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Improved delayed detached-eddy simulations of sawtooth spoiler control before supersonic cavity Kunyu Luo a, Weng Zhe b, Zhixiang Xiao a,∗, Song Fu a a b

School of Aerospace Engineering, Tsinghua University, Beijing 100084, China AVIC Shenyang Aircraft Design & Research Institute, Shenyang 110035, China

a r t i c l e

i n f o

Article history: Available online xxx Keywords: Sawtooth spoiler Cavity ﬂow Flow control IDDES

a b s t r a c t Flows at Ma = 1.5 over a cavity with different leading-edge sawtooth spoilers were numerically studied using improved delayed detached-eddy simulation based on the two-equation shear stress transport model, coupled with the adaptive dissipation scheme. Transonic ﬂow past M219 cavity was chosen as the validation case for numerical methods and mesh convergence. Comparison of predicted sound pressure levels and spectra against the measurements has proved that the major oscillation dynamics inside the cavity is captured numerically. Five sawtooth spoilers were then evaluated before the leading edge of a complex irregular cavity in a supersonic ﬂow. It is found that the spoiler lifts the shear layer, preventing its reattachment on the cavity ﬂoor, and completely changes the cavity ﬂow type. The presence of sawtooth is necessary to promote instability more upstream. All spoilers show signiﬁcant reduction in pressure ﬂuctuation. Pressure decrease on the cavity ﬂoor and rear wall contributes to overall drag reduction, despite the extra drag by the spoiler. The spoiler with height equal to the local boundary layer thickness and a 90° tooth angle achieves optimal and balanced performance in both drag reduction and noise suppression among the evaluated spoilers. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Dynamic load or acoustic noise within a cavity at transonic or supersonic speed is an important research area in the ﬁeld of ﬂuid mechanics, especially in the weapon bay design of a stealth aircraft. When exposed to high speed free-stream ﬂow, the cavity experiences an intense aero-acoustic environment. The ﬂow past a cavity is accompanied by extremely unsteady and complex features, including boundary layer separation, shear layer instabilities, pressure oscillations, impingement to the rear wall and acoustic noise. Interactions among these features can result in damage to the cavity’s internal equipment or the structure, or cause unexpected inﬂuence on store releasing. High acoustic noise levels can also limit the ﬂight envelope when releasing stores inside the cavity. To this effect, it is extremely important to understand the ﬂow mechanisms at work throughout the cavity and to ﬁnd ways to reduce dynamic loads. Cavity ﬂows can be divided into three categories based on the length-to-depth ratio and free-stream Mach number: open, closed

∗

Corresponding author. E-mail address: [email protected] (Z. Xiao).

and transitional. Open cavity ﬂow occurs for deep cavities such as a bomber’s weapon bay, while closed cavity ﬂow occurs for shallow cavities like a ﬁghter’s weapon bay. Transitional ﬂows occurs between the boundaries of open and closed ﬂow. Shear layer behavior differs most among the three categories. In an open cavity ﬂow, the shear layer stretches straight across the cavity opening. In a closed cavity ﬂow, the shear layer reattaches on the cavity ﬂoor and then separates again downstream. In a transitional cavity ﬂow, shear layer behavior could be an alternate combination of the two above. For transonic or supersonic ﬂows, different shear layer behaviors can lead to complex wave systems, both inside and outside the cavity. Though cavity ﬂows have been under research for more than sixty years, the ﬂow mechanisms of complex geometries and related ﬂow control strategies remain the focus of the scientiﬁc and industrial community. Numerical methods which provide both qualitative and quantitative insight into ﬂow details are a satisfactory choice to facilitate cavity ﬂow studies without the use of expensive wind tunnel tests. Interested readers may refer to the comprehensive review by Lawson and Barakos (2011) who systematically report on over 60 experimental and computational cavity ﬂow studies before 2010. Early studies on cavity ﬂows were typically conducted by solving Reynolds-Averaged Navier-Stokes (RANS) or unsteady RANS (URANS) equations with various turbu-

http://dx.doi.org/10.1016/j.ijheatﬂuidﬂow.2017.01.012 0142-727X/© 2017 Elsevier Inc. All rights reserved.

Please cite this article as: K. Luo et al., Improved delayed detached-eddy simulations of sawtooth spoiler control before supersonic cavity, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatﬂuidﬂow.2017.01.012

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Fig. 1. Distributions of eddy viscosity and adaptive dissipation function in the cavity with a spoiler.

lence models. But recent studies (Peng, 2006; Liggett and Smith, 2011) have conﬁrmed that they are incapable of simulating the high-frequency small scale turbulence that dominates in cavity ﬂows, not to mention the acoustic tones or broadband noise. Thus only very few studies (Aradag et al., 2010) are still using traditional URANS now. On the other hand, direct numerical simulation (DNS) which resolves all the scales on a very ﬁne grid, offers the highest accuracy. However it has limited application to cavity ﬂows at high Reynolds number with computation resources currently available. Few studies using DNS (Gloerfelt et al., 2003; Bres and Colonius, 2007; Sun et al., 2014) were restricted to low Reynolds numbers or two-dimensional ﬂows. Large-eddy simulation (LES) is affordable nowadays thanks to the growing computation power, and is being used in studies of cavity ﬂows (Aybay et al., 2010; Li et al., 2013). Some of them are quite impressive. For example, Morton et al. (2012) studied a 1/15-scale F-22 main weapon bay model with interior details such as door hinges and rods by LES on an unstructured mesh. However its purpose was more of a code benchmark than a scientiﬁc investigation with ﬂow details. But to accurately resolve a large range of turbulent scales in wall-bounded cavity ﬂows, pure LES still suffers the same problem as DNS. For high Reynolds-number wall-bounded ﬂow, special near-wall treatments like wall functions are usually necessary in LES computations. Hybrid RANS-LES methods (HRLMs) use RANS for the nearwall ﬂow and LES for the separated ﬂow. Using RANS in nearwall regions is sometimes regarded as a unique LES near-wall treatment and saves computation resources. Typical HRLMs include detached-eddy simulation (DES; Spalart et al., 1997; Spalart, 2009), constrained LES (CLES; Chen et al., 2012), zonal DES (ZDES; Deck, 2005; 2012), partially-averaged Navier-Stokes (PANS; Girimaji, 2006; Chang et al., 2015a; 2015b) and scale-adaptive simulation (SAS; Menter and Egorov, 2010; Egorov et al., 2010; Davidson, 2006), as well as their combinations (Davidson and Peng, 2013; Davidson, 2014). Among those, DES and its variants are still the most popular in cavity ﬂow simulation. In addition to the HRLM cavity studies summarized in the review of Lawson and Barakos (2011), a lot of HRLM studies have emerged since 2010 (Liggett and Smith, 2011; Temmerman et al., 2012; Wang et al.,

