Biglobal instabilities of compressible open-cavity flows
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Biglobal instabilities of compressibleopen-cavity flows
Yiyang Sun † , Kunihiko Taira , Louis N. Cattafesta III , andLawrence S. Ukeiley Department of Mechanical Engineering, Florida State University, Tallahassee, FL 32310, USA Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL32611, USA(Received xx; revised xx; accepted xx)
The stability characteristics of compressible spanwise-periodic open-cavity flows areinvestigated with direct numerical simulation (DNS) and biglobal stability analysis forrectangular cavities with aspect ratios of
L/D = 2 and 6. This study examines thebehavior of instabilities with respect to stable and unstable steady states in the laminarregimes for subsonic as well as transonic conditions where compressibility plays animportant role. It is observed that an increase in Mach number destabilizes the flow inthe subsonic regime and stabilizes the flow in the transonic regime. Biglobal stabilityanalysis for spanwise-periodic flows over rectangular cavities with large aspect ratiois closely examined in this study due to its importance in aerodynamic applications.Moreover, biglobal stability analysis is conducted to extract 2D and 3D eigenmodes forprescribed spanwise wavelengths λ/D about the 2D steady state. The properties of 2Deigenmodes agree well with those observed in the 2D nonlinear simulations. In the analysisof 3D eigenmodes, it is found that an increase of Mach number stabilizes dominant 3Deigenmodes. For a short cavity with
L/D = 2, the 3D eigenmodes primarily stem fromcentrifugal instabilities. For a long cavity with
L/D = 6, other types of eigenmodesappear whose structures extend from the aft-region to the mid-region of the cavity,in addition to the centrifugal instability mode located in the rear part of the cavity. Aselected number of 3D DNS are performed at M ∞ = 0 . L/D = 2 and 6.For
L/D = 2, the properties of 3D structures present in the 3D nonlinear flow correspondclosely to those obtained from linear stability analysis. However, for
L/D = 6, the 3Deigenmodes cannot be clearly observed in the 3D DNS, due to the strong nonlinearity thatdevelops over the length of the cavity. In addition, it is noted that three-dimensionality inthe flow helps alleviate violent oscillations for the long cavity. The analysis performed inthis paper can provide valuable insights for designing effective flow control strategies tosuppress undesirable aerodynamic and pressure fluctuations in compressible open-cavityflows.
Key words:
1. Introduction
Flow over an open rectangular cavity has been a fundamental research topic for decadesbecause of its ubiquitous nature in many engineering settings, including landing gear † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] J un Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley wells, gaps between plates, and aircraft weapon bays (Cattafesta et al. et al. et al. M ∞ . Heller & Bliss (1975) later modified the formula to better match the experimentalmeasurements. This modified formula is used as reference in the present work. The modeassociated with the resonance is referred to as the Rossiter mode and its frequency f n can be predicted in terms of the Strouhal number as St L = f n Lu ∞ = n − α /κ + M ∞ / (cid:112) γ − M ∞ / , (1.1)where L is the length of cavity, empirical constant κ (= 0 .
57) is the average convectivespeed of disturbance in shear layer, α (= 0 .
25) (Rossiter 1964) corresponds to phase delay, γ (=1.4) is specific heat ratio, and n = 1 , , . . . leads to the n th Rossiter mode. We alsodefine a Strouhal number based on the cavity depth ( St D = f n D/u ∞ ) to quantify thefrequencies of 3D modes of open-cavity flows.The oscillations associated with open-cavity flows are generally undesirable, becausethey may damage the cavity contents due to the unsteady aerodynamic forces andintense pressure fluctuations. During the past few decades, researchers have developedtechniques to suppress the oscillations through various passive and active flow controlstrategies. A comprehensive review on active control of high Reynolds number cavityflow for a wide range of Mach numbers is given by Cattafesta et al. (2008). Both openand closed-loop control techniques have demonstrated the ability to significantly reducethe pressure fluctuations and noise emission (Samimy et al. et al. et al. et al. et al. et al. et al. et al. et al. b ),three dimensionality can affect the dominant oscillation characteristics. Although thethree dimensionality discussed in these experiments and simulations mostly relates tothe significance of spanwise end effects on the flow, such end effects can modify spanwise iglobal instabilities of compressible open-cavity flows L/D = 2and M ∞ = 0 . . (cid:54) M ∞ (cid:54) .
6) cavity flow instabilities at low Reynoldsnumbers. They also performed 3D linear simulations on compressible open-cavity flowwith
L/D = 2 and 4 and identified the spanwise wavelength of the most-unstable/least-stable modes via examination of the most amplified disturbances with respect to thesteady base flow. They found that the 3D modes have an order-of-magnitude lowerfrequency than those of the 2D resonant modes in the cavity, with the wavelengthof the most-unstable mode being λ/D ≈
1. Yamouni et al. (2013) performed globalstability analysis to investigate the interaction between feedback aeroacoustic mechanismand acoustic resonance in the flow over cavity with
L/D = 1. Moreover, de Vicente et al. (2014) conducted global stability analysis on incompressible open-cavity flows andobserved that 3D instability modes can split into different branches depending on theirspanwise wavelengths. They also compared their numerical results to experiments andreported that their numerical results resembled the fully saturated nonlinear flow featuresseen in experiments. Considering lateral wall effects on the 3D structures present in finite-span cavity flows, Liu et al. (2016) performed triglobal instability analysis to unravelthe transition of steady laminar flow over a three-dimensional cavity for incompressibleflow. All of these studies provide insights into the characteristics of spanwise instabilitiesassociated with open-cavity flows. Nonetheless, there is still a gap in the literaturewith respect to instabilities of open-cavity flow in the transonic regime. Moreover, theinstabilities of flows over the cavities with large
L/D = 6, which are particularly relevantin aircraft bays, have been rarely studied. Furthermore, three-dimensional flow controlstrategies applied on realistic compressible open cavity flows (Lusk et al. et al. et al.
L/D = 2 and 6 to characterize the effects of free stream Mach number M ∞ ,Reynolds number Re θ and aspect ratio L/D on the 2D flow instabilities. Furthermore,the stable/unstable steady states obtained from 2D simulations serve as base statesin the biglobal stability analysis to reveal characteristics of 2D ( λ/D = ∞ ) and 3D Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley
L/D M ∞ β (= 2 π/λ )Br`es & Colonius (2008) 1, 2, 4 0.1-0.6 3.14 - 12.56Yamouni et al. (2013) 1 & 2 0.0-0.9 0Meseguer-Garrido et al. (2014) 1 - 3 0.0 0 - 22de Vicente et al. (2014) 2 0.0 0 - 22Present 2 & 6 0.1-1.4 0 & 3.14 - 12.56 Table 1.
A summary of biglobal stability analysis of laminar open-cavity flows studied in pastliterature and the present work. Mach number 0 represents incompressible flow. ( λ/D = 0 . − .
