# Precise drag prediction of airfoil flows by a new algebraic model

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## Abstract

We report the results of accurate prediction of lift (\(C_L\)) and drag (\(C_D\)) coefficients of two typical airfoil flows (NACA0012 and RAE2822) by a new algebraic turbulence model, in which the eddy viscosity is specified by a stress length (SL) function predicted by structural ensemble dynamics (SED) theory. Unprecedented accuracy of the prediction of \(C_D\) with error of a few counts (one count is \(10^{-4}\)) and of \(C_L\) with error under 1%-2% are uniformly obtained for varying angles of attack (AoA), indicating an order of magnitude improvement of drag prediction accuracy compared to currently used models (typically around 20 to 30 counts). More interestingly, the SED-SL model is distinguished with fewer parameters of clear physical meaning, which quantify underlying turbulent boundary layer (TBL) with a universal multi-layer structure, and is thus promising to be more easily generalizable to complex TBL. The use of the new model for the calibration of flow condition in experiment and the extraction of flow physics from numerical simulation data of aeronautic flows are discussed.

## Keywords

Lift and drag coefficients Airfoil Flows Stress Length Structural ensemble dynamics## 1 Introduction

Drag prediction continues to be the main challenge for computational fluid dynamics (CFD) simulation of aerodynamic applications, which has inspired such major international effort as drag prediction workshop (DPW) over more than a decade. Among all factors affecting the drag prediction accuracy, turbulence model in the Reynolds-averaged Navier-Stokes (RANS) framework plays a central role [1], due to the lack of a complete turbulent boundary layer (TBL) theory based on physical understanding of wall turbulence. Current models typically involve the modeling of correlation terms by dimensional argument, which is too simplistic and then inevitably introduces numerous coefficients/functions having no physical interpretation. Consequently, the determination of the parameters rely heavily on the experience of engineers, so that high-quality CFD becomes rather “artistic” and the generalization of a working program to new flows (with new features) becomes increasingly a formidable task (since model formulation becomes increasingly more complex). Theoretically speaking, the bottleneck lies on the lack of a understanding of wall turbulence, in terms of its universal structure. The mixing length theory of Prandtl [2] and the log law of von Karman [3], and later the similarity hypothesis of Townsend, represent the milestones in this pursuit, but unfortunately they have not yet been developed to form a complete TBL theory, able to guide the development of turbulence model. The present work explores such a path, namely, developing accurate algebraic model based on well-founded physics theory of wall turbulence.

Recently proposed structural ensemble dynamics (SED) theory [4, 5, 6] is a symmetry-based approach, which expresses the (universal) wall constraint on turbulence eddies in terms of a (Lie-group) dilation invariance of a stress length (SL), \(\ell _{12}\), quantifying eddies’ scales. The theory predicts a multi-layer analytic form (see below) for \(\ell _{12}\) [4, 5, 6] which includes a linear region corresponding to the log law in the mean velocity, but extends to the entire flow domain with four layers including viscous sublayer, buffer layer and a bulk flow region. The four-layer structure characterizing the entire TBL is physically sound and then capture the true similarity form of wall turbulence, for increasing Reynolds numbers (*Re*). The SED theory is able to predict its universal analytic form, based on a dilation invariance assumption, which is presumably universal whenever a wall is present. More interestingly, the variation of the (multi-layer) parameters may represent the variation among different TBLs of a variety of physical conditions. Indeed, the theory has yielded unified description of mean velocity as well as mean kinetic energy profiles in a variety of wall flows [4, 5], from channel/pipe (which is internal flow) to TBL (which is external flow), from incompressible to supersonic and hypersonic TBL, and from smooth to rough wall. On the other hand, the theory, at the very beginning, aimed at developing a closure for the RANS equations: an order-function-based model was reported in 2009 [6], and a SED-*k*-\(\omega \) model in 2017 [7]. The latter yields a uniformly accurate description of mean velocity and streamwise kinetic energy profiles in turbulent channel and pipe, covering a wide range of *Re*.