2013; Arunajatesan et al., 2014; Luo and Xiao, 2015; Babu et al., 2015; Sheta et al., 2015; Hassan et al., 2016). Many researches have started dealing with complex geometries like realistic weapon bays and store release problems (Lawson and Barakos, 2010b; 2010a; Kannepalli et al., 2011; Khanal et al., 2011; Chaplin and Birch, 2012; Kim et al., 2015; Barone and Arunajatesan, 2016). Many researchers have also explored methods of cavity ﬂow control and improving the cavity environment. These control strategies can be roughly categorized as passive and active ﬂow controls, according to the need for external energy input (Cattafesta et al., 2008). Passive control methods often have advantages of weight and complexity over active control methods, but are not as versatile over various ﬂight conditions. Lawson and Barakos (2011) and Cattafesta et al. (2008) have summarized passive and active ﬂow control studies respectively. Wind tunnel tests have been employed to test a wide array of control devices, most of which are leading edge modiﬁcations such as serration (Gai et al., 2015), block (Shaaban and Mohany, 2015), transverse rods (Dudley and Ukeiley, 2014) and sawtooth spoilers (Saddington et al., 2016a), as well as active blowing (Zhang et al., 2015; George et al., 2015) and plasma (Yugulis et al., 2014; de Jong and Bijl, 2014) controls. Leading edge devices generally are designed to drive the shear layer up and away from the cavity trailing edge, or to increase its instability and weaken large-scale impingement. Trailing edge modiﬁcations like the traditional trailing edge ramp control (Vikramaditya and Kurian, 2009) are less common, but have also proven effectiveness in suppressing pressure ﬂuctuations. Recent experimental study (Saddington et al., 2016b) of both leading-edge and trailing-edge modiﬁcations on a cavity in transonic ﬂow has reported that leading-edge control techniques are more effective at suppressing cavity tone amplitudes than trailingedge modiﬁcations. Although wind tunnel tests are still currently the most popular method of researching, it is not easy for experimental studies to reach a balance of spatial and temporal resolution. A limited number of computations using advanced CFD methods (like LES or hybrid RANS-LES) are emerging, including leading edge rod (Comte et al., 2009), injection (Wang et al., 2013) and blowing (Zhang et al., 2015). Das Gupta and Roy (2014) designed a plasma actuated receptive channel that performs like a

Please cite this article as: K. Luo et al., Improved delayed detached-eddy simulations of sawtooth spoiler control before supersonic cavity, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatﬂuidﬂow.2017.01.012

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Fig. 2. Meshes of M219 cavity. Left: coarse; right: dense.

Table 1 M219 cavity mesh details. Mesh

Coarse Dense

2. Turbulence modeling and numerical methods

Cell number [Million] Total

Cavity vicinity

Outside

8.4 13.3

2.7 6.7

5.7 6.6

x+

y+

z+

1400 400

0.4 0.4

400 400

rounded trailing edge. Regrettably, evaluations of control devices on real cavity or weapon bay geometries (such as Panickar et al.’s (Panickar et al., 2013) experimental study on an F-35 weapon bay model) are rarely reported. To avoid further system complexity, a well-designed passive control device is surely a more ideal choice. As was pointed out by Lawson and Barakos (2011), though many passive control devices have been proposed for the cavity ﬂow, comparisons between multiple passive ﬂow devices are not often reported. The leadingedge sawtooth spoiler is a typical passive control device for cavities that has signiﬁcant reduction in overall sound pressure level (Saddington et al., 2016b), better than the square-tooth spoiler. In a DES computation (Ashworth, 2008) for a sawtooth spoiler applied to the M219 cavity, the lifting of shear layer over the front part of the cavity was the primary mechanism and the sawteeth was of no signiﬁcance. However, the mesh used in the computations seemed somehow too coarse (1.4 million in total and only 4 cells on each sawtooth edge) to capture the ﬂow past the sawteeth. Lawson and Barakos (2009) also evaluated sawtooth spoiler together with other control devices for the M219 cavity, but only 25% of the total 6 million grid points were located within the cavity. Besides, the geometry used in previous investigations was mostly a simple rectangular cavity, but in real applications both the cavity itself and the incoming ﬂow are complex. In this study, the authors set out to numerically examine the control mechanism and effectiveness of sawtooth spoilers before the leading edge of a complex irregular cavity in a more realistic scenario. Computations were performed on a much ﬁner mesh to resolve small turbulent structures from the teeth, and the drag reduction was also accounted for as an important aspect to spoiler evaluation, as well as noise reduction. The remaining sections begin with a brief description of the numerical methods used in this paper and the M219 cavity validation. Then the control mechanism is discussed by comparing the baseline and controlled cavities. Five sawtooth spoilers of different heights and tooth angles are evaluated, where both drag reduction and noise suppression are measured and compared to determine the optimal design.

