Acoustic response of turbulent cavity flow using resolvent analysis
11 Acoustic response of turbulent cavity flow usingresolvent analysis
Qiong Liu † and Datta Gaitonde Department of Mechanical and Aerospace Engineering, The Ohio State University, ColumbusOH 43220, USA
Fluid-acoustic interactions are important in a variety of applications, and typically resultin adverse effects. We analyze the influence of Mach number on such interactions andtheir input-output characteristics by combining resolvent analysis with Doak’s momentumpotential theory. The specific problem selected is the flow over an open cavity of
L/D = 6 at Re = 10 , and M ∞ = 0 . and . , respectively. The resolvent forcing and responsemodes of the time- and spanwise-averaged Large Eddy Simulations are first obtained at eachMach number. The response modes are then decomposed into their hydrodynamic, acousticand thermal components. Although the results depend quantitatively on Mach number,some trends remain consistent. In particular, at lower frequencies, the acoustic componentappears primarily at the trailing edge of the cavity. When the frequency is increased, theprimary acoustic response moves towards the leading edge and overlaps with its hydrodynamiccomponent, indicating greater influence on the flowfield. Inspired by actual cavity flow control,the forcing is then localized to two regions – the leading-edge and front wall of the cavity –and also filtered to consider notional actuators that can separately introduce each componentof velocity, density, and temperature forcing, respectively. Among these different types ofactuation perturbations, regardless of Mach number, streamwise velocity forcing achieves thelargest energy amplification when placed at the leading edge, with considerable reduction ineffectiveness when placed on the front wall. A suitable ratio is defined to assess the relativeacoustic versus hydrodynamic modification. The frequency where this ratio is largest is oftenslightly higher than that associated with the energy amplification peak. For both subsonic andsupersonic cavity flows, beyond a certain forcing frequency threshold value, the nature of theacoustic versus hydrodynamic response becomes independent of the forcing type; however,the amplification continues to be strongly impacted by the forcing frequency. This reinforcesthe importance of the frequency for energy amplification mechanisms. Overall, this workprovides an alternative approach to examine input-output flow-acoustic characteristics whereresonance is dominant and provides a means to evaluate the relative effectiveness of differenttypes and locations of actuation.
1. Introduction
The wide range of applications where cavity flows are important has motivated nu-merous efforts to understand their physics and to design effective flow controls. Theseinquiries started from 1950’s (Krishnamurty 1955) with both experimental (Roshko 1955;Rossiter 1964; Heller et al. et al. et al. et al. et al. et al. et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] F e b Q. Liu & D. Gaitonde edge, vortical structures are formed over the shear-layer and impinge on the cavitytrailing edge, scattering acoustic waves. These disturbances then propagate upstreamand interfere with the unsteadiness of vortical structures. Such interactions between flowand acoustic waves produce a feedback loop type of oscillation (Rowley & Williams2006; Cattafesta et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. (2000) performed an experimental controlstudy using a zero-net-mass type of actuation at the leading edge of the cavity. Theexperiment was conducted using forcing jet directions of 0 ◦ , 45 ◦ , and 90 ◦ , respectively,to the free-stream flow. The control result showed that forcing in-line with the free-streamwas more effective at suppressing resonance tones, than the other directions. Ukeiley et al. (2007) sought the optimal angle for leading-edge blowing in a supersonic ( M ∞ = 1 . ◦ , 45 ◦ , and90 ◦ with the reference direction of free-stream flow. The most effective control case wasthe mass injection actuation with a blowing direction of 90 ◦ . Other types of actuatorsemployed include those using plasma effects (Chan et al. ◦ to the free- coustic response of turbulent cavity flow et al. et al. et al. ρ u ) into itsacoustic (irrotational-isentropic), hydrodynamic or vortical (rotational) and a thermal orentropic (irrotational-isobaric) component. To distinguish this decomposition from theKovasznay approach, we designate these as fluid-thermodynamic (FT) components. Thesplitting is exact regardless of non-linearities or variations in the underlying mean flowand has been successfully applied to model problems (Daviller et al. et al. Q. Liu & D. Gaitonde behavior of the acoustic mode and its interactions with the other components that resultin the observed near-field sound signature and the features inside the cavity and the endwall. The use of MPT with input-output analysis to provide insightful information on thefluid-thermodynamic response to forcing has been illustrated by Houston et al. (2020),who separated disturbances in a hypersonic boundary layer over blunt cone into theirvortical, thermal and acoustic components. A key finding was that the vortical output ismore sensitive to the wall-normal forcing.The results from the underlying flow-acoustic interaction feed naturally into the answerto the second, more practical, question above regarding actuator placement. Usually, theforcing location is chosen to be close to the leading edge, which is considered to be themost sensitive region to alter the shear layer. Consequently, we restrict the input forcingto two distinct locations: leading-edge and front wall. The other important propertyconcerns the nature of the perturbation introduced (mass or energy injection for exam-ple), which varies with the actuator employed. To isolate the influence of this variable,we consider idealized inputs corresponding to different types of perturbations, one ata time. Thus, the independent effects of velocity components, density and temperatureperturbations are evaluated on the overall desired flow-acoustic interaction.Finally, the response and effectiveness of perturbations depends substantially on theflow parameters, particularly Mach number. Control of supersonic ( M ∞ = 1 .
