Experimental Validation of a Viscous-Inviscid Interaction Model

Experimental validation of a viscous-inviscid interaction model

Mikaela Lokatt and David Eller Aeronautical and Vehicle Engineering Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract

A combined numerical and experimental study aimed at evaluating the performance of a relatively recently developed three- dimensional viscous-inviscid interaction model is presented. Nu- merical predictions are compared to experimental data and aspects such as accuracy and computational cost are discussed. An experimental study of the flow over a constant chord wing model with a natural laminar flow airfoil profile is performed in order to provide suitable validation data. The experimental wing model is equipped with a trailing edge flap and a significant variation in boundary layer characteristics is observed when the flap deflection and the angle of attack is varied. The experimental results, to which the numerical predictions are compared, include chordwise pressure distributions and oil flow photographs for a range of angles of attack and flap deflections. Numerical predictions are also compared to experimental data for a three-dimensional wing-body configuration. A comparison of the chordwise pressure distributions at a number of spanwise locations as well as of integrated force coefficients is made. The numerically predicted skin friction field is also studied, with particular focus on the area near the wing-fuselage junction where a notable interaction between the flow over the wing and the flow over the fuselage is observed. It is found that the flow model can provide relatively accurate predictions of boundary layer characteristics and pressure distribution in attached flow fields as well as in flow fields including small to moderate separated flow regions. The computational cost is found to be sufficiently low for the model to be considered to be useful for aircraft design.

1 1 Introduction

Prediction of aerodynamic loads is an important part of aerodynamic design and aircraft performance analysis. As such, mathematical models which can provide accurate flow field predictions at a low computational cost are desirable. Because of their relatively high accuracy and relatively low computational cost, viscous-inviscid interaction (VII) models have long received interest in aeronautical research [1]. Significant ef- forts aimed at developing two-dimensional, three-dimensional as well as combined two and three-dimensional VII models have been made, see for example [2, 3, 4, 5, 6, 7, 8, 9].

In a relatively recent paper by the present authors [10] the develop- ment of a fully three-dimensional VII model aimed at application to aircraft configurations was described. The model is applicable to three- dimensional geometries discretized by unstructured meshes. In a set of validation cases focusing on the flow over different wing models the VII model has been found to robustly provide relatively accurate predictions of flow field properties in attached as well as mildly separated flow fields including transitional flow [10]. As such, it appears to be promising for use in aeronautical engineering applications.

A main purpose of the present study is to perform a thorough evaluation of the performance of the above described ﬂow model [10]. Applications to ﬂow cases with relatively complex boundary layer characteristics as well as to a more complex aircraft-related geometry are considered and comparisons are made to experimental data as well as to predictions of another, two-dimensional, VII model.

In particular, an experimental study of the flow over a wing model which has a natural laminar flow airfoil and is equipped with a trailing edge flap has been conducted and is described in some detail. The experimental wing model has interesting boundary layer characteristics. A laminar separation bubble and a trailing edge flow separation are present in some of the flow cases and the transition location varies significantly depending on the angle of attack and flap deflection. The experimental data is considered to be suitable for validation of flow models aimed at prediction of aerodynamic characteristics of laminar flow airfoils, the study of which has received renewed attention in recent years [11]. As such, a main contribution of the present work is that it not only evaluates the accuracy of the present flow model but also presents data which can be useful for

2 validation of other ﬂow models aimed at aeronautical applications.

In a second validation study numerical predictions of the present flow model are compared to experimental data for the DLR-F4 model presented by Redeker [12]. The DLR-F4 model is a three-dimensional wing body configuration which was designed to resemble a transport aircraft. As such, this test case allows to evaluate the performance of the VII model when applied to a more complex aircraft-related geometry. The focus is not only on the accuracy of the predictions of the flow model but computational efficiency is also discussed.

2 Flow model

The flow model employed in the present study is based on a fully three- dimensional VII model previously developed by the present authors [13, 14, 10]. The inviscid part of the VII model, known as PHI [13], is based on an equation governing potential flow which is discretized by a Galerkin finite-element scheme with linear or quadratic Lagrange elements. It is applicable to subsonic flow cases which may include small pockets of supersonic flow [13].

The viscous part of the VII model is based on a set of three-dimensional integral boundary layer (IBL) equations consisting of the momentum loss equations in two wall tangential directions and an equation for the kinetic energy loss [14, 10]. These are combined with a transition model based on a transport equation for the Tollmien-Schlichting amplification factor [15] as well as with a transport equation for the turbulent shear stress lag [15] intended to account for turbulent non-equilibrium effects. The viscous model is discretized by a finite-volume scheme which is ro- tationally invariant and stabilized by an upwind scheme applicable to hyperbolic and mixed hyperbolic systems [14, 10].

