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Zonal Detached Eddy Simulation (ZDES) of the ﬂow around the AVT-183 diamond wing conﬁguration Sébastien Deck a,∗ , James M. Luckring b a b

ONERA, The French Aerospace Lab, F-92190 Meudon, France NASA Langley Research Center, Hampton, VA, 23861, USA

a r t i c l e

i n f o

Article history: Received 23 October 2015 Received in revised form 9 February 2016 Accepted 16 February 2016 Available online xxxx Keywords: Diamond wing Vortex ﬂow Unsteady aerodynamics ZDES

a b s t r a c t The onset and progression of vortex separation from a round leading edge on a moderately-swept diamond wing was investigated using the Zonal Detached Eddy Simulation (ZDES). Calculations were performed using mode 2 of ZDES that is appropriate for pressure-gradient-induced separation and that includes automated switching between the RANS and LES regions of the ﬂow. The computations were performed early in conceptual design of the AVT-183 project to contribute unsteady ﬂow assessments to other RANS-based steady ﬂow simulations. The salient features of the roll-up of two vortex sheets are investigated together with a spectral analysis of the ﬂow dynamics. © 2016 Elsevier Masson SAS. All rights reserved.

1. Introduction The AVT-183 program [1] was focused on creating a simple wing shape, perhaps even a unit problem, to study bluntleading-edge vortex separation at conditions relevant to current UCAV concepts. Early conceptual design and numerical design-byanalysis activity [1] resulted in a simpliﬁed diamond wing, with 53◦ leading-edge sweep, that developed the desired ﬂow; the onset of blunt leading edge vortex separation occurred about half way down the wing leading edge at a nominal angle of attack of 12◦ . However, all of the numerical design-by-analysis work was performed with steady RANS solvers, and it was understood for the AVT-183 diamond wing, with its moderate leading-edge sweep, that unsteady vortical effects would likely occur within angles of attack of interest to the program. As compared to highly-swept slender wings, much less is known about the unsteady vortex ﬂows for these moderately swept wings. Gursul et al. [2] have provided a very useful overview of low-speed unsteady vortex ﬂows on nonslender delta wings, and a selection of these unsteady vortical phenomena are noteworthy for the present work. Primary vortex shear-layer instabilities can readily form at moderate angles of attack, and a signiﬁcant unsteady behavior of the secondary vortex can accompany these shear-layer instabilities. This collective unsteady behavior can then lead to a signiﬁcant wandering of the

*

Corresponding author. E-mail address: [email protected] (S. Deck).

http://dx.doi.org/10.1016/j.ast.2016.02.020 1270-9638/© 2016 Elsevier Masson SAS. All rights reserved.

vortex core about a mean location. The unsteady secondary vortex and the wandering of the primary core seem to be vortical features of the nonslender wing. Vortex breakdown is, of course, another source for unsteady vortex ﬂow effects, and this too occurs at practical, low-to-moderate angles of attack for the nonslender wing. It is noteworthy that the basic topological structure, as well as ﬂow details, of vortex breakdown is distinctly different from the breakdown characteristics observed with slender wings. For example, the vortex core ﬂow exhibits three distinct longitudinal zones for the nonslender wing under vortex breakdown conditions, whereas the slender wing would typically exhibit two. Finally, the primary vortex ﬂow reattachment line is situated further outboard on the nonslender wing, and remains so for a wider angle of attack range, as compared to the slender wing. This provides yet another mechanism for unsteady effects to be manifested on the nonslender wing. The reader is referred to Gursul et al. [2] for many details of these unsteady vortex ﬂows. All of these fundamental features can occur at low-to-moderate angles of attack (e.g., α < 15◦ ) for the nonslender delta wings, and would be manifested on nonslender diamond wings as well. Thus, higher-accuracy numerical simulations (i.e., simulations with unsteady ﬂow physics) were needed early in the program to assess unsteady effects. The aerodynamic objective for the AVT-183 program was to isolate, as much as the ﬂow physics permits, the separation-induced blunt-leading-edge vortical ﬂow on a simple wing shape. The conceptual ﬂow ﬁeld, along with some critical ﬂow physics regions, is shown in the sketch of Fig. 1. The sketch shows an isolated bluntleading-edge vortex separation for the notional 53◦ swept diamond wing, and identiﬁes ﬁve ﬂow phenomena. This would represent

