 sinð2waÞ ð1Þ
fðaÞ ¼ p1 a5 þ p2 a4 þ p3 a3 þ p4 a2 þ p5 a þ p6
ð2Þ
where a0, a1, a2, b1, b2, p1, p2, p3, p4, p5 and p6 are regression function coefficients and w is the frequency of the signal in the Fourier model. To obtain the coefficient estimates, the least-squares method minimizes the summed square of the residuals. The residual for the ith data point ri is defined as the difference between the observed response value fi and the
Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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Fig. 2 Values of aerodynamic coefficients CL, CD and Cm at different angles of attack from 90° to 90° for various helicopters.
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fitted response value f^i , and is identified as the error associated with the data.23 ri ¼ fi f^i
ð3Þ
The summed square of residuals is given by n n X X 2 S¼ r2i ¼ ðfi f^i Þ i¼1
ð4Þ
i¼1
where n is the number of data points included in the fit and S is the sum of squares error estimate. Because the least-squares fitting process minimizes the summed square of the residuals, the coefficients are determined by differentiating S with respect to each parameter, and setting the result equal to zero.23 Nonlinear models are more difficult to fit than linear models because it is impossible to estimate their coefficients using simple matrix techniques. Instead, an iterative approach is required which involves these steps: (A) start with an initial estimate for each coefficient. For some nonlinear models, a heuristic approach is provided which produces reasonable starting values. For other models, random values on the interval [0, 1] are provided; (B) produce the fitted curve for the current set of coefficients. The fitted response value fb is given by f^ ¼ fða; bÞ
ð5Þ
and involves calculation of the Jacobian of fða; bÞ, which is defined as a matrix of partial derivatives taken with respect to the coefficients; (C) adjust the coefficients and determine whether the fit is improved, the direction and magnitude of the adjustment depends on the fitting algorithm such as Trust-region and Levenberg–Marquardt algorithms; (D) iterate the process by returning to step 2, until the fit reaches the specified convergence criteria.23 In Fig. 3, Fourier model is fitted on CFD data of the mentioned helicopters, and the general engineering model of aerodynamic coefficients of a helicopter with 95% prediction
Fig. 3 Fourier models fitted on aerodynamic coefficients with 95% prediction bounds.
bounds is also shown. In order to obtain specific engineering model for each helicopter, the proposed regression functions should be fitted on the same CFD data, which results in a higher accuracy of the estimates concerning the considered helicopter. Due to the aforementioned issues, the following general engineering models are suggested for estimating the aerodynamic coefficients of a helicopter fuselage of medium and intermediate weights.
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CL ¼ 0:2008 0:279 cosð0:01025aÞ 0:2161 sinð0:01025aÞ þ 0:07152 cosð0:0205aÞ þ 0:1893 sinð0:0205aÞ
ð6Þ
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CD ¼ 0:006114 0:005387 cosð0:03422aÞ 0:000515 sinð0:03422aÞ þ 0:0002531 cosð0:06844aÞ þ 0:00006993 sinð0:06844aÞ
ð7Þ
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Cm ¼ 0:002452 þ 0:003519 cosð0:01728aÞ þ 0:001641 sinð0:01728aÞ 0:002447 cosð0:03456aÞ þ 0:005776 sinð0:03456aÞ
ð8Þ
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3. Engineering-statistical model building strategy and validation method
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The engineering models presented in Section 2 are perfect tools for quick estimation of the aerodynamic coefficients of a helicopter fuselage in arbitrary angles of attack. One of the downsides of engineering models is that the model predictions may
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not be accurate enough when compared with data from wind tunnel test.1,8 Strategies to improve the models generally fall into two categories: (A) model refinement and (B) model updating.2,14,16 Model refinement involves changing the physical principles in modeling or the use of other means in the construction of a model. It is more sophisticated and better represents the physics of the problem. Model updating utilizes mathematical concepts, such as calibration parameters and bias correction of model to match predictions of the model with physical observations. Although model refinement is desirable to improve the predictive capability, however, its strategy is often restricted by the available knowledge and computing resources. In contrast, model updating is a cheaper means which can be practical and useful when carried out correctly. Several model updating strategies have already been proposed in literature.1,2,14,16 In this paper, the modified version of the JM method (Fig. 4) was used to update the engineering models of aerodynamic coefficients of a helicopter fuselage. The advantage of this method when compared with other methods is that, it has minimal changes related to the engineering model, and it prevents the additional adjustments which could lead to a model that differs from the concept of physical phenomena.1 One of the flaws of the JM method is the lack of a validation process in the modeling strategy. Therefore, in order to determine the interpolation and extrapolation capability of the model and its validation after the engineering-statistical modeling, the division of the wind tunnel test data into two
Fig. 4
5 groups is proposed. The first set of data is used to build the engineering-statistical model and the second dataset is used to validate the proposed model. Let Ck be the wind tunnel test output for k ¼ L; D; m. Due to the presence of noise (uncontrollable) factors and measurement error, the output is random and is denoted as: Ck ¼ lðaÞ þ e
ð9Þ
where lðaÞ is the mean of Ck at a given a and e Nð0; r2 Þ. The objective is to find the unknown function lðaÞ. What is obtainable in the engineering model is fða; gk Þ function, and the outputs of wind tunnel tests are ða1 ; Ck1 Þ; ða2 ; Ck2 Þ; . . . ; ðan ; Ckn Þ, where g ¼ fgk1 ; gk2 ; . . . ; gkq g0 denotes the unknown calibration parameters. Note that the argument of gk is omitted from the lðaÞ function, because the Ckj ’s in Eq. (9) are generated with gk fixed at its true value gk .1 The first step in sequential model building strategy involves the investigation of the usefulness of an engineering model. For this purpose graphical plots such as a Ckj vs fðaj Þ can be used. A positive correlation suggests the usefulness of the engineering model in prediction. Otherwise, assumptions that engineering model has been derived based on them, should be revised and corrected.1 If the engineering model is known to be useful, then this usefulness can be assessed. Let the predictor at this stage be ^E ðaÞ, which is equal to fðaÞ.1 The model inadedenoted by l quacy can be measured using different indicators. In this paper, due to the presence of positive and negative coefficients,
Sequential model building strategy (a modified version of JM method1).
Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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the index of mean percentage positive error (Eq. (10)) was suggested as an indicator of the model inadequacy as: MPPE ¼
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n
j¼1
jCkj j
ð10Þ
^Ej l
where ¼ fðaj Þ. In engineering calculations, the mean percentage positive error (MPPE) is often considered to be less than 10%. With the MPPE less than 10%, the engineering model is appropriate. Otherwise, to reduce the error, statistical adjustments are used. For this purpose, a simple location-scale adjustment (Eq. (11)) is applied in the model.1 ^C ðaÞ fðaÞ ¼ b0 þ b1 ðfðaÞ fÞ l ð11Þ Pn fðaj Þ where f ¼ j¼1n . Due to the fact that two constants are used for the adjustments, this method is known as the constant adjustment model.1 Upon obtaining the constant adjustment model, the model inadequacy measure (Eq. (12)) is computed again. MPPE ¼
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n 1X jj^ lE ðaÞj jCkj jj
n 1X jj^ lC ðaÞj jCkj jj n i¼1 jCkj j
ð12Þ
^0 þ b ^1 ðfðaj Þ fÞ and b ^j denote b esti^Cj ¼ fðaj Þ þ b where l j mated from the data. If MPPE is small enough (less than the maximum error appropriate for engineering), the constant adjustment model is used for prediction. Otherwise, in order to improve the adequacy of the model, more advanced adjustments are applied to the engineering model.1 Let lðaÞ lC ðaÞ ¼ dða; hÞ, where dða; hÞ is used for capturing the inadequacy of the constant adjustment model. This method is called functional adjustment model because the form of the predictor function differs from the engineering model. The form of the function d is obtained by analyzing the residual from the constant adjustment model. One simple method involves the use of a linear model given by P dða; hÞ ¼ m i¼0 bi ui ðaÞ, where ui ’s are known functions in a. The initial value for b is chosen to make Eðdða; hÞÞ ¼ 0. Therefore, it is compulsory to have EðhÞ ¼ 0 in a linear model. Now, lC ðaÞ and h can be estimated from the data. Thus, the functional adjustment predictor is achieved as:1 ^F ðaÞ ¼ l ^C ðaÞ þ dða; ^hÞ l
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4. Numerical examples
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In this section, to illustrate the application of the proposed models and the effectiveness of the proposed engineeringstatistical model for estimating the aerodynamic coefficients of a helicopter fuselage, three engineering-statistical models were developed to estimate the drag coefficients CD of F2 (Bell 212), F4 (Bell 205) and F7 (UH-60) fuselages at the desired angles of attack. Also, the accuracy of estimation of these models was compared with CFD method. Fig. 5 shows the drag coefficients of F2, F4 and F7 fuselages obtained from wind tunnel test and the values obtained by CFD simulation. If wind tunnel test data are available for limited angles of attack, and there is a need to carry out predictions in points away from data, the predictions will be poor.