To accurately predict high Reynolds number turbulent ﬂows past a cavity, two issues are very important; ﬁrst, the turbulence simulation model, and second, the numerical scheme especially the numerical dissipation level. Proper combination of advanced turbulence modeling methods and adaptive dissipation scheme is essential. In this study, the authors applied improved delayed detachededdy simulation (IDDES) (Shur et al., 2008) based on the twoequation shear stress transport (SST) turbulence model (Menter, 1994) is applied. 2.1. Improved delayed detached-eddy simulation IDDES is an advanced DES-type approach developed from the delayed detached-eddy simulation (DDES) (Spalart et al., 2006). It

Fig. 3. Instantaneous ﬁeld of hybrid function f˜d from the coarse M219 mesh.

Fig. 4. Velocity proﬁles inside the M219 cavity. LES data derived from Ref. (Larcheveque et al., 2004).

Please cite this article as: K. Luo et al., Improved delayed detached-eddy simulations of sawtooth spoiler control before supersonic cavity, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatﬂuidﬂow.2017.01.012

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Fig. 5. OASPL and BISPL on M219 cavity ﬂoor.

150

150

Exp. Coarse Dense

140

140

130

PSD [dB]

PSD [dB]

130

120

110

100

90

100

K20 200

400

600

800

Freq. [Hz]

1000

1200

90

1400

PSD [dB]

PSD [dB]

110

600

800

Freq. [Hz]

1000

1200

1400

Exp. Coarse Dense

120

110

100

K24 200

400

600

800

Freq. [Hz]

1000

1200

150

90

1400

K26 200

400

600

800

Freq. [Hz]

1000

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150

Exp. Coarse Dense

140

1400

Exp. Coarse Dense

140

130

PSD [dB]

130

PSD [dB]

400

130

120

120

110

90

200

140

130

100

K21

150

Exp. Coarse Dense

140

90

120

110

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100

Exp. Coarse Dense

120

110

100

K28 200

400

600

800

Freq. [Hz]

1000

1200

1400

90

K29 200

400

600

800

Freq. [Hz]

1000

1200

1400

Fig. 6. PSD at six typical sample points on M219 cavity ﬂoor.

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successfully ameliorates a few of the major shortcomings inherent to the original DES method including the log-layer mismatch and the separation induced by grid, and has proven beneﬁcial in previous studies (Xiao et al., 2013). Readers interested in the development of DES-type approaches can refer to the review article by Spalart (2009). The construction of IDDES method based on the Menter’s SST model (SST-IDDES) is fairly straightforward. See the turbulence kinetic energy (TKE) equation:

∂ (ρ k ) ∂ (ρUi k ) ˜ ρ k3/2 ∂ ∂k + = Pk − + (μ + σk μt ) ∂t ∂ xi Lhybrid ∂ xi ∂ xi

(1)

where k is the modeled TKE, and the length scale Lhybrid is introduced:

Lhybrid = LIDDES

= f˜d (1 + fe ) × LRANS + (1 − f˜d ) × LLES √

Fig. 7. Baseline “clean” cavity geometry and surface mesh.

(2)

β∗ω

in which LRANS = k/( ) and LLES = CDES are the turbulence length scales of the RANS portion and ﬁlter length scale of the LES portion respectively. ω is the speciﬁc turbulence dissipation rate and β ∗ is a constant in the SST model that equals 0.09. CDES is a DES constant calibrated by the decay of homogeneous isotropic turbulence. In SST-IDDES, CDES takes form of the blending of CDES,k− = 0.61 and CDES,k−ω = 0.78 according to the blending function F1 in SST model. The grid scale is redeﬁned as follows

= min [max (Cw max , Cw d, min ), max ]

(3)

where Cw = 0.15 and d is the distance from the cell center to the nearest wall. The minimum grid scale min is deﬁned as the minimum grid size in x, y and z directions, i.e., min = min (x, y, z ); the max grid scale is deﬁned similarly as max = max (x, y, z ). For the sake of simplicity, we used the minimum grid scale min instead of the grid step in the wallnormal direction hwn used in the original paper (Shur et al., 2008). For a typical mesh set generated for hybrid RANS-LES computation, the wall-normal direction size is the smallest grid scale in most cases, especially for mesh cells near the wall. However, this simpliﬁcation does slightly increase the mesh quality requirement for the near-wall grid in mesh generation. Function f˜d is deﬁned as max [(1 − fdt ), fB ], which is determined by both the geometry-related fB and the ﬂow-related (1 − fdt ). When fe = 0, LIDDES can be rewritten as:

Lhybrid = LIDDES = f˜d × LRANS + 1 − f˜d × LLES

(4)

at which point IDDES reverts to DDES. When fe > 0 and f˜d = fB , LIDDES can be rewritten as:

Lhybrid = LIDDES = fB (1 + fe ) × LRANS + (1 − fB ) × LLES

(5)

and IDDES acts in wall-modeled LES (WMLES) mode near the wall in the transition from RANS to LES. Because 0 < f B = f˜d < 1, we have LIDDES = LW MLES > LLES , which increases the eddy viscosity and avoids insuﬃcient modeling of the Reynolds stress. Though Gritskevich et al. (2012) has proposed some modiﬁcation and optimization for SST-IDDES, the impact on the results are minor. Therefore the detailed formulations of the functions mentioned above are kept the same as in the original proposal (Shur et al., 2008) and are not elaborated here. DES-type methods are known to suffer from the slow switch from RANS to LES due to the convection of the upstream RANS eddy viscosity. It can lead to a possible delay in the formation of instabilities in shear layers. Recent studies are trying to improve this issue (Mockett et al., 2015b). One way is to identify the initial stage of the shear layer and decrease the eddy viscosity by changing the deﬁnition of turbulence length scale (Mockett et al., 2015a; Strelets et al., 2016); another is to generate upstream perturbations to trigger the resolving of turbulence (Shur et al., 2011; 2014). In