0) eigenmodes associated with the flows. In the linear stability analysiscomponent of this study, 2D and 3D global eigenmodes are identified for M ∞ = 0 . − . L/D = 2, 6. These global eigenmodes are also compared to the flow fields from the 2Dand 3D nonlinear simulations. We will show that most of the linear stability predictionsof flow properties are in a good agreement to those captured from the nonlinear flows,which could serve a foundation for parameter choice in flow control designs.In what follows, the computational approach and numerical validation are presented in §
2. In §
3, the characteristics of the 2D instabilities in open-cavity flows are investigatedvia DNS. With the stable/unstable steady states obtained from 2D DNS, 2D eigenmodescaptured via biglobal stability analysis are discussed and compared to the flow char-acteristics revealed in the nonlinear simulations in § § § §
2. Computational approach
Direct numerical simulation setup and validation
Two-dimensional DNS of compressible flows over a rectangular cavity are performedusing a high-fidelity compressible flow solver
CharLES (Khalighi et al. a , b ; Br`es et al. x i , time t ,density ρ , velocity u i , energy e , pressure P , temperature T , are non-dimensionalized as x i = x id D d , t = t d a d ∞ D d , ρ = ρ d ρ d ∞ , u i = u id a d ∞ , e = e d ρ d ∞ ( a d ∞ ) , P = P d γP d ∞ , T = T d T d ∞ , where variables with superscript d refer to the dimensional quantities and those withthe subscript ∞ denote the free stream values. The x -, y -, and z -directions represent thestreamwise, wall-normal, and spanwise directions, respectively. A structured mesh withnon-uniform spacing in both x - and y -directions is used for the simulations. Open-cavityflows are specified by L/D , where L and D represent the length and depth of the cavity,respectively, initial boundary layer momentum thickness θ at the leading edge of thecavity, and free stream Mach number M ∞ ≡ u d ∞ /a d ∞ . The free stream sonic speed isdenoted by a d ∞ . The Reynolds number based on the initial momentum boundary layer iglobal instabilities of compressible open-cavity flows Sponge RegionAcoustic Waves M Shear Layer Instability
L D x y (0 , Figure 1.
Computational setup for open-cavity flow (not to scale). thickness θ and the Prandtl number are respectively defined as Re θ ≡ ρ ∞ u ∞ θ µ ∞ and P r ≡ c p µ ∞ k , where µ ∞ is the dynamic viscosity, c p is the specific heat, and k is the thermal conduc-tivity.In the present investigation, we consider two-dimensional cavities with L/D = 2 and6. The former geometry serves as the basis for comparison with those reported in theliterature, while the latter is representative of a prototypical cavity application on aircraft.As illustrated in figure 1, the origin is placed at the leading edge of the cavity. The initialmomentum boundary layer thickness θ is prescribed at the leading edge, and the distancebetween the upstream wall boundary and the leading edge is adjusted accordingly forthe chosen Re θ . The outflow boundary is placed 7 D from the trailing edge of the cavity.The normal distance from the cavity surface to the top boundary of the computationaldomain is maintained at 9 D . The size of computational domain follows the work ofColonius et al. (1999), in which they studied the effects of the computational domainon the flows and identified the appropriate size of the domain. No-slip and adiabaticboundary conditions are specified at the upstream and downstream floor as well as thewalls of the cavity. To damp out exiting acoustic waves and wake structures, spongezones (Freund 1997) are applied to the outlet and top boundaries spanning a length of2 D from computational boundaries. The computational domain for three-dimensionalDNS extends the two-dimensional setup with spanwise periodicity for a width-to-depthratio of W/D = 2 (which is suitable for λ/D (cid:54)
2) with 64 grid points spaced uniformlyin the spanwise direction.The effect of Mach number is analyzed from the subsonic regime to the transonicregime with Reynolds number Re θ from 5 to 144 for flows over cavities with L/D = 2and 6. To initialize the flow field, an incompressible Blasius boundary layer profile isimposed over the entire computational domain above the floor, while the flow insidethe cavity is set to be quiescent. Consistent with the chosen Reynolds number range ofthis study, the incompressible Blasius boundary layer profile is utilized as the variationin boundary layer thickness for the range of Mach numbers from 0 to 1 . D/θ = 26 . L/D = 2 and 6) is
Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley ( a ) L/D = 2, M ∞ = 0 . Re θ = 144 ( b ) L/D = 6, M ∞ = 0 . Re θ = 46 Figure 2.
Comparison of the v -velocity at the midpoint ( x = L/ y = 0) of the cavity. Thesolid line shows the baseline case with half a million grid points and the dashed line representsthe refined case with 1.1 million grid points. St L St L Rossiter (1964) 0.321 0.750Br`es (2007) 0.404 0.698Present 0.412 0.715
Table 2.
Comparison of Rossiter modes I and II frequencies for open-cavity flow with
L/D = 2, Re θ = 56 . M ∞ = 0 . performed. Presented in figure 2 are two grid convergence comparisons performed with Re θ = 144 and M ∞ = 0 . L/D = 2, and Re θ = 46 and M ∞ = 0 . L/D = 6.The baseline computation is conducted on a structured mesh with approximately halfa million grid points. A finer mesh with one million grid points is also performed forcomparison. The v -velocity history at the midpoint location ( x = L/ y = 0) over thecavity is shown in figure 2. The baseline mesh of half a million grid points is shown to besufficient to achieve numerical convergence. Moreover, the frequencies of the oscillationsin the flow for L/D = 2, Re θ = 56 . M ∞ = 0 . M ∞ = 0 . et al. (2002) in figure 3. In the experiments conducted by Krishnamurty(1956), the cavity width is almost 40 times the depth ( W/D ≈ M ∞ = 0 .
8, the acoustic waves generated at the trailing edge ofcavity propagate upstream, and for the M ∞ = 1 . § iglobal instabilities of compressible open-cavity flows Krishnamurty, M ∞ = 0 .
