Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 854308, 31 pages

http://dx.doi.org/10.1155/2015/854308

## An Extended Assessment of Fluid Flow Models for the Prediction of Two-Dimensional Steady-State Airfoil Aerodynamics

^{1}Wind Energy Group, Department of Physics, Instituto Tecnológico y de Estudios Superiores de Monterrey, Eugenio Garza Sada 2501 Sur, 64849 Monterrey, NL, Mexico^{2}Solar Energy and Thermosciences Group, Department of Mechanical Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, Eugenio Garza Sada 2501 Sur, 64849 Monterrey, NL, Mexico^{3}Department of Mechanical Engineering, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA^{4}School of Mechanical and Electrical Engineering, Universidad Autónoma de Nuevo León, Avenida Universidad s/n, Ciudad Universitaria, 66451 San Nicolás de los Garza, NL, Mexico

Received 30 August 2014; Revised 13 January 2015; Accepted 13 January 2015

Academic Editor: Shaofan Li

Copyright © 2015 José F. Herbert-Acero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work presents the analysis, application, and comparison of thirteen fluid flow models in the prediction of two-dimensional airfoil aerodynamics, considering laminar and turbulent subsonic inflow conditions. Diverse sensitivity analyses of different free parameters (e.g., the domain topology and its discretization, the flow model, and the solution method together with its convergence mechanisms) revealed important effects on the simulations’ outcomes. The NACA 4412 airfoil was considered throughout the work and the computational predictions were compared with experiments conducted under a wide range of Reynolds numbers () and angles-of-attack (). Improvements both in modeling accuracy and processing time were achieved by considering the RS LP-S and the Transition SST turbulence models, and by considering finite volume-based solution methods with preconditioned systems, respectively. The RS LP-S model provided the best lift force predictions due to the adequate modeling of the micro and macro anisotropic turbulence at the airfoil’s surface and at the nearby flow field, which in turn allowed the adequate prediction of stall conditions. The Transition-SST model provided the best drag force predictions due to adequate modeling of the laminar-to-turbulent flow transition and the surface shear stresses. Conclusions, recommendations, and a comprehensive research agenda are presented based on validated computational results.

#### 1. Introduction

The measurement and prediction of aerodynamic forces on two-dimensional airfoils is a problem that has been widely investigated since the early 1930s and its development has produced important improvements in the aerospace, automotive, and wind-based sciences, among others [1–4]. Prior to the experimental assessment of aerodynamic forces, the state-of-the-art procedures [1, 2] impose major prerequisites such as the detailed manufacture of the tested airfoil [5], the setup of expensive wind tunnel facilities [6], and the use of special sensing equipment to characterize both the aerodynamic behavior of the airfoil and the disturbances it produces on the free stream (e.g., streamlines, flow attachment/detachment, flow compression dynamics, and wake aerodynamics). In addition, correction factors [1, 4, 6–10] are often applied to account for nonideal inflow conditions (e.g., buoyancy, solid blockage, wake blockage, or streamline curvature corrections). These prerequisites and issues, together with the overall propagation of uncertainty, turn the experimentation procedures into daunting tasks. A useful, inexpensive, and faster alternative to perform aerodynamic characterizations involves the implementation of computational methods for the theoretical estimation of aerodynamic forces, which are predicted through the numerical solution of the governing equations of fluid mechanics. This approach is formally known as Computational Fluid Dynamics (CFD).