Fig. 8. Geometries and meshes of leading-edge sawtooth spoilers. Table 2 Geometric parameters of sawtooth spoilers. Spoiler name

h1

h2

H = h1 + h2

Tooth angle

Big Sharp Smallow Obtlow Obtlower

0.7δ 0.7δ 0.4δ 0.7δ 0.2δ

1.2δ 1.2δ 0.6δ 0.3δ 0.3δ

1.9δ 1.9δ 1.0δ 1.0δ 0.5δ

90° 60° 90° 120° 120°

the present study, IDDES resolves most of the turbulence around the spoiler due to the disturbances from the boundary layer that develops on the fore fuselage before the cavity. Details will be discussed in Section 3.1 2.2. Spatial scheme and adaptive dissipation function The Roe scheme (Roe, 1981) is favored for its ability to capture discontinuities in ﬂow-ﬁelds. The inviscid ﬂux of NavierStokes equations discretized in the Roe scheme, in i-direction for

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Fig. 9. Percentage of the resolved TKE (left) & Reynolds stress proﬁle downstream the “Obtlower” spoiler (right). Y/ = 0 is aligned with the tip of the spoiler.

Fig. 10. Spanwise slices of time-averaged ﬂow-ﬁeld inside the cavity.

Fig. 11. Time-averaged streamlines inside the cavity.

instance, is:

Fi+1/2 =

1 2

FR + F

L i+1/2

1 ˜ R −φ × Ainv q − qL i+1/2 2

(6)

The superscript R and L mark values on the right and left side of the cell interface i + 1/2 computed by 3rd-order MUSCL interpolation. When DES-type methods are applied to simulate turbulent ﬂows, the original Roe scheme is too dissipative to keep the small turbulent ﬂow structures from being smoothed out. In fact, an ideal numerical dissipation should be low enough in the separated region with locally clustered grids to resolve the appropriate turbulence scales, and high enough near the wall and in the irrotational region to avoid any spurious oscillation where the grids are not ﬁne enough. The adaptive dissipation scheme, in which dissi-

pation becomes large near the wall and in the irrotational region while very small in the separation region, is an appropriate choice here. The second term on the right hand side of Eq. (6), which functions as dissipation, is multiplied by an adaptive function φ . This ensures that the scheme dissipation dynamically adjusts to ﬂow characteristics. The following adaptive dissipation function has a hyperbolic tangent form and provides a smooth and quick switch of numerical dissipation in different ﬂow zones:

φ = max φmin , φmax · tanh ACH1

(7)

φ varies from minimum cutoff value φ min to maximum value φmax = 1 depending on the ﬂow characteristics. For the ﬂow separation zones calculated by LES, φ = φmin and the scheme has very low dissipation, which is preferable for resolving small scale ﬂow

Please cite this article as: K. Luo et al., Improved delayed detached-eddy simulations of sawtooth spoiler control before supersonic cavity, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatﬂuidﬂow.2017.01.012

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Fig. 12. Time-averaged pressure coeﬃcients on the cavity ﬂoor.

Fig. 13. Instantaneous isosurface of Q criterion.

structures. As for the far-ﬁeld and the near-wall zones where RANS is used, large dissipation is necessary to suppress the numerical oscillation, then φ = φmax , and Eq. (6) reverts to the original Roe scheme. Details of the adaptive dissipation function available in our references (Strelets, 2001; Mockett, 2009) are not re-illustrated here. In this study, we set φ min to 0.1. Distributions of eddy viscosity and φ shown in Fig. 1 clearly demonstrate that the adaptive dissipation function approaches the minimum φ min =0.1 where the turbulence dominants, while in the near-wall and irrotational regions it remains 1 to maintain an upwind scheme. Low dissipation

near the spoiler and inside the cavity helps IDDES resolve the turbulent structures. 2.3. Other numerical methods The in-house code UNITs (Unsteady NavIer-STokes equation solver) was invoked in our computations. The UNITs code is a density-based Navier-Stokes solver that uses the ﬁnite volume method (FVM) on multi-block structured grids and performs well with the implemented DDES or IDDES in various cases including

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Fig. 14. Streamwise vorticity contours showing spoiler perturbation development.

tandem cylinders (Xiao et al., 2012; Xiao and Luo, 2015), rudimentary landing gear (Xiao et al., 2013), OAT15A airfoil (Huang et al., 2012) and cavity-induced transition in hypersonic boundary layer (Xiao et al., 2015). The turbulence model equations are solved decoupled with the mean ﬂow equations. The solver is parallelized using domain-decomposition and message-passing-interface (MPI) strategies on computer clusters.

For unsteady simulations, a modiﬁed fully implicit lowupper symmetric Gauss-Seidel (LU-SGS) method is employed with Newton-like sub-iterations in pseudo time when solving the mean ﬂow and the turbulence model equations. Global non-dimensional time stepping is also implemented to capture the unsteady properties of the massively separated ﬂows. All IDDES computations were initialized from URANS ﬂow-ﬁelds.

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Fig. 15. Sawteeth affecting on downstream shear layer development.

Fig. 16. Spatial OASPL distributions on the central plane of the cavity.

