Dec 7, 2007 - The flow around an F/A-18 aircraft at high angle-of-attack is modelled with .... case, the diverter duct was shortened and a pressure outlet was.
16th Australasian Fluid Mechanics Conference Crown Plaza, Gold Coast, Australia 2-7 December 2007
Prediction of Aircraft Empennage Buffet Loads Stefan Schmidt and Oleg Levinski Flight Systems Branch, Air Vehicles Division, Defence Science and Technology Organisation, Fishermans Bend, Victoria, 3207 AUSTRALIA
method (FVM, Fluent [6]) with a SIMPLE pressure-correction algorithm [12] to enforce mass balance. The accuracy is of second order in both space and time. Excluding the momentum equations, convective terms are discretised by upwindbased schemes. The momentum equations are discretised with a bounded central differencing scheme—reducing 2∆-wiggles1 forming in poorly resolved areas—which are better suited for the direct resolution of turbulence. The linearised equations system is solved by an efficient Gauss-Seidel algorithm combined with an algebraic multigrid method [4].
Abstract
The flow around an F/A-18 aircraft at high angle-of-attack is modelled with the aim of predicting the bending moments of the vertical tail resulting from differential buffet pressures on either side of the tail. In order to resolve the unsteady vortex breakdown and its subsequent interaction with the vertical tail, the method of detached-eddy simulation (DES) is used. The results obtained on a fairly coarse mesh show that the main flow dynamics of the vortex breakdown is well captured. A comparison of the fluctuating buffet pressure distribution on both sides of the tail with experimental data shows that reasonable agreement is obtained, although some detail around the leading edge of the tail is missing. However, the prediction of the bending moment is very encouraging considering the relatively coarse mesh used in the computation.
Turbulence Modelling
Detached-eddy simulation (DES, [20]) is one approach to overcome deficiencies of statistical turbulence models typically used in conjunction with the Reynolds-Averaged Navier–Stokes equations (RANS) for the prediction of unsteady flow patterns and coherent structures. DES models near-wall regions using a RANS method, while applying a modified-RANS model (like a subgrid-scale model used in large-eddy simulation (LES)) in off-wall areas such as separation zones, vortical and wake regions. Although DES in its initial form [20] was based on the Spalart–Allmaras (SA) model, it has now become a more generalised term for hybrid turbulence modelling that exploits the benefits of RANS and LES in one integral approach. This hybrid modelling technique can be in principle applied to almost any existing RANS model by changing the turbulent length scale Lt expressed in terms of turbulent quantities.
Introduction
The highly unsteady buffet loading on the empennage of modern twin-tail fighter aircraft is caused by the burst of longitudinal vortices originating from the leading-edge extension (LEX) of the wing [8, 23, 24]. These highly turbulent vortices interact with the vertical tails and promote a low-frequency dynamic response of the entire empennage, which, over a long period of time, results in structural fatigue [5]. The design of buffetaffected structures therefore requires detailed knowledge of the temporal and spatial characteristics of unsteady buffet loads. The modelling of such flows is a complex task and relies on accurate predictions of the turbulent structures and their interaction with the aircraft surfaces.
In this paper the realizable k-ε model [18] is used as a baseline model, because it determines the RANS-LES border using a turbulent length scale rather than the wall distance. Therefore, it is more suitable to massively-separated flows than the SA-model [22]. In its current formulation the model prevents unphysical realisations of the turbulent quantities during the course of the simulation, by ensuring that all normal stresses are positive (ui ui 2 ≥ 0), and for all shear stresses the Schwartz inequality ui u j 2 ≤ u2i u2j
Over the last decade some attempts have been made to tackle this numerically-challenging problem [2, 9, 10, 11]. The difficulty arises from the very high Reynolds number (of the order of Re ≈ 107 ) and the subsequent need for adequate nearwall resolution in a viscous flow simulation. Additionally, the fine detail of vortical structures which scale with Re3/4 require fine meshes in the crucial vortex-dominated areas that affect the flow around the entire aircraft. Hence, for previous studies mentioned above, very fine meshes of the order of 106 −107 cells are used with adaptive grid refinement to make effective use of the computational resources. However the most important factor determining the computing time is the need to resolve the flow in an unsteady fashion to accurately predict the buffet pressure, which is not possible with steady methods.
is met [14]. This equation can potentially be violated in poorlymeshed areas leading to false estimates of the production terms in the turbulence equation. This can be equally ensured in a post-iteration stage where the dissipation rate ε is locally adjusted to fulfil √ 3 ε≥ Cµ k |S| 2 with the anisotropy parameter Cµ = 0.09, the turbulent kinetic p energy k and the strain-rate magnitude |S| = 2 Si j Si j calculated from the rate-of-strain tensor Si j =(∂ui /∂x j + ∂u j /∂xi )/2.