Recently, this theory is further extended to derive an algebraic model [8] for compressible turbulent boundary layers (CTBL), with accurate prediction of mean profiles of velocity, temperature and density, valid from incompressible to hypersonic flow regimes (thus achieving a Mach number (*Ma*) invariant description). More recently, a transition model [9] for transitional flat plate flow is shown to yield accurate prediction, for the first time, of the entire streamwise profile of the friction coefficient, \(C_f\), of all seven sets of experimental (e.g., of T3 series) and simulations data with varying incoming turbulence intensities (*Tu*) (or scales), superior to existing closure models. This work explores its applicability to industrial turbulent boundary layers such as airfoil flows, to assess its potential merit for aeronautic applications.

Here, we focus on two transonic airfoil flows, NACA0012 and RAE2822, which stands as common benchmark flow for testing turbulence model for aeronautic applications. In these two airfoil flows, shock may form and interact with TBL, with notable pressure gradient. Two quantities of interest are the coefficients of lift and drag, \(C_L\) and \(C_D\); the latter is particularly challenging for currently used turbulence models, which have limited accuracy, due to inadequate description of turbulent boundary layer in the presence of varying pressure gradient of airfoil flow, compared to flat plate TBL for which the model parameters are calibrated. Specifically, the drag coefficient predicted by popular models for the two standard airfoil configurations always have (absolute) error of 20 to 30 counts (each count is \(10^{-4}\)), compared to experimental measurements, despite extensive effort by the international CFD community. The remarkable message of the present work is that, with a refined physical (multi-layer) description of wall turbulence, this bottleneck has been overcome.

## 2 New algebraic model

### 2.1 Governing equations

*R*is the gas constant) and entropy \(H = E + \frac{p}{\rho }\), where total energy \(E = {C_\nu }T + k\) with \(C_{v}\) the specific heat at constant volume,

*T*the static temperature and

*k*the kinetic energy of the flow. The laminar Prandtl number

*Pr*is 0.7, and the turbulent Prandtl number \(Pr _{\tau }\) is set to be 0.9. Meanwhile, the the Boussinesq approximation is expressed by

### 2.2 SED-SL model

*r*, and \(\delta \) is defined as the position of 0.99\({{U}_{e}}\) for flat plate TBL (\({{U}_{e}}\) is free stream velocity), but generalized here for airfoil flows, following Baldwin-Lomax model [11] by

*x*, with respect to the leading edge location (i.e. \(x=0\)):

Finally, the SED-SL model has two parameters, \(\ell _0^\infty \) and \(y_{buf}^\infty \), characterizing the fully developed TBL, which may be slightly different for different physical conditions such as wall curvature, pressure gradient, or *Ma*. For flat plate TBL (presented in Sect. 3.1), they are set to be 1.1 and 41, respectively, (yielding a Karman constant \(\kappa \approx 0.45\)), according to the original SED theory [4, 5]. For the two airfoil flows (presented in Sects. 3.2 and 3.3), we empirically determine that \(\ell _0^\infty \approx 0.3\), while \(y_{buf}^\infty \approx 85 \) (for yielding a smaller \({\kappa }\) of 0.27), which are fixed for all cases and all angle of attack (AoA). In the future, more simulation studies need to be conducted to establish a systematic variation of \(\ell _0^\infty \) and \(y_{buf}^\infty \) with *Ma* and wall curvature, in order to develop a wide range of engineering predictions.