2.4. Code validation The M219 cavity is a test case often used to provide validation data for cavity ﬂow predictions, for example, in the European DESider project (Haase et al., 2009). The M219 cavity wind-tunnel conﬁguration is a three-dimensional rectangular cavity mounted into a testing rig. The cavity length-to-depth ratio (L/D) and lengthto-width ratio (L/W) are both 5. The computation was performed under transonic conditions of Ma∞ = 0.85, P∞ = 6.3 × 104 Pa, T∞ = 266.53 K and Re = 1.35 × 107 /m (or ReD = 1.36 × 106 based on the cavity depth D). The geometry includes the cavity, a testing rig and wind tunnel walls but not the support sting. Two mesh sets were tested: the coarse one with 8.4 million cells and the dense one with 13.3 million. Due to the carefully designed mesh topology, the outer part of both mesh sets are completely identical. Differences between those two meshes are restricted inside the cavity and around its vicinity.

Fig. 17. Divergence of time-averaged velocity on the central plane of the cavity.

The ﬁrst grid layer is located 2×10−3 away from the cavity walls and y+ is approximately 0.4. Fig. 2 shows the coarse and dense surface meshes. The origin is located at the middle of the cavity leading edge, with the y axis pointing up. Details regarding the two meshes are listed in Table 1. x+ and z+ are taken from the middle of the cavity. Mesh cells inside the cavity are kept as isotropic and orthogonal as possible. The inﬂow/outﬂow boundaries are situated suﬃciently far from the cavity, with a distance more than 5 times the rig length. The rig and the cavity surfaces are set to adiabatic no-slip walls, and the wind tunnel walls are treated as inviscid walls. Fig. 3 shows the

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Fig. 18. Instantaneous density gradient magnitude inside the cavity.

Fig. 19. Locations of sample points at the cavity opening and on the walls.

Fig. 20. Streamwise evolution of pressure PSD at the cavity opening.

instantaneous ﬁeld of IDDES hybrid function f˜d from the coarse mesh simulation. In the attached boundary layer approaching the cavity, the model is in DDES mode, while on the ﬂoor the model works as WMLES, which is just as designed for IDDES. Experimental data (Henshaw, 20 0 0; Stanek et al., 20 0 0) and full LES data (Larcheveque et al., 2004) were compared against current IDDES prediction; time-averaged velocity proﬁles on the cen-

tral plane of the cavity are shown in Fig. 4. Predictions on both coarse and dense meshes generally match with the full LES prediction, though Different mean ﬂow patterns predicted inside the cavity cause some discrepancy. Since the LES computation was performed on a simpliﬁed geometry of a cavity mounted on a ﬂat plate, such discrepancies are reasonable and acceptable. Results on the dense mesh are closer to the LES results.

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The overall sound pressure level (OASPL) and band-integrated pressure ﬂuctuation levels (BISPL) distributions on the cavity ﬂoor are shown in Fig. 5. The integral bandwidth is 100 Hz centered around the spectral peaks (see Fig. 6). It is clear that the spatial mode shape are globally well predicted. Both meshes yield low error level on Rossiter mode energy, lower than < 4 dB at all locations. The results on the dense mesh match with the measure-

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ments especially well. Even though there are some small discrepancies, we can conclude that the current numerical methods and meshes are able to compute the pressure ﬂuctuation energy related to the Rossiter oscillation phenomenon. Pressure power spectral density (PSD) spectra at six typical points on the cavity ﬂoor are estimated by the Burg algorithm (Burg, 1975) and presented in Fig. 6 where results on both meshes

Fig. 21. OASPL distributions on the cavity surface.

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Fig. 22. Pressure PSD at sample points on the cavity surfaces.

successfully capture the Rossiter modes (Rossiter, 1966). The predicted frequencies of the mode peaks seem uniformly larger than the experimental data on both meshes and the coarse mesh generates slightly higher PSD levels than the dense mesh. Higher predicted frequencies are often observed in computations with a selfdeveloping boundary layer on the test rig, which might alter the boundary layer thickness at the cavity leading-edge and therefore change the oscillation frequencies in the cavity. Spectra estimation algorithms may also cause discrepancy. Nevertheless, the current numerical results are qualitatively and quantitativeness comparable to the measurements as well as other hybrid RANS-LES predictions for the M219 cavity. The major oscillation ﬂow dynamics are believed to be correctly captured. In general, the two sets of meshes used here are suitable for cavity ﬂow prediction. The numerical methods used are capable of accurately predicting the unsteady ﬂow past a cavity on a mesh with such density. The dense mesh performs better in terms of dynamic quantities like pressure ﬂuctuations, suggesting that the dense mesh is suﬃcient and more applicable than coarse mesh for unsteady IDDES prediction of cavity ﬂows. In the following section, the authors will use the same numerical strategies and mesh scales described above to investigate the sawtooth spoiler at the leading edge of a complex irregular cavity. 3. Flows past a complex irregular cavity with leading edge sawtooth spoilers Though many previous studies have investigated cavity ﬂows and cavity ﬂow control methods, most of them focus on simple ge-

ometries. In most actual cases, however, cavities do not have regular shape. In this section, we will discuss the ﬂow mechanism of sawtooth spoilers before a complex irregular cavity, and present the results of our evaluation of the effects of spoiler geometric parameters on cavity ﬂow. The ﬂow conditions for the following cases are identical: Ma∞ = 1.5, Re∞ = 2.3 × 106 /m, P∞ = 4.7 × 104 Pa, and T∞ = 202.53 K. Angles of incidence, sideslip and yaw are all zero. The non-dimensional time step is 0.01, which corresponds to a physical time step of 23 μs. 3.1. Description of the geometries The irregular shaped cavity model is embedded in a simpliﬁed aircraft fuselage, where only the lower fuselage surfaces are reserved to let the boundary layer develop itself and thus provide more reasonable incoming ﬂow information, as in most cases the ﬂow at the cavity leading edge is quite different from a uniform ﬂow or a ﬂat-plate ﬂow. The fuselage after the cavity is simpliﬁed as a ﬂat plate. The cavity has serrated leading & trailing edges, and a curved bottom ﬂoor & outer wall, which makes it diﬃcult to deﬁne the length-to-depth or length-to-width ratio, though generally we could say that the cavity is somewhat a shallow one. Fig. 7 shows the surface mesh for the “clean” cavity without spoiler. The total cell number is approximately 9 million, out of which 4.5 million cells are placed within the cavity. The ﬁrst layer of the grids near the wall has a height of 0.002 mm and corresponding y+ less than 0.5. Cell size inside the cavity is kept between 1 mm and 2 mm, leading to x+ and z+ inside the cavity