44) andsubsonic ( M ∞ = 0 .
6) cavity flow were experimentally conducted by Lusk et al Lusk et al. (2012) and Zhang et al Zhang et al. (2019), respectively. The results showed thatthe most effective control actuation for the subsonic flow with the slot length was threetimes longer than that for supersonic control. The Mach number is thus clearly influential,not only because of the change in the nature of resonance and acoustic emission, but alsobecause of the differences in response to control perturbations. To address this aspect, inthis work, we analyze the results for two Mach numbers, one subsonic ( M ∞ = 0 .
6) andthe other supersonic ( M ∞ = 1 .
2. Problem description and approach
Problem setup
We consider the turbulent flow over a rectangular cavity at a subsonic ( M ∞ ≡ u ∞ /a ∞ = 0 .
6) and a supersonic (1 .
4) Mach number, where a ∞ is the sound speedand u ∞ is the free-stream velocity. The cavity length-to-depth ratio is L/D = 6. FIG. 1shows an instantaneous snapshot of the compressible flow over the cavity as well asa schematic of flow-acoustic feedback loop in compressible cavity flow. The Reynoldsnumber is Re ≡ ρ ∞ u ∞ D/µ ∞ = 10 , µ ∞ and ρ ∞ are the free-stream dynamicviscosity and density, respectively. All variables are non-dimensionalized, lengths by thecavity depth D , temperature by T ∞ , pressure by ρ ∞ u ∞ , density by ρ ∞ , and time by D/u ∞ . coustic response of turbulent cavity flow Figure 1.
Problem description: (a) compressible flow over a rectangular cavity with
L/D = 6and
W/D = 2 at Re = 10 , Q − criterion = 10 colored by pressure fluctuation at M ∞ = 1 .
4. The density gradient magnitudeis shown in gray color. (b) Schematic of flow-acoustic feedback loop in compressible cavity flow(not to scale).
The numerical setup (Sun et al. a ; Liu et al. et al. b ) has beenvalidated with the companion experimental study (Zhang et al. CharLES (Khalighi et al. et al. et al. × ×
128 points is used in the x , y , and z directions around the cavity region( x, y, z ) /D ∈ [ − , × [ − , × [ − , δ /D = 0 .
167 based on the companion experiments George et al. (2015); Zhang et al. (2019). A one-seventh power law velocity profile is imposed, with random superimposedFourier modes to simulate unsteady fluctuations entering the cavity interaction (Bechara et al.