### 2.3 Numerical method and model implementation

The above model is implemented on a so-called OpenCFD-EC platform, which is a parallel, multi-blocks, structured-grid, finite-volume compressible CFD software developed for engineering computation by Li et al. [17] at the Institute of Mechanics of Academy of Science of China. In this code, a third-order MUSCL-scheme and a van-Leer method are adopted to discretize the inviscid terms and the viscous terms, respectively; an implicit LU-SGS method is used for time advance. Our model is also successfully implemented in CFL3D platform [18], which is a structured-grid, cell-centered, upwind-biased RANS code, developed in the early 1980’s in the Computational Fluids Laboratory at NASA Langley Research Center used to support numerous NASA programs since then and continues to be used today. The results (e.g. accuracies) are similar using both codes, with slightly adjustment of the model parameter \(\ell _0^\infty \) (e.g. changing from 0.3 to 0.5 owing to different numerical viscosity associated with different numerical schema).

In order to assess the quality of the predictions, we compare our model with the popular \(\gamma - {{\mathop {Re}\nolimits } _{\theta t}}\) transition model [12] for the flat plate transitional flow case, and with two well-known turbulence models, namely the Spalart-Allmaras (SA) one-equation model [13] and the shear stress transport (SST) two-equation model [14], for the airfoil flows; the results clearly show that the SED-SL model perform the best. But we do not claim the general superiority of the SED-SL model over the popular models, since more flows need to be tested.

## 3 Results and discussion

*Re*, and for the latter, the computation is carried out for different

*AoA*, with significant improvement of the prediction accuracy of the lift (\(C_L\)) and drag (\(C_D\)) coefficients compared to the two popular models (e.g. SA and SST). Note that for latter, experimentally reported incoming flow conditions (

*Ma*and

*AoA*) are slightly modified in a systematic way, which is explained in Appendix.

### 3.1 Transitional flow over flat plate

*Re*by a simple correlation formula, following [9]. The flow is computed on an \(500\times 150\) grid over the calculation domain \(1.0~\mathrm{m}\times 0.1\) m, with a free stream boundary condition applied at the upper, inlet and outlet boundaries, and no-slip condition at the (bottom) flat plate wall. The grids are homogeneous in the streamwise direction, while stretched exponentially in the wall-normal direction with a small value for the closest mesh point to the wall: \(y_1^+ \approx 0.3\).

*x*denotes the streamwise coordinate and \(\nu \) represents kinetic viscosity) for T3B flat-plate test case [19] with zero pressure gradients, showing much better agreement of the present model, compared with the \(\gamma - {{\mathop {Re}\nolimits } _{\theta t}}\) transition model [12]. As shown in Fig. 1(b), the errors of the predictions by the SED-SL model are bounded within 5%, which is clearly below that by Menter’s model, especially in the transition region. Figure 2 compares the predictions of the SED-SL model on the mean velocity

*U*and Reynolds stress profiles \(-uv^+\) along wall normal direction \(y^+\), at different streamwise locations for T3B transitional flat plate flow with our DNS data. Note that the incoming turbulence intensity is 6.5%, which is rather large, so that a bypass transition takes place, which is difficult to describe for most models. Nevertheless, the predicted mean velocity profiles by the SED-SL model are in almost perfect agreement with DNS results; the Reynolds stress profiles have slight deviations, only near the transition region.

### 3.2 Transonic flow over RAE2822 airfoil

Then, consider flow passing RAE2822 airfoil for case 6, whose experimental condition is well documented in Ref. [20]. Note that previous studies showed that the experimentally reported flow conditions, mainly *Ma* and *AoA*, need to be rectified, in order for the computation to reproduce accurately the measured lift and drag coefficients. For case 6, the used flow conditions was suggested in NASA turbulence validation and verification website: \(Re=6.5\times {{10}^{6}}, Ma=0.729, AoA={2.31^ \circ }\), while for case 9, we use following flow conditions: \(Re=6.5\times {{10}^{6}}, Ma=0.732, AoA={2.65^ \circ }\), as discussed in Appendix. Using these values, we compute the flow passed RAE2822 airfoil, with a standard mesh of \(395\times 65\) with 306 points lying on the airfoil surface, provided by NASA turbulence modeling validation page. Grid convergence is fully achieved as shown in Appendix (in Fig. 6(a)).