82 Rowley et al., M ∞ = 0 . M ∞ = 0 . Re θ = 56 . Re θ = 67Krishnamurty, M ∞ = 1 .
38 Present, M ∞ = 1 . Re θ = 56 . Figure 3.
Comparison of schlieren images by Krishnamurty (1956), DNS by Rowley et al. (2002)(reproduced with permission from Cambridge University Press) and the present work ( ∂ρ/∂x )at M ∞ = 0 . Biglobal stability analysis setup and validation
Biglobal stability analysis is performed with respect to the base flow obtained fromthe 2D nonlinear simulation to uncover the compressible spanwise-periodic global insta-bilities. The state vector q = [ ρ, ρu, ρv, ρw, e ] T is decomposed into base state ¯ q ( x, y ) andperturbations q (cid:48) ( x, y, z, t ) as q ( x, y, z, t ) = ¯ q ( x, y ) + q (cid:48) ( x, y, z, t ) . (2.1)The steady solution (base state) ¯ q satisfies the Navier–Stokes equations and q (cid:48) is small inits magnitude compared to the base state ( | q (cid:48) | (cid:28) | ¯ q | ). In the present work, the base state ¯ q , obtained from the 2D DNS, is either a time-invariant stable state or an unstable steadystate calculated by the selective frequency damping method (˚Akervik et al. q (cid:48) which Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley yields the following set of linear equations for q (cid:48) ∂ρ (cid:48) ∂t + ∂∂x j ( ρ (cid:48) ¯ u j + ¯ ρu (cid:48) j ) = 0 ,∂∂t ( ρ (cid:48) ¯ u i + ¯ ρu (cid:48) i ) + ∂∂x j (¯ ρ ¯ u i u (cid:48) j + ¯ ρu (cid:48) i ¯ u j + ρ (cid:48) ¯ u i ¯ u j + P (cid:48) δ ij )= 1 Re ∂∂x j (cid:18) ∂u (cid:48) i ∂x j + ∂u (cid:48) j ∂x i − ∂u (cid:48) k ∂x k δ ij (cid:19) ,∂e (cid:48) ∂t + ∂∂x j ((¯ e + ¯ P ) u (cid:48) j + ( e (cid:48) + P (cid:48) )¯ u j )= 1 Re ∂∂x j (cid:20) ¯ u i (cid:18) ∂u (cid:48) i ∂x j + ∂u (cid:48) j ∂x i − ∂u (cid:48) k ∂x k δ ij (cid:19) + u (cid:48) i (cid:18) ∂ ¯ u i ∂x j + ∂ ¯ u j ∂x i − ∂ ¯ u k ∂x k δ ij (cid:19)(cid:21) + 1 Re P r ∂ T (cid:48) ∂x k ∂x k (2.2)along with the linearized equation of state P (cid:48) = R ( ρ (cid:48) ¯ T + ¯ ρT (cid:48) ) , (2.3)where R is the gas constant.The above linear governing equations permit modal perturbations of the form q (cid:48) ( x, y, z, t ) = ˆ q ( x, y ) e i ( βz − ωt ) + complex conjugate . (2.4)Upon substitution of this modal expression into the linearized Navier–Stokes equations(2.2), we can transform the instability analysis from solving an initial value problem toan eigenvalue problem of A ( ¯ q ; β ) ˆ q = ω ˆ q . (2.5)Temporal instability is examined by inserting a real wavenumber β with its correspondingwavelength λ/D = 2 π/β . The corresponding eigenmodes consist of eigenvectors ˆ q ( x, y ) = ˆ q r ( x, y ) + i ˆ q i ( x, y ) and their complex eigenvalues ω = ω r + iω i , where ˆ q r and ˆ q i representreal and imaginary components of eigenvectors; ω r and ω i are the modal frequency andgrowth ( ω i >
0) or decay ( ω i <
0) rate, respectively. While solving for the eigenmodes,the linear operator
A ∈ C n × n and eigenvector ˆ q ∈ C n for n = O (10 ) grid points, canbecome extremely large. For this large-scale eigenvalue problem, the ARPACK library(Lehoucq et al. A (¯ q ; β )ˆ q repeatedly to thesolver, which avoids requesting large memory space to store matrix entries while solvingthe eigenvalue problem. All the eigenmodes reported in this paper are converged with || − iω ˆ q − A ˆ q || (cid:54) O (10 − ). Along the cavity walls, velocity perturbations and the wall-normal gradient of pressure perturbation are set to zero. According to the governingequations, the boundary condition for density perturbation is not required because themomentum perturbation flux is zero due to zero velocity along the wall. For the inlet,density and velocity perturbations, as well as the pressure gradient are prescribed to bezero. For the outflow and far field boundaries, gradients of density, velocity and pressureare prescribed as zero. Moreover, an adiabatic condition is assumed for all boundaries.The base state is interpolated on a coarse mesh for the eigenvalue problem (Bergamo iglobal instabilities of compressible open-cavity flows Figure 4.
Comparison of dominant 3D modes properties (growth/decay rate ω i D/u ∞ andfrequency St D ) between the present biglobal analysis and linear simulation by Br`es (2007) at Re θ = 56 .
8. Eigenvectors of the most-unstable mode with λ/D = 1 . u r /u ∞ and ˆ u i /u ∞ ∈ [ − . , . a ) Present: M ∞ = 0 .
3; ( b ) Br`es (2007): M ∞ = 0 .
325 (reproduced with permission from Cambridge University Press). et al. et al. (2016 a ).The validation of global instability analysis is performed on the flow at M ∞ = 0 . Re θ = 56 . L/D = 2. As shown in figure 4, the dominant 3D modes from this studyare compared to the results from Br`es & Colonius (2008), in which they resolved thedominant 3D modes by solving an initial value problem based on the linearized Navier–Stokes equations. The growth/decay rate, frequencies and eigenvector of the dominanteigenmode obtained from present instability study agree well with those from Br`es &Colonius (2008).
3. 2D direct numerical simulation and analysis
We focus on examining the effects of
L/D , Mach number, and Reynolds number on 2Dflow oscillation mechanisms in this section. These parameters are known to significantlyinfluence cavity flows (Krishnamurty 1956; Rowley et al. et al. et al. a ). It was also observed by Gharib &Roshko (1987) and Zhang & Naguib (2011) for incompressible axisymmetric cavity flow0 Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley in their experiments. The primary feature of the wake mode is the shedding of largevortices leading to the interaction between the vortex and the trailing edge, causingviolent fluctuations in the cavity.Here, we discuss the flow characteristics, instabilities, and behavior of the Rossitermode by performing 2D DNS. The simulations were conducted over a sufficiently longconvective time (with a minimum of 150 convective units
D/u ∞ ) for the flow to reacha steady state. Such flow in this study is categorized as stable (asymptotically stable)if the flow is devoid of any oscillation and otherwise unstable. A number of cases thatspan the range of Mach numbers from M ∞ = 0 . Re θ ) upto 144 with L/D = 2 and 6 are analyzed in detail. For
L/D = 2, the parameters of M ∞ and Re θ are chosen to greatly expand upon the subsonic stability analysis performed byBr`es & Colonius (2008). We extend their analysis to the transonic Mach number regimeand determine unstable steady states of oscillatory cavity flows in preparation for thebiglobal stability analysis in §
4. Flow over a long cavity with
L/D = 6 is also examinedin detail as this configuration is representative of long cavities used in aircraft.3.1.
Flow field characteristics
To examine how compressibility affects flow features, the instantaneous density gradi-ent field, the instantaneous vorticity field, and the time-averaged streamlines are shownin figures 5 and 6 for cavities with
L/D = 2 and 6, respectively.For a cavity with
L/D = 2, the instantaneous density gradient fields for Mach numbersfrom 0.6 to 1.4 are presented in figure 5 at Re θ = 46. At M ∞ = 0 .
6, the compressionwaves in the flow field are not prominent compared to those at higher Mach numbers.When M ∞ increases to 0.8, the acoustic radiation emitted from the trailing edge becomesnoticeable and its wavelength becomes smaller, which has also been discussed by Rowley et al. (2002). For M ∞ (cid:62) .
0, the acoustic waves are more prominent and its structureover the cavity becomes directional. This phenomenon is observed in experiments byKrishnamurty (1956) as well. Part of the compression waves are generated at the rearcorner of cavity. At the trailing edge, for the oncoming flow with a large deflection angle,a shock is formed due to the impingement of the shear layer on the trailing edge. Theremaining waves are generated from the periodic expansion and compression of shear-layer oscillations in the supersonic flow. Inside the cavity, animations (not shown) revealthat waves travel back and forth due to reflections. Outside of the cavity, compressionwaves propagate upstream in the subsonic cases ( M ∞ < . M ∞ (cid:62) . ϕ of the beam composed of these compression waves above the cavity is measured in thedensity gradient flow fields. This angle ϕ approximates the corresponding Mach waveangle sin − (1 /M ∞ ) for M ∞ >
1. For M ∞ = 1 . ϕ = 60 . ◦ and 46 . ◦ with sin − (1 /M ∞ ) = 56 . ◦ and 45 . ◦ , respectively.As shown in figure 6, for a cavity with L/D = 6 at Re θ = 19, large density gradientsare caused by expansion and compression of the shear layer for the subsonic regime( M ∞ = 0 . M ∞ = 0 .
9, the acoustic waves become observablewith their wavelength corresponding to the length of the cavity. At M ∞ = 1 .
2, the flowbecomes stable with weak Mach waves emanating from the leading and trailing edges.For a cavity with
L/D = 2, we only observe the shear-layer mode for unstable flows,with a large clockwise-rotating vortex present in the rear part of the cavity as shownin figure 5. In all cases considered for the short cavity, the streamlines reveal twomajor recirculation zones inside the cavity. The lack of variation in the time-averaged iglobal instabilities of compressible open-cavity flows M ∞ ∂ρ/∂x ω z D/u ∞ Time-averaged streamlines0.60.81.01.21.4 ' Figure 5.