Previous work [2, 9, 10, 11] concentrated on the application of DES and the prediction of dominating frequencies as a means to assess the capability of the numerical method to simulate these problems. In contrast, this work attempts to predict the unsteady differential pressures, and to quantify the aerodynamic loads (bending moments) that the tail has to withstand. This is even more challenging because long simulation times are required to accumulate statistical values. This has, to the authors’ knowledge, only been achieved previously in costly experiments [8].
In case of the DES-k-ε model, the length scale Lt is modified from Lt = k3/2 /ε to
LDES = MIN (Lt ,CDES ∆)
Numerical Method 1 oscillations
The governing equations are discretised using the finite-volume
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of solution due to large Peclet numbers.
with ∆ = MAX (∆x , ∆y , ∆z ). The value of CDES = 0.61 [22] is adjusted to ensure the correct removal of turbulent kinetic energy into the subgrid level (l ≤ ∆). The modified destruction term of the k-equation is then DDES ≈ k3/2 /LDES . k Various flavours of this method exist with the √ main difference being the length scale (i.e. k-ω model: Lt ≈ k/ω [22]). Similar strategies such as VLES [21] and SAS [7] have been developed, but the rather easy-to-implement modification necessary in DES to produce a single set of modelling equations is very appealing for practical applications. Consequently, DES— mostly in its original formulation—has been applied to a wide range of aerodynamic flows and other flows where unsteady flow features had to be resolved [15, 16, 17].
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Figure 1: F/A-18: Focus region for buffet modelling DES at different locations: (a) at LEX fence; (b) at vertical tail.
Computational Domain and Boundary Conditions
In this study, only a half model is considered because the dominant vortical structures occur in pairs on both sides of the aircraft. This compromise disregards the interaction of vortices on the port and starboard side, but does not limit the validity of the results, because these interactions occur at much lower frequencies than the shedding frequency of the counterrotating leading-edge vortices themselves. The flow conditions (Ma = 0.15, Angle-of-Attack α = 30.0◦ , Re∞ = 12.3 × 106 ) allow for a detailed comparison with the experimental data [8, 13] as well as other numerical studies [9, 11].
task [5] than running a very fine model over a short period of time [9], because the unsteady simulation has to be run for a very long physical time of up to TS ≈ 300s [5]. Its only after this period that statistically useful bending moments results are obtained. This can be regarded as an upper limit to ensure statistically converged results, but remains currently infeasible to accomplish numerically with a reasonable effort. Even shorter simulation times in the order of TS ≈ 30s should be able to give reasonable answers [5] and it will be assessed how much physical time is actually ‘good enough’. Hence, the current simulation was integrated over TS ≈25s at a time step of ∆t =5 ×10−4 s to accommodate multiple vortex burst cycles to interact with the vertical tail. The time history of the integral pressure on both sides of the tail fin indicates this phenomenon (figure 2).
The outer boundary of the computational domain is defined by a hemisphere of diameter roughly the size of 40 aircraft wing spans. At this location, far-field pressure boundary conditions are applied that require the Mach-number, flow angles and temperatures (T = 300K) to be specified. At the centre-plane, symmetry of all quantities was assumed. At the engine intake, Area-Weighted Average Static Pressure 0 pressure-outlet boundary conditions are used because the enInboard gine itself is not included in the model. The engine exhaust is modelled as a far-field pressure boundary with normal exit flow -200 at the free-stream Mach number. The F/A-18 has an inclined rectangular duct to divert parts of the turbulent boundary layer through the fuselage towards the top of the aircraft. This vent was closed in the experimental setup [8], resulting in an upstream shift of the vortex breakdown location [9]. In order to quantify this shift simulations with an open and closed diverter were carried out. In the closed-diverter case, the diverter duct was shortened and a pressure outlet was placed at the exit plane of the duct.
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The hybrid computational mesh contained around Ncv = 1.8 × 106 cells including 6 prism layers with a near-wall resolution of y+ = 80−120 (δ = 0.025in = 635µm). This mesh was intentionally kept rather coarse compared to previous simulations [9] to minimise computing time. For this reason, special care was taken to resolve the areas of interest [19] from the leading-edge extension (LEX) downstream past the vertical tail using a conical mesh density. This provided a nearly constant cells size that was optimal for capturing the downstream development of the vortical structures (cf. figure 1). The size of this focus region [19] was determined in a preliminary RANS simulation with the k-ε model.