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CDðF2Þ ¼ 0:004811 0:004352 cosð0:03541aÞ 0:001113 sinð0:03541aÞ þ 0:0004172 cosð0:07082aÞ þ 0:0001692 sinð0:07082aÞ
ð14Þ
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CDðF4Þ ¼ 0:001833 þ 0:01452 cosð0:01347aÞ 0:0001867 sinð0:01347aÞ 0:01176 cosð0:02694aÞ þ 0:00008333 sinð0:02694aÞ
ð15Þ
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CDðF7Þ ¼ 0:00006577 þ 0:01302 cosð0:01383aÞ 0:0001406 sinð0:01383aÞ 0:0119 cosð0:02766aÞ þ 0:00003828 sinð0:02766aÞ
ð16Þ
ð13Þ
This two-stage model building procedure prevents additional adjustments on the engineering model and helps to identify an engineering-statistical model with minimal adjustments.1
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In order to build and validate the proposed model, wind tunnel test data were divided into two categories. The first category includes drag coefficient values of F2, F4 and F7 fuselages, respectively at angles of attack given as {15, 10, 6, 0}, {20, 15, 8, 6, 4, 2, 0, 2, 4, 8, 10, 15, 20, 25, 40} and {25, 30, 40, 50, 60, 70}, which were used in the construction of the model, and the second category includes drag coefficient values of F2, F4 and F7 fuselages respectively at angles of attack given as {8, 4, 2, 4, 6, 15}, {90, 50, 10, 6, 30, 90} and {80, 25, 35, 45, 65, 80}, which were used in the validation process.19,24 In this section, based on Section 2 in which the engineering model building method for aerodynamic coefficients of helicopter fuselage was described, firstly, the engineering model to estimate the drag coefficient of F2, F4 and F7 fuselages were achieved (Eqs. (14)–(16)) by fitting Fourier function (Eq. (1)) on CFD data, of which the shape of function and the prediction bounds at 95% was provided in Fig. 6.
Fig. 5 Drag coefficient values of F2, F4 and F7 fuselages related to wind tunnel test and CFD method.
Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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7 than the maximum error appropriate for engineering practice, which is 10%. Therefore, to improve the precision of the engineering model estimation, an engineering-statistical model based on sequential model building strategy is referred. At the first stage, by using the data from wind tunnel test drag coefficient of F2, F4 and F7 fuselages at the aforementioned angles of attack, the constant adjustment can be applied to engineering models. Thus, by using simple linear regression the values of Be0 and Be1 were ^C ðaÞ fðaÞ on fðaÞ f, for l obtained. So, constant adjustment models will be in the form of Eqs. (17)–(19). F2: ^C ðaÞ fðaÞ ¼ 0:0002888 þ 0:02125ðfðaÞ 0:00124Þ l
ð17Þ
F4: l ^ ðaÞ fðaÞ ¼ 0:0004287 þ 0:25778ðfðaÞ 0:00166Þ
ð18Þ
F7: ^ ðaÞ fðaÞ ¼ 0:001151 þ 0:03556ðfðaÞ 0:00689Þ l
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Then, the sequential model building strategy presented in the previous section was used to create engineering-statistical models to estimate the drag coefficient CD of F2, F4 and F7 fuselages. Fig. 7 shows the plot of CD (the drag coefficient value from wind tunnel test) vs fðaÞ of F4 fuselage (the drag coefficient value obtained from the engineering model). From this plot, there exists evidence of a positive correlation of data obtained from wind tunnel test and engineering model. Also, the Pearson correlation coefficient was +1 which indicated a strong positive correlation between them. This subject is similar for F2 and F7 fuselages. Therefore, the proposed engineering model is useful for prediction. Table 1 presents the inadequacy of the engineering model and its comparison with CFD method on the basis of the MPPE. The MPPE of the CFD method and the engineering model are, respectively equal to 19.99476% and 19.48765% for F2 fuselage, 20.62754 and 20.88905% for F4 fuselage and 16.78131% and 16.49641% for F7 fuselage, which are greater
ð19Þ
The inadequacies of the model for constant adjustment models are calculated in Table 1, which are equal to 0.70327%, 1.10185% and 2.63700% for F2, F4 and F7 fuselages respectively, and are far smaller than the inadequacy of the engineering model and the CFD method. The inadequacies of the engineering models obtained at this step, by applying constant adjustments, were greatly reduced and predictions derived from these models were of high accuracy. However, in order to show the full process of sequential model building, the second stage of the model improvement implied that the functional adjustment was applied. To improve the accuracy of the constant adjustment model, the functional adjustment model is: Ck l ðaÞ ¼ dða; hÞ þ e c
ð20Þ
So that: dða; hÞ ¼ b0 þ
m X
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bi ui ðaÞ
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ð21Þ 431
i¼1
By plotting the residuals of the constant adjustment model of F4 fuselage vs the angles of attack (Fig. 8) and after some trial and error, u1 ðaÞ ¼ a; u2 ðaÞ ¼ a2 ; u3 ðaÞ ¼ a3 and u4 ðaÞ ¼ a4 are selected. Through regression of the residuals ~0 ¼ 1:70 105 ; b ~1 ¼ 1:54 107 ; against a; a2 ; a3 and a4 ; b ~2 ¼ 1:39 107 ; b ~3 ¼ 1:45 109 and b ~4 ¼ 1:37 1010 b values are obtained. As a result:
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^F ðaÞ l ^C ðaÞ ¼ 0:000017 þ 0:0000001537a l 0:0000001386a2 0:000000001448a3 þ 0:0000000001367a4
Fig. 7 Drag coefficient of wind tunnel test vs engineering model in F4 fuselage example.
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C
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C
Fig. 6 Engineering model of drag coefficient of F2, F4 and F7 fuselages with 95% prediction bounds.
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ð22Þ
^ C ðaÞ is given in Eq. (18). This model gives a good fit to where l the data (see Fig. 9). The inadequacies of the model for functional adjustment models of F2, F4 and F7 fuselages are 0.00741%, 0.78936%, and 0.54670% (Table 1), which are smaller than the inadequacies of constant adjustment models, with values of 0.70327%, 1.10185% and 2.63700%. For comparison of the models inadequacy, in addition to the criterion of the MPPE, other criteria such as the mean absolute error (MAE), mean bias error (MBE) and root mean square error (RMSE) can also be used.
Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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H. Sheikhi, A. Saghaie Table 1 Comparison of accuracy of CFD method with engineering model, constant adjustment model and functional adjustment model in drag coefficient estimation of F2, F4 and F7 fuselages. Fuselage
a (°)
Drag coefficient
Percentage positive error compared with wind tunnel test
Functional adjustment model
Wind tunnel test
CFD
Engineering model
CFD
Engineering model
Constant adjustment model
0.00197 0.00163 0.00137 0.00116
0.00169 0.00132 0.00109 0.00086
0.00168 0.00132 0.00110 0.00088
14.36826 19.17069 20.75066 25.68944
14.85325 18.90565 19.67972 24.51200
0.27138 1.08581 1.15105 0.30483
0.01701 0.00862 0.00398 0.00001
Mean percentage positive error F4 20 0.00260 15 0.00200 8 0.00140 6 0.00130 4 0.00126 2 0.00120 0 0.00119 2 0.00120 4 0.00125 8 0.00140 10 0.00150 15 0.00199 20 0.00260 25 0.00330 40 0.00640
0.00207 0.00158 0.00113 0.00104 0.00099 0.00095 0.00094 0.00095 0.00098 0.00112 0.00122 0.00157 0.00204 0.00264 0.00501
0.00208 0.00158 0.00112 0.00103 0.00098 0.00094 0.00093 0.00094 0.00097 0.00111 0.00122 0.00157 0.00206 0.00267 0.00504
19.99476 20.38462 21.00000 19.28571 20.00000 21.42857 20.83333 21.00840 20.83333 21.60000 20.00000 18.66667 21.10553 21.54820 20.00000 21.71875
19.48765 20.14208 20.82425 20.20714 20.41834 22.60040 21.72179 22.10084 21.81203 22.15596 20.52990 18.88312 20.89894 20.68528 19.10238 21.25338
0.70327 0.47722 0.37074 0.42412 0.16328 2.57915 1.47059 1.94675 1.58410 2.01960 0.01816 2.08530 0.46446 0.20601 1.77782 0.94046
0.00741 0.16730 0.60489 1.00992 1.05390 1.44461 0.12460 0.51818 0.18880 0.79243 0.67371 2.39170 0.95924 1.17043 0.71694 0.02377
Mean percentage positive error F7 25 0.00415 30 0.00495 40 0.00670 50 0.00845 60 0.01080 70 0.01317
0.00295 0.00374 0.00560 0.00769 0.00976 0.01154
0.00299 0.00379 0.00564 0.00769 0.00971 0.01149
20.62754 28.81335 24.40353 16.41919 9.03179 9.65457 12.36543
20.88905 27.85359 23.47618 15.82686 8.99103 10.08316 12.74764
1.10185 3.45300 2.45118 0.68315 4.96961 1.50295 2.76208
0.78936 0.23681 0.03893 0.86761 1.27112 0.70813 0.15759
16.78131
16.49641
2.63700
0.54670
F2
15 10 6 0
Mean percentage positive error
Fig. 8 Plotting residuals of constant adjustment model vs angles of attack of F4 fuselage and fitting a function on them. Fig. 9 Comparison of engineering model, constant adjustment model and functional adjustment model trends of F4 fuselage. 452 453 454 455 456 457 458 459 460 461 462 463 464 465
In Table 2, CFD method, engineering model, the constant adjustment and functional adjustment models of F4 fuselage have been compared according to the aforementioned criteria. From the table, functional adjustment model based on all criteria was better than the other models. However, according to Fig. 8, it is evident that although the functional adjustment model in the observed points had a better fit to the data and had a higher accuracy than the constant adjustment model, the function form changed and demonstrated a different trend compared to the engineering model. In particular, the more the increase in the angle of attack the more the difference. According to the engineering model, an increase in the angle of attack of a fuselage resulted in a reduction in the slope of the drag coefficient curve of the fuselage, however this was not the case
in the functional adjustment model. Therefore, in choosing the appropriate model for predicting the aerodynamic coefficient of a fuselage, in addition to the consideration of the criteria for measuring the model inadequacy, it is better to compare the prediction model function with the engineering model. As a general conclusion, it can be said that a prediction model having appropriate model adequacy measuring criteria and which follows the same trend as the engineering model, is more appropriate. Also, the second set of wind tunnel test data can be used as a tool to select a more suitable model for estimating the
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Table 2 Comparison of estimation accuracy of CFD method, engineering model, constant adjustment model and functional adjustment model in drag coefficient of F4 fuselage. Method
Root mean square error (RMSE)
Mean bias error (MBE)
Mean absolute error (MAE)
Mean percentage positive error (MPPE)
CFD Engineering model Constant adjustment model Functional adjustment model
0.000512898 0.000505589 0.000027458 0.000016845
0.000424022 0.000423378 0.000005940 0.000000001
0.000424022 0.000423378 0.000020696 0.000013489
20.62754 20.88905 1.10185 0.78936
Table 3 Comparison of CFD method, engineering model, constant adjustment and functional adjustment models on the basis of percentage positive error for data within the range of date modeling. Fuselage
a (°)
Percentage positive error compared with wind tunnel test CFD
Engineering model
Constant adjustment model
Functional adjustment model
F2
8 4 2 Mean
20.99813 22.61703 24.17254 22.59590
20.09995 21.50270 22.94808 21.51691
1.00224 0.49579 0.24508 0.58104
0.87243 1.59025 1.66676 1.37648
F4
10 6 30 Mean
18.00000 20.00000 20.23810 19.41270
18.49726 20.67400 19.51112 19.56079
2.57063 0.15829 1.25805 1.32899
2.66667 0.76923 0.47619 1.30403
F7
35 45 65 Mean
18.44663 12.44325 11.93205 14.27398
18.34041 12.30344 11.95903 14.20096
0.37818 2.75526 1.33386 1.48910
0.77236 0.23321 1.43836 0.81464
Table 4 Comparison of CFD method, engineering model, constant adjustment and functional adjustment models on the basis of percentage positive error for data outside the range of data modeling. Fuselage
477 478 479 480 481 482 483 484 485 486
a (°)
Percentage positive error compared with wind tunnel test CFD
Engineering model
Constant adjustment model
Functional adjustment model
F2
4 6 15 Mean
24.33812 20.85488 13.50930 19.56743
23.46728 20.22537 14.14251 19.27838
3.46876 8.58005 13.50930 8.51937
18.89592 40.21373 195.30380 84.80448
F4
90 50 90 Mean
19.52818 19.09197 19.51872 19.37962
19.74987 19.18825 19.74582 19.56131
0.94273 1.65355 0.94793 1.181403
59.30537 9.77881 46.52406 38.53608
F7
80 25 80 Mean
11.30505 22.56366 12.51696 15.46189
10.63788 21.23783 11.74691 14.54087
1.22484 5.03590 2.35921 2.87331
302.92412 62.66847 10.22403 125.27220
aerodynamic coefficients of a fuselage and also to validate the proposed models. The interpolation and extrapolation of models can be determined using a validation process. The MPPE of F2, F4, and F7 fuselages for a number of angles of attack inside and outside the data range, such that modeling is complete on their basis, are provided in Tables 3 and 4, respectively. As evident in Table 4, for the data inside the range, the predictability for both constant adjustment models and functional adjustment models were appropriate, and the inadequacies of both models on the basis of the MPPE
are close to each other. Consequently, constant adjustment and functional adjustment models offered a good interpolation capability. According to Table 4, for data outside the range, the functional adjustment models had a lot of errors and it can be concluded that these models exhibit poor extrapolation ability. In contrast, the inadequacies of the constant adjustment models were small and this issue indicated the appropriateness of the extrapolation ability for the constant adjustment model. Considering the aforementioned points, to estimate the drag coefficients of F2, F4, and F7 fuselages, constant
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adjustment model is proposed as the better model, because in addition to having appropriate engineering precision and similar trend with the engineering model, it has good interpolation and extrapolation functionality.
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Models which describe the performance of a physical phenomenon are critical to design, predict, control and optimize. The analysis of fluid flow around a helicopter, in a macroscopic scale, so as to calculate the aerodynamic coefficients of helicopter fuselage, such as the drag CD, lift CL and pitching moment Cm, is one of the key issues in the design of helicopters. To obtain fuselage aerodynamic coefficients, wind tunnel test and computational fluid dynamics simulation are used. Wind tunnel tests are very costly, time consuming and limited. Although the CFD method is less expensive, owing to the limitation of numerical solution, the number of nodes and the used solution, it is often not accurate and also has its own cost and time. In addition, carrying out a wind tunnel test at certain angles of attack is necessary for the validation of CFD results. This paper aims to develop models which predicted the aerodynamic coefficients of helicopter fuselage at any desired angle of attack with appropriate engineering precision. This objective has been realized by combining and integrating computational fluid dynamics simulation and physical wind tunnel test. With such models, the design process can be accelerated, a tool is provided to aid designers with the prediction and optimization, and also there is significant reduction in the design costs. Firstly, by modeling the helicopter fuselage in the intermediate and medium weights (the Bell 412, Bell 212, Bell 214, Bell 205, Agusta A109, Dauphin SA365N and UH-60), and using CFD simulation with Fluent software, aerodynamic coefficients behavior of fuselage was analyzed. Then, using the CFD results and nonlinear regression, engineering models were developed which were able to predict the aerodynamic coefficients of the fuselage and demonstrate their trend. The JM method was modified and a validation step was included in order to eliminate its shortcomings. The proposed engineering-statistical models were obtained by combining the engineering models, modified version of the JM method, and a few numbers of wind tunnel test data. These models were able to estimate the aerodynamic coefficients of the fuselage at any desired angle of attack with accuracy much higher than CFD and engineering models, in the shortest possible time. Three numerical examples were provided to illustrate the model building strategy and demonstrate the effectiveness of the proposed engineering-statistical models. So, the engineering models were obtained to estimate the drag coefficients of the F2 (Bell 212), F4 (Bell 205), and F7 (UH-60) fuselages by fitting Fourier model on CFD data. The engineeringstatistical models (constant adjustment and functional adjustment models) were developed based on the modified JM method. The accuracy of the proposed models of the F2, F4, and F7 fuselages were compared with the results of wind tunnel tests, CFD simulations and engineering models, based on MPPE. The inadequacies of the model for constant adjustment models of F2, F4 and F7 fuselages were equal to 0.70327%, 1.10185% and 2.637%, and were far smaller than the inade-
H. Sheikhi, A. Saghaie quacies of CFD simulations (19.99476%, 20.62754%, and 16.78131%) and engineering models (19.48765%, 20.88905%, and 16.49641%) which show the appropriate engineering accuracy (a value usually less than 10%). The inadequacies of the model for functional adjustment models of F2, F4 and F7 fuselages were 0.00741%, 0.78936%, and 0.54670%, which were smaller than the inadequacies of constant adjustment models. Also, the MAE, MBE, and RMSE criteria were used for comparison of the model inadequacies for engineering model, CFD simulation, constant adjustment and functional adjustment models of F4 fuselage. The results of the evaluation were similar according to all criteria. The results show the accuracy and efficiency of the introduced engineering-statistical model. It was also observed that, in addition to considering the criteria of the model inadequacy, the function form and trend were also important and should have a similar trend to the engineering model. Regarding the drag coefficient plots, it was observed that although the functional adjustment model on the basis of criteria of the model inadequacy was preferable to the constant adjustment model, due to its different trend when compared with the engineering model, the constant adjustment model was chosen as better engineering-statistical model to predict the drag coefficient of the fuselages. This was proved by the evaluation of the interpolation and extrapolation ability of the proposed models. The interpolation capability of both constant adjustment models and functional adjustment models of F2, F4, and F7 fuselages were appropriate and the inadequacies of both models on the basis of the MPPE were less by 1.5%. The functional adjustment models exhibited poor extrapolation capabilities and had a lot of errors. The inadequacies of the functional adjustment models of the F2, F4 and F7 fuselages were 84.80448%, 38.53608%, and 125.27220%. In contrast, the extrapolation capability of the constant adjustment models was appropriate and contained little errors as compared to wind tunnel test data (8.51937%, 1.181403%, and 2.87331%, respectively). Consequently, the superiority of the constant adjustment model was demonstrated. In future researches, the development of the engineering model and engineering-statistical model for aerodynamic coefficients of the fuselage on the basis of a helicopter yaw angle is suggested. Also, the development of such models for attack helicopters will be very valuable and important. On the other hand, the described model building strategy can also be used to model other physical phenomena.