Instantaneous numerical schlieren ∂ρ/∂x ∈ [ − . , . ω z D/u ∞ ∈ [ − . , .
5] contours and time-averaged streamlines are shown for M ∞ = 0 . L/D = 2, and Re θ = 46. The arrows in numerical schlieren images indicate the propagationdirections of compression waves. Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley M ∞ ∂ρ/∂x ω z D/u ∞ Time-averaged streamlines0.30.60.91.2
Figure 6.
Instantaneous numerical schlieren ∂ρ/∂x ∈ [ − . , . ω z D/u ∞ ∈ [ − . , .
4] contours and time-averaged streamlines are shown for M ∞ = 0 . L/D = 6, and Re θ = 19. streamlines versus Mach number suggests that compressibility does not affect the meanflow inside the cavity significantly.However, the flow features in the L/D = 6 cases vary greatly with Mach number. At M ∞ = 0 .
3, instantaneous vorticity flow fields shown in figure 6, reveal a shear-layer modewith a clockwise rotating vortex sitting near the trailing edge. An increase in M ∞ from0.3 to 0.6 causes the flow oscillate more violently with a transition from the shear-layermode to the wake mode. The wake mode (at M ∞ = 0 .
6) dominates the flow with alarge-scale vortex rolling up with an opposite sign vortex sheet being engulfed betweenthe vortices. Details on the appearance of the wake mode are reported in Sun et al. (2016 a ). For M ∞ = 0 .
9, the opposite sign vorticity patch pulled between the sheddingvortices decreases, and the fluctuations over the cavity weaken. In the transonic regime, iglobal instabilities of compressible open-cavity flows ( a ) L/D = 2, Re θ = 46 ( b ) L/D = 6, Re θ = 19 Figure 7.
Vorticity thickness normalized by its value δ ω at the leading edge is shown as afunction of streamwise location. Different regions are indicated by A ( x/L ∈ [0 , . B ( x/L ∈ [0 . , . L/D = 2
L/D = 6 M ∞ A B M ∞ A B
Table 3.
Spreading rate of vorticity thickness d δ ω / d x of flows over L/D = 2 cavity for M ∞ = 0 . Re θ = 46 and L/D = 6 cavity for M ∞ = 0 . Re θ = 19 as illustrated in figure7. Different regions are indicated by A ( x/L ∈ [0 , . B ( x/L ∈ [0 . , . increasing the Mach number from 0.9 to 1.2 leads to a steady flow. Compressibilityapparently affects the flow instabilities by first destabilizing at subsonic speeds and theneventually stabilizing at transonic speeds.A large difference between flow states versus Mach number is revealed in the time-averaged streamlines. When the free stream Mach number increases from 0.3 to 0.6, thecenter of the main recirculation region moves towards the leading edge, then moves backtowards to the trailing edge for M ∞ = 0 .
9. The features of flows inside the longer cavityare somewhat more complex than those of the shorter length.The complexity of open-cavity flow results from the feedback process associated withthe shear-layer development. Let us further investigate the characteristics of the shearlayer over the cavity. Several researchers have noted the similarities between the shearlayer spanning the cavity and a free shear layer, such as their linear spreading rate(Sarohia 1975; Cattafesta et al. et al. δ ω = M ∞ (cid:20)(cid:18) ∂ ¯ u∂y (cid:19) max (cid:21) − . (3.1)In figure 7, the vorticity thickness is normalized by its value at the leading edge δ ω . Weexamine the spreading rate dependence on different regions in the shear layer and onlyreport d δ ω / d x for the cases with linear spreading rate, which are summarized in table 3.4 Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley
As shown in figure 7 for cavity with
L/D = 2, for the cases ( M ∞ = 0 . et al. (2002) found d δ ω / d x = 0 .
05 for
D/θ = 26 . Re θ = 56 . M ∞ = 0 .
6, and the value is close to the spreading rate d δ ω / d x = 0 .
062 (region B ) for D/θ = 26 . Re θ = 46 and M ∞ = 0 . M ∞ (cid:62) .
0, a double-hump distribution of vorticity thickness is observed in figure 7 (a).As the numerical schlieren reveals in figure 5, strong compression waves are formed insupersonic flows. When these waves propagate upstream, they interact with shear layerover the cavity, which distorts the mean profile of the shear layer and results in thedouble-hump feature. However, for subsonic flows, the compression waves are not strongenough to affect the mean profile, resulting in a linear spreading rate. Near the leadingedge of cavity (region A ), the spreading rate is increased as Mach number increases until M ∞ = 1 .
2. Further increasing Mach number to M ∞ = 1 . § L/D = 6, except for the wake-mode case at M ∞ = 0 .
6, all the other cases (shear-layer modes) reveal approximately linear spreading rates. We note that the growth rateshere (region B ) are in agreement with the values reported by Gharib & Roshko (1987),which are almost constant d δ ω / d x = 0 .
124 when
L/θ > M ∞ = 1 . B is reduced compared to the other shear-layer cases.3.2. Stability diagram
In the present work, the parameters of M ∞ and Re θ are selected to expand upon thesubsonic stability analysis performed by Br`es & Colonius (2008). Through an extensiveparametric study, the influence of M ∞ and Re θ on the stability of 2D open-cavity flowsare revealed, as shown in figure 8. We find the approximate neutral stability curve vianonlinear flow simulations to separate the stable and unstable zones in terms of M ∞ and Re θ . For both L/D = 2 and 6, when the Mach number decreases towards theincompressible limit, the flow becomes more stable for a wider range of Reynolds numbersas shown in figure 8. In the subsonic regime below M ∞ = 0 .
6, an increase in Machnumber destabilizes the flow, which was also documented by Br`es & Colonius (2008).However, in the present study, as the Mach number increases above M ∞ = 0 . L/D = 2 and 6, the slope of the neutral stability curve increases, which correspondsin the simulations to a reduction in the amplitude of the observed oscillations. Thisdestabilization ( M ∞ (cid:54) .
6) and subsequent stabilization ( M ∞ > .
6) effects of Machnumber are also reported by Yamouni et al. (2013) for short cavities with
L/D = 1 and2. Compared to
L/D = 2, the neutral stability curve for
L/D = 6 shifts downwards.The flows over cavities with larger aspect ratio are thus more unstable because of theincreased spatial extent for the shear layer to develop and amplify disturbances.3.3.
Rossiter modes
The primary oscillation mechanism in open-cavity flow is associated with the Rossitermodes. For all cases considered herein, the time history of the vertical velocity at themid-point of the cavity shear layer ( x = L/ , y = 0) is recorded. A discrete Fouriertransform is performed on the probe data collected after a minimum of 50 convectiveunits to eliminate the initial transients. For the unstable cases, the extracted oscillationfrequencies are compared with Rossiter’s prediction based on the modified formula (Heller& Bliss 1975), Eq. (1.1). The dominant and subdominant Rossiter modes as a function iglobal instabilities of compressible open-cavity flows Figure 8.
Stability diagram of 2D open-cavity flow for
L/D = 2 and 6 obtained from DNS.Stable cases: (cid:52) , ◦ ; unstable cases: (cid:78) , • . The triangles are from Br`es & Colonius (2008) and thecircles are from the present study. The dashed lines represent the approximate neutral stabilitycurves. For the cavity with L/D = 6, the wake-mode dominated cases are indicated by +. of Mach number are presented in figure 9. The frequencies (Strouhal numbers) of theRossiter modes show a decreasing trend with increasing M ∞ , which follows the predictionof the semi-empirical formula.From the previous work by Br`es (2007) on L/D = 2, it was inferred that Rossitermode I dominates the flow oscillation in the subsonic regime (0 . (cid:54) M ∞ (cid:54) . et al. (2004)for cavity with L/D = 2 at a much higher Reynolds number. As shown above in figure 5at Re θ = 46, the acoustic radiation becomes stronger at M ∞ = 0 .