The viscous and inviscid flow models are coupled by velocity and velocity gradient terms, defined as first and second order differentials of the potential field variable, as well as with wall transpiration boundary conditions [10]. The coupled VII system is solved by a Newton-based algorithm using a direct coupling scheme [10]. To enable application also to flow cases including separated flow regions the viscous model has been modified to avoid the boundary layer singularity [16, 17], com-

3 monly known as the Goldstein singularity [17]. The numerical stability of the, modiﬁed, direct coupling scheme has been illustrated in test cases including both quasi two-dimensional and three-dimensional separated ﬂow [10].

A detailed description of the ﬂow model and its discretization can be found in the papers [13, 14, 10]. In the present study a few updates have been made in order to improve the accuracy, robustness and computational eﬃciency. These are described in the appendix.

3 Natural laminar ﬂow experiment

This section provides a description of an experimental study of a wing model with a natural laminar ﬂow airfoil. In the experimental study both steady and unsteady ﬂow cases have been investigated. The present study presents results from the steady measurements whereas the unsteady part will be discussed in a follow up paper. Also, numerical predictions of the VII model are compared to experimental data as well as to predictions of a two-dimensional VII model.

It may be noted that results from an earlier experimental campaign con- cerning measurements performed with the same wing model but with a diﬀerent experimental setup have been presented in a previous paper [10], where they were used for validation of an earlier version of the three-dimensional VII model. While these experimental results resemble the ones presented in this paper, they result from a diﬀerent set of measurements and, as such, are not the same as the ones presented here.

3.1 Experimental setup

The experiment was performed in the low-speed wind tunnel L2000 at the Royal Institute of Technology KTH. The experimental model, shown in Figures 1, 2, consists of a straight, constant chord wing model with the natural laminar flow airfoil profile ED36F128. The model has a span of 2 m and a chord of 0.5 m and is equipped with a trailing edge flap spanning 15 percent of the chord. When deflecting the flap a gap appears along the hinge axis. The gap is covered by a polyester strip, seen as a white line in the figures. On the lower side of the wing model a

4 transition trip is placed at 93 percent of the chord. Regarding the upper side of the wing model, the transition was free in one set of experiments whereas a transition trip was placed at 19% of the chord in another set of experiments.

The wing model is mounted vertically on a turntable which can be rotated to adjust the angle of attack. Since the same experimental set up was used for both steady and unsteady measurements, the wing model is connected to a mechanism which can be rotated to set the model into a periodic motion. For the steady measurements the mechanism is kept in a constant position. To cover the mechanism, a boundary layer splitter plate is placed 15 cm above the wind tunnel floor. The flap setting is adjusted manually by moving the flap to a desired position where it is clamped by a set of aluminum plates.

The wing model and its attachment can deform elastically when sub- jected to aerodynamic loading. As such, both the angle of attack and the flap deflection can vary depending on the flow conditions. For this reason, a camera system [18] is used to record the position of the wing. By measuring the position of three reflective markers placed at the lower side of the wing model it is possible to determine the angle of attack and the flap deflection for each flow condition. Position measurements are made at a sampling frequency of 150 Hz.

The wing model is equipped with pressure taps along the chord on both the upper and lower side of the airfoil section, positioned at 25% of the span from the wind tunnel ceiling. The pressure taps are connected to a set of pressure scanners [19] from which the pressure can be read with help from a data acquisition system [20] and a graphical programming software [21]. The pressure scanners have a nominal full-scale range of 5kPa and are declared to have an accuracy of 0.06% of this value. A sampling frequency of 500 Hz is used for the pressure measurements. An oil ﬂow visualization technique [22] is used to obtain information about the boundary layer character.

5 Figure 1: Experimental model setup.

Figure 2: Transition trip and reﬂective markers.

6 3.2 Test conditions

All results presented in the following sections were obtained for an air- speed of 30 m/s which, assuming sea level standard atmospheric properties, yields a chord-based Reynolds number of 1.0 · 106 and a dynamic pressure of 550 Pa.

A sequence of experiments was performed for different nominal flap de- ◦ ◦ ◦ ◦ ◦ flections of δn = −4 , 0 , 8 , 11 , 14 and a range of angles of attack. Patterns of interest were identified resulting in the the 18 specific cases listed in Table 1 being selected for validation. For all of these cases, the transition on the upper side of the wing model is free. As became ap- parent from the optical position measurements, the actual observed flap deflections differed from the nominal target values as indicated in Table 1. The deviation displayed can be attributed to the limited accuracy of the flap adjustment and to a small elastic deformation caused by the aerodynamic hinge moment.