It is generally acknowledged that there is no universal model/method that ultimately describes the complete characteristics of a fluid flow and its interactions with objects with reasonable accuracy while employing a reasonable amount of computational resources. This modeling problem becomes more complex as more physical phenomena are considered (e.g., if turbulent, compressible, and multiphase flows are considered, among other relevant conditions). Therefore, depending on the case study conditions and the assumptions made, different CFD-based approaches with different levels of sophistication can be employed. Some of the most important fluid flow modeling techniques are briefly presented next: (1) the potential flow theory [11], considered the coarsest modeling approach, does not account for turbulence or vorticity effects in its basic formulations. Nonetheless, recent advances in potential flow theory and Boundary Layer (BL) modeling have led to the development of the vortex modeling approach [12], in which viscous and vorticity effects have been successfully integrated into the fluid flow modeling, resulting in improved aerodynamic predictions. (2) The turbulence modeling approach, considered as the industry standard approach for design purposes, is much more complex and computationally demanding since both small-scale and large-scale turbulence effects are modeled by solving either the Reynolds-Averaged Navier-Stokes (RANS) or the Favre-Averaged Navier-Stokes (FANS) equations complemented with turbulence models [13, 14]. (3) Advanced techniques that solve large-scale turbulence effects and model only the small-scale turbulence effects are based on large eddy simulations (LES) complemented with subgrid-scale models [15]. Finally, (4) more advanced techniques based on Direct Numerical Simulations (DNS) [16], which are typically implemented for theoretical research purposes, have the ability to solve the whole range of spatial and temporal scales of the turbulence and predict all the effects and interactions between fluids and solids at the cost of an extraordinary large amount of computational resources.

Currently, there is no consensus regarding the minimum level of flow modeling required to accurately predict airfoil aerodynamics, while considering different flow regimes/states (e.g., laminar, transitional, and turbulent flows) and different inflow conditions (e.g., Reynolds numbers and angles of attack). Despite this issue, in works mainly focused on airfoil shape development and optimization [3, 17] it is often considered sufficient to solve the compressible Euler equations [18] or solve a set of potential flow equations that are coupled with integral BL formulations, some of which are implemented in popular public domain and commercial codes such as XFOIL [19, 20] or VisualFoil [21], respectively, to estimate aerodynamic forces under subsonic, transonic, or supersonic flows. The selection of these approaches, however, was mostly based on the convenient amount of computational resources they require rather than their performance for predicting aerodynamic forces.

Only a limited number of research works have attempted to determine the accuracy of different fluid flow modeling techniques for predicting the aerodynamic behavior of two-dimensional airfoils undergoing different inflow conditions. Wolfe and Ochs [22] presented a laminar/turbulent flow analysis, considering an airflow at a Reynolds number (Re) of and the range of angles of attack (, AOA) , to determine the asymmetric S809 airfoil aerodynamics. They employed the commercial code CFD-ACE, which solves the FANS equations coupled with the Standard - turbulence model. Wolfe and Ochs contrasted the computed pressure coefficient distributions and the computed aerodynamic coefficients with experimental measurements obtained under laminar inflow conditions and observed a drag force overprediction when fully turbulent computations were considered. To address the issue of simulating transitional flows, they developed a mixed laminar/turbulent calculation method, in which the computational domain was split into one laminar and one turbulent region at a guessed transition point, which in turn improved the drag force predictions. They concluded that more research on both the determination of the laminar-to-turbulent flow transition point [23] and the accurate modeling of turbulent effects under stall conditions was necessary to reduce observed discrepancies.

Some of the discrepancies observed by Wolfe and Ochs are related to BL modeling issues. In their work, the modeled dimensionless wall distance ( [24], which is a parameter typically used to determine what sublayers of the BL are solved) was of the order of , thus limiting the probed sample volume of the BL to the logarithmic and outer layers. Therefore, the modeling of the near-wall flow dynamics and the calculation of wall shear stresses depended on the use of wall functions. The standard wall functions [24, 25], such as the ones used by Wolfe and Ochs, have proven to be inaccurate while modeling BLs subject to large adverse pressure gradients (like the ones encountered on airfoils undergoing inflow conditions at large AOA), which in turn induce flow detachment conditions. The appropriate description of the complex BL, from which aerodynamic forces are calculated, requires the accurate modeling of the viscous, turbulent, and rotational properties of the flow found within the airfoil’s vicinity. Therefore, the better the airfoil’s BL is modeled, with special emphasis on the viscous sublayer (or laminar sublayer, which is located at the inner part of the BL), the better the agreement between the computed and the measured aerodynamic forces and the observed flow dynamics. In order to model the viscous sublayer, a solved is required over the entire airfoil surface [24, 26]. The interested reader is directed to [1, 2, 27] for additional and comprehensive descriptions of the physics of airfoil aerodynamics.