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cussing the Reynolds-averaged data and in the ﬂuctuating pressure and velocity ﬁelds. 2. Test case

Fig. 1. Sketch of ﬂow features.

the most basic vortical ﬂow ﬁeld, although the details of this ﬂow are by no means simple. The ﬁrst phenomenon is incipient separation. Very little is known about the ﬂow details of this incipient separation process that leads to the initiation of the blunt-leading-edge vortical ﬂow. The second phenomenon is the blunt-leading-edge vortex itself. Because of the blunt edge and low sweep, the properties of this vortex will be different from those known in association with the slender sharp-edged delta wing. The third phenomenon is the secondary vortex. This vortex is counter-rotating with respect to the primary vortex and can signiﬁcantly affect primary vortex attributes. The fourth phenomenon is the attached ﬂow on the inboard portion of the wing. Blunt-leading-edge vortex separation can also spawn a small, additional inner vortex near the incipient separation region that is corotating with respect to the primary vortex. Much less is known about this vortex, but it represents a possible ﬁfth ﬂow phenomenon for investigation. It has only recently been studied as part of an RTO project, AVT-113 [3], which included blunt-leadingedge vortical studies for a 65◦ delta wing, Vortex Flow Experiment 2 (VFE-2). Eight summary VFE-2 articles have been published in a special issue [4] of Aerospace Science and Technology. It must be observed that any turbulence model needs to be able to simulate the ﬂow physics of all these phenomena just discussed. All of these vortical separation phenomena are sources of unsteadiness, not only from any one particular phenomenon, but also from the interactions among the phenomena. On the numerical side as the need for higher accuracy simulations has increased, the computational ﬂuid dynamics (CFD) community has in turn put emphasis on assessing the quality of the results and now focuses a great deal of its effort on validation of advanced methods. Note that the validation of inviscid calculations has been primarily focused on the capability to evaluate the wall pressure distribution while the validation of steady viscous calculations has been mainly based on the correct assessment of the boundary layer integral quantities. Now the ﬂow-ﬁeld model has to include a comprehensive, unsteady description of turbulence including ﬂuctuations both in pressure and velocities (see the discussion by Sagaut & Deck [5]). In the AVT-183 program, one of the objectives of this preliminary, unsteady simulation is precisely to get a ﬁrst insight into the spatial organization of the ﬂuctuating aerodynamic ﬁeld. Such knowledge may be especially helpful to the experimentalist in positioning unsteady Kulite sensors and estimating the frequencies of interest. This article is organized as follows. The salient features of the unsteady calculation, including the ZDES approach as well as the computational description, are ﬁrst brieﬂy presented before dis-

This study is focused on the diamond wing designed in the AVT-183 STO Task Group [1]. The model has a root chord equal to cr = 1.2 m. The leading edge sweep is 53◦ and the trailing edge sweep −26.5◦ . The common test conditions for the NACA64006 wing have been retained, namely an incidence α = 12 deg., a free-stream Mach number M = 0.15 and a Reynolds number Re = 2.74 × 106 (based on a m.a.c. = 0.8 m, a total pressure of 1 bar and a total temperature of 288 K). Measurements in the AVT-183 experimental campaign included forces and moments, static and dynamic surface pressures, and detailed ﬂow-ﬁeld measurements for all three mean and ﬂuctuating velocity components. Uncertainty quantiﬁcation and test section ﬂow characterization were also included in the work. Additional information for this experimental campaign has been given by Hövelmann et al. [6]. A structured multi-block mesh was designed based on the common CAD ﬁle deﬁned by NLR [1]. This grid is made of 40 blocks and is based on an O–H topology in order to maintain, as far as possible, square-shaped cells on the wing. The size of the domain is [−10 < Lx/cr < 6] × [−0.075 < Ly/cr < 7] × [−5.5 < Lz/cr < 5.5]. The total number of points is Nxyz = 18 × 106 points. The extent of the computational domain as well as the surface mesh is shown in Fig. 2. In addition, the peniche as well as the ﬂoor boundary layer have been taken into account to simulate the planned experimental conditions. A wall slip condition is applied for −10 < X /cr < −2 and an adiabatic non-slip condition is applied for X /cr > −2. 3. Computational description 3.1. Numerical method Results in the present study are obtained with the FLU3M code developed at ONERA. This code solves the compressible Navier– Stokes equations with a low-dissipation AUSM + (P) convective ﬂux scheme [11] on multiblock structured grids. The time integration is carried out by means of the second order accurate backward scheme of Gear. Further details on the numerical method can be found in Refs. [12,13]. Two turbulence models are assessed. The ﬁrst one is the original Spalart–Allmaras (SA) [14] turbulence model which solves an equation for the pseudo eddy viscosity υ . The production term in the SA model depends on the vorticity, leading to very high values in the core of the vortices. In order to limit very large values in the core of the vortex, Daclès-Mariani et al. [16] modiﬁed the production term (initially for the Baldwin–Barth one-equation equation model) by introducing the strain rate in the production term. This rotation correction has been included in the Spalart– Allmaras model and is referred to as SARC in the following. The time-step of the calculation is ﬁxed to t CFD = 10−6 s which corresponds to a non-dimensional time step t = t CFD (U 0 /cr ) = 4.25 × 10−5 . Temporal accuracy of the calculation was checked during the inner iteration process (four Newton-type inner-iterations are used to reach second order time accuracy). A decrease of the inner-residuals of at least one order is obtained. The simulations are performed on 32 quad-core Nehalem X5560 processors. The CPU cost per cell and per inner iteration is about 3 × 10−6 s. 3.2. Zonal Detached Eddy Simulation (ZDES) The ZDES was ﬁrst proposed by Deck [8] and the complete formulation has been recently published in Ref. [9]. This method