◦ δn =0 α 4.3◦ 5.3◦ 6.3◦ 7.3◦ 8.3◦ 9.4◦ δ -0.4◦ -0.5◦ -0.5◦ -0.5◦ -0.5◦ -0.6◦ ◦ δn =8 α 1.3◦ 2.3◦ 3.4◦ 4.3◦ 5.4◦ 6.5◦ δ 7.5◦ 7.4◦ 7.3◦ 7.2◦ 7.2◦ 7.1◦ ◦ δn =14 α 0.2◦ 1.2◦ 2.2◦ 3.3◦ 4.3◦ 5.4◦ δ 13.3◦ 13.3◦ 13.2◦ 13.1◦ 13.0◦ 13.1◦

Table 1: Subset of angle of attack and ﬂap deﬂections used for validation.

3.3 Numerical model

The numerical model has a span to chord ratio of 1.0, which appears to be large enough for spanwise variations to be negligible over the main part of the wing span. Side walls are included in the geometry. The model employs a tetrahedral mesh with approximately 37 000 elements of which 7 000 are triangular elements on the airfoil surface. Quadratic Lagrange elements are used in the inviscid discretization.

Computations are also performed using XFoil [2], a two-dimensional VII

7 model aimed at analysis of flow over airfoil sections. XFoil employs a fully simultaneous coupling scheme and, as opposed to the present model, does not modify the boundary layer closure equations in separated flow regions. It is thus considered to provide a suitable reference for comparison with the results of the present flow model.

While XFoil allows to specify transition locations at the respective upper and lower sides of the airfoil section, the present flow model has a somewhat limited capacity of specifying transition locations. As such, the transition trip on the lower side of the airfoil section is not easily accounted for in the present flow model. In order to use the same conditions for both flow models, free transition is assumed on both the lower and the upper side of the airfoil section in the computations with the present VII model as well as in the computations with XFoil.

The trailing edge flap deflections are set to 0.0◦, 8.0◦ and 14.0◦ degrees for the respective flow cases.

3.4 Experimental results

A set of measured pressure distributions for the respective configurations are shown in Figures 3 - 20. The points shown for experiments are median values of 5000 samples which were measured while keeping the angle of attack constant. To get a perception of the variation between the different samples, the 25th and 75th percentiles [23] are included in the form of the shaded blue regions. It is seen that the percentiles nearly coincide except for some of the pressure taps near the trailing edge. It was found that the silicone tubes from two pressure taps near the trailing edge detached during the experimental campaign and the measured, nearly zero, values at these positions are thus considered to ◦ be uncertain. For the flow cases with δn =14 one of the pressure taps was found to be hidden below the polyester strip and, since they are believed to be disturbed by the strip, the values measured in this point have been removed from the presented results. For the flow cases with ◦ ◦ δn =0 , 8 the pressure taps near the hinge axis were found to be close to the strip. It is possible that these measurements have been disturbed by the presence of the strip and the presented values near 85% of the chord are thus regarded to be uncertain.

Figures 3 - 20 also contain graphs of the skin friction coeﬃcient predicted

8 by the two computational models. Below the friction graphs, slices of the oil photographs are included. In these pictures, light regions indicate areas with low skin friction whereas dark areas indicate regions with high skin friction. Regions with transition from laminar to turbulent flow are indicated by areas with a steep increase in skin friction and hence a notable lightness gradient. Regions with separated flow are identified by very low skin friction and reversed oil flow near a separation line, indicated by a very light region. Laminar separation bubbles typically show as a very light area in the laminar region which is followed by a transition region. The markings in the photographs highlight where the oil patterns have been interpreted to indicate laminar-turbulent transition or a turbulent separation line.

The transition locations are summarized in Figures 24, 25, 26. Normal force coeﬃcients Cz, obtained by integrating the pressure along the airfoil contour, are presented in Figures 21, 22, 23.

3.5 Numerical results

The numerical predictions are presented together with the experimental results in Figures 3 - 26. The results for the present model represent values at the mid span cross section.

3.6 Discussion

Because of the uncertainty regarding the pressure measurements near the hinge axis as well as near the trailing edge, there is some uncertainty regarding the value of the experimental normal force coefficients. As such, it is difficult to make a detailed comparison between the numerical and experimental normal force coefficients and the results in Figures 21, 22, 23 may be considered to primarily be suitable for a qualitative comparison. It is seen that both numerical models predict the normal force coefficients to show a clearly non-linear variation with respect to the angle of attack, which is similar to the non-linear variation observed in the experimental results. However, regarding the magnitude of the normal force coefficient, there is some difference between the numerical and the experimental results. The relatively large variation between the predictions of the inviscid models and the predictions of the VII models

9 indicate that the aerodynamic forces appear to be significantly affected by viscous effects. For both the inviscid and the VII results, the values of the normal force coefficient predicted by XFoil are consistently higher than the ones predicted by the present flow model.