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less than 500. According to Section 2.4, this mesh scale is capable of capturing the dominant ﬂow dynamics in a cavity. No-slip wall condition is applied on all surfaces, and outﬂow condition is applied to the outlet and the boundary beside the fuselage nose, to let the ﬂow freely exit the domain. Symmetry condition is set for the z = 0 boundary. Outer boundaries including the inlet are set to the far-ﬁeld condition. As shown in Fig. 8, ﬁve types of sawtooth spoilers at the cavity leading edge were applied for comparison. The spoilers are mounted on the baseline “clean” cavity mentioned above, and most of the mesh is identical except in the vicinity of spoilers. Each side of each tooth contains at least seven cells to resolve the small turbulent scales created by the spoiler teeth. The total cell number exceeds ten million, in which about 1 million additional cells are placed near the spoiler to ensure the local grid resolution. Those spoilers had different heights and tooth angles, and were accordingly marked “big”, “sharp”, “smallow” (small + low), “obtlow”(obtuse + low) and “obtlower” (obtuse + lower). Details of the spoiler geometries are listed in Table 2. Due to the speciﬁc nose

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shape of the fuselage, local boundary layer thickness δ is not uniform and a nominal value is taken from the baseline geometry approximate value of 10 mm. The total heights of the spoilers varied from 0.5δ to 1.9δ . Since the spoiler height is comparable with the boundary layer, LES branch of IDDES must be switched on to correctly resolve the ﬂow near the spoiler. Previously Fig. 1 has demonstrated the partition of RANS and LES by eddy viscosity and adaptive dissipation function distributions. In fact, as the boundary layer develops itself on the geometry of the forebody, the ﬂow disturbance has been generated from shockwaves and separations and therefore triggered the LES branch upstream of the spoiler location. Even for the lowest spoiler “Obtlower” which is only 0.5δ high, Fig. 9 shows that over 80% of the turbulence kinetic energy is well resolved near the spoiler. The resolved part is also dominant for the normal and shear stress. Although in the core of the shear layer the resolved portion slightly drops, it quickly recovers downstream. This assures that IDDES well resolves the ﬂow around the spoiler and the results are reliable.

Fig. 23. Mean y-vorticity at half tooth-height of ﬁve spoilers.

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Fig. 24. Spanwise & streamwise correlation of vertical velocity v at the cavity opening. Table 3 Total drag reduction by different spoilers (drag coeﬃcients presented in count). Wall location Forebody Spoiler Front Floor Rear Outer Inner Tail Total %

Clean 7.60 – 8.60 23.37 45.66 0.11 0.14 1.16 86.44 100.00

Big

Sharp

Smallow

Obtlow

Obtlower

+10.54 +24.16 −4.54 −8.33 −31.86 +0.24 +0.15 −0.25 −9.69 −11.21

+9.71 +22.23 −6.44 −11.22 −28.01 +0.04 +0.09 −0.35 −13.75 −15.91

+9.19 +12.64 −6.15 −13.20 −28.26 +0.04 +0.07 −0.40 −25.86 −29.92

+8.47 +14.31 −5.50 −10.86 −25.79 +0.08 +0.13 −0.42 −19.36 −22.39

+8.01 +9.61 −5.87 −12.26 −25.50 +0.04 +0.10 −0.28 −25.94 −30.01

3.2. Effects of a sawtooth spoiler In order to facilitate clear and comprehensive discussion, in this subsection we compare the “clean” model and the “big” spoiler as an example; other sawtooth spoilers are believed to share a similar control mechanism. Previous studies with regular cavities have shown that spoilers (whether sawtooth or ﬂat top) work by lifting up the incoming shear layer at the leading edge. A similar conclusion is reached here, as evidenced by Figs. 10 and 11. In our case, this leads to a signiﬁcant change of cavity ﬂow type from closed to open. The sawtooth spoiler combs the incoming ﬂow and makes the ﬂow above the cavity parallel to the ﬂow direction, pushing the high speed ﬂow outside of the cavity. Before the spoiler, a separation bubble can be observed near the corner of spoiler and the fuselage surface, due to the blockage effect. The vortex at the rear corner exists in both cases. The thin tornado-like vortex in the “clean” cavity turns into a large recirculation zone after the spoiler is applied. The existence of such a large recirculation zone decreases the pressure inside the cavity as shown in Fig. 12 where the pressure drop on the rear wall of the cavity signiﬁcantly reduces the drag on the cavity itself. There was high stagnation pressure before the spoiler, which introduces extra drag on the spoiler. Considering the area difference between the spoiler and the rear wall of the cavity, however, an overall drag reduction of 11.2% is achieved (detailed data and comparison are shown in Table 3 and Fig. 26). The line plot in Fig. 12 also shows the decreased pressure gradient on the cavity ﬂoor in the presence of a spoiler, which is important because slowed pressure increase leads to lower nose-up pitching moment and has potential positive inﬂuence on store release. Observing the instantaneous ﬂow-ﬁeld offers insight into the ﬂow control mechanisms. Fig. 13 shows the isosurface of the Q criterion, which demonstrates the spatial vortex structures. In the “clean” cavity, ﬂow structures move into the cavity and ﬂood outside downstream. The serrated leading edge only generates structures along the cavity side edges. The “big” spoiler again lifts the shear layer above the cavity. The sawteeth also inﬂuence the shear