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Figure 2: Time history of integral pressure on both sides of the vertical tail. Results
The simulation was started from a non-converged steady-state RANS solution which reduces the initial transient stage until a statistically steady solution develops. A total of N = 50000 time steps were run, each taking about t = 40s on 8 nodes of an Opteron Cluster to reduce the maximum residual three orders of magnitude. This led to a total computing time of about T = 556 hrs = 23 days.
Time Advancement
Rather than attempting to resolve every small detail of the vortex breakdown (which has been achieved by other authors using the same methodology [9, 11]), this paper focuses on the overall prediction of the buffet pressures and the resulting buffet loads acting on the vertical tail. This is a much more challenging
The assessment of the solution cannot be solely be based on the buffet loads. Hence, the first thing to investigate is whether the main flow features are reasonably well resolved.
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Figure 4: F/A-18: Location of vortex breakdown: △, closed diverter DES-k-ε ; ▽, open diverter DES-k-ε ; Experimental data obtained with closed diverter [24].
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During the simulation 32 pressure signals on opposite sides of the vertical fin (see figure 5) were recorded for analysis using standard signal-processing techniques. In this post-processing step the mean pressure P, root-mean-square (RMS) pressures PRMS and power-spectral densities (PSD) from the buffet pressures PB = Pout − Pin were derived. All pressures were normalised with the dynamic pressure of and the mean values were subtracted from the pressure values before calculating the spectra. The calculation of the spectra followed the method of Meyn and James [8] in order to be consistent with previous experimental data sets. Data was split into blocks of 512 points with a 50% overlap (256 points). A Hanning window was applied to each window to reduce bandwidth leakage and all local PSD values were averaged to give the time-averaged values. This procedure was implemented in a MATLAB script [1].
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Statistical Results
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The location of the vortex breakdown in relation to the airframe is one key indicator in the assessment of the simulation quality. Figure 4 displays streamlines released from the tip of the LEX where the longitudinal vortex originates. At x/L ≈ 0.5 they escape the low-pressure region associated with the vortex core and large scale vortical structures develop downstream. The area between the vertical dashed lines indicates range of experimental and numerical data [9, 24]. As mentioned earlier, the flow through the diverter duct shifts the location of the vortex breakdown 5-10% downstream. This can be confirmed by the present results shown in figure 4.
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In figure 3 the vortical structures dominating the flow on the upper side of the aircraft are visualised by a representative quantity called the Okubo-Weiss criterion (Q = − 12 {S : S − Ω : Ω}2 ; Q = 2000) which includes the second invariants of the rate-ofstrain S and rotation tensors Ω [3]. The LEX vortices in figure 3 can be identified by their conical shape. Further downstream, the LEX vortices burst and split into smaller vortices which interact with the vertical tail. As mentioned earlier, the computational mesh was kept homogeneous in this vortex-dominated region to promote a natural evolution of the vortical structures without distortion by excessive cell stretching (see figure 1).
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Figure 3: F/A-18: Isosurface of Okubo-Weiss Criterion.
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Figure 5: F/A-18 with closed diverter: Distribution of mean pressure coefficient on vertical tail based on recorded pressure locations only. Pressure Distribution
Figure 5 shows the mean pressure coefficient (CP = P/Pdyn ) distribution on either side of the vertical tail. From figure 5 it is evident that the largest mean pressure difference occurs at the root of the leading edge, whereas the rear of the tail experiences a much weaker mean pressure difference.
product in tensor notation: A : A = Ai j Ai j
The distribution of the RMS pressure coefficient shown in fig-
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Figure 6: F/A-18 with closed diverter: Distribution of RMS pressure coefficient on either side of vertical tail: DES vs. EXP [8]. DES
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Figure 7: F/A-18 with closed diverter: Distribution of differential RMS pressure coefficient on either side of vertical tail: DES vs. EXP [8].