From Figures 24, 25, 26, it is found that the numerically predicted transition locations correspond well with the experimental results for the ◦ case of a neutral (δn =0) flap setting. For the cases of a positive flap deflection, the shape of the experimental transition curves resembles the numerically computed transition curves. However, while the predictions of the two different numerical models are similar, it appears that the computational predictions match the experimentally observed transition locations at an about 2◦ lower angle of attack. At first sight, this appears to indicate that both computational models predict transition to occur ◦ ◦ too late for the cases with δn =8 , 14 .

It has been found that for flap deflections of 8◦ and 14◦, the numerical and experimental pressure distributions and skin friction fields look fairly different at a given value of the angle of attack. As for the transition locations, what resembles an angle of attack offset of approximately 2◦ can be identified. It is presumed that a better comparison of the devel- opment with respect to the angle of attack can be made if adjusting for the observed offset and this is the reason for the angle of attack offset applied in Figures 3 - 20.

After applying the offset it is found that, overall, the present model provides accurate predictions of the magnitude of the leading edge suction peak. For the flow cases in which there is a trailing edge flow separation it is found that the line representing the pressure on the suction side of the airfoil section is consistently somewhat below the experimentally measured values. It is credible that this difference would decrease if applying a slightly larger angle of attack offset. XFoil on the other hand consistently over-predicts the magnitude of the leading edge suction peak whereas the predicted pressure distribution over the mid part of the airfoil section agrees well with the experimental results. As discussed above, the flow field appears to be strongly affected by viscous effects. The test case is thus considered to be rather challenging and, as such, while there is a visible difference between the experimental and the numerical results both numerical models are considered to perform reasonably well in predicting the pressure distribution.

10 Comparing the numerically predicted skin friction fields to the oil flow pictures it is found that the numerical predictions, from both of the VII models, and the experimental visualizations correspond well both regarding the boundary layer character and the occurrence of separated flow regions. It is thus concluded that the experimental measurements and the numerical predictions agree well regarding the relation between the pressure distribution and the boundary layer character.

It should be noted that the experimental data has not been adjusted to account for wind tunnel wall interference effects. Using first order wind tunnel corrections [22], the wall effects are estimated to correspond to an angle of attack offset of the order of 0.1◦. Thus, while the wind tunnel wall interference effect could explain part of the offset between the numerical computations and the experimental results it appears as if the observed shift in angle of attack is too large to be explained by such effects only.

From Table 1 it is seen that the flap deflections used in the numerical computations are slightly larger than the flap deflections in the experiment. Since a positive flap deflection increases the curvature of the wing, an increased flap deflection could be expected to have a similar effect on the flow field quantities as an increased value of the angle of attack. Such a variation is opposite to what is observed in the results and, as such, the deviation in flap deflection is not considered to be a likely explanation for the observed offset.

Instead, the observed shift in angle of attack could well be an indication that there is some aspect of the ﬂow physics which is not correctly captured by the ﬂow models. It is noted that both numerical models are showing a similar behavior. However, the reason for the deviation has not yet been understood.

1 4 Experiment . XFoil VII XFoil inviscid PHI VII 1.2 PHI inviscid

z 1 C

0.8

0.6

456789 α (degrees)

◦ Figure 21: Normal force coeﬃcient for δn =0 .

Experiment 1.6 XFoil VII XFoil inviscid PHI VII 1.4 PHI inviscid

z 1.2 C

0.8

0.6 2468 α (degrees)

◦ Figure 22: Normal force coeﬃcient for δn =8 .

Experiment 1.8 XFoil VII XFoil inviscid 1 6 PHI VII . PHI inviscid

z 1.4 C

1.2

0.8 02468 α (degrees)

◦ Figure 23: Normal force coeﬃcient for δn =14 . 21 1 Experiment XFoil VII 0.8 PHI VII

0.6 tr c x 0.4

0.2

0 456789 α (degrees)

◦ Figure 24: Upper side transition for δn =0 .

1 Experiment XFoil VII 0.8 PHI VII

0.6 tr c x 0.4

0.2

0 2468 α (degrees)

◦ Figure 25: Upper side transition for δn =8 .