Eleni et al. [28] presented a work focused on determining which turbulence model, among the Spalart-Allmaras, the Realizable -, and the SST - turbulence models, was the best performer for predicting the symmetric four-digit National Advisory Committee for Aeronautics (NACA) 0012 airfoil aerodynamics, while considering an airflow at a and for the range of AOA . Similar to the study performed by Wolfe and Ochs, Eleni et al. found drag overpredictions when comparing fully turbulent computations with experimental measurements that considered laminar inflow conditions. To improve the drag force predictions, they conducted mixed laminar/turbulent simulations, similar to the procedure proposed by Wolfe and Ochs. However, in order to determine the laminar-to-turbulent flow transition point, Eleni et al. developed an iterative method that depends on already measured data. In addition, they conducted simulations, considering five different Reynolds numbers (), at a zero AOA by assuming both fully turbulent and mixed laminar/turbulent conditions. In such simulations, the laminar-to-turbulent flow transition took place at the same axial point at both the top and the bottom surfaces of the airfoil due to its symmetry. The resultant drag coefficients were compared satisfactorily with the experimental measurements. They concluded that for both fully turbulent and mixed laminar/turbulent flows the best performer turbulence model was the SST -.

Kumar et al. [29] presented a work in which the asymmetric NACA 4412 airfoil was simulated in a turbulent airflow, at a and for the range of AOA , while considering the Spalart-Allmaras and the Standard - turbulence models. The computed predictions were contrasted with the experimental data (at a ) provided by Abbott and von Doenhoff [1]. Kumar et al. reached the same conclusions as Wolfe and Ochs about the importance of the accurate determination of the laminar-to-turbulent flow transition points at both the upper and the lower surfaces of the airfoil, since a notable overprediction of the drag coefficient was observed by simulating a fully turbulent flow over the airfoil’s vicinity. They concluded that the Standard - model was the best performer turbulence model.

Villalpando et al. [30] presented an assessment of the ability of different turbulence models to predict the asymmetric NACA 63-415 airfoil aerodynamics, while considering an airflow at a , the range of AOA , and an inlet turbulence intensity (TI) of 1%. The contrasted turbulence models were the Spalart-Allmaras, the RNG -, the SST -, and the Reynolds Stress Low-Re S-. The computational results were compared with experimental measurements performed under laminar inflow conditions, which were provided by the Risø National Laboratory for Sustainable Energy. Unlike the observations of the previously described works, the contrasted turbulence models accurately predicted the experimental drag coefficient for . Only at large AOA (i.e., under stall conditions) all the tested models overpredicted both the lift and drag coefficients. Villalpando et al. did not report the numerical convergence criteria of their finite-volume-based solution method but reported oscillatory convergence, an observation pointing to numerical instabilities, for moderately large AOA (e.g., ). In such cases, an averaging procedure was performed to estimate the aerodynamic forces. The approach, however, is believed to be inadequate since such convergence issues are often related to the quality of the domain discretization and/or to the numerical approach used to solve the problem. They concluded that the SST - model was the best performer model, with the Reynolds Stress Low-Re S- being the worst performer model.

Other studies related to the prediction of airfoil aerodynamics (e.g., [31–34]), in which laminar/transitional flows are involved, may be affected by the already noted overprediction of the drag coefficient, which is an issue directly related to the lacking ability of full-turbulent models to simulate the laminar-to-turbulent flow transition. In order to deal with this issue, improved turbulence models, named Transition-based models [24, 26, 35], have been developed to accurately estimate the laminar-to-turbulent flow transition zones and solve the flow as turbulent at downstream locations. So far, improved drag predictions have been obtained as described by Yuhong and Congming [36], Yao et al. [37], Aranake et al. [38], and Khayatzadeh and Nadarajah [26].