*x*is the streamwise locations of surface points, \(x_c\) denotes the chord length of airfoil), and the friction coefficient \(C_f\) distribution along the upper surface. It is seen that the shock position is well captured, significantly closer to experimentally measured values than the prediction by (standard) SA and SST model; the latter show a too early shock formation with a under-estimation of pre-shock pressure level. Same results can be seen for case 9, as shown in Fig. 4, where the results predicted by CKDO model [21] with \(Ma=0.73, AoA={3.19^ \circ }\) (same as experimental flow condition without correction) are also included for comparison (black solid lines). The prediction of pre-shock skin friction on upper surface by the CKDO model are yet much higher than experimental measurements, while pressure coefficient distribution is captured better than SA and SST models.

Then, remarkably, the predicted lift and drag coefficient by the SED-SL almost exactly matches the experimental measurements, sharply distinct from the two other popular models, as shown in Table 1. The deviation of the SED-SL model for case 6 and case 9 are bounded within 1% for \(C_L\), and within 2 counts for \(C_D\) (below 1 count in case 6). The drag accuracy is clearly superior to the SA, SST and CKDO models (whose maximum deviation up to nearly 30, 10 and 36 counts, respectively); the errors are reduced by nearly one order of magnitude. Note that our reported results by the SA and SST model are obtained using the same OpenCFD software with the same grid resolution, so the comparison is meaningful.

*AoA*further enhance the conclusion that the SED-SL model delivers significantly better predictions than the existing models.

Results of case 6 and case 9 for airfoil flow passing RAE2822

Results | Exp.\(_1\) | SED-SL | SA | SST | CKDO | |
---|---|---|---|---|---|---|

case 6 | \(C_L\) | 0.743 | 0.7434 | 0.6737 | 0.6892 | – |

Err(%)\(_2\) | – | 0.05 | \(-\) 9.33 | \(-\)7.24 | – | |

\(C_D\) | 0.0128 | 0.01278 | 0.01558 | 0.01259 | – | |

Err(counts)\(_3\) | – | \(-\) 0.2 | 27.8 | \(-\)2.1 | – | |

case 9 | \(C_L\) | 0.803 | 0.8100 | 0.7433 | 0.7532 | 0.832 |

Err(%) | – | 0.87 | \(-\)7.43 | \(-\)6.20 | 3.61 | |

\(C_D\) | 0.0168 | 0.01692 | 0.0187 | 0.01586 | 0.0204 | |

Err(counts) | – | 1.2 | 19 | \(-\)9.38 | 36 |

### 3.3 Flow over NACA0012 airfoil

Flows passing NACA0012 airfoil model were performed in the ONERA S3MA transonic wind tunnel by Thibert et al. [22]. Test cases selected for a range of *AoA* under \(Ma \approx 0.7\), are computed. The computation is performed on \(449 \times 129\) grids with 257 points lying on the airfoil surface, provided also by NASA Langley Research Center Turbulence Modeling Resource [23]. For this set of grids, the far field is located approximately 500 chords from the airfoil surface. Grid convergence is shown in Fig. 6(b) (see Appendix) under flow condition \(Ma=0.74\), \(Re=3.8 \times 10^6\) and \(AOA={2.26^\circ }\).

Under a systematic adjustment of *AoA* explained in the Appendix, the predictions of \(C_L\) and \(C_D\) are compared to experimental data, as shown in Fig. 5. For all cases, the relative errors of \(C_L\) for the SED-SL model is bounded within \(2\%\), while the absolute errors of \(C_D\) is within 6 counts, which is close to the experimental uncertainty [22]. On the other hand, the SA and SST model have errors for \(C_L\) up to \(20\%\), with, in addition, a systematic variation with increasing *AoA*, showing that both models do not capture the right similarity TBL structure for varying *AoA*. The errors of SA and SST for \(C_D\) also widely spread in the range of 10 to hundreds counts, with again a wild variation of *AoA*, with mostly significant over-predictions. In sharp contrast, the SED-SL model’s errors remain unchanged for increasing *AoA*, indicating that the multi-layer description by the SED theory, and hence the SED-SL model, capture the right similarity property of the TBL in airfoil flows, versus *AoA*. Note that the SA and SST model used here are the standard version without any refinement, e.g. with no modeling of streamwise transition [17], and so they may not be the best version used in the industry. But it is suffice here for the purpose to demonstrate the effectiveness of the SED-SL model.