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Fig. 2. Computational domain and grid details.

Fig. 3. Classiﬁcation of typical ﬂow problems and corresponding ZDES modes. Mode 1: separation ﬁxed by the geometry, Mode 2: separation induced by a pressure gradient on a gently curved surface, Mode 3: separation strongly inﬂuenced by the dynamics of the incoming boundary layer (adapted from Ref. [9]).

belongs to the family of multiresolution approaches and is initially based on the Spalart–Allmaras (SA) RANS model but can be extended to any eddy viscosity model. This approach takes full advantage of its zonal nature, not only to allow the user to specify RANS and LES regions, but also to make possible the use of various formulations within the same calculation. ZDES also provides an ‘automatic’ operating option (referred to as mode 2 in the following) for which the switch between RANS and LES regions is dynamically set by the model itself. As shown in Fig. 3, mode 2 of ZDES provides a speciﬁc deﬁnition of the subgrid length scale according to the ﬂow resolution. In attached regions, = max as in standard DDES the subgrid lengthscale reads as to permit a safe RANS treatment of the attached boundary layer while in separated ﬂow areas, the subgrid length scale becomes = vol or ω to permit a rapid development of the LES con tent. The switch between the two subgrid length scales is done automatically using a threshold value f d0 = 0.8 for the f d function. Thus, ZDES is ﬂexible in the treatment of turbulent ﬂows in

technical applications and has been applied often with good results over a range of Mach numbers and conﬁgurations (see Refs. [9,19]). To guide the aerodynamicist through the simulation process, a system based around ﬂow taxonomies is contained in the framework of ZDES. Three speciﬁc hybrid length scale formulations (see Eq. (1)), also called modes, are optimized to be employed on three typical ﬂowﬁeld topologies as illustrated in Fig. 3. Mode 1 concerns ﬂows where the separation is triggered by a relatively abrupt variation in the geometry; mode 2 is retained when the location of separation is induced by a pressure gradient on a gently curved surface, and mode 3 for ﬂows where the separation is strongly inﬂuenced by the dynamics of the incoming boundary layer (see Fig. 3). All these ﬂow cases may be treated by the same ZDES technique in its different modes. An example where the three modes of ZDES are used at the same time on a curvilinear geometry can be found in Ref. [20].

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Fig. 4. Skin friction lines on the wing model and on the wind tunnel ﬂoor and eddy viscosity ﬁelds (RANS-SARC calculation).

Fig. 5. Mean pressure distribution C p on the wing. From left to right: RANS-SA/RANS-SARC/ZDES.