Yuhong and Congming [36] presented a work in which the asymmetric S814 airfoil aerodynamics was predicted using the Transition SST model, considering an airflow at a and for the range of AOA . They concluded that the Transition SST model predicts the lift and drag coefficients more accurately than full-turbulent models only in prestall conditions. Under stall conditions, both tested turbulence models (the Transition SST and the SST -) failed to predict the aerodynamic behavior of the tested airfoil.

Yao et al. [37] performed a similar work by predicting the symmetric NACA 0018 airfoil aerodynamics for an airflow at a , a relative Mach number of 0.023, and for the range of AOA . The contrasted turbulence models were the Standard -, the RNG -, the four-equation Transition SST model, and a five-equation Reynolds Stress model. For all the considered inflow conditions, all turbulence models overpredicted the drag coefficient, as concluded in previous works. The magnitude of the lift coefficient was systematically underpredicted for both large negative AOA and large positive AOA. The Reynolds Stress model was the top performer in that study.

Aranake et al. [38] presented an evaluation of the RANS-based Transition model, which was coupled with the Spalart-Allmaras turbulence model (named the Transition model), for predicting the asymmetric S827 and the S809 airfoil aerodynamics. For the S827 airfoil aerodynamics prediction, an airflow at a , a Mach number of 0.1, an inlet , and the range of AOA were considered. The same conditions, but considering a , were assumed for the evaluation of the S809 airfoil aerodynamics. After comparing fully turbulent computations performed with the Spalart-Allmaras model, laminar/transitional computations performed with the Transition model, and experimental measurements conducted under laminar inflow conditions, Aranake et al. found improved lift, drag, and pressure coefficients predictions under both prestall and stall conditions. Nevertheless, significant discrepancies were found at intermediary AOA, where a characteristic double-stall condition arises for both tested airfoils. They concluded that in situations for which the flow remains completely attached or massively separated, the Transition model qualitatively and quantitatively exhibits the same behavior as the fully turbulent SA model. But when moderate flow separation occurs, the coupled model significantly improves the quality of the predictions.

Khayatzadeh and Nadarajah [26] presented a comprehensive evaluation of the RANS-based Transition model, which was coupled with the SST - turbulence model, for predicting the asymmetric NLF(1)-0416 and the S809 airfoil aerodynamics. They provided modifications for both the SST - and the Transition models in order to allow an appropriate gradual activation of the SST - model along the BL upon the onset of transition from laminar-to-turbulent flow conditions. The proposed modifications improved the aerodynamic predictions of the original models, which were compared with experimental measurements performed under laminar inflow conditions. The improved model predicted the laminar-to-turbulent transition locations, the skin friction coefficient distribution, the pressure coefficient distribution, and the lift, drag, and moment coefficients with reasonable accuracy for different inflow conditions, which included the ranges and . Khayatzadeh and Nadarajah highlighted the importance of using an adequate domain discretization (considering average values below 1) and the importance of developing robust and well-calibrated relations for the prediction of the laminar-to-turbulent flow transition, which were shown to be very sensitive to the considered inflow condition.