Resolvent analysis
The input-output characteristics of the turbulent cavity flows at each Mach numberare first discussed, as obtained from a resolvent analysis. For this, the flow variablesare decomposed into the spanwise- and time-averaged base state ¯ q ( x, y ) ≡ [¯ ρ, ¯ u, ¯ v, ¯ w, ¯ T ]and statistically stationary fluctuating components q (cid:48) ( x, y, z, t ) ≡ [ ρ (cid:48) , u (cid:48) , v (cid:48) , w (cid:48) , T (cid:48) ]. Theresulting fluctuation Navier-Stokes equation can be expressed as an input-output sys-tem (Jovanovi´c & Bamieh 2005; McKeon & Sharma 2010; Schmid & Henningson 2012), ∂ q (cid:48) ∂t = L ( ¯ q ) q (cid:48) + M f (cid:48) (2.1)where L (¯ q ) is the Navier-Stokes operator linearized about the base state ¯ q . The finite-amplitude nonlinear terms are incorporated in f (cid:48) . M is the coupling matrix as furtherdiscussed below. Q. Liu & D. Gaitonde
The spanwise-homogenous property of the base state enables the use of a modal ansatzwith spanwise wavenumber β and temporal frequency ω : q (cid:48) ( x, y, z, t ) = ˆ q ( x, y ) e − i ωt − βz , f (cid:48) ( x, y, z, t ) = ˆ f ( x, y ) e − i ωt − βz (2.2)Inserting these into equation (2.1), we obtainˆ q = [ − i ω I − L ( ¯ q , β )] − M ˆ f (2.3)where R = [i ω I − L ( ¯ q , β )] − M is the resolvent operator, which serves as a transferfunction between the input ˆ f and the corresponding output ˆ q for a given flow state ( ¯ q )and modal parameters ( β and ω ).The energy amplification of the system may be evaluated from the ratio of outputto input energy || ˆ q || E || ˆ f || E , where || · || E is an energy norm. A singular value decomposition(SVD) of the resolvent operator facilitates a ranking of the energy amplification ratioin descending order. For compressible flow, the compressible energy norm (Chu 1965) isused: E = (cid:90) S (cid:20) ¯ a ρ γ ¯ ρ + ¯ ρ ( u + v + w ) + ¯ ρC v T ¯ T (cid:21) d s where S is the domain of interest. This yields W R W − = Q Σ F ∗ , (2.4)in terms of the weight matrix, W , based on the compressible energy norm E . The matrix Q = [ ˆ q , ˆ q , . . . , ˆ q n ] holds the set of optimal response directions and F = [ ˆ f , ˆ f , . . . , ˆ f n ]contains the corresponding forcing directions, where ˆ q i = (ˆ ρ r , ˆ u r , ˆ v r , ˆ w r , ˆ T r , ) and ˆ f i =(ˆ ρ f , ˆ u f , ˆ v f , ˆ w f , ˆ T f , ) with n is number of solved singular values. The superscript ∗ denotesthe Hermitian transpose. The singular values Σ = diag( σ , σ , . . . , σ n ) represent theenergy amplification (gain) between response and forcing modes.For computational efficiency, the resolvent analysis is performed on the computationaldomain which has a downstream and farfield extent of 5 D and grid size of 50 ,
358 cells.This is reasonable because the primary flow physics of interest occur over the shear layerregion and inside the cavity. A smaller size computational domain and grid balancesthe computational efficiency and flow dynamics of the desired investigation. The SVD isperformed using the ARPACK package with a Krylov space of 12 vectors and a residualtolerance of 10 − . The results converge to at least 7 significant digits and are verified tobe O (1%) of accuracy with respect to the domain size and mesh resolution.Since the base flow is unstable, it becomes crucial to highlight amplifications that occuron a shorter time scale than those associated with the asymptotic behavior observed withclassical instability theory. This consideration aids in achieving the main objective offinding preferred energy transfer mechanisms from the mean flow to the fluctuation field,which are necessary to provide physical insight into potential flow control strategies. Toachieve this objective, the discounting technique (Jovanovi´c 2004; Yeh & Taira 2019) isemployed to obtain forcing and response modes. The method introduces a free parameter,denoted the discounting parameter κ , the choice of which is predicated on informationabout the most unstable growth rate as obtained from the stability analysis. Other, morephysical techniques, to address the unstable linear operator may be found in (Schmidt et al. et al. coustic response of turbulent cavity flow Doak’s momentum potential theory