Though the method can be adapted to any turbulence model, in the underlying SA model [14], d w is replaced with dZDES in the model according to:

⎧d w ⎪ ⎪ ⎪ I ) ⎨ dDES ( dZDES = II ) ⎪ dDES ( ⎪ ⎪ ⎩ III ) dDES (

if mode = 0 (i.e. RANS) if mode = 1 if mode = 2

(1)

if mode = 3

is the new length scale entering ZDES. In ZDES, the where is not a minor adjustproposal of a new subgrid length scale ment of the detached eddy simulation (DES) formulation, because the modiﬁed subgrid length scales depend not only on the grid (x, y , z) (e.g. = max(x, y , z) in DDES [15]), but also on the velocity gradients (U i , j ) and eddy viscosity ﬁelds (νt ), because

≡ ( x, y , z, d w , U i , j , υt )

(2)

It is argued in [9] that this modiﬁcation signiﬁcantly improves the issue of delay in the formation of instabilities in shear layers that may be observed with standard DDES [10]. Note that this aforementioned delay (i.e. slow LES content formation) can have a large effect on the pressure ﬁeld. As the location of separation is not known in advance on this round leading-edge, mode 2 of ZDES is retained for the unsteady calculation as it permits operation in an ‘automatic’ manner. A similar approach was adopted by Riou et al. [10] for a delta wing ﬂows.

4. Results/discussion 4.1. Reynolds averaged data and aerodynamic loads A ﬁrst glimpse into the ﬂow structure is presented in Fig. 4 that shows the skin friction lines as well as the eddy viscosity ﬁeld at several streamwise locations. The ﬂow is mainly organized around a main vortex sheet named V I in the following. The onset of separation is located around halfway down the wing leading edge. It is also observed that due to the peniche a low inﬂuence of the thick wind tunnel ﬂoor boundary layer is observed on the ﬂow around the wing as no ﬂow separation is observed at the junction between the peniche and the wind tunnel ﬂoor. The pressure coeﬃcient on the wing is displayed in Fig. 5 for both RANS (SA [14] & SARC [16]) and ZDES calculations. The most downstream location of separation onset is provided by the ZDES calculation where a much more spread out aspect of the C p distribution characterizing ﬂow unsteadiness is observed. Of interest, the footprint of a second inner vortex V II is only obtained with the ZDES calculation. It is noteworthy that the unsteady ZDES simulation did not fundamentally alter the prior design objective for vortex separation to occur near the middle of the leading edge at these conditions analyzed. As mentioned in the introduction, much less has been known about this second, inner vortex (discussed in association with Fig. 1). However, the recent accomplishments from numerical studies of AVT-183 have begun to establish understanding of this vortex ﬂow. The inner vortex seems to have its origin near the emer-

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Fig. 6. Total pressure loss 1 − P t / P t0 ﬁelds.

Fig. 7. Mean pressure distribution C p at several sections along the wing (b is the local span of the wing).

gence of the blunt leading-edge vortex from the incipient leadingedge separation region. The computations from Hitzel et al. [17] provide an explanation of a mechanism for the inner vortex separation process, while those from Frink [18] provide a very detailed view of the incipient separation and blunt-leading-edge vortex emergence phenomena. It is noteworthy that these results come from a variety of numerical formulations (structured and unstructured grids, various turbulence models) and yet the fundamental features of the blunt-leading-edge separation (incipient separation, blunt leading edge vortex, inner vortex) were always present. Many other simulations from AVT-183 also demonstrated these vortex separation features. However, details of these vortices vary among the results, and further research will be needed to achieve a quantitative predictive capability for this aggregate blunt-leading-edge vortex ﬂow. The total pressure loss ﬁeld, 1 − P t / P t ,o , is shown in Fig. 6 and further demonstrates that the inner vortex V II is not captured

with the present RANS-SARC calculation. Though not shown here, a similar ﬂowﬁeld is observed with the SA calculation. One can nevertheless notice that the separation occurs earlier when the rotation correction (SARC) is active (see Fig. 5). To achieve a more quantitative insight, Fig. 7 displays the C p distribution in different sections along the wing. While no differences are observed between the different calculations for x/c ≤ 0.305, both RANS calculations indicate a nearly constant pressure level at the most downstream location (e.g. x/c = 0.6). In other words, the size of the separated area is overestimated with the present RANS calculations. Though the pressure distribution on the wing is then dramatically modiﬁed, only minor differences are observed on the lift and drag coeﬃcients (denoted as C D and C L ) as summarized in Table 1. Indeed, the values of the lift coeﬃcient (C L ) are nearly the same and a 8% lower value for the drag coeﬃcient (C D ) is observed for the ZDES calculation.