Fig. 25. Indication of cavity model surfaces.

layer development, promote the instability more upstream, comb the ﬂow, and generate spanwise λ-shape structures right after the spoiler (Fig. 14). These structures travel and develop downstream, then quickly break down into smaller scale structures (Figs. 14 and 15). There are also small-scale structures before the spoiler generated by unsteady separation bubble and shockwaves which do not exist in the “clean” cavity. With the signiﬁcant change of the ﬂow inside the cavity, the spoiler shows excellent suppression of the near-ﬁeld pressure ﬂuctuations. Spatial OASPL distributions on the central plane of the cavity (Fig. 16) demonstrate signiﬁcant noise reduction. The high OASPL region almost ﬁlls the cavity behind Line A (the mean location of the shockwave created by the reattachment of the shear layer on the cavity ﬂoor, Fig. 17), and extends far outside the “clean” cavity without a spoiler. After the spoiler is applied and the shear layer no longer attaches to the ﬂoor, the cavity ﬂow type changes and thus the shockwave along Line A is eliminated. The high OASPL area turns smaller into a region corresponding to the wake of the shear layer (Fig. 18) and the corner vortex back in Fig. 11; we also notice a small high OASPL region right before the spoiler that is produced by the small buffeting shockwave and small recirculation bubble located before the spoiler. Signals at some typical sample points were recorded during the computation. Their locations are displayed in Fig. 19. K1–K6 are located on the cavity surfaces. Among these points, K1, K2 and K5

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Fig. 26. Drag contribution of different wall parts of the cavity model.

Fig. 27. Comparison of surface OASPL for different spoilers.

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Fig. 28. Comparison of OASPL at typical sample points.

are located on the cavity ﬂoor, K3 on the inner wall, K4 on the outer wall, and K6 on the rear wall. Burg algorithm (Burg, 1975) is used again to estimate the pressure PSD from 8192 records. Evolution of PSD spectrum at the cavity opening along the shear layer is shown in Fig. 20. Pressure ﬂuctuation lower than 1.6 kHz rises after x/L = 0.4 due to the shockwave in the baseline “clean” cavity. The spoiler eliminates the shockwave and suppresses the pressure ﬂuctuation between x/L = 0.4 and x/L = 0.8, but also enhances the high frequency (up to 8 kHz) ﬂuctuation near the cavity leading edge. It is believed that the high frequency ﬂuctuation is related to the small scale structures generated by the spoiler teeth. OASPL reduction on the cavity surfaces (Fig. 21) is particularly impressive. High ﬂuctuation areas are greatly reduced on both the cavity ﬂoor and rear wall. In the “clean” cavity, the OASPL increases in the streamwise direction until x/L = 0.6, then stays near 181 dB throughout the rest of the cavity. The sawtooth spoiler case exhibits a signiﬁcant reduction in pressure, especially in the rear half of the cavity. The high OASPL region is greatly reduced by the spoiler, where OASPL decreases as much as 9 dB. The widespread high OASPL values all over the rear cavity wall are also reduced by 5 dB–6 dB. Spectra at the six wall sample points (Fig. 22) can be divided into the following three categories according to their streamwise locations. 1. K1 is the most upstream sample point, and comprises the ﬁrst category alone. The spoiler is weakly effective at K1. Minor energy reduction is observed almost for all frequencies, though the spoiler is beneﬁcial at K1 for higher frequencies. A small peak about 140 dB at 4 kHz can be seen in the “clean” cavity, which is moved down to about 2 kHz by the spoiler. 2. Sample points K2 and K4 are located in the middle of the cavity, where pressure ﬂuctuation is suppressed signiﬁcantly with the spoiler. The spectrum of point K2 drops about 5 dB along the negative PSD-axis globally, and the spoiler ﬂattens the peak at 4 kHz of point K4. This highly eﬃcient noise suppression effect is achieved by the spoiler lifting up of the shear layer and changing the ﬂow-ﬁeld inside the cavity. 3. Sample points K3, K5 and K6 are located in the rear part of the cavity. The spoiler is quite effective in the low frequency range (<1 kHz) at these points, but does not work at all at high frequencies, due to the inevitable breakdown of the shear layer with or without a spoiler. 3.3. Evaluation of spoilers of different geometric parameters Though the sawtooth spoilers all have a similar ﬂow control mechanism, their disparate geometric parameters do notably impact on their effectiveness. Fig. 23 shows the mean y-vorticity at half tooth-height (h1 + 0.5h2 ) of the spoilers. Higher spoilers lead to larger separation bubbles before them. Between spoilers of the