The evidence presented in the above-described works indicates that by employing Transition-based models, or by performing mixed laminar/turbulent procedures, improved predictions can be obtained when comparing computational results with experiments conducted under laminar/transitional inflow conditions. However, for the mixed laminar/turbulent procedures proposed in [22, 28], three unsolved issues have been identified: (1) no insights of the boundary conditions used between the split regions were provided; thus, the procedure may have resulted in an unphysical representation of the flow field in the vicinity of that boundary, (2) the determination of the transition point was guessed or computed from already measured data, and (3) the laminar/turbulent domain segmentation must be changed for different scenarios, resulting in a complex task when considering asymmetric airfoils undergoing nonhomogeneous inflow conditions. Moreover, grid-independence tests must be performed for each considered scenario. If a mixed laminar/turbulent procedure is to be implemented, instead of guessing the transition location, the authors of the present work recommend computing the airfoil’s top transition point () and bottom transition point () by first using specialized software such as XFOIL that incorporates the method for transition prediction [39], which has proven to be accurate [40], thus avoiding possible metastability issues. It should be noted that one of the key advantages of employing Transition-based turbulence models is that they implicitly solve the three above-described issues, at the expense of increased computational requirements and the incorporation of complex formulations for the accurate prediction of the laminar-to-turbulent flow transition [24, 26, 35].

As it becomes apparent from this literature review, most of the research works have focused on validating the effectiveness of a limited number of fluid flow models for predicting two-dimensional airfoil aerodynamics, while considering a limited range of inflow conditions. None of them provided a justification for the selection of specific full-turbulence models (e.g., while coupling with Transition models [26, 38]) during the validation process and their scope was limited mainly due to the availability of experimental measurements. Moreover, only a few works have investigated the sensitivity of the simulations’ outcomes to the different free parameters, and the majority of the works have not provided detailed descriptions of the numerical solution process and its convergence mechanisms. As consequence, only limited conclusions can be drawn about the performance of the tested models due to the impossibility of extrapolating the findings to situations different from the original case studies. It should be noted that a wide range of conflicting findings arise in the literature with regard to which models, or combination of models, are the most effective in terms of solution quality and computational efficiency. Thus, in order to overcome these issues, this work reports on an extended assessment of the accuracy of thirteen state-of-the-art fluid flow models applied to the problem of quantifying aerodynamic forces on two-dimensional airfoils for a wide range of inflow conditions. The outcomes of the assessment allowed to (1) identify the best performing fluid flow models, (2) understand the modeling pitfalls for different conditions (e.g., stall conditions), (3) determine which free parameters are the most important during the computational evaluation, (4) identify strategies for improved numerical convergence, and (5) identify key research needs and provide a comprehensive research agenda.

A full comparison of two different CFD-based methodologies was performed; the first one is based on the potential flow modeling approach [11, 41] complemented with an transition model and a set of integral BL formulations, which are implemented in the XFOIL 6.96 software. The second methodology is based on the turbulence modeling approach, where twelve different turbulence models were tested. Transition-based turbulence models were adopted when laminar/transitional inflow conditions were considered. For the remaining turbulence models, the free-stream flow was considered to be turbulent (i.e., no attempts were made to separate the laminar and the turbulent regions within the computational domain) and, therefore, an overprediction of the drag coefficient was expected when comparing the computational outputs with experimental measurements that were typically conducted under laminar inflow conditions. The asymmetric four-digit NACA 4412 airfoil was considered as a test case. The rationale for selecting this airfoil is twofold: on the one hand, abundant information of the NACA 4412 airfoil can be found in the literature [1, 42–45], containing experimental measurements for up to 23 different Reynolds numbers ranging from . On the other hand, the asymmetric features of the NACA 4412 airfoil pose a reasonable challenge to the flow modeling techniques.

The remainder of this work is structured as follows: Section 2 presents the different setups of the computational study. Section 3 presents the computational results, their validation, and the corresponding physical interpretations. Finally, Section 4 presents the overall conclusions and outlines future research.

#### 2. Computational Study

The NACA 4412 is an airfoil that has a maximum camber of 4%, which is located at 40% from the leading edge, and has a maximum thickness of 12%, all percentages measured with respect to the airfoil’s chord length. The NACA 4412 ordinates were obtained from [20, 45]. The airfoil’s trailing edge was smoothed, with the aid of the XFOIL 6.96 software, on the last 5% of the chord to produce a sharp closed profile since the original ordinates had an open section at the tip of the trailing edge (i.e., blunt shape). This is a common issue for analytically developed airfoils and affects all the NACA four-digit airfoils.