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Table 1 Force coeﬃcient with the wing alone. Model

CD

CL

RANS-SA RANS-SARC ZDES

0.0942 0.0969 0.0877

0.607 0.597 0.609

To gain further physical insight into this ﬂow, the unsteady properties of the aerodynamic ﬁeld are investigated in the next section. 4.2. Velocity ﬂuctuations Fig. 8 shows the turbulent structures by computing a positive value of the Q criterion where the second invariant of velocity gradient tensor Q = 1/2(||Ω|| − || S ||) is positive and S and Ω denote respectively the strain and the rotation tensor. The roll-up of two main vortex sheets named V I and V II can be clearly identiﬁed together with the wake shown by the total pressure loss. This dual vortex structure had also been demonstrated in the earlier EARSM-RANS-based design-by-analysis work [1]. Of interest, the unsteady content of the ZDES simulation did not fundamentally alter this overall vortex topology. In addition, one may notice a crossﬂow instability near the root of the wing, the study of which is beyond the scope of this paper. While the second inner-vortex V II remains stable and attached to the wing, the main vortex around the wing loses its coherence. The resolved non-dimensional turbulent kinetic energy is next investigated, where k/U 02 is given by

k U 02

=

u 2rms + v 2rms + w 2rms 2U 02

Fig. 8. Turbulent structures displayed by the Q criterion and colored by the streamwise velocity with the total pressure loss shown in the background. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

(3)

The second inner-vortex V II features low-to-moderate levels of turbulent kinetic energy as well as a very quick growth as highlighted in Fig. 9. These levels of turbulent kinetic energy are induced by the interaction between the vortical structures embedded in the shear layer and the leading edge vortex (see Ref. [22]). Nevertheless at the most downstream locations, the maxima are located closer to the wall and not within the separating shear layer. The core of the main leading edge vortex V I is the place of especially intense turbulent phenomena. Indeed, one can notice in Fig. 10 that the ﬂow in the leading-edge vortex core strongly decelerates to reach negative values near x/cr ≈ 0.6. This rapid decrease of the streamwise velocity is also associated with a rapid increase of the streamwise turbulent kinetic energy which can reach values as high as 13%. These two features are classically acknowledged to characterize the vortex breakdown [10] and are in line with the experimental ﬁndings by Hövelmann et al. [7]. Indeed, these authors have shown, thanks to stereo PIV measurements, that the main vortex features bursting tendencies after its formation. This particular feature observed on this moderate leading-edge wing sweep and rounded leading contour differs from the typical vortex burst characteristics known from classical sharp delta wing conﬁgurations. To get a better description of the unsteady ﬁeld, several sensors have been deﬁned inside the ﬂowﬁeld as shown in Fig. 11. Note that the frequency range is given both in Hertz (relevant for the design of the experiment) and normalized by the free-stream velocity U 0 and the root chord c (i.e. Stc = f · c /U 0 ) in order to better identify physical phenomena. One can notice that spectra of streamwise velocity display a very broadband aspect since energy is observed up to frequencies of Stc ≈ 200. The spectra of sensors V60 and V61 clearly highlight the vortex merging dynamics (Kelvin–Helmholtz instability) at St c ≈ 7–14 in the mixing layer surrounding the vortex sheet V I . As an example, the spectrum for

Fig. 9. Turbulent kinetic energy ﬁelds.

Fig. 10. Evolution of turbulent kinetic energy and mean streamwise velocity along the leading-edge vortex core.

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Fig. 11. Left: iso surface of the Q criterion and location of sensors (the blue symbols indicate wall pressure sensors and the red symbols indicate sensors located in the ﬂowﬁeld). Right: Power Spectral Density (PSD) of the streamwise velocity component u at discrete locations. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

sensor V61 displays a peak near f ≈ 600 Hz (or Stc ≈ 14). This frequency can be compared with the one given linear stability theory [21] which indicates that the two-dimensional linearly most unstable mode has a streamwise wave number nearly equal to 7δω where δω is the vorticity thickness deﬁned by:

δω =

U

(4)

max N ( dU ) dN

where U = U high − U low denotes the difference between the local maximum and minimum velocities and N the shear-normal direction. The most ampliﬁed frequency for a spatially developing mixing layer between two streams with respective velocity U high and U low may be given by

f ω = (U high + U low )/7δω /2

(5)

One can then compute the local vorticity thickness δω from the local wall-normal velocity proﬁle crossing sensor V61 and one gets δω /c ≈ 6 × 10−3 leading to Stc = f ω c /U 0 ≈ 12 which is not so far from the observed frequency content in Fig. 11 since the present mixing layer differs from a planar one. According to Menke et al. [23,24], this frequency range (i.e., Stc > 10) is characteristic of the Kelvin Helmholtz instability and higher than the frequency range classically characterizing the helical mode (0.9 < Stc < 4) and the vortex shedding (0.2 < Stc < 0.6). Sensor V50 is located in the second vortex V II , whose separation onset occurs close to the leading edge of the wing. This vortex grows close to the wall and the ﬁne scale structures feature much higher frequency dynamics since a spectral hump is observed near Stc ≈ 30 (Kelvin Helmholz instability mode). Recall that this second vortex is not simulated with the present SA or SARC calculations (see Fig. 5). Nevertheless, the energy content of vortex V II is small compared with V I . 4.3. Pressure ﬂuctuations Fig. 12 highlights the distribution of pressure ﬂuctuations C p ,rms = P rms /(1/2γ M 02 P 0 ) on the upper-side of the wing. Recall that in mode 2 of ZDES, the attached boundary layers are treated in URANS mode so that the unsteady separated ﬂow effects can more readily contribute to the unsteady character of the wall pressure ﬁeld. First note that the footprint of the unsteady ﬂow over the wing is highly three-dimensional and though qualitative, the extent of the “dynamically active area” is of interest to focus the area of experimental investigation. The onset of separation occurring near the middle of the leading edge is clearly visible. The highest levels of pressure ﬂuctuations which can reach up to 40%

Fig. 12. C p ,rms distribution on the wing.

of the free-stream dynamic pressure are located in the impingement region of the main vortex sheet V I . The Power Spectral Density (PSD) function of pressure ﬂuctuations, named G ( f ) and expressed in Pa2 /Hz describes how the mean squared-value of the wall pressure previously described is distributed in frequency since:

∞ 2 P rms

=

G ( f )df

(6)

0

Several sensors along four lines named L 1 to L 4 have been deﬁned and are plotted in Fig. 13 together with a snapshot of the wall pressure distribution. As an example, the spectral map ( f · c /U 0 , x/c) of pressure ﬂuctuation for line L 2 located under the main vortex sheet is given in Fig. 13. Two slices named respectively a) and b) at stations x/c = 0.62 and 0.68 are extracted from this spectral map. The bandwith of the pressure signal is observed for normalized frequencies f · c /U 0 ≤ 35. In addition, the spectrum at x/c = 0.62 displays sharps peaks that emerge from the broadband content. The main peak is observed at Stc ≈ 14 together with its ﬁrst sub-harmonic at Stc ≈ 7. Further downstream at station x/c = 0.68, the relative intensity of the fundamental frequency decreases to the beneﬁt of its subharmonics. This behavior is characteristics of the merging process of the large-scale structures populating the mixing layer in the

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Fig. 13. PSD of wall pressure ﬂuctuations. Top left: spectral map along line L2; Top right: instantaneous wall pressure. Bottom: PSD at x/c = 0.62 (rake a) and x/c = 0.68 (rake b).

5. Conclusions

Fig. 14. PSD of Lift and Drag components.

main vortex sheet. Note that the corresponding physical length scale is given by:

λ c

=

U0 f ·c

=

1 Stc

≈

1 14

≈ 0.071

(7)

which corresponds to the length λ ≈ 0.066c identiﬁed on the instantaneous footprint of the wall pressure. This dynamics characterized by spectral humps emerging from the broadband spectral content affects the whole aerodynamic ﬁeld as can be seen in the PSD of the lift and drag coeﬃcients, Fig. 14. The knowledge of the spectral content of the dynamic loads may be of interest for both aero-structural and aerodynamics/ﬂight mechanics coupling. Lift and Drag coeﬃcients feature a similar spectral content, and this content is characterized by contributing frequencies for Stc ≤ 10. As discussed earlier, this frequency range does not correspond to the Kelvin–Helmholtz instability which occurs at a higher frequency. Conversely, the spectral hump near Stc ≈ 2 indicates that large scale phenomena as helical modes and vortex shedding contribute mostly to the ﬂuctuating pressure load.