8, where the dominantRossiter mode shifts from mode I to II. Thus, it appears that the strong acoustic waveemission can be correlated with Rossiter mode II rather than mode I for the short cavity.For flow over a cavity with
L/D = 6, the frequencies (Strouhal numbers) of Rossitermode for Re θ = 10 and 19 are shown in figure 10. At Re θ = 10, Rossiter mode II isthe dominant mode and it is the sole mode present. As Re θ increases to 19, the flow isunstable for M ∞ = 0 . M ∞ ∈ [0 . , . Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley M ∞ S t L Rossiter mode III M ∞ S t L Mode III ( a ) Re θ = 46 ( b ) Re θ = 56 . M ∞ S t L Mode III M ∞ S t L Mode III ( c ) Re θ = 67 ( d ) Re θ = 77 Figure 9.
Comparison of St L (= St D · L/D ) from the classic Rossiter semi-empirical formula,Eq. (1.1) (solid line) and the present work for
L/D = 2. Rossiter mode I: • ; mode II: ◦ . Thedominant modes are indicated by (cid:3) . The shaded grey line shows the estimate of the Machnumber where the dominant mode shifts. M ∞ S t L IIIIIIVMode I M ∞ S t L IIIIIIVMode I Wake mode ( a ) Re θ = 10 ( b ) Re θ = 19 Figure 10.
Comparison of St L (= St D · L/D ) from the classic Rossiter semi-empirical formula,Eq. (1.1) (solid line) and present work for
L/D = 6. The wake-mode dominated cases areindicated in the shaded region with M ∞ ∈ [0 . , . • , ◦ , (cid:78) and (cid:77) represent wake mode and itsharmonics. In the shear-layer mode, these symbols denote Rossiter mode I to IV. The dominantmodes are indicated by (cid:3) . iglobal instabilities of compressible open-cavity flows L/D = 6 at Re θ = 19, we alsofind that the flow exhibits a shear-layer mode near the neutral stability boundary. Forlonger cavity flows with L/D = 6, there is no dominant Rossiter mode shifting observedin the shear-layer mode cases; and the wake mode dominates the flow at M ∞ ∈ [0 . , . §
4, biglobal stability analysis is employed togain deeper insights into the flow instabilities for a range of spanwise wavenumbers, β .Two-dimensional global stability can be uncovered with β = 0, while spanwise-periodic3D global eigenmodes can be revealed by specifying β >
0. Because 2D base statesare required for performing the biglobal stability analysis, the results from the abovenonlinear simulations serve as the foundation for the subsequent stability analysis, inaddition to the unstable steady states discussed below.3.4.
Unstable steady states
A steady-state flow is a time-invariant solution of the Navier–Stokes equations, whichcan be used as base state in linear stability analysis. This base state might be stable orunstable depending on the flow conditions. Time-averaged flow is obviously time-invariantbut is not necessarily a solution of the Navier–Stokes equation. Numerical techniques canbe used to determine the unstable steady state even if the flow is naturally unstable.In the present work, the selective frequency damping method (˚Akervik et al. χ = 0 .
1. It has been verified that the numerically found time-invariant solutionsfrom the selective frequency damping method are indeed the unstable steady states bysubstituting them into the Navier–Stokes equations and ensuring that | ˙ q | < − . Asshown in figure 11, we find that the unstable steady states remain similar regardless ofthe Mach number variation in the short cavity ( L/D = 2), exhibiting the same featuresof the time-averaged streamlines as shown in figure 5. For the cavity with
L/D = 6,all the stable and unstable steady states share similar features regardless of the Machnumber, while the time-averaged flow fields vary significantly depending on the Machnumber as shown in figure 6. In the present paper, we use these steady states (in figure11) as the base states to perform global stability analysis.As we discuss in the next section, we find that the use of the stable/unstable steadystates shown in figure 11 as the base states in the biglobal stability analysis yieldsexcellent agreement in the oscillatory features in most of the cases except for a verylimited number of cases where the wake mode dominates the flow. For the wake-modedominated flow (at Re θ = 19), the mean flow can also be considered for its use as thebase flow, as discussed in our companion work (Sun et al. a ).Using the stable and unstable steady states, we perform extensive 2D and 3D linearglobal stability analysis. The 2D and 3D results are provided below in § §
4. Biglobal stability analysis
2D eigenmodes ( β = 0)In the biglobal stability analysis, 2D global eigenmodes can be found by specifying β = 0 in Eqs. (2.4) and (2.5), which theoretically translates to the perturbation beingassociated with infinite wavelength in the spanwise direction. In this section, the stabilityproperties are found for the base states. The eigenvalues ω = ω r + iω i obtained are8 Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley
Figure 11.
Streamlines of stable (S)/unstable (U) steady state of flows at various Machnumber.
L/D = 2: Re θ = 56 . L/D = 6: Re θ = 19. Contours represent vorticity ω z D/u ∞ ∈ [ − , reported in the non-dimensional form of the growth/decay rate ω i D/u ∞ and frequency St L = ω r L/ πu ∞ for 2D eigenmodes while St D = ω r D/ πu ∞ for spanwise periodic 3Deigenmodes.The leading eigenmodes are shear-layer modes whose spatial structures are mainlylocated in the shear layer region. Realizing that not all shear-layer modes can be calledRossiter modes, we further check that the frequencies of these shear-layer modes matchthe Rossiter mode frequencies predicted by the semi-empirical formula. Hence, all theleading eigenmodes herein are associated with Rossiter modes and correspond to shear-layer modes (i.e., Kelvin–Helmholtz instabilities) reported in the works by Meseguer-Garrido et al. (2014) and Yamouni et al. (2013). The eigenvalues of Rossiter modes arelisted in table 4 and their eigenvectors are shown in figures 12 and 13 for L/D = 2and 6, respectively. As predicted by the semi-empirical formula, an increase in Machnumber reduces the flow oscillation frequency. The spatial structure size of Rossitermodes becomes larger as Mach number increases. The spatial structures of higher-ordermodes exhibits finer spatial scales than those of lower-order modes. The major spatialstructures of the velocity eigenmodes ˆ u r have the largest magnitude in the shear layer,which indicates that Rossiter modes are driven by the shear layer of the flow. In thepressure eigenvector contours, there are beam-shaped structures that develop above thecavity. The geometric features of these beams parallel the compression waves previouslydiscussed in figure 5. Based on the results from both 2D nonlinear simulations andbiglobal stability analysis with β = 0, it indicates that Rossiter modes are indeed two-dimensional, shear-layer driven instabilities.From the discussion in § ω i D/u ∞ over M ∞ is presented in figure 14. For cavity with L/D = 2,the growth/decay rates of Rossiter modes I and II increase as Mach number changes from0.3 to 0.6. However, when M ∞ reaches the transonic regime, the values of growth rateshow a decreasing trend. The growth rate for Rossiter mode I peaks around M ∞ = 0 . M ∞ = 0 .
7, which reachesits peak near M ∞ = 1. This crossover point corresponds to the shift from Rossiter modeI to mode II seen in figure 9 (b). This peaking trends of growth/decay rates of Rossitermodes are also observed for the flows over cavity with L/D = 6. The growth rate ofthe eigenmode negatively correlating to the increase in free stream Mach number can be iglobal instabilities of compressible open-cavity flows L/D = 2 M ∞ I II0.3 0 . − . i . − . i .