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1. Joseph V Roshan, Melkote SN. Statistical adjustments to engineering models. J Qual Technol 2009;41(4):362–75. 2. Chang CJ, Roshan Joseph V. Model calibration through minimal adjustments. Technometrics 2014;56(4):474–82. 3. Kang L, Roshan Joseph V. Kernel approximation: from regression to interpolation. J Uncertainty Quantification 2016;4(1): 112–29. 4. Qian Z, Seepersad CC, Roshan Joseph V, Allen JK, Wu CFJ. Building surrogate models based on detailed and approximate simulations. J Mech Des 2006;128(4):668–77. 5. Plumlee M, Roshan Joseph V, Wu CFJ. Comment: alternative strategies for experimental design. Technometrics 2013;55(3): 289–92.
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6. Ba S, Jain N, Roshan Joseph V, Singh RK. Integrating analytical models with finite element models: an application in micromachining. J Qual Technol 2013;45(2):200–12. 7. Roshan Joseph VR, Kang L. Regression-based inverse distance weighting with applications to computer experiments. Technometrics 2011;53(3):254–65. 8. Bayarri MJ, Berger JO, Paulo R, Sacks J, Cafeo J, Cavendish J, et al. A framework for validation of computer models. Technometrics 2007;49(2):138–54. 9. Plumlee M. Fast prediction of deterministic functions using sparse grid experimental designs. J Am Stat Assoc 2014;109(508):1581–91. 10. Biava M, Vigevano L. Simulation of a complete helicopter: A CFD approach to the study of interference effects. Aerosp Sci Technol 2012;19(1):37–49. 11. Steijl R, Barakos GN. CFD analysis of complete helicopter configurations — lessons learnt from the GOAHEAD project. Aerosp Sci Technol 2012;19(1):58–71. 12. Antoniadis AF, Drikakis D, Zhong B, Barakos G, Steijl R, Biava M, et al. Assessment of CFD methods against experimental flow measurements for helicopter flows. Aerosp Sci Technol 2012;19 (1):86–100. 13. Higdon D, Kennedy M, Cavendish JC, Cafeo JA, Ryne RD. Combining field data and computer simulations for calibration and prediction. J Sci Comput 2004;26(2):448–66. 14. Xiong Y, Chen W, Tsui KL, Apley DW. A better understanding of model updating strategies in validating engineering models. Comput Method Appl M 2009;198(15–16):1327–37. 15. Hung Y, Joseph V Roshan, Melkote SN. Analysis of computer experiments with functional response. Technometrics 2015;57 (1):35–44. 16. Dasgupta T, Weintraub B, Joseph V Roshan. A physical-statistical model for density control of nanowires. IIE Trans-Quality Reliability Eng 2011;43(4):233–41. 17. White FM. Fluid mechanics. 7nd ed. New York: McGraw-Hill; 2011. p. 150–249.
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11 18. Wang Q, Zhao Q, Wu Q. Aerodynamic shape optimization for alleviating dynamic stall characteristics of helicopter rotor airfoil. Chin J Aeronaut 2015;28(2):346–56. 19. Hilbert KB. A mathematical model of the UH-60 helicopter. Washington, D.C.: NASA; 1984. Report No.: NASA/TM-85890 and USAAVSCOM TM-84-A-2. 20. Brand AG, McMahon HM, Komerath NM. Surface pressure measurements on a body subject to vortex wake interaction. AIAA J 1989;27(5):569–74. 21. Kim JM, Komerath NM. Summary of the interaction of a rotor wake with a circular cylinder. AIAA J 1995;33(3):470–8. 22. Biava M, Khier W, Vigevano L. CFD prediction of air flow past a full helicopter configuration. Aerosp Sci Technol 2012;19(1):3–18. 23. Bates DM, Watts DG. Nonlinear regression analysis and its applications. 2nd ed. New York: Wiley; 2007. p. 33–67. 24. Prouty RW. Helicopter performance, stability, and control. 2nd ed. Florida: Krieger Publishing Company; 2001. p. 515–698. Hossein Sheikhi is a Ph.D. candidate at Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran. He received his M.Sc. degree from Department of Industrial Engineering, Qazvin Branch, Islamic Azad University in 2010. His main research interests are design of experiments, computer experiments, aerodynamic, modeling, optimization, and operation research. Abas Saghaie is a professor and Ph.D. supervisor at Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran. He received the Ph.D. degree from Iran University of Science & Technology in 2004. His main research interests are design of experiments, advanced quality control, data mining, engineering probability and statistics, time series analysis, and operation research.