1 Experiment XFoil VII 0.8 PHI VII

0.6 tr c x 0.4

0.2

0 02468 α (degrees)

◦ Figure 26: Upper side transition for δn =14 . 22 4 Three-dimensional validation on DLR-F4 experiments

In the present test case numerical predictions are compared to experimental data for the DLR-F4 model, presented by Redeker [12]. The DLR-F4 model is a wing-body configuration designed to resemble a transport aircraft [12]. The experimental study [12] was conducted to provide experimental data suitable for validation of computational models. Experiments were performed in three different wind tunnels (NLR, ONERA and DRA) so as to allow comparison of experimental data from different testing facilities [12].

Detailed information about the geometry of the DLR-F4 model is pro- vided by Redeker [12]. In particular, the model has an aerodynamic mean chord of 141.2 mm and a semi-span of 585.7 mm. Transition trips are placed at chordwise positions ranging from 5 to 15% of the local chord on the upper side of the wings and on 25% of the local chord on the lower side of the wing as well as 15 mm from the nose of the wing-body conﬁguration.

The numerical model employs a tetrahedral mesh, which has previously been employed for validation of the inviscid flow model PHI [13], with approximately 382 000 quadratic 10-node tetrahedra for the inviscid outer region and 99 100 unconstrained nodes in the boundary layer domain. Transition is artificially enforced at 15 percent of the aerodynamic mean chord from the inflow nodes.

While the experimental study [12] focuses on transonic flow cases, experimental data for a Mach number of 0.6 is presented, for which the flow can reasonably be regarded as subsonic since only small regions show supersonic flow. In the pressurized wind-tunnels where the experiments were performed, this flow condition corresponds to a Reynolds number of 3.0 million based on the aerodynamic mean chord [12]. In order to match the Reynolds number, computational conditions were adapted to make use of a fluid density of about 1.8 kgm−3.

23 4.1 Results

Numerically computed pressure distributions for angles of attack ranging from -3.0 to 7.0 degrees are shown in Figures 27 - 32. Blue marks values below the sonic limit (cp = −1.3) whereas red marks values above the free stream pressure (cp =0.0).

Experimentally measured and numerically computed lift and drag coefficients are shown in Figures 33, 34 and lift to drag polars are shown in Figure 35. For reference, inviscid predictions by PHI [13] are included in the figures. As previously described, the transition locations employed for the experimental model and for the numerical model are somewhat different. To provide a reference on how much a variation in the transition location could affect the results, fully turbulent VII computations by the present flow model are also included in the figures.

Chordwise pressure distributions at a number of spanwise stations at flow conditions corresponding to a value of the lift coefficient CL equal to 0.5 are presented in Figures 36 - 38. The normalized spanwise coordinate η, which is part of the figures, is equal to 0 at the mid span section and 1 at the wing tip. A contour plot of the friction coefficient magnitude overlaid with skin friction lines for the same flow condition is shown in Figure 39. Here, blue indicates negative skin friction (cf < 0) whereas −3 red indicates a higher skin friction (cf > 3.5 · 10 ). In Figure 40, a close-in image of the wing-fuselage junction is shown.

24 Figure 27: Pressure coeﬃcient for α = −3.0 degrees.

Figure 28: Pressure coeﬃcient for α = −1.0 degrees.

Figure 29: Pressure coeﬃcient for α =1.0 degrees. 25 Figure 30: Pressure coeﬃcient for α =3.0 degrees.

Figure 31: Pressure coeﬃcient for α =5.0 degrees.

Figure 32: Pressure coeﬃcient for α =7.0 degrees. 26 4.2 Computational eﬀort

In order to provide an indication of the computational eﬀort required to solve the VII problem, values for the DLR-F4 case for ﬂow conditions corresponding to an angle of attack of 1.0 degrees are shown in Table 2.

The ﬁrst column corresponds to a mesh of about 1 million linear (P1) 4-node tetrahedra shown in Figure 41, while the second column is for the mesh used to generate the results displayed in the ﬁgures, which consists of about 380 000 quadratic (P2) 10-node tetrahedra and is shown in Figure 42.

Run-times for two computers are given, where the ﬁrst is a laptop with a 2.6 - 3.6 GHz quad-core processor, while the second is a workstation equipped with a 3.8 GHz six-core processor. The inviscid solution alone accounts for 10.5 seconds (P1) and 25 seconds (P2) on the laptop computer.

P1 P2 Tetrahedra 1 316 304 382 220 Variables 460 251 1 147 792 Potential variables 263 561 652 212 Memory 2.8 GByte 7.2 GByte Run-time laptop 572 s 1100 s Run-time workstation 328 s 662 s

Table 2: Computational eﬀort.