426 + 0 . i .
726 + 0 . i .
377 + 0 . i .
687 + 0 . i . − . i .
613 + 0 . i . − . i .
566 + 0 . iL/D = 6 M ∞ I II III IV0.3 0 . − . i .
878 + 0 . i .
410 + 0 . i . − . i . − . i .
784 + 0 . i .
175 + 0 . i . − . i . − . i .
708 + 0 . i .
085 + 0 . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i Table 4.
Eigenvalues ω = ω r + iω i associated with Rossiter mode I to IV for β = 0 are reportedin a form of St D · ( L/D ) + i ( ω i D/u ∞ ) at Re θ = 56 .
8. Frequency ω r is normalized as Strouhalnumber St L = ω r L/ πu ∞ , and growth/decay rate ω i is normalized as ω i D/u ∞ . related to the behavior of the neutral stability curve in the nonlinear stability diagramsshown in figure 8. Both stability diagrams indicate that an increase of Mach number inthe subsonic regime ( M ∞ < .
6) destabilizes the flow, but over the transonic regime, anincrease of Mach number ( M ∞ > .
6) stabilizes the flow. Furthermore, in the cases of
L/D = 2 and M ∞ = 0 .
3, as shown in figure 14, the values of ω i D/u ∞ are both negative.In other words, disturbances related to Rossiter modes I and II decay in the base flow,which matches the result from the stability diagram in figure 8 that the flow is stable.This agreement is also noticed in the cases of cavity with L/D = 6 at M ∞ = 1 . § M ∞ = 0 . L/D = 6, the dominant Rossiter mode from biglobalstability analysis deviates from that in the DNS in which the wake mode is dominant.Moreover, the eigenvector shown in figure 13 at M ∞ = 0 . et al. a ).By the observation that the mean (time-averaged) flow and unstable steady state aresignificantly different in this wake-mode dominated flow, biglobal stability analysis wasperformed using both states (mean flow and unstable steady state) as base states. Whenthe mean flow is prescribed as the base state, the wake-mode eigenmode is captured bythe linear stability analysis, which can also predict its frequency with only 4% differencebetween that and the wake-mode frequency determined from the 2D DNS. It should bementioned that in the linear stability theory, the nonlinear interactions among modesare neglected. There is however a strong nonlinear dynamical process in the wake-modedominated flow, making the mean greatly deviate from the unstable steady state. Hence,the use of the mean flow as the base flow is necessary for uncovering the wake mode.0 Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley . . . . u r ˆ p r M Rossiter mode I Rossiter mode I Rossiter mode I Rossiter mode IRossiter mode IRossiter mode IRossiter mode IRossiter mode I
Rossiter mode IIRossiter mode IRossiter mode IIRossiter mode I
Figure 12.
Spatial structures of the real component of the eigenmodes ˆ u and ˆ p for cavity of L/D = 2 at Re θ = 56 .
3D eigenmodes ( β (cid:54) = 0)In this section, we discuss characteristics of the 3D global instability modes with finitewavelength λ/D = 2 π/β in the spanwise direction prescribed in Eq. (2.4). The analysesperformed for the 3D instabilities are analogous to those for the 2D cases shown in theprevious section. In what follows, the spanwise wavelength λ/D is set in a range of 0.5– 2.0 following Br`es & Colonius (2008). Note that the 3D instabilities discussed in thissection represent spanwise-periodic 3D instabilities with selected wavenumbers β .The eigenspectra of the 3D eigenmodes for L/D = 2 and 6 are presented in figure 16.It should be noted that the eigenspectra are only shown in the vicinity of the origin.For the
L/D = 2 cavity flows at Re θ = 56 .
8, unstable 3D modes are only observed inthe subsonic cases of M ∞ = 0 . λ/D = 1 .
0, while all other 3D eigenmodes withhigher Mach number ( M ∞ (cid:62) .
6) are stable. For
L/D = 6 cases at Re θ = 19, all the3D eigenmodes captured are stable. Below, the effects of cavity geometry, Mach numberand spanwise wavelength λ/D on the most-unstable/least-stable (dominant) 3D modesare further examined.Based on eigenspectra shown in figure 16, the eigenvalue of the most-unstable/least- iglobal instabilities of compressible open-cavity flows . . . M Rossiter mode IRossiter mode IIRossiter mode IIIRossiter mode IVRossiter mode IRossiter mode IIRossiter mode IIIRossiter mode IV ˆ u r ˆ p r Figure 13.
Spatial structures of the real component of the eigenmodes ˆ u and ˆ p for cavity of L/D = 6 at Re θ = 19. stable eigenmode for each Mach number is extracted and plotted as a function of spanwisewavelength λ/D in figures 17 and 18. For both cavity geometries, the overall trends ofgrowth/decay rates are similar regardless of M ∞ . An increase in Mach number stabilizesall the dominant 3D eigenmodes. For each Mach number, the growth/decay rate ω i D/u ∞ Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley
Figure 14.
Growth/decay rates of Rossiter modes as a function of free stream Mach numberdetermined from biglobal stability analysis with β = 0. Left: L/D = 2, Re θ = 56 .
8; right:
L/D = 6, Re θ = 19. Rossiter mode I: • ; mode II: ◦ ; mode III: (cid:78) and mode IV: (cid:77) . M ∞ S t L M ∞ S t L IIRossiter mode I
Biglobal Nonlinear sim.
IIMode I
Biglobal Nonlinear sim.
IIIIV
Figure 15.
Dominant Rossiter modes captured from the 2D nonlinear simulations and thebiglobal stability analysis. Left:
L/D = 2, Re θ = 56 .
8; right:
L/D = 6, Re θ = 19. Rossiter modeI: • ; mode II: ◦ ; mode III: (cid:78) and mode IV: (cid:77) . The dominant Rossiter modes from the biglobalstability analysis and the nonlinear simulations are indicated by (cid:3) and ∗ , respectively. of the dominant 3D eigenmode is a function of λ/D . To identify the 3D eigenmodeproperties, their frequencies and spatial structures can distinguish the modes based ontypes of instabilities.We follow the nomenclature used by Br`es & Colonius (2008) for the leading modes infigures 17 and 18. As shown in figure 17 for cavity with L/D = 2, the leading eigenmodewith λ/D = 0 . i with St D = 0 (referred to as mode i by Br`es & Colonius (2008))except at M ∞ = 1 .
4. The dominant 3D mode at M ∞ = 1 . ii , a traveling modediscussed next. However, its decay rate is close to that of a stationary mode as shown infigure 16. The eigenmodes with λ/D ∈ [0 . , .
25] are traveling modes with St D ≈ . ii by Br`es & Colonius (2008)). For each of λ/D (cid:54) .
25, the frequency, St D , of all the dominant 3D modes are independent of Mach number, but these 3D modescan be stabilized by increasing M ∞ as mentioned above. However, the frequencies of thedominant 3D modes exhibit a sudden decrease near λ/D = 1 .
5, and the modes exhibitdifferent branches for larger λ/D ( (cid:62) .
75) depending on Mach numbers. As shown infigure 17, for the flows at 0 . (cid:54) M ∞ (cid:54) .
6, the dominant modes are traveling modes with St D ≈ .
015 (referred to as mode iii by Br`es & Colonius (2008)), while for the transonicflows at 0 . (cid:54) M ∞ (cid:54) .
4, the dominant modes are stationary. Moreover, we indicatethe transitions of dominant modes over the spanwise wavelength in figure 17 by grey iglobal instabilities of compressible open-cavity flows Figure 16.