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage
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Hossein Sheikhi, Abas Saghaie *
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Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
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Received 2 February 2016; revised 6 May 2016; accepted 4 July 2016
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KEYWORDS
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Aerodynamic coefficients; Computational fluid dynamics (CFD); Engineering-statistical model; Helicopter fuselage; Wind tunnel test
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Abstract The design of the geometric shape of a helicopter fuselage poses a serious challenge for designers. The most important parameter in determining the shape of the helicopter fuselage is its aerodynamic coefficients. These coefficients are determined using two methods: wind tunnel test and computational fluid dynamics (CFD) simulation. The first method is expensive, time-consuming and limited. In addition, estimates in regions away from data can be poor. The second method, due to the limitations of numerical solution, the number of nodes and the used solution, is often inaccurate. In this paper, with the aim of accelerating the design process and achieving results with reasonable engineering accuracy, an engineering-statistical model which is useful for estimating the aerodynamic coefficients was developed, which mitigated the drawbacks of these two methods. First, by combining CFD simulation and regression techniques, an engineering model was presented for the estimation of aerodynamic coefficients. Then, by using the data from a wind tunnel test and implementation of statistical adjustment, the engineering model was modified and an engineering-statistical model was obtained. By spending less time and cost, the final model provided the aerodynamic coefficients of a helicopter fuselage at the desired angles of attack with reasonable accuracy. Finally, three numerical examples were provided to illustrate the application of the proposed model. Comparative results demonstrate the effectiveness of the engineering-statistical model in estimating the aerodynamic coefficients of a helicopter fuselage. Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author. E-mail addresses: [email protected] (H. Sheikhi), [email protected] (A. Saghaie). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
1. Introduction
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Models which describe the performance of a physical process are essential for prediction, process control and optimization.1–3 In general, two approaches exist for the development of these models. The first approach involves the development of models based on engineering/physical laws governing the process, which includes analytical models and numerical
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http://dx.doi.org/10.1016/j.cja.2016.12.015 1000-9361 Ó 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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simulation.4–7 These kinds of models are called engineering models.1,8 The other development approach involves the postulation of statistical models and their estimation on the basis of generated data from physical experiments.9 These models are known as statistical models.1,2 Both modeling approaches possess some shortcomings and limitations.1,2,8 Predictions derived from the engineering models are often not accurate due to some assumptions in modeling,1,8 limitations in numerical solution, number of nodes and used solution method.10,11 Statistical models can provide good predictions at points close to the observed data, however, in an attempt to carry out predictions in regions away from the data, the predictions are usually poor. Moreover, the preparation of experimental data required to estimate the statistical models is costly and time consuming.1,8,10–12 Some researchers began to study various methods in order to discover models by combining engineering models with statistical models, which are useful in adjusting the shortcomings and limitations of the aforementioned models.1,2,8,13–16 These models are called engineering-statistical models.1 Engineering-statistical models are expected to provide more realistic predictions than the engineering models, and they are less expensive to estimate than the statistical models.1,2 In helicopters, one important characteristic of quality is the shape of the fuselage. The shape of the fuselage is impressive on flight endurance, cruise speed, stability and controllability, altitude, fuel consumption, maneuverability, etc. One of the technical characteristics which determines the suitability of the shape of the fuselage is its aerodynamic coefficients, which is achieved based on macroscopic fluid flow analysis around the helicopter, using Newton’s laws of motion and the basic principles of the laws of conservation of mass, momentum, energy and chemical species.17 Based on the nature of the problem and the desired parameters, these basic concepts can be described as algebraic, differential or integral equations.17,18 With regards to a body with complex geometric shape, such as helicopter fuselage, there is no exact solution to these equations, and physical experiments (wind tunnel tests) or numerical simulation techniques (computational fluid dynamics) are used to obtain these coefficients.12,19–22 Both wind tunnel test and CFD methods used in the analysis of fluid flow around helicopters have shortcomings and limitations expressed in engineering and statistical models. The main objective of this study is to construct an engineering-statistical model in order to accelerate the design process, thereby achieving results with appropriate engineering precision and cost reduction in computing the aerodynamic coefficients of helicopter fuselage which are: the drag coefficient CD, lift coefficient CL and pitching moment coefficient Cm. The engineering-statistical model is far more superior to the statistical model and engineering model because it behaves like the engineering model and has values those are close to the data.1 Therefore, by firstly using the CFD technique to model different well known helicopter fuselage in medium and intermediate weights, including the Bell 412, Bell 212, Bell 214, Bell 205, Agusta A109, Dauphin SA365N and UH-60 (here, for the ease of reference, they are called F1, F2, F3, F4, F5, F6, and F7, respectively), with the aid of a Fluent software, and running it, the aerodynamic coefficients of a helicopter fuselage are obtained for a number of angles of attack. Therefore, the trend of changes in these coefficients can be understood in
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H. Sheikhi, A. Saghaie terms of angle of attack of a helicopter. Since the observed trends are nonlinear, by using nonlinear regression, a function obtains fitting on the data. At this stage, an engineering model is achieved which estimates the aerodynamic coefficients of helicopters of medium and intermediate weights at the desired angles of attack. To achieve the considered engineeringstatistical model, a modified version of the sequential model building strategy of Joseph and Melkote1 was applied (the JM method). The JM method provides a sequential model building strategy, which helps to identify a prediction model that introduces minimal changes to the engineering model. This prevents additional adjustments which could cause the formation of a model that differs from the concept of the physical phenomena.1 Lack of a validation process is a shortcoming of the JM method. Hence, the modified version of the JM method is proposed. Therefore, the resultant model will be one which overcomes the shortcomings and limitations of wind tunnel test and CFD methods, and which can also estimate the aerodynamic coefficients with reasonable engineering accuracy. The rest of this paper is organized as follows. Section 2 presents the data of various helicopter fuselage modeling in Fluent software and engineering modeling method for estimation of the aerodynamic coefficients. In Section 3, the strategy of sequential engineering-statistical model building of Joseph and Melkote1 is introduced and a validation method of the model is proposed. Section 4 presents three numerical examples to demonstrate the application of the proposed models and the effectiveness of engineering-statistical model in estimating the aerodynamic coefficients of a helicopter fuselage. The conclusions and directions of future work are provided in Section 5.
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2. Engineering model develop by combining CFD simulation and regression techniques
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The modern helicopter industry requires design tools which are able to accurately and efficiently predict aerodynamic flow.10,18 In recent years, CFD methods have been increasingly used in the design and analysis of helicopters. This tendency was made by advances in CFD algorithms and the access to more powerful affordable computers. CFD is a virtual simulation technique. A flow can be fully simulated using CFD.12 In this paper, at the first step, by CFD technique, helicopters fuselage of F1, F2, F3, F4, F5, F6 and F7 (Fig. 1), were modeled by Fluent 6.3 software. Simulations were implemented in 3D and steady state, using implicit second order method, density based turbulent flow ((with aduption model) K-x shear stress transport (SST)), the ideal gas model, the Sutherland viscous model, at Mach number Ma = 0.2 and the number of meshes for different angles of attack a = 3.70 106 to 5.43 106. The software output values for aerodynamic coefficients of CD, CL and Cm are presented in Fig. 2. As shown in the figure, aerodynamic coefficients in terms of angle of attack, show nonlinear behavior and this trend is similar in almost all helicopters. Fitting a function to a particular data requires a parametric model that relates the response data to the predictor data with one or more coefficients. The result of the fitting process is an estimate of the model coefficients.23 Through trial and error, and also through the survey of various nonlinear
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Fig. 1 Shape of Bell 412 (F1), Bell 212 (F2), Bell 214 (F3), Bell 205 (F4), Agusta A109 (F5), Dauphin SA365N (F6) and UH-60 (F7) helicopters. 157 149 150 151 152 153 154
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regression models, the best fit to the data is achieved for the Fourier model (Model 1) and fifth-order polynomial model (Model 2), respectively. Hence, each of the Models 1 and 2 can be fitted on the CFD data in order to obtain the engineering model of aerodynamic coefficients CD, CL and Cm. fðaÞ ¼ a0 þ a1 cosðwaÞ þ b1 sinðwaÞ þ a2 cosð2waÞ þ b2 sinð2waÞ ð1Þ
fðaÞ ¼ p1 a5 þ p2 a4 þ p3 a3 þ p4 a2 þ p5 a þ p6
ð2Þ
where a0, a1, a2, b1, b2, p1, p2, p3, p4, p5 and p6 are regression function coefficients and w is the frequency of the signal in the Fourier model. To obtain the coefficient estimates, the least-squares method minimizes the summed square of the residuals. The residual for the ith data point ri is defined as the difference between the observed response value fi and the
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Fig. 2 Values of aerodynamic coefficients CL, CD and Cm at different angles of attack from 90° to 90° for various helicopters.
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fitted response value f^i , and is identified as the error associated with the data.23 ri ¼ fi f^i
ð3Þ
The summed square of residuals is given by n n X X 2 S¼ r2i ¼ ðfi f^i Þ i¼1
ð4Þ
i¼1
where n is the number of data points included in the fit and S is the sum of squares error estimate. Because the least-squares fitting process minimizes the summed square of the residuals, the coefficients are determined by differentiating S with respect to each parameter, and setting the result equal to zero.23 Nonlinear models are more difficult to fit than linear models because it is impossible to estimate their coefficients using simple matrix techniques. Instead, an iterative approach is required which involves these steps: (A) start with an initial estimate for each coefficient. For some nonlinear models, a heuristic approach is provided which produces reasonable starting values. For other models, random values on the interval [0, 1] are provided; (B) produce the fitted curve for the current set of coefficients. The fitted response value fb is given by f^ ¼ fða; bÞ
ð5Þ
and involves calculation of the Jacobian of fða; bÞ, which is defined as a matrix of partial derivatives taken with respect to the coefficients; (C) adjust the coefficients and determine whether the fit is improved, the direction and magnitude of the adjustment depends on the fitting algorithm such as Trust-region and Levenberg–Marquardt algorithms; (D) iterate the process by returning to step 2, until the fit reaches the specified convergence criteria.23 In Fig. 3, Fourier model is fitted on CFD data of the mentioned helicopters, and the general engineering model of aerodynamic coefficients of a helicopter with 95% prediction
Fig. 3 Fourier models fitted on aerodynamic coefficients with 95% prediction bounds.
bounds is also shown. In order to obtain specific engineering model for each helicopter, the proposed regression functions should be fitted on the same CFD data, which results in a higher accuracy of the estimates concerning the considered helicopter. Due to the aforementioned issues, the following general engineering models are suggested for estimating the aerodynamic coefficients of a helicopter fuselage of medium and intermediate weights.