27 1 L C

0.5 Experiment NLR Experiment ONERA Experiment DRA PHI VII transition PHI VII turbulent 0 PHI inviscid −4 −20246810 α (degrees)

Figure 33: Lift coeﬃcient.

Experiment NLR Experiment ONERA 0.15 Experiment DRA PHI VII transition PHI VII turbulent PHI inviscid 0.1 D C

5 · 10−2

0 −4 −20246810 α (degrees)

Figure 34: Drag coeﬃcient.

1 L C

0.5 Experiment NLR Experiment ONERA Experiment DRA PHI VII transition PHI VII turbulent 0 PHI inviscid 00.05 0.10.15 CD

Figure 35: Lift-to-drag polar. 28 2 Experiment NLR Experiment ONERA Experiment DRA PHI VII transition 1 p c −

−1 00.20.40.60.81 x (m)

Figure 36: Chordwise pressure coeﬃcient at η =0.185.

2 Experiment NLR Experiment ONERA Experiment DRA PHI VII transition 1 p c −

−1 00.20.40.60.81 x (m)

Figure 37: Chordwise pressure coeﬃcient at η =0.409.

2 Experiment NLR Experiment ONERA Experiment DRA PHI VII transition 1 p c −

−1 00.20.40.60.81 x (m)

Figure 38: Chordwise pressure coeﬃcient at η =0.844. 29

4.3 Discussion

From Figures 33 - 35 it is clear that, overall, the predictions of the VII model are closer to the experimental results as compared to the predictions of the inviscid model. Studying the lift and drag curves corresponding to transition at 15 percent of the aerodynamic mean chord it is seen that the numerical and experimental results agree well for the lower values of the angle of attack. For the higher values of the angle of attack the numerical model over predicts the lift and under predicts the drag.

Comparing the results for a transition location at 15 percent of the aerodynamic mean chord to the fully turbulent results it is seen that the difference between the results is relatively small. If assuming that the variation in aerodynamic force coefficients related to the difference between the experimental and the numerical transition locations are of a comparable magnitude, it appears as if the variation is not large enough for a difference in transition location to be a likely explanation for the difference between the numerical and experimental results observed at the higher values of the angle of attack.

From the pressure distributions shown in Figure 27 - 32 it is seen that at an angle of attack of 1 degree, a small region of supersonic flow has appeared on the upper side of the wing. The size of the supersonic flow region increases with increasing angle of attack. It is thus credible that the main reason for the variation between the experimental and numerical results is a shock wave forming near the leading edge, which is not captured by the flow model. A growing strength of the shock wave could explain the increasing difference between the numerical and experimental results for an increasing value of the angle of attack. As such, it appears as if the flow cases at the higher values of the angle of attack are outside the range where the subsonic assumption in the VII model is accurate.

Regarding the chordwise pressure distributions for CL equal to 0.5, shown in Figures 36 - 38, the experimental and the numerical results are found to agree well. From the skin friction plot in Figures 39, 40 it is seen that the skin friction is low along most of the trailing edge and around the blunt end of the tail cone. High skin friction is observed in the areas where the boundary layer is thin, such as the nose and leading edge, and where there is a notably favorable pressure gradient. For example, a

32 small area on the fuselage near the wing junction is visibly affected by the strong favorable pressure gradient produced near the leading edge, caus- ing a more rectangular boundary layer velocity profile and hence higher skin friction. From the close-in image of the wing-fuselage junction it is seen that the computed friction coefficient on the wing side of the junction is significantly lower than just a short distance further outboard. The surface friction lines reveal that the boundary layer in this narrow wedge-shaped region originates on the fuselage and is hence thicker than that on the wing next to it. The lower values of the skin friction coefficient indicates a more pronounced risk for separation, which corresponds well with the wider blue (zero or negative skin friction) patch near the trailing edge junction.

Considering the computational time, it is found that the speed-up from four to six cores corresponds to a factor of 1.7, which is in the expected range considering the higher sustained clock frequency of the workstation. Since the problem size may be considered to be comparable to the size which would be obtained for analysis of complete aircraft con- ﬁgurations, the computational times presented here indicate that the computational cost is low enough for the VII model to be useful for preliminary design or conﬁguration studies.

5 Conclusions

From the experimental study of the wing model with a natural laminar flow airfoil it has been found that the VII model appears to be able to provide relatively accurate predictions of variations in boundary layer characteristics as well as variations in pressure distribution for transitional flow cases including mildly to moderately separated flow regions. From the study of the DLR-F4 test case, it is concluded that the flow model also appears to be able to robustly provide relatively accurate predictions for subsonic flow cases over more complex three-dimensional aircraft-related geometries. In transonic flow regions the accuracy deteri- orates, as is expected since the flow model does not account for transonic effects such as shock waves. Because of the relatively low computational cost, the VII model is considered to be suitable for use in aircraft design.