One may wonder how much the transition model contributes to the accurate drag prediction. We have investigated this question by removing the transition from the SED-SL model; only very small changes in \(C_L\) and \(C_D\) are observed. For instance, for RAE2822 airfoil, a full turbulence SED-SL model yields changes of 0.2% and 0.3 counts in \(C_L\) and \(C_D\), respectively. Therefore, adding the transition model is not a major factor for the accurate drag prediction in the present set of computations. Instead, the simultaneously accurate lift and drag predictions are due to refined multi-layer description of TBL structure over the airfoil surface, much better than existing models.

## 4 Conclusion

In summary, the SED-SL yields accurate prediction of \({{C}_{L}}\) and \({{C}_{D}}\) for airfoil flows over RAE2822 and NACA0012 in a range of *Re* and *AoA*. The improvement of drag prediction accuracy by an order of magnitude over current popular models (approaching the limit of experimental uncertainty) is rather striking, in two aspects. First, it is a sign that the essence of wall turbulence is captured, namely, the universal multi-layer description of the stress length defines the invariant (in *x*) wall-normal distribution of the eddy viscosity, which is behind the success of the computation. It is remarkable that the extension from flat plate to airfoil flow is so straightforward that only two parameters slightly change: \(y^{+\infty }_{buf}\) from 41 to 85, and \(\ell _0^\infty \) from 1.1 to 0.3. Secondly, it offers a more optimistic view for the future of RANS model: accurate prediction is possible when fundamental understanding of wall turbulence is taken into consideration.

Furthermore, three specific merits of the present algebraic model are noteworthy. First, it may produce an alternative and better (more smooth and accurate) wall constraint for unsteady simulations such as detached-eddy or large-eddy simulations (LES) (e.g. to be used in the recently developed constrained LES [24, 25]. Second, it may provide a tool to calibrate the experimental flow conditions. Our study shows that the suggested corrections of *AoA* for RAE2822 and NACA0012 are reasonable (see Appendix). Thirdly, for new applications such as flows at higher *Ma* (e.g. hypersonic flow) or flows over new geometrical configurations, for which simulation data (from DNS or LES) is available, the present model may be used to determine the corresponding multi-layer parameters (i.e. \(\ell _0^\infty \) and \(y^{+\infty }_{buf}\)) which specify flow physics of TBL. Note that the *Ma*-dependence of \(\ell _0^\infty \) and \(y^{+\infty }_{buf}\) is generally smooth, due to the fact that the multi-layer structure is universal. This consideration highlights the interest of carrying out a combined DNS and LES study with the SED-SL model: it would yield a refined description of the corresponding TBL structures, and, at the same time, an accurate turbulence model. Thus, it is highly recommended to carry out such study for a larger class of wall flows. Note that some very different wall flows may display new multi-layer structure. An example is flow with separation under strong adverse pressure gradient, for instance, the recently concerned dynamic stall problem [26]; our study shows that additional layers need to be added, so that stall at higher *AoA* and unsteady separation can be correctly predicted, the results of which will be reported elsewhere.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11452002 and 11521091). ZSS is specially grateful to Professor Zhuang Fenggan, and to Professor Zhang Hanxin for lasting encouragement for the pursuit of this project over the past fifteen years. MJX likes to thank Xin-Liang Li for the help in implementing the model to OpenCFD. We also thank Xi Chen for helpful discussions, and to Shiyao Li for assistance.

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