204 205 206 207 208 209 210 211 212 213
CL ¼ 0:2008 0:279 cosð0:01025aÞ 0:2161 sinð0:01025aÞ þ 0:07152 cosð0:0205aÞ þ 0:1893 sinð0:0205aÞ
ð6Þ
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CD ¼ 0:006114 0:005387 cosð0:03422aÞ 0:000515 sinð0:03422aÞ þ 0:0002531 cosð0:06844aÞ þ 0:00006993 sinð0:06844aÞ
ð7Þ
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Cm ¼ 0:002452 þ 0:003519 cosð0:01728aÞ þ 0:001641 sinð0:01728aÞ 0:002447 cosð0:03456aÞ þ 0:005776 sinð0:03456aÞ
ð8Þ
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3. Engineering-statistical model building strategy and validation method
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The engineering models presented in Section 2 are perfect tools for quick estimation of the aerodynamic coefficients of a helicopter fuselage in arbitrary angles of attack. One of the downsides of engineering models is that the model predictions may
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not be accurate enough when compared with data from wind tunnel test.1,8 Strategies to improve the models generally fall into two categories: (A) model refinement and (B) model updating.2,14,16 Model refinement involves changing the physical principles in modeling or the use of other means in the construction of a model. It is more sophisticated and better represents the physics of the problem. Model updating utilizes mathematical concepts, such as calibration parameters and bias correction of model to match predictions of the model with physical observations. Although model refinement is desirable to improve the predictive capability, however, its strategy is often restricted by the available knowledge and computing resources. In contrast, model updating is a cheaper means which can be practical and useful when carried out correctly. Several model updating strategies have already been proposed in literature.1,2,14,16 In this paper, the modified version of the JM method (Fig. 4) was used to update the engineering models of aerodynamic coefficients of a helicopter fuselage. The advantage of this method when compared with other methods is that, it has minimal changes related to the engineering model, and it prevents the additional adjustments which could lead to a model that differs from the concept of physical phenomena.1 One of the flaws of the JM method is the lack of a validation process in the modeling strategy. Therefore, in order to determine the interpolation and extrapolation capability of the model and its validation after the engineering-statistical modeling, the division of the wind tunnel test data into two
Fig. 4
5 groups is proposed. The first set of data is used to build the engineering-statistical model and the second dataset is used to validate the proposed model. Let Ck be the wind tunnel test output for k ¼ L; D; m. Due to the presence of noise (uncontrollable) factors and measurement error, the output is random and is denoted as: Ck ¼ lðaÞ þ e
ð9Þ
where lðaÞ is the mean of Ck at a given a and e Nð0; r2 Þ. The objective is to find the unknown function lðaÞ. What is obtainable in the engineering model is fða; gk Þ function, and the outputs of wind tunnel tests are ða1 ; Ck1 Þ; ða2 ; Ck2 Þ; . . . ; ðan ; Ckn Þ, where g ¼ fgk1 ; gk2 ; . . . ; gkq g0 denotes the unknown calibration parameters. Note that the argument of gk is omitted from the lðaÞ function, because the Ckj ’s in Eq. (9) are generated with gk fixed at its true value gk .1 The first step in sequential model building strategy involves the investigation of the usefulness of an engineering model. For this purpose graphical plots such as a Ckj vs fðaj Þ can be used. A positive correlation suggests the usefulness of the engineering model in prediction. Otherwise, assumptions that engineering model has been derived based on them, should be revised and corrected.1 If the engineering model is known to be useful, then this usefulness can be assessed. Let the predictor at this stage be ^E ðaÞ, which is equal to fðaÞ.1 The model inadedenoted by l quacy can be measured using different indicators. In this paper, due to the presence of positive and negative coefficients,
Sequential model building strategy (a modified version of JM method1).
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the index of mean percentage positive error (Eq. (10)) was suggested as an indicator of the model inadequacy as: MPPE ¼
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299 300 301 302 303
306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 326
n
j¼1
jCkj j
ð10Þ
^Ej l
where ¼ fðaj Þ. In engineering calculations, the mean percentage positive error (MPPE) is often considered to be less than 10%. With the MPPE less than 10%, the engineering model is appropriate. Otherwise, to reduce the error, statistical adjustments are used. For this purpose, a simple location-scale adjustment (Eq. (11)) is applied in the model.1 ^C ðaÞ fðaÞ ¼ b0 þ b1 ðfðaÞ fÞ l ð11Þ Pn fðaj Þ where f ¼ j¼1n . Due to the fact that two constants are used for the adjustments, this method is known as the constant adjustment model.1 Upon obtaining the constant adjustment model, the model inadequacy measure (Eq. (12)) is computed again. MPPE ¼
305
n 1X jj^ lE ðaÞj jCkj jj
n 1X jj^ lC ðaÞj jCkj jj n i¼1 jCkj j
ð12Þ
^0 þ b ^1 ðfðaj Þ fÞ and b ^j denote b esti^Cj ¼ fðaj Þ þ b where l j mated from the data. If MPPE is small enough (less than the maximum error appropriate for engineering), the constant adjustment model is used for prediction. Otherwise, in order to improve the adequacy of the model, more advanced adjustments are applied to the engineering model.1 Let lðaÞ lC ðaÞ ¼ dða; hÞ, where dða; hÞ is used for capturing the inadequacy of the constant adjustment model. This method is called functional adjustment model because the form of the predictor function differs from the engineering model. The form of the function d is obtained by analyzing the residual from the constant adjustment model. One simple method involves the use of a linear model given by P dða; hÞ ¼ m i¼0 bi ui ðaÞ, where ui ’s are known functions in a. The initial value for b is chosen to make Eðdða; hÞÞ ¼ 0. Therefore, it is compulsory to have EðhÞ ¼ 0 in a linear model. Now, lC ðaÞ and h can be estimated from the data. Thus, the functional adjustment predictor is achieved as:1 ^F ðaÞ ¼ l ^C ðaÞ þ dða; ^hÞ l
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4. Numerical examples
331
In this section, to illustrate the application of the proposed models and the effectiveness of the proposed engineeringstatistical model for estimating the aerodynamic coefficients of a helicopter fuselage, three engineering-statistical models were developed to estimate the drag coefficients CD of F2 (Bell 212), F4 (Bell 205) and F7 (UH-60) fuselages at the desired angles of attack. Also, the accuracy of estimation of these models was compared with CFD method. Fig. 5 shows the drag coefficients of F2, F4 and F7 fuselages obtained from wind tunnel test and the values obtained by CFD simulation. If wind tunnel test data are available for limited angles of attack, and there is a need to carry out predictions in points away from data, the predictions will be poor.
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344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362
CDðF2Þ ¼ 0:004811 0:004352 cosð0:03541aÞ 0:001113 sinð0:03541aÞ þ 0:0004172 cosð0:07082aÞ þ 0:0001692 sinð0:07082aÞ
ð14Þ
364 365
CDðF4Þ ¼ 0:001833 þ 0:01452 cosð0:01347aÞ 0:0001867 sinð0:01347aÞ 0:01176 cosð0:02694aÞ þ 0:00008333 sinð0:02694aÞ
ð15Þ
367 368
CDðF7Þ ¼ 0:00006577 þ 0:01302 cosð0:01383aÞ 0:0001406 sinð0:01383aÞ 0:0119 cosð0:02766aÞ þ 0:00003828 sinð0:02766aÞ
ð16Þ
ð13Þ
This two-stage model building procedure prevents additional adjustments on the engineering model and helps to identify an engineering-statistical model with minimal adjustments.1
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In order to build and validate the proposed model, wind tunnel test data were divided into two categories. The first category includes drag coefficient values of F2, F4 and F7 fuselages, respectively at angles of attack given as {15, 10, 6, 0}, {20, 15, 8, 6, 4, 2, 0, 2, 4, 8, 10, 15, 20, 25, 40} and {25, 30, 40, 50, 60, 70}, which were used in the construction of the model, and the second category includes drag coefficient values of F2, F4 and F7 fuselages respectively at angles of attack given as {8, 4, 2, 4, 6, 15}, {90, 50, 10, 6, 30, 90} and {80, 25, 35, 45, 65, 80}, which were used in the validation process.19,24 In this section, based on Section 2 in which the engineering model building method for aerodynamic coefficients of helicopter fuselage was described, firstly, the engineering model to estimate the drag coefficient of F2, F4 and F7 fuselages were achieved (Eqs. (14)–(16)) by fitting Fourier function (Eq. (1)) on CFD data, of which the shape of function and the prediction bounds at 95% was provided in Fig. 6.
Fig. 5 Drag coefficient values of F2, F4 and F7 fuselages related to wind tunnel test and CFD method.
Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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7 than the maximum error appropriate for engineering practice, which is 10%. Therefore, to improve the precision of the engineering model estimation, an engineering-statistical model based on sequential model building strategy is referred. At the first stage, by using the data from wind tunnel test drag coefficient of F2, F4 and F7 fuselages at the aforementioned angles of attack, the constant adjustment can be applied to engineering models. Thus, by using simple linear regression the values of Be0 and Be1 were ^C ðaÞ fðaÞ on fðaÞ f, for l obtained. So, constant adjustment models will be in the form of Eqs. (17)–(19). F2: ^C ðaÞ fðaÞ ¼ 0:0002888 þ 0:02125ðfðaÞ 0:00124Þ l
ð17Þ
F4: l ^ ðaÞ fðaÞ ¼ 0:0004287 þ 0:25778ðfðaÞ 0:00166Þ
ð18Þ
F7: ^ ðaÞ fðaÞ ¼ 0:001151 þ 0:03556ðfðaÞ 0:00689Þ l
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Then, the sequential model building strategy presented in the previous section was used to create engineering-statistical models to estimate the drag coefficient CD of F2, F4 and F7 fuselages. Fig. 7 shows the plot of CD (the drag coefficient value from wind tunnel test) vs fðaÞ of F4 fuselage (the drag coefficient value obtained from the engineering model). From this plot, there exists evidence of a positive correlation of data obtained from wind tunnel test and engineering model. Also, the Pearson correlation coefficient was +1 which indicated a strong positive correlation between them. This subject is similar for F2 and F7 fuselages. Therefore, the proposed engineering model is useful for prediction. Table 1 presents the inadequacy of the engineering model and its comparison with CFD method on the basis of the MPPE. The MPPE of the CFD method and the engineering model are, respectively equal to 19.99476% and 19.48765% for F2 fuselage, 20.62754 and 20.88905% for F4 fuselage and 16.78131% and 16.49641% for F7 fuselage, which are greater
ð19Þ
The inadequacies of the model for constant adjustment models are calculated in Table 1, which are equal to 0.70327%, 1.10185% and 2.63700% for F2, F4 and F7 fuselages respectively, and are far smaller than the inadequacy of the engineering model and the CFD method. The inadequacies of the engineering models obtained at this step, by applying constant adjustments, were greatly reduced and predictions derived from these models were of high accuracy. However, in order to show the full process of sequential model building, the second stage of the model improvement implied that the functional adjustment was applied. To improve the accuracy of the constant adjustment model, the functional adjustment model is: Ck l ðaÞ ¼ dða; hÞ þ e c
ð20Þ
So that: dða; hÞ ¼ b0 þ
m X
392 393 394 395 396 397 398 399 400 401 403
407
bi ui ðaÞ
411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 427 428 429
ð21Þ 431
i¼1
By plotting the residuals of the constant adjustment model of F4 fuselage vs the angles of attack (Fig. 8) and after some trial and error, u1 ðaÞ ¼ a; u2 ðaÞ ¼ a2 ; u3 ðaÞ ¼ a3 and u4 ðaÞ ¼ a4 are selected. Through regression of the residuals ~0 ¼ 1:70 105 ; b ~1 ¼ 1:54 107 ; against a; a2 ; a3 and a4 ; b ~2 ¼ 1:39 107 ; b ~3 ¼ 1:45 109 and b ~4 ¼ 1:37 1010 b values are obtained. As a result:
432 433 434 435 436 437 438 439
^F ðaÞ l ^C ðaÞ ¼ 0:000017 þ 0:0000001537a l 0:0000001386a2 0:000000001448a3 þ 0:0000000001367a4
Fig. 7 Drag coefficient of wind tunnel test vs engineering model in F4 fuselage example.
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408 409
C
371
390
404 405
C
Fig. 6 Engineering model of drag coefficient of F2, F4 and F7 fuselages with 95% prediction bounds.
389
ð22Þ
^ C ðaÞ is given in Eq. (18). This model gives a good fit to where l the data (see Fig. 9). The inadequacies of the model for functional adjustment models of F2, F4 and F7 fuselages are 0.00741%, 0.78936%, and 0.54670% (Table 1), which are smaller than the inadequacies of constant adjustment models, with values of 0.70327%, 1.10185% and 2.63700%. For comparison of the models inadequacy, in addition to the criterion of the MPPE, other criteria such as the mean absolute error (MAE), mean bias error (MBE) and root mean square error (RMSE) can also be used.
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H. Sheikhi, A. Saghaie Table 1 Comparison of accuracy of CFD method with engineering model, constant adjustment model and functional adjustment model in drag coefficient estimation of F2, F4 and F7 fuselages. Fuselage
a (°)
Drag coefficient
Percentage positive error compared with wind tunnel test
Functional adjustment model
Wind tunnel test
CFD
Engineering model
CFD
Engineering model
Constant adjustment model
0.00197 0.00163 0.00137 0.00116
0.00169 0.00132 0.00109 0.00086
0.00168 0.00132 0.00110 0.00088
14.36826 19.17069 20.75066 25.68944
14.85325 18.90565 19.67972 24.51200
0.27138 1.08581 1.15105 0.30483
0.01701 0.00862 0.00398 0.00001
Mean percentage positive error F4 20 0.00260 15 0.00200 8 0.00140 6 0.00130 4 0.00126 2 0.00120 0 0.00119 2 0.00120 4 0.00125 8 0.00140 10 0.00150 15 0.00199 20 0.00260 25 0.00330 40 0.00640
0.00207 0.00158 0.00113 0.00104 0.00099 0.00095 0.00094 0.00095 0.00098 0.00112 0.00122 0.00157 0.00204 0.00264 0.00501
0.00208 0.00158 0.00112 0.00103 0.00098 0.00094 0.00093 0.00094 0.00097 0.00111 0.00122 0.00157 0.00206 0.00267 0.00504
19.99476 20.38462 21.00000 19.28571 20.00000 21.42857 20.83333 21.00840 20.83333 21.60000 20.00000 18.66667 21.10553 21.54820 20.00000 21.71875
19.48765 20.14208 20.82425 20.20714 20.41834 22.60040 21.72179 22.10084 21.81203 22.15596 20.52990 18.88312 20.89894 20.68528 19.10238 21.25338
0.70327 0.47722 0.37074 0.42412 0.16328 2.57915 1.47059 1.94675 1.58410 2.01960 0.01816 2.08530 0.46446 0.20601 1.77782 0.94046
0.00741 0.16730 0.60489 1.00992 1.05390 1.44461 0.12460 0.51818 0.18880 0.79243 0.67371 2.39170 0.95924 1.17043 0.71694 0.02377
Mean percentage positive error F7 25 0.00415 30 0.00495 40 0.00670 50 0.00845 60 0.01080 70 0.01317
0.00295 0.00374 0.00560 0.00769 0.00976 0.01154
0.00299 0.00379 0.00564 0.00769 0.00971 0.01149
20.62754 28.81335 24.40353 16.41919 9.03179 9.65457 12.36543
20.88905 27.85359 23.47618 15.82686 8.99103 10.08316 12.74764
1.10185 3.45300 2.45118 0.68315 4.96961 1.50295 2.76208
0.78936 0.23681 0.03893 0.86761 1.27112 0.70813 0.15759
16.78131
16.49641
2.63700
0.54670
F2
15 10 6 0
Mean percentage positive error
Fig. 8 Plotting residuals of constant adjustment model vs angles of attack of F4 fuselage and fitting a function on them. Fig. 9 Comparison of engineering model, constant adjustment model and functional adjustment model trends of F4 fuselage. 452 453 454 455 456 457 458 459 460 461 462 463 464 465
In Table 2, CFD method, engineering model, the constant adjustment and functional adjustment models of F4 fuselage have been compared according to the aforementioned criteria. From the table, functional adjustment model based on all criteria was better than the other models. However, according to Fig. 8, it is evident that although the functional adjustment model in the observed points had a better fit to the data and had a higher accuracy than the constant adjustment model, the function form changed and demonstrated a different trend compared to the engineering model. In particular, the more the increase in the angle of attack the more the difference. According to the engineering model, an increase in the angle of attack of a fuselage resulted in a reduction in the slope of the drag coefficient curve of the fuselage, however this was not the case
in the functional adjustment model. Therefore, in choosing the appropriate model for predicting the aerodynamic coefficient of a fuselage, in addition to the consideration of the criteria for measuring the model inadequacy, it is better to compare the prediction model function with the engineering model. As a general conclusion, it can be said that a prediction model having appropriate model adequacy measuring criteria and which follows the same trend as the engineering model, is more appropriate. Also, the second set of wind tunnel test data can be used as a tool to select a more suitable model for estimating the
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Table 2 Comparison of estimation accuracy of CFD method, engineering model, constant adjustment model and functional adjustment model in drag coefficient of F4 fuselage. Method
Root mean square error (RMSE)
Mean bias error (MBE)
Mean absolute error (MAE)
Mean percentage positive error (MPPE)
CFD Engineering model Constant adjustment model Functional adjustment model
0.000512898 0.000505589 0.000027458 0.000016845
0.000424022 0.000423378 0.000005940 0.000000001
0.000424022 0.000423378 0.000020696 0.000013489
20.62754 20.88905 1.10185 0.78936
Table 3 Comparison of CFD method, engineering model, constant adjustment and functional adjustment models on the basis of percentage positive error for data within the range of date modeling. Fuselage
a (°)
Percentage positive error compared with wind tunnel test CFD
Engineering model
Constant adjustment model