33 6 Acknowledgements

The work presented in this paper was ﬁnanced by the VINNOVA project UMTAPS (Dnr. 2014-00933) and coordinated by Roger Larsson at Saab AB.

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36 Appendix

This appendix summarizes a few updates made in order to improved the accuracy and robustness as well as the computational eﬃciency of the three-dimensional VII model.

A.1 Accuracy

In order to improve the predictional accuracy some of the closure relations [24, 6, 10] (which relate integral boundary layer quantities to a set of primary variables) have been updated. In particular, the approximate relation between the kinetic shape parameter Hk and the kinetic energy shape parameter H∗ which was introduced in the previous study [10] has been replaced by two different relations, one for laminar flow and one for turbulent flow. The updated relations are summarized below whereas the full set of closure relations are found in a previous paper by the present authors [10].

Shape factor

=105 + 2 2 · (−5.0(H∗−1.5)) Hklaminar . . e (A.1) =105 + 1 8 · (−7.0(H∗−1.5)) Hkturbulent . . e (A.2)

Skin friction

1 c = −0.07 + 3.2 · e−1.15(Hk−1) f1laminar (A.3) Reθ c = β · c f2laminar tan w f1laminar (A.4)

37 2 Fc = 1.0+0.2M (A.5) 0.3 · e−1.33Hk c = (A.6) f0a 1.74+0.31Hk Reθ log10 Fc H c =0.00011 tanh 4 − k − 1 (A.7) f0b 0.875 1 c = c + c f1turbulent f0a f0b (A.8) Fc c = β · c f2turbulent tan w f1turbulent (A.9)

Turbulent shear stress lag (source term)

3 0.015 (H − 1.0) C = H∗ k τeq (1 0 − ) 2 (A.10) . Us Hk H √ 2.8 C g = τ C − C (A.11) c δ τeq τ

Laminar and turbulent mixing

fN =0.5(1.0+tanh (10.0(N − Ncrit))) (A.12)

In the above equations Hk denotes the kinetic shape parameter while ∗ H represents the kinetic energy shape parameter and Reθ denotes the Reynolds number based on the streamwise momentum loss thickness θ11. Further, βw denotes the cross flow angle while M represents the Mach ∗ number. Also, δ2 denotes the cross wise displacement thickness, δ represents the boundary layer thickness and Ac denotes a cross flow parameter while Cτ represents the turbulent shear stress lag coefficient with equilibrium value Cτeq . N denotes the Tollmien-Schlichting amplification factor and Ncrit its critical value.

38 A.2 Robustness

The following updates have been made in order to improve the robustness of the solution.

A.2.1 Velocity gradient bound

The IBL equations contain velocity and velocity gradient terms, defined as first and second order differentials of the potential field variable [10]. Since it has been found that unrealistically large values of the velocity gradients in the IBL system can deteriorate the convergence of the numerical solution [10], a bound limiting the magnitude of the velocity gradients is part of the flow model [10].

In the present study the bound has been reformulated to be in the form of a weighting function, so as to obtain a smooth transition between bounded and unbounded velocity gradient values. As such, the bounding is performed by multiplying each velocity gradient value by the function

k2 fφ = γ +(1− γ) (A.13) mg where k2 γ =0.5 1+tanh k1 1 − . (A.14) mg

In the above equation k1 is a constant, mg is a non-dimensional measure of the magnitude of the velocity gradients (deﬁned in the previous study [10]) and k2 is a suitably chosen reference value for the gradient bound (which may be a function of the viscous and inviscid variables).

A.2.2 Wall transpiration

The wall transpiration is related to the growth of the displacement thickness [6, 10]. In certain areas, typically where a relatively thick boundary layer encounters steep pressure gradients, very large growth rates can be obtained.

Unrealistically large values of the wall transpiration terms can deteriorate the convergence of the numerical solution. To ensure a robust solution of

39 the coupled system, a bound on the transpiration velocity w is introduced

|w|≤kw. (A.15)

The value of the bound kw can be adapted to suit the ﬂow case at hand so as to not notably aﬀect the accuracy of the solution.