The distribution of the response modes into their fluid-thermodynamic content is per-formed using Doak’s momentum potential theory (Doak 1989) in frequency-wavenumberdomain. The corresponding time-domain implementation for fluctuations obtained fromLES has been discussed Unnikrishnan & Gaitonde (2016); Prasad & Morris (2020) forvarious free jet flows. The frequency-wavenumber domain is more suitable for applicationto the response mode. A crucial feature of the approach is to adopt a vector quantity,the momentum density, ρ u , as the primary dependent field on which to perform thedecomposition. This is expressed as a sum of solenoidal and irrotational componentsaccording to Helmholtz’s theorem, ρ u = B − ∂ψ∂ x , ∂ B ∂ x = 0 (2.5)where ρ is the density and u is the velocity vector. The solenoidal and irrotationalcomponents are B and − ∂ψ/∂ x , respectively.Upon substitution of equation (2.5) into the continuity equation, the following rela-tionship is obtained ∂ρ∂t − ∂ ψ∂ x = 0 . (2.6)By Reynolds decomposing the flow variables into a time-averaged state and fluctuationcomponents, the retained fluctuation density and scalar momentum potential gradientbecome ∂ρ (cid:48) ∂t = ∂ ψ (cid:48) ∂ x (2.7)For the single-chemical-component flow, the density is considered as a function of pressure p and entropy S . Hence, ∂ρ (cid:48) /∂t can be splitted as the sum of 1 /c ∂p (cid:48) /∂t and ρ s ∂S (cid:48) /∂t ,where c is local sound speed. This effectively splits the irrotational field into its acousticand thermal components. Equation (2.7) then becomes ∂ ψ (cid:48) ∂ x = 1 c ∂p (cid:48) ∂t + ρ s ∂S (cid:48) ∂t . (2.8)Thus ψ (cid:48) can be written as the sum of acoustic ψ (cid:48) A and thermal ψ (cid:48) T components, where ∂ ψ (cid:48) A /∂ x = 1 /c ∂p (cid:48) /∂t and ∂ψ (cid:48) T /∂ x = ρ s ∂S (cid:48) /∂t .By again considering the spanwise homogeneous nature of the flow problem, modalexpressions (2.2) with spanwise wavenumber β and temporal frequency ω may be intro-duced into (2.8) and its acoustic component. The fluctuation scalar momentum potentialgradient and its acoustic components are related through a Poisson equation:( ∂ ∂x + ∂ ∂y − β ) ˆ ψ = − iω ˆ ρ (2.9)( ∂ ∂x + ∂ ∂y − β ) ˆ ψ A = − iω ˆ pc , (2.10)where ˆ ρ is density and ˆ p is the pressure, respectively, of the response mode. The Poissonequations for ˆ ψ and ˆ ψ A are solved by prescribing Dirichlet boundary conditions alongthe cavity wall boundaries. For simplicity, the thermal component is calculated usingˆ ψ T = ˆ ψ − ˆ ψ A . As shown below, this successful decomposition of response momentumpotential into hydrodynamic, acoustic and thermal components provides a much deeperunderstanding of the flow-acoustic input-output characteristics. Q. Liu & D. Gaitonde
Figure 2.
Mean flow stability analysis for estimating the discounting parameter κ for subsonicand supersonic flows. The dashed line indicates the chosen value of κ =0.15 and 0.12 forsubsonic and supersonic cases, respectively. Power spectral density of instantaneous pressureat [ x, y, z ] /D = [3 , ,
0] from nonlinear LES simulations.
3. Results
Stability analysis of mean flow linear operator
The discounting technique (Jovanovi´c 2004; Yeh & Taira 2019) used to highlight theshort time scale of energy amplification, introduces a free parameter κ into the resolventoperator: R (cid:48) = [ − i( ω + i κ ) I − L ( ¯ q , β )] − M (3.1)Since κ is selected to be slightly larger than the most unstable growth rate Yeh et al. (2020), a reasonable choice is obtained from stability analysis, after solving the associatedeigenvalue problem. FIG. 2 shows the eigenspectra of the mean flow stability analysis forboth subsonic and supersonic cavity flows. The power spectral densities of instantaneouspressure at [ x, y, z ] /D = [3 , , β = 0. This asymptotic analysis shows that the meanflow is unstable for both flow conditions, with β = 0 yielding the most unstable modes,except at very low frequencies.Using this mean flow stability analysis information, we select the same offset parameter (cid:15) = 0 .