Functional adjustment model
F2
8 4 2 Mean
20.99813 22.61703 24.17254 22.59590
20.09995 21.50270 22.94808 21.51691
1.00224 0.49579 0.24508 0.58104
0.87243 1.59025 1.66676 1.37648
F4
10 6 30 Mean
18.00000 20.00000 20.23810 19.41270
18.49726 20.67400 19.51112 19.56079
2.57063 0.15829 1.25805 1.32899
2.66667 0.76923 0.47619 1.30403
F7
35 45 65 Mean
18.44663 12.44325 11.93205 14.27398
18.34041 12.30344 11.95903 14.20096
0.37818 2.75526 1.33386 1.48910
0.77236 0.23321 1.43836 0.81464
Table 4 Comparison of CFD method, engineering model, constant adjustment and functional adjustment models on the basis of percentage positive error for data outside the range of data modeling. Fuselage
477 478 479 480 481 482 483 484 485 486
a (°)
Percentage positive error compared with wind tunnel test CFD
Engineering model
Constant adjustment model
Functional adjustment model
F2
4 6 15 Mean
24.33812 20.85488 13.50930 19.56743
23.46728 20.22537 14.14251 19.27838
3.46876 8.58005 13.50930 8.51937
18.89592 40.21373 195.30380 84.80448
F4
90 50 90 Mean
19.52818 19.09197 19.51872 19.37962
19.74987 19.18825 19.74582 19.56131
0.94273 1.65355 0.94793 1.181403
59.30537 9.77881 46.52406 38.53608
F7
80 25 80 Mean
11.30505 22.56366 12.51696 15.46189
10.63788 21.23783 11.74691 14.54087
1.22484 5.03590 2.35921 2.87331
302.92412 62.66847 10.22403 125.27220
aerodynamic coefficients of a fuselage and also to validate the proposed models. The interpolation and extrapolation of models can be determined using a validation process. The MPPE of F2, F4, and F7 fuselages for a number of angles of attack inside and outside the data range, such that modeling is complete on their basis, are provided in Tables 3 and 4, respectively. As evident in Table 4, for the data inside the range, the predictability for both constant adjustment models and functional adjustment models were appropriate, and the inadequacies of both models on the basis of the MPPE
are close to each other. Consequently, constant adjustment and functional adjustment models offered a good interpolation capability. According to Table 4, for data outside the range, the functional adjustment models had a lot of errors and it can be concluded that these models exhibit poor extrapolation ability. In contrast, the inadequacies of the constant adjustment models were small and this issue indicated the appropriateness of the extrapolation ability for the constant adjustment model. Considering the aforementioned points, to estimate the drag coefficients of F2, F4, and F7 fuselages, constant
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adjustment model is proposed as the better model, because in addition to having appropriate engineering precision and similar trend with the engineering model, it has good interpolation and extrapolation functionality.
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5. Conclusion
497 498 499
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Models which describe the performance of a physical phenomenon are critical to design, predict, control and optimize. The analysis of fluid flow around a helicopter, in a macroscopic scale, so as to calculate the aerodynamic coefficients of helicopter fuselage, such as the drag CD, lift CL and pitching moment Cm, is one of the key issues in the design of helicopters. To obtain fuselage aerodynamic coefficients, wind tunnel test and computational fluid dynamics simulation are used. Wind tunnel tests are very costly, time consuming and limited. Although the CFD method is less expensive, owing to the limitation of numerical solution, the number of nodes and the used solution, it is often not accurate and also has its own cost and time. In addition, carrying out a wind tunnel test at certain angles of attack is necessary for the validation of CFD results. This paper aims to develop models which predicted the aerodynamic coefficients of helicopter fuselage at any desired angle of attack with appropriate engineering precision. This objective has been realized by combining and integrating computational fluid dynamics simulation and physical wind tunnel test. With such models, the design process can be accelerated, a tool is provided to aid designers with the prediction and optimization, and also there is significant reduction in the design costs. Firstly, by modeling the helicopter fuselage in the intermediate and medium weights (the Bell 412, Bell 212, Bell 214, Bell 205, Agusta A109, Dauphin SA365N and UH-60), and using CFD simulation with Fluent software, aerodynamic coefficients behavior of fuselage was analyzed. Then, using the CFD results and nonlinear regression, engineering models were developed which were able to predict the aerodynamic coefficients of the fuselage and demonstrate their trend. The JM method was modified and a validation step was included in order to eliminate its shortcomings. The proposed engineering-statistical models were obtained by combining the engineering models, modified version of the JM method, and a few numbers of wind tunnel test data. These models were able to estimate the aerodynamic coefficients of the fuselage at any desired angle of attack with accuracy much higher than CFD and engineering models, in the shortest possible time. Three numerical examples were provided to illustrate the model building strategy and demonstrate the effectiveness of the proposed engineering-statistical models. So, the engineering models were obtained to estimate the drag coefficients of the F2 (Bell 212), F4 (Bell 205), and F7 (UH-60) fuselages by fitting Fourier model on CFD data. The engineeringstatistical models (constant adjustment and functional adjustment models) were developed based on the modified JM method. The accuracy of the proposed models of the F2, F4, and F7 fuselages were compared with the results of wind tunnel tests, CFD simulations and engineering models, based on MPPE. The inadequacies of the model for constant adjustment models of F2, F4 and F7 fuselages were equal to 0.70327%, 1.10185% and 2.637%, and were far smaller than the inade-
H. Sheikhi, A. Saghaie quacies of CFD simulations (19.99476%, 20.62754%, and 16.78131%) and engineering models (19.48765%, 20.88905%, and 16.49641%) which show the appropriate engineering accuracy (a value usually less than 10%). The inadequacies of the model for functional adjustment models of F2, F4 and F7 fuselages were 0.00741%, 0.78936%, and 0.54670%, which were smaller than the inadequacies of constant adjustment models. Also, the MAE, MBE, and RMSE criteria were used for comparison of the model inadequacies for engineering model, CFD simulation, constant adjustment and functional adjustment models of F4 fuselage. The results of the evaluation were similar according to all criteria. The results show the accuracy and efficiency of the introduced engineering-statistical model. It was also observed that, in addition to considering the criteria of the model inadequacy, the function form and trend were also important and should have a similar trend to the engineering model. Regarding the drag coefficient plots, it was observed that although the functional adjustment model on the basis of criteria of the model inadequacy was preferable to the constant adjustment model, due to its different trend when compared with the engineering model, the constant adjustment model was chosen as better engineering-statistical model to predict the drag coefficient of the fuselages. This was proved by the evaluation of the interpolation and extrapolation ability of the proposed models. The interpolation capability of both constant adjustment models and functional adjustment models of F2, F4, and F7 fuselages were appropriate and the inadequacies of both models on the basis of the MPPE were less by 1.5%. The functional adjustment models exhibited poor extrapolation capabilities and had a lot of errors. The inadequacies of the functional adjustment models of the F2, F4 and F7 fuselages were 84.80448%, 38.53608%, and 125.27220%. In contrast, the extrapolation capability of the constant adjustment models was appropriate and contained little errors as compared to wind tunnel test data (8.51937%, 1.181403%, and 2.87331%, respectively). Consequently, the superiority of the constant adjustment model was demonstrated. In future researches, the development of the engineering model and engineering-statistical model for aerodynamic coefficients of the fuselage on the basis of a helicopter yaw angle is suggested. Also, the development of such models for attack helicopters will be very valuable and important. On the other hand, the described model building strategy can also be used to model other physical phenomena.
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Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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Please cite this article in press as: Sheikhi H, Saghaie A Developing an engineering-statistical model for estimating aerodynamic coefficients of helicopter fuselage, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.12.015
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