Eigenspectra of the 3D eigenmodes for cavity with
L/D = 2 ( Re θ = 56 .
8) and 6( Re θ = 19) with M ∞ ∈ [0 . , .
4] and λ/D ∈ [0 . , . λ /D -0.18-0.12-0.0600.02 ω i D / U ∞ . M ∞ M increase λ /D S t D M o d e i M o d e ii M o d e iii M o d e i M o d e ii M o d e iii (de Vicente et al.)(Brès & Colonius) ! i D / u Figure 17.
Eigenvalues of the dominant 3D modes as a function of M ∞ and spanwise wavelength λ/D for cavity with L/D = 2 at Re θ = 56 .
8. Left: growth/decay rate ω i D/u ∞ ; right: frequency St D . Results from Br`es & Colonius (2008) and de Vicente et al. (2014) are also compared tothe present results. The gray regions represent the transitions of dominant modes. regions. As the wavelength of the 3D eigenmode is increased from 0.5 to 2, in general,the dominant 3D mode shifts from stationary to traveling mode and back to stationarymode. This phenomenon was also reported by de Vicente et al. (2014) on incompressiblecavity flows that the dominant 3D mode can arise from different instabilities in terms ofits spanwise wavelength.For the eigenvalues in the case of a cavity with L/D = 6 presented in figure 18, thedominant 3D mode with λ/D = 0 . ii ( St D ≈ . Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley λ /D -0.5-0.3-0.10 ω i D / U . M ∞ increase M λ /D S t D M o d e ii M o d e ii M o d e ii ⇤ M o d e ii ⇤ ! i D / u Figure 18.
Eigenvalues of the dominant 3D modes as a function of M ∞ and spanwise wavelength λ/D for cavity with L/D = 6 at Re θ = 19. Left: growth/decay rate ω i D/u ∞ ; right: frequency St D . The gray regions represent the transitions of dominant modes. has characteristics of mode ii type, as we later discuss in the spatial structures of theeigenmodes (in figure 19). The dominant 3D modes with λ/D = 0 .
75 and 1.0 have higherfrequencies compared to that of smaller wavelenghths, but their eigenvectors still sharesimilarities as shown later. There is another mode having frequency St D ∈ [0 . , . λ/D = 1 .
25 and 1.5, which is denoted as mode ii ∗ due to the variation in bothfrequency and eigenvector compared to mode ii . As λ/D increases from 1.5 to 1.75, thedominant mode changes from a traveling mode to a stationary mode. With λ/D = 1 . . (cid:54) M ∞ (cid:54) .
6, these 3D modespossess nonzero frequencies, but with relatively small St D compared to the other travelingmodes. A decreasing trend in the growth/decay rates at λ/D = 1 . . (cid:54) λ/D (cid:54) . λ/D . For the short cavity with L/D = 2, the eigenvectors obtained forall Mach numbers considered are similar in terms of the spatial structures. For all valuesof λ/D considered, the eigenvectors show variations in the aft part of the cavity wherethe major recirculation zone is located as shown in the unstable steady states shown infigure 11, while their structures are also present in the shear-layer region of the flows.Br`es & Colonius (2008) examined the 3D instabilities as well at M ∞ = 0 . L/D = 6, the spatial structures of the dominant 3D eigenmodes alsoappear independent of the Mach number, as in the case of
L/D = 2. As shown in figure 19,in the cases of λ/D = 0 . ii are located in the rearpart of the cavity, which exhibit similar features to those presented in the shorter cavity,but with the eigenmodes extending upstream along the shear layer. The spatial structuresof mode ii ∗ ( λ/D = 1 .
5) align along the floor from the rear towards the mid-region ofthe cavity, which is significantly different from the mode ii that stems from centrifugalinstabilities. The streamwise-stretched recirculation pattern in the unstable steady stateshown in figure 11 appears due to the large aspect ratio of the long cavity. This elongatedrecirculating flow is likely the reason for the formation of the mode ii ∗ , which explains iglobal instabilities of compressible open-cavity flows Figure 19.
Iso-surface of eigenvectors of the least-stable 3D eigenmodes at M ∞ = 0 . λ/D ∈ [0 . , . u r /u ∞ areindicated below each subplot. Red and blue colors represent the indicated positive and negativevalues, respectively. the absence of mode ii ∗ in the short cavity flows. For the stationary mode for λ/D = 2 . λ/D = 2 . M ∞ = 0 . − .
6) is not exactly zero, theeigenvectors still share similar features to the stationary mode. The spatial structures of3D instability modes appear to be strongly influenced by their spanwise wavelengths inthe long cavity cases, and the leading eigenmode can be a traveling or stationary modedepending on λ/D .In the work by Liu et al. (2016) that sidewall effects are considered with triglobalstability analysis for incompressible cavity flow with
L/D = 6, they concluded thatshear-layer instability is dominant in finite-span cavity flow. This is also observed in thepresent study on spanwise homogeneous cavity flow at M ∞ = 0 . Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley St D λ/D
2D DNS 0.207 Rossiter mode I ∗ ∞ Biglobal stability 0.213 Rossiter mode I ∗ ∞ ii ∗ ∞ ii Table 5.
Comparison of the frequencies of the dominant modes obtained from DNS and biglobalstability analysis for
L/D = 2, Re θ = 56 . M ∞ = 0 .
6. Dominant mode is denoted with ∗ . Comparison with DNS
Next, let us compare our findings from linear stability analysis with those from DNS.Three-dimensional DNS with
W/D = 2 are performed at M ∞ = 0 . L/D = 2 ( Re θ = 56 . St D , obtained fromDNS and biglobal stability analysis are listed in table 5. In the 3D DNS of L/D = 2,the dominant 2D mode is Rossiter mode II. However, in 2D DNS and biglobal stabilityanalysis with λ/D = ∞ , the Rossiter mode I is the dominant mode. By performingthe simulation over a sufficiently long time, the traveling mode with St D = 0 .
027 at λ/D = 1 . ii with λ/D = 1 . ω i D/u ∞ = − . λ/D considered. Although this mode hasnegative growth rate, in the 3D nonlinear simulation, unstable 2D Rossiter mode and3D modes could interact via nonlinearities and result in the existence of the 3D modein the nonlinear flows, especially when the eigenvalue is close to the neutral stabilityline ( ω i D/u ∞ = 0). The comparison of the 3D spatial structures of nonlinear flow andtraveling mode ii are displayed in figure 21, where instantaneous flow fields and eigenmodewith a quarter period interval are presented. Mode ii shown in figure 20 has qualitativelysimilar spatial structures to the eigenfunction of Mode I found by Meseguer-Garrido et al. (2014) at a similar incompressible flow condition. In their work, this mode is unstablein incompressible conditions. In contrast, mode ii is slightly stable in the present workat M ∞ = 0 .
6, which agrees with the discussion in section 4.3 that an increase in Machnumber can stabilize 3D eigenmodes. The iso-surface of spanwise velocity w/u ∞ fromnonlinear simulation exhibits comparable structures to those of mode ii . In particular,the 2D Rossiter mode also appears in the nonlinear DNS, in which streamwise distortionsof the spatial structures are observable in the shear-layer region.For the cavity with L/D = 6 ( Re θ = 19), the violent wake mode present in 2D DNS isnot observed in 3D DNS. Instead, Rossiter mode II dominates the flow. When the flow hasreached steady oscillations, there is no evident low-frequency peaks of the 3D structuresin the nonlinear flows. In the comparison of the 3D spatial structures revealed by the 3DDNS and 3D eigenmodes as shown in figure 22, we find that for the 3D nonlinear flows,the spatial structures located in the rear region of the cavity stay almost stationary inthe spanwise direction with wavelength of λ/D = 2, while those obtained from the linearstability analysis are distributed in the front and rear regions of the cavity. As shown infigure 18, though all the 3D eigenmodes have negative values of growth rate, the spatialstructures of 2D Rossiter modes (figure 13) and 3D eigenmodes (figure 19) overlap in the iglobal instabilities of compressible open-cavity flows Figure 20.