The work with ZDES provided the ﬁrst unsteady simulations of the blunt-leading-edge vortex separation about the AVT-183 diamond wing conﬁguration. This work was performed early in the program to assess the unsteady ﬂow effects on a preliminary design that had been achieved with steady RANS methodology. The ZDES simulations demonstrated signiﬁcant unsteady content in the primary leading edge vortex (V I ) as well as an inner vortex (V II ). This is the ﬁrst documentation of unsteady content in an inner vortex of which the authors are aware. While the second inner vortex V II remains stable and attached to the wing, the main vortex around the wing features bursting tendencies after its formation. This particular feature observed on this moderate leading-edge wing sweep and rounded leading contour differs from the typical vortex burst characteristics known from classical sharp delta wing conﬁgurations. Despite this unsteady content in the vortices, the ZDES results demonstrated that the overall location and topology the vortical ﬂow system on this blunt-edge diamond wing was similar, in a preliminary design sense, to the vortex ﬂows developed from the initial RANS-centric design-by-analysis activity. The ZDES results thus contributed to advancing the AVT-183 work from conceptual development to program execution. These results also provided the initial guidance for the placement of unsteady pressure sensors in planning a suite of wind tunnel investigations. Conﬂict of interest statement No conﬂict of interest. Acknowledgements The work was part of a NATO/STO program entitled “Reliable Prediction of Separated Flow Onset and Progression for Air and Sea Vehicles”, also known as AVT-183. The authors appreciate the opportunity to work within the Applied Vehicle Technology (AVT) sector of the STO. The work has been supported by a number

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of program and project oﬃces. These include the NASA Revolutionary Computational Aerosciences (RCA) and the Environmentally Responsible Aircraft (ERA) projects, and the French Aerospace Laboratory ONERA. All of this support is appreciated. References [1] O.J. Boelens, J.M. Luckring, C. Breitsamter, A. Hövelmann, F. Knoth, D.J. Malloy, S. Deck, Objectives, approach, and scope for the AVT-183 diamond-wing investigations, Aerosp. Sci. Technol. (2016), submitted for publication. [2] I. Gursul, R. Gordnier, M. Visbal, Unsteady aerodynamics of nonslender delta wings, Prog. Aerosp. Sci. 41 (2005) 515–557. [3] RTO, Understanding and modeling vortical ﬂows to improve the Technology readiness level for military aircraft, Oct. 2009, RTO-TR-AVT-113. [4] D. Hummel, R. Cummings (Eds.), Special Issue, VFE-2, Aerosp. Sci. Technol. 24 (1) (2013). [5] P. Sagaut, S. Deck, Large eddy simulation for aerodynamics: status and perspectives, Philos. Trans. R. Soc. A 367 (2009) 2849–2860. [6] A. Hövelmann, F. Knoth, C. Breitsamter, AVT-183 diamond wing ﬂow ﬁeld characteristics Part 1: varying leading-edge roughness and the effects on ﬂow separation onset, Aerosp. Sci. Technol. (2016), http://dx.doi.org/10.1016/j.ast. 2016.01.002. [7] A. Hövelmann, M. Grawunder, A. Buzica, C. Breitsamter, AVT-183 diamond wing ﬂow ﬁeld characteristics Part 2: experimental analyses on leading-edge vortex formation and progression, Aerosp. Sci. Technol. (2016), http://dx.doi.org/ 10.1016/j.ast.2015.12.023. [8] S. Deck, Zonal detached eddy simulation of the ﬂow around a high-lift conﬁguration, AIAA J. 43 (2012) 2372–2384. [9] S. Deck, Recent improvements in the Zonal Detached Eddy Simulation (ZDES) formulation, Theor. Comput. Fluid Dyn. 26 (6) (2012) 523–550, http:// dx.doi.org/10.1007/s00162-011-0240-z. [10] J. Riou, E. Garnier, C. Basdevant, Compressibility effects on the vertical ﬂow over a 65◦ sweep delta wing, Phys. Fluids 22 (2010) 035102.

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