A.2.3 Jacobian evaluation

The Jacobian matrix consists of two different parts, one relating to flow terms (flow Jacobian) and one relating to source terms (source Jacobian). In a previous paper [10] it has been discussed how the boundary layer singularity [17, 16, 7, 10] supposedly can be related to ill-conditioning of the flow Jacobian. For two-dimensional flow fields it appears as if the ill-conditioning relates to a zero slope of the relation between H∗ and ∗ ∗ Hk, H (Hk), and thus could be avoided by choosing H (Hk) to be a monotonically decreasing function. For three-dimensional flow fields the relation which is believed to govern the ill-conditioning is more complex and also depends on the cross flow parameter Ac as well as on the angle σ between the external streamline and the line along which the char- acteristic lines converge. It has been found [10] that when σ is small, i.e. the convergence line is almost parallel with the external inviscid streamline, numerical ill-conditioning could possibly be obtained for rel- ∗ atively small values of Ac and Hk,evenifH (Hk) is chosen to be a monotonically decreasing function. However, if the cross-flow parameter Ac is equal to 0 the relation which is believed to govern the stability in three-dimensional flow fields becomes equivalent to its two-dimensional counterpart. As such, it is independent on the convergence angle σ and ∗ numerical ill-conditioning is supposedly avoided if choosing H (Hk) to be a monotonically decreasing function [10].

In the present implementation the flow terms in the Jacobian matrix are evaluated for Ac equal to 0, irrespective of the true value of Ac. Numerical ill-conditioning should thus presumably be avoided also in complex three-dimensional flow fields. The true value of Ac is used in the residual evaluation.

Also the source Jacobian is evaluated for Ac equal to 0, whereas the source terms in the residual are evaluated for the true value of Ac.It should be noted that this modiﬁcation is not related to the boundary layer singularity. Instead, it is primarily a choice by the authors who,

40 for consistency, prefer to use the same value of Ac in both the ﬂow and source Jacobian parts.

A.2.4 Residual evaluation

The convergence criterion of the inviscid system is related to the change in inviscid variables while the criterion for the viscous part is related to the change in wall transpiration velocity. The criteria are described in a previous paper [10].

There can exist, relatively small, parts of the computational domain in which it is diﬃcult to obtain convergence. Such regions are typically found on complex geometries in areas with a poor mesh quality. However, the exact solution in these areas may not notably aﬀect the solution of the overall system. To prevent a poor convergence in a small region from deteriorating the convergence of the overall coupled system, an upper bound rmax on the residual contribution from each node rn is introduced.

When evaluating the residual each nodal residual value rn is multiplied by rmax fr = Min 1, . (A.16) rn

The bound rmax is chosen as

r = i (A.17) imax k

r = v (A.18) vmax k where denotes the convergence limit, the subscripts i, v denote respective inviscid and viscous values and k denotes a suitably chosen constant. Global convergence is determined by comparing the 1-norm of the respective viscous and inviscid residuals with user-supplied threshold values. As a consequence, if k is chosen to 0.01, the convergence criteria i, v can be reached if less than 1 percent of the nodes experience convergence problems.

41 A.3 Eﬃciency of the linear sub-problem solution

As described in a previous paper, the coupled viscous-inviscid system is solved by a direct coupling scheme in a sequence of inner and outer iterations. Previously, the parallel sparse direct solver PARDISO [25] from the Intel Math Kernel Library [26] was used to factorize the global Jacobian matrix in each inner and outer solution step [10]. In order to improve the computational eﬃciency, the solution algorithm has been updated.

The multithreaded direct sparse solver is used to factorize the Jaco- bian matrix in each of the inner solution steps - wherein an un-coupled boundary-layer problem is solved - whereas in the outer solution steps, the linear system is solved using the iterative solver GMRES preconditioned with a previous factorization of the Jacobian for the block of inviscid equations [27]. GMRES is restarted after 16 iterations. Whenever the convergence rate of the iterative solver slows down, the preconditioner is updated with a new numerical re-factorization of the most recent Ja- cobian for the inviscid system. The slow-down threshold is not a ﬁxed number of iterations, but rather chosen such that GMRES is permitted to accumulate twice the wall-clock time needed for a full re-factorization on application of the preconditioner before it is updated.

The reason for applying GMRES to the inviscid sub-problem but not for the viscous sub-problem is related to properties of the corresponding Ja- cobian matices. The finite-volume scheme applied in the discretization of the integral boundary layer equations results in relatively weak coupling between nodes as the viscous sub-problem only couples nodes that share a mesh edge on the surface. Therefore, a factorization of the Jacobian for the block of isolated boundary layer equations can be obtained at a comparatively low cost even if the number of variables in the viscous problem may be large. The corresponding inviscid Jacobian is charac- terized by much stronger coupling and can thus be significantly more expensive to factorize, in particular with the quadratic finite elements normally used. However, the changes in the inviscid variables between outer iterations are typically moderate. As such, the variation in the inviscid Jacobian matrix between different outer iterations are expected to be small, indicating that preconditioned GMRES [27] could provide an efficient option.