same height, such as “big” & “sharp” and “smallow” & “obtlow”, a sharper tooth angle leads to a smaller streamwise bubble size. The mean ﬂow combed by the spoiler merges into two major counterrotating vortices downstream. One originates from the outer two or three teeth and the other from the inner ones. Spanwise and streamwise correlation of vertical velocity v (Fig. 24) also reveal that the spoilers generates spanwise irregularity and trigger the instability in the shear layer. However, there seems no obvious laws of the tooth geometric parameters and their effects. The following evaluations of the spoilers by geometry will focus on their respective performance in terms of both drag reduction and noise suppression. Table 3 lists the total drag coeﬃcients on different parts of the model wall of the “clean” cavity and different spoilers’ contributions. The reference area is 1 m2 . The inner and outer walls of the cavity and the tail ﬂat plate after the cavity (see in Fig. 25) are all parallel to the ﬂow and thus only have viscous drag. Though the skin-friction on those three parts is altered by the spoiler’s presence, it is negligible against the pressure drag, which is the case for all other parts. Clearly, every spoiler reduces the drag coeﬃcients of the cavity model. The sawtooth spoilers themselves do introduce extra pressure drag as they act as an obstacle to the incoming ﬂow. Deﬁnitely, the size or height of the spoiler is the decisive parameter for the drag. The bigger the spoiler, the larger the extra drag. Large spoilers like the “big” and the “sharp” have the highest drag while the small “obtlower” has the lowest. The “big” and the “sharp” spoiler share the same height parameters (h1 , h2 & H), but have different tooth angles and numbers. The “big” spoiler has about 8% higher drag than the “sharp” spoiler, which indicates that a sharper tooth may help reduce the extra spoiler drag. Also, comparing the “smallow” and “obtlow” spoilers of the same height, the drag of the latter is 12% lower than the former. Clearly, the height of the spoiler base which blocks the ﬂow should be as low as possible. The rear cavity wall makes the largest contribution to drag and also beneﬁts the most from the spoiler, as is shown in Fig. 26. The ﬂoor also has substantial drag reduction. The drag reduction on the rear wall reaches as much as 70% with the “big” spoiler. Overall drag reduction varies from 11% to 30%, where the lowest “obtlower” spoiler shows the best drag performance, closely followed by the “smallow” spoiler. All the spoilers tested also show remarkable performance in terms of suppressing pressure ﬂuctuation inside the cavity, as shown in Figs. 27 and 28. High OASPL regions are limited to the small area near the trailing edge of the cavity after the addition of a spoiler, rather than spreading all over on the rear wall and the rear portion of the ﬂoor which is the case of the “clean” cavity. Higher spoilers like “big”, “sharp” and “smallow” perform better in regard to the high OASPL area on the cavity ﬂoor. The maximal reduction is 9 dB at K4 by the “big” and “sharp” spoilers, followed by the “smallow” spoiler.

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In short, both height and tooth angle of the spoiler are important factors affecting acoustic suppression. Control devices are more effective if the total height is greater than or equal to the local boundary layer thickness and the tooth angle is 90° or less. Among the spoilers investigated in this study, the “smallow” spoiler shows the optimal performance in both drag reduction and noise suppression. 4. Conclusions In this study, IDDES based on SST turbulence model is used to study the control mechanism and effectiveness of sawtooth spoilers in cavity ﬂows. Results on two sets of meshes of the M219 cavity prove that the numerical methods applied here are capable of capturing the unsteady oscillation dynamics in cavity ﬂows, and that the dense mesh is quantitatively and visually better in resolution of pressure oscillation and turbulence structures than the coarse mesh. The baseline conﬁguration includes a fore fuselage, cavity and a ﬂat plate as the after fuselage. Spoilers are mounted before the cavity leading edge. The “big” spoiler (1.9δ high and tooth angle is 90°) is analyzed and compared to the baseline cavity in detail. The spoiler effectively lifts up the incoming shear layer, preventing it from directly impinging on the cavity wall and changing the cavity ﬂow type from closed to open. Spanwise structures are also generated into the shear layer to promote more upstream instability. Though the spoiler itself brings additional drag, it changes the pressure distribution inside the cavity especially on the ﬂoor and rear wall, which leads to overall drag reduction. After adding a spoiler, the spatial and surface high OASPL regions shrink significantly and the amplitude of pressure ﬂuctuation also decreases. Frequency components lower than 1 kHz prove more sensitive to the presence of the control device and contribute most to noise suppression. Five sawtooth spoilers of different heights and tooth angles are evaluated on the same complex irregular cavity model and all ﬁve spoilers are effective in both drag reduction and noise suppression. The “smallow” spoiler, which has a height equal to local boundary layer thickness and tooth angle of 90°, exhibits optimal and balanced performance in both drag reduction and noise suppression among the ﬁve investigated designs. However, considering the complex geometry and incoming ﬂow condition, the current study is unable to isolate and investigate the quantitative effects of the tooth geometric parameters, which should be of interest in some future works. Acknowledgment The authors would like to thank the National Natural Science Foundation of China (Grant No. 11372159), the National Key Research and Development Program of China (Grant No. 2016YFA0401200) and the EU Horizon 2020 Research & Innovation Program IMAGE (Grant No. 688971) for their ﬁnancial support, and express our gratitude to Tsinghua National Laboratory for Information Science and Technology for computation resources. References Aradag, S., Kim, H.J., Knight, D.D., 2010. Two and three dimensional simulations of supersonic cavity conﬁgurations. Eng. Appl. Comput. Fluid Mech. 4 (4), 612–621. Arunajatesan, S., Barone, M.F., Wagner, J.L., Casper, K.M., Beresh, S.J., 2014. Joint experimental/computational investigation into the effects of ﬁnite width on transonic cavity ﬂow. In: 32nd AIAA Applied Aerodynamics Conference, June 16–June 20, 2014. AIAA, Atlanta, GA, United states. Ashworth, R., 2008. DES of a cavity with spoiler. In: Peng, S.-H., Haase, W. (Eds.), Advances in Hybrid RANS-LES Modelling: Papers contributed to the 2007 Symposium of Hybrid RANS-LES Methods, Corfu, Greece, 17–18 June 2007. Springer Berlin Heidelberg, pp. 162–171.

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Please cite this article as: K. Luo et al., Improved delayed detached-eddy simulations of sawtooth spoiler control before supersonic cavity, International Journal of Heat and Fluid Flow (2017), http://dx.doi.org/10.1016/j.ijheatﬂuidﬂow.2017.01.012

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