025 for both M ∞ = 0 . M ∞ = 1 . κ =max( λ i ) + (cid:15) , where max( λ i ) is the most unstable growth rate from the stability analysis.Thus, the discounting parameters are κ = 0 .
15 and 0 .
12 for subsonic and supersonic cases,respectively, as indicated by the black dashed lines in FIG. 2. Based on this discountedresolvent operator, we now consider the compressibility effect on the energy amplificationof flow systems.3.2.
Compressibility effect on input-output characteristics
In the cavity flow, the most important flow-acoustic resonance is linked to the evolutionof the shear layer oscillations Krishnamurty (1955); Rossiter (1964); Rowley & Williams(2006); Beresh et al. (2016). The relevant frequencies may be calculated using themodified empirical formula Rossiter (1964); Heller et al. (1971), which reads St L = f Lu ∞ = n − α /k + M ∞ / (cid:112) γ − M ∞ / coustic response of turbulent cavity flow Figure 3.
Primary ( σ ) and second ( σ ) energy amplification at (a) M ∞ = 0 . M ∞ = 1 .
4. The dominant oscillation detected from LES and higher energy amplification relatedfrequency in resolvent analysis are indicated by dashed lines. where k (=0.65) and α (=0.38) Zhang et al. (2019) are the average convective speed ofdisturbances in shear layer and phase delay, respectively. The cavity length L is usedto define dimensionless Strouhal number St L , γ = 1 . n =1 , , ... denotes the n th resonant frequency tone of the Rossiter mode. The resolventanalysis, and subsequent FT decomposition, performed below examine the input-outputcharacteristics at various frequencies from this spectrum to understand the correspondingmodal features.Two-dimensional forcing displays the largest growth in the stability analysis of thelinearized solution about the mean turbulent state. The response for the M = 0 . β (cid:54) = 0) at M ∞ = 1 .
4. Severalexperimental campaigns (Zhuang et al. et al. et al. β (cid:54) = 0.The overall energy amplification is a strong function of Mach number. FIG. 3 displaysresults for the primary and second gain ( σ and σ ) for three spanwise wavenumbers, β = π , 2 π and 3 π . As the Mach number increases from 0 . .
4, the overall energyamplification decreases. The peak energy amplification is an order of magnitude largerfor the subsonic flow relative to the supersonic case. This observation is consistent withseveral studies of the effect of compressibility in turbulent shear flow. Sarkar (1995)showed that the reduced turbulent growth due to compressibility effects is primarilyassociated with reduced levels of turbulence production. In cavity flows also, Beresh et al. (2016) showed a substantial drop in all three components of the turbulence intensity aswell as the turbulent shear stress with an increase in Mach number.In general, both flows are more receptive to high-frequency forcing. At M ∞ = 0 . St = 4 .
54, lies in the vicinity of the10 th Rossiter mode frequency. This is much higher than the dominant frequency of St L = 0 .
76 detected in the turbulent cavity LES Sun et al. (2019 a ). A similar scenario isobserved in the supersonic cavity flow; the frequency associated with the largest energyamplification, St = 2 .
43, while lower than the corresponding value for the subsonic case, isalso much higher than the strong oscillation frequency tone St L = 1 .
33 detected naturallyin the turbulent flow Liu et al. (2020). The non-normal nature of the operator and non-linear effects thus influence the energy amplification magnitude and its associated forcingfrequency in the flow response.0
Q. Liu & D. Gaitonde St L = 0 . St L = 2 . n th mode St L Primary modeSecondary mode 0.760.76Primary mode 4.54 Forcing mode ˆ u f Response mode ˆ u r Reynolds stress ˆ τ z -0.200.2 Figure 4.
Streamwise velocity component of forcing and response modes at β = π for the casesof M ∞ = 0 .