Figure 7 depicts the distribution of the RMS differential pressure coefficient on the fin, which is the cause of the bending moment. The overall agreement of DES and experiment is quite satisfactory, despite the under-predictions of buffet pressure at the upstream root of the tail and the failure to predict the local pressure extrema in the centre of tail. However, considering the pressure fluctuations (inboard EXP, fig. 6) the quality of the experimental results could be questionable.
ure 6, contains information on the dynamics of the flow for both the DES and experimental data EXP [8]. On the outboard side of the tail, the lowest pressure fluctuations occur right in the path of the main vortex at the 50% spanwise position towards the top rear corner of the tail. On the opposite side, the upstream central part is the most fluctuating part, particularly at the leading edge, which is missed by the DES. However, most areas, especially the rear part, are in favourable agreement. Remarkably, even the odd spot at 70% chord and 30% span in the DES data showing high levels of pressure fluctuations is present to a slightly lesser degree in the experimental data. This provides some confidence in the DES results.
The relative error for the pressure fluctuations at the 32 pressure locations is of the order of 10%-20% (the maximum is about 50% at lower left corner), but remarkably the relative errors in
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the differential pressures are smaller than the maximum errors of the RMS pressures at the same location. This happens because absolute errors of the pressure from the current simulation and the experiment are of opposite sign and partly cancel each other out. This suggests that, despite an over- and underprediction of the pressure fluctuations, the differential pressure is reasonably well captured. Hence, this might suggest that the model is able to predict the correct dynamics and vortex-tail interaction of the flow despite some errors in the estimation of the buffet pressure levels.
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Bending Moments
The bending moment is directly affected by the differential pressures acting on the vertical fin. Each pressure signal represents a part on the surface of the fin, and the resulting forces and moments associated with these local pressures can be calculated for each time step to give a time series of the bending moment on the fin. These moments are calculated by integrating the the N=16 local moments
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N Mb (t) = ∑ mi (t) , with mi (t) = li Ai Pout,i (t) − Pint,i (t) .
Figure 9: Spectral distribution of bending moment coefficient on vertical tail; EXP [8].
The surface areas Ai are sub-divided by the blue solid lines shown in figure 8 and li is the distance between the centroid of the surface area Ai and the root of the tail.
The area under each curve in the power-spectral density (figure 9) represents the RMS of the bending moment coefficient and are summarised in table 1. The differences appear to be large. However, they clearly show the drop of the bending moment due to the diverter channel, which was closed in the experiment, but present in the normal operation of the F/A-18. Comparison with later results using more than 16-pressure pairs revealed that the approximation of the tail surface area has a strong impact on the estimation of the bending moment [5]. It can be expected that a large number of pressure points will improve the quality of the results without adding extra costs to the simulation.
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Table 1: RMS of bending moment coefficients.
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Conclusions
Figure 8: Bending moments: reference surface areas on the vertical tail.
This paper documents the use of numerical modelling to estimate buffet loads on a vertical tail of an F/A-18 twin-tail fighter aircraft. The method of detached-eddy simulation (DES) was adopted to predict the evolution of vortex breakdown and vortex-tail interaction in a time-accurate manner. During the simulation, the unsteady pressures were recorded and analysed. Despite the fact that a fairly coarse mesh was used to allow for reasonable computing times, the results show that the leadingedge vortex, as well as its breakdown, was accurately captured by the simulation. Satisfactory agreement of the surface pressure distributions on both sides of the vertical tail was obtained. The spectral distribution of the differential pressure time history revealed that the dynamics of the flow was well resolved, but the bending moment itself was under-estimated. This can be attributed to the insufficient number of pressure points on the tail surface used to calculate the bending moment. Although a larger number of pressure locations will improve the force approximation and the prediction of the bending moment, further simulations will be carried out with a finer mesh to resolve more detail of the complex vortex structure, which is most likely to produce more accurate results as the current model.
Figure 9 displays the power-spectral density of the bending moment as a function of the reduced frequency F = f c/U. The peak of the bending moment for almost all curves is in the range of F = 0.50 − 0.60, which is in good agreement with values reported [8]. Even though the current DES results under-predict the experimental value by roughly a factor of two, valuable conclusions can still be drawn. The results depend on the duration of the simulation and they start to converge towards a single solution at about t ≈ 20s, as indicated earlier. Selecting shorter time periods lead to larger values of the bending moment, which is consistent with findings by Levinski [5]. The effect of the diverter duct is also clearly visible, but no evidence to support this result could be found in the literature. As reported, the diverter slot shifts the vortex breakdown downstream, and the pressure distribution changes as a consequence. The diverter acts as an active flow control device, and stabilises the flow similar to a wall jet or ducts in multi-component highlift airfoils to enhance turbulent momentum transfer. In the present case, this reduces the differential pressure and the bending moment decreases.
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