The present work reproduced the experimental tests reported in [1, 10, 42, 43, 45] in which transonic and supersonic flows were avoided since the experiments were performed in the Langley two-dimensional low-turbulence pressure tunnel [6], which compressed the airflow up to an absolute pressure of 4 atmospheres in order to increase the air density. Therefore, the simulated airflow was considered to be incompressible and standard sea level air properties ( (kg/m^{3}), (kg/m-s), (K), and (kPa)) were considered in all case studies. Furthermore, the earth’s gravitational force was neglected.

The computer used to perform the computational assessment was a customized machine containing a water-cooled Intel Core i7-2600k processor operating at 5.2 GHz, 16 GB of memory DDR3 @1600 MHz, a Corsair Force GT solid state drive operating at 6 Gb/s, and an ASUS Maximus IV GENE-Z motherboard based on the Intel Z68 chipset. Double-precision parallelized simulations were performed for all case studies using an Intel 64 Message Passing Interface (MPI-2).

##### 2.1. Computation of the NACA 4412 Airfoil Aerodynamics Using XFOIL 6.96

XFOIL [19, 20] is Fortran-based software, created by Drela in 1986 at the Massachusetts Institute of Technology (MIT) during the MIT Daedalus project, for the analysis of the subsonic aerodynamics of isolated airfoils. The XFOIL computations are based on the panel method [11, 19, 41, 46], which is combined with an laminar-to-turbulent transition method [35, 39] and a set of integral boundary layer formulations. Given an initial inflow condition, the flow velocity distribution around the airfoil is computed from the panel method while accounting for viscous forces and the induced vorticity from the airfoil surface. The resultant boundary layer and wake are interacted with a surface transpiration model. The resultant flow field is incorporated into the fluid mechanics viscous equations, yielding a nonlinear elliptic system of equations which is solved by a Newton-Raphson algorithm, resulting in both a complete pressure and velocity distributions in the airfoil vicinity. The lift force coefficient () is calculated by direct surface pressure integration, as viscous contributions to the lift force are often neglected, and the pressure coefficient () is calculated using the Karman-Tsien compressibility correction. The drag force coefficient () is determined from the wake momentum thickness at a location far downstream of the airfoil and calculated with use of the Squire-Young formulation. The methods, corrections, and the boundary layer formulation used in XFOIL are extensively described in [19, 46].

For panel-based methods, the first set of free parameters is related to the discretization of the airfoil’s geometry. In all the XFOIL-based simulations, a constant number of panel nodes (160) were considered. The panel nodes were concentrated towards both the leading and the trailing edges of the airfoil, as shown in Figure 1, with the aim of increasing the density of nodes in these sensitive zones. The trailing edge to the leading edge panel density ratio was of 0.15. The panel density ratio at the leading edge was 0.2 and the maximum panel angle was 7.87°. The second set of free parameters is related to the definition of the specific inflow conditions to be studied. In XFOIL, the laminar-to-turbulent flow transition begins when one of the following two scenarios occur: (1) a free transition occurs when the criterion is met or (2) a forced transition occurs when a trip or the airfoil’s trailing edge is encountered. For this Transition-based model, a free parameter named the critical amplification factor (), which affects the laminar-to-turbulent flow transition location, must be defined. A suitable value of this parameter depends on the ambient disturbance, or Turbulence Intensity (TI), in which the airfoil operates and mimics the effect of such disturbances on the flow state transition. A value of = 2.6232 was set to simulate a free-stream TI of 1%, which was the same flow condition of the experimental measurements obtained from the literature. Finally, it was observed that the XFOIL computations were usually very fast, in the order of milliseconds for one simulated case, and consume almost negligible computational resources.