Time histories of streamwise velocity u/u ∞ collected by a probe located at x/D = 1, y/D = 0, z/D = 1 are shown on the left. The corresponding Fourier spectra are presented onthe right. (a) 2D DNS and (b) 3D DNS ( W/D = 2) for
L/D = 2, Re θ = 56 . M ∞ = 0 . W/D = 2)( t − t ) /T = 0 1 / / /
43D mode ii ( λ/D = 1 . φ − φ ) / π = 0 1 / / / Figure 21.
Comparison of the iso-surface of the spanwise velocity w/u ∞ = ± .
013 from the3D nonlinear simulation and ˆ w/u ∞ = ± .
03 from the dominant 3D eigenmode with λ/D = 1 . L/D = 2 at M ∞ = 0 . Re θ = 56 .
8. Represented by T is the time periodof the 3D mode captured by the nonlinear simulation, t and φ are reference time and phase,respectively. Red and blue colors represent positive and negative values, respectively. Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley ( a ) 3D nonlinear simulation ( b ) 3D stationary mode ( λ/D = 2 . Figure 22.
Comparison of the iso-surface of the instantaneous spanwise velocity w/u ∞ = ± .
083 from the 3D nonlinear simulation and ˆ w/u ∞ = ± . × − from thedominant 3D eigenmode (stationary mode) with λ/D = 2 . L/D = 6 and M ∞ = 0 . Re θ = 19. Red and blue colors represent positive and negative values, respectively. rear part of the cavity, which can lead to modal interactions and modify the instabilitiespredicted from biglobal stability analysis.In the 3D DNS, the large vortex roll-ups of wake mode present in the 2D DNS is nolonger observable. Meanwhile, the spanwise motion becomes evident in the 3D flows. Touncover these spanwise effects on the long cavity, further analysis has been performedin our companion work (Sun et al. a ), in which we found that the presence ofspanwise motion (or three-dimensionalities) in the flow can preclude the formation of thewake mode, reducing the intense fluctuations of the flow. This observation suggests thepotential to reduce cavity flow oscillations by introducing spanwise variations to the baseflows. In order to weaken the strength of the impingement of shear layer on the trailingedge of the cavity, we can trigger the emergence of 3D global modes and achieve energyredistribution from spanwise vortices to streamwise vortical structures. To retain theenergy inside the cavity, the spanwise wavelength of perturbation introduced to the flowshould be in the range of 0 . − .
5, since the spatial structures of 3D modes associatedwith these wavelengths mainly stay inside the cavity, which has been illustrated in figure19.To verify this concept of using 3D instability to reduce cavity flow oscillations, we notein passing that 3D slot-jets can be introduced along the leading edge of the cavity toalter the flow features. The performance of these control strategies via 3D steady blowingfor cavity flows is assessed by Zhang et al. (2015) and George et al. (2015) through alarge number of experiments and some representative large-eddy simulations. They findthat the wavelengths for effective 3D flow control to reduce cavity flow oscillations arein good agreement with the findings of present global stability analysis. Although theidea of 3D flow control is not new (Zdravkovich 1981), the present work particularlyuncovers the temporal instabilities affected by compressibility in the transonic regime.Moreover, the spanwise wavelength and frequency associated with 3D instabilities areidentified, which can be useful in examining the influence of flow parameters for 3D flowcontrol designs. We believe that the insights from the global instability characteristics ofcompressible open-cavity flows can provide valuable information for designing physics-based flow control strategies. iglobal instabilities of compressible open-cavity flows
5. Conclusions
Two- and three-dimensional biglobal instabilities of two-dimensional compressibleopen-cavity flows are examined for rectangular cavities with aspect ratios of
L/D = 2and 6. Noteworthy in the present work is the focus on the global instability analysis forlong cavity (
L/D = 6) flows in the transonic regime, which has not been examined indetail in the past, despite their importance in being representative of practical cavityapplications on aircraft. Direct numerical simulations and biglobal stability analysisare used to uncover the influence of Mach number, Reynolds number, and spanwisewavelength on the 2D and 3D global instability modes with respect to the 2D steadystates. In the 2D DNS, we find that an increase in Mach number in the subsonic regimedestabilizes the flow but stabilizes the flow in the transonic regime at low Reynoldsnumber for both short (
L/D = 2) and long (
L/D = 6) cavities, which agrees with thefindings from the work by Yamouni et al. (2013) on flows over short cavities (
L/D = 1and 2). Moreover, the dominant Rossiter mode can shift due to the variation in the freestream Mach number.The 2D eigenmodes obtained from biglobal stability analysis exhibit excellent agree-ment with the properties of the 2D flow characteristics uncovered from 2D DNS, indicat-ing that these 2D global modes are Rossiter modes driven by the shear layer. Throughthe 2D eigen-analysis with β = 0, destabilization and subsequent stabilization effects viaincreasing Mach number is captured. Moreover, a shift in the dominant Rossiter mode isobserved with Mach number variation, which explains the mode shifting behavior in 2Dnonlinear simulation results.In contrast to the 2D modes, the dominant 3D modes unraveled in the present work areinstabilities that are largely unaffected by a change in Mach number. Although an increasein Mach number stabilizes all 3D eigenmodes, the overall behavior of 3D eigenmodes asa function of the spanwise wavelength λ/D is almost independent of the Mach number.However, the type of dominant 3D mode is strongly dependent on λ/D . For the shortcavity with L/D = 2, the most-unstable/least-stable 3D modes have λ/D ∈ [1 . , . ii ) stemming from centrifugal instabilities for almost all Mach numbers considered,as reported by Br`es & Colonius (2008). For the long cavity with L/D = 6, the least-stable3D modes have λ/D = 1 .
25 and 2.0 for cases of M ∞ (cid:46) . M ∞ (cid:38) .
6, respectively.Moreover, there are additional types of eigenmodes for this long cavity flow possessingvarious spatial structures compared to those of the shorter cavity.In 3D DNS, both frequencies and spatial structures of the leading 3D mode seen inthe DNS correspond closely to the results of the linear biglobal stability analysis ofthe dominant 3D eigenmode for
L/D = 2. However, for
L/D = 6, 3D DNS revealslarge roll up of the cavity shear layer, which indicates strong nonlinearities that departfrom the linear analysis. Nevertheless, the insights from global stability analysis providea potential pathway to reduce flow oscillations by introducing spanwise variations tothe base flows, which has been demonstrated in the experiments by Lusk et al. (2012);Zhang et al. (2015) and George et al. (2015). The analyses in this study provide valuableknowledge on global instabilities of 2D and 3D compressible open-cavity flows, which canbe leveraged in designing physics-based flow control strategies in upcoming studies.
Acknowledgement
This work was supported by the U.S. Air Force Office of Scientific Research (Grantnumber: FA9550-13-1-0091, Program Managers: Drs. D. Smith and I. Leyva). The highperformance computing resource was provided by the Research Computing Center at the0
Y. Sun, K. Taira, L. N. Cattafesta and L. S. Ukeiley
Florida State University. The authors also acknowledge the insightful discussions withDr. Guillaume A. Br`es.
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