6. The corresponding energy amplifications are indicate as blue circles in figure 3(a).
Next, we examine the influence of forcing frequency St L on the features of the responseand forcing modes. Results for the subsonic case are shown in FIG. 4. For the sake ofconciseness, we elaborate on cases at two frequencies of special interest. The first, alsodesignated the natural frequency, St L = 0 .
76, corresponds to the dominant oscillationsobserved in the pressure fluctuation PSD of the LES; for this, both the primary andsecondary modes are shown. The second frequency, St L = 4 .
54 is the forcing frequencythat displays peak gain; only the primary mode is shown here. At the lower frequency, thedominant regions of both primary and secondary forcing and response modes extend overmuch of the shear layer. In contrast, the primary forcing mode at the higher frequency iscompactly restricted to the region near the lip. The response mode has smaller structures,as anticipated, and are limited to the region around the shear layer. The region inside thecavity is primarily a region of support for the secondary mode at St L = 0 .
76. At higherfrequency, the secondary modes (not shown) become variants of the primary modeswith little presence inside the cavity. Meanwhile, the gap in the energy amplificationsignificantly enlarges between the primary and secondary modes, as shown in FIG. 3(a).The spanwise Reynolds stress of the response mode may be obtained from ˆ τ z = R (ˆ u ∗ r ˆ v r ),where ∗ is a complex conjugate and R denotes the real component of the argument.Whereas the highest values occur closer to the downstream cavity face, the primarydistribution of the Reynolds stress moves upstream in the shear layer, indicating thatmixing in the shear layer occurs earlier as frequency increases.FIG. 5 displays forcing and response modes at St L = 1 .
33 and 2 .
43 for M ∞ = 1 . M ∞ = 0 .
6, the β = 2 π resultsare shown since this wave number shows larger amplification (see FIG. 3). Some of thefeatures evident in the subsonic flow when frequency is varied are also evident at thissupersonic speed in the forcing and response modes. For instance, the primary responsemode is prominent in the shear layer region and its streamwise length scale shortens withincreasing forcing frequency. The structures of secondary modes extend inside the cavity.At the higher frequencies pertinent to the supersonic case, the structures of responsemodes extend over the cavity shear layer and do not show much support inside thecavity. Although a similar structure observed for the secondary mode (not shown), itsenergy amplification is much lower than that of the primary modes, as shown in FIG. 3(b).Changing the spanwise wavenumber has a similar effect on the structures of modes acrosssubsonic and supersonic cases; specifically, as the spanwise wavenumber increases, themodal structures become more compact in the wall-normal extent.3.3. Fluid-thermodynamic content of resolvent modes
As mentioned earlier, since the flow-acoustic interaction plays a critical role in self-sustained oscillations, a characterization of the modal fluctuations into their hydrody- coustic response of turbulent cavity flow St L = 0 . St L = 2 . n th mode St L Primary modeSecondary mode 1.331.33Primary mode 2.43 Forcing mode ˆ u f Response mode ˆ u r Reynolds stress ˆ τ z -0.200.2 Figure 5.
Streamwise velocity component of forcing and response modes at β = 2 π for the casesof M ∞ = 1 .
4. The corresponding energy amplifications are indicate as red circles in figure 3(b). St L = 0 . St L = 4 . B x ˆ ψ Ax ˆ ψ Tx ˆ B y ˆ ψ Ay ˆ ψ Ty ˆ B x ˆ ψ Ax ˆ ψ Tx ˆ B y ˆ ψ Ay ˆ ψ Ty Figure 6.
Hydrodynamic, acoustic and thermal components of the primary response modesfor β = π at M ∞ = 0 . namic, acoustic and thermal (FT) components is helpful to understand the interactionmechanism and the response of actuator based forcing. Doak’s decomposition is nowemployed to characterize the FT content of the primary response mode for the differentforcing frequencies.The three FT components are shown in FIG. 6 for the M ∞ = 0 . St L = 0 .
76 and4 .
54. Since the decomposition is performed on a vector quantity, ρ u , the FT componentsare also vectors. Therefore, they are plotted separately for the x and y directions;the notation uses subscripts so that, for example, ˆ ψ Ax = ∂ ˆ ψ A /∂x . The hydrodynamiccomponents, ˆ B , of the structure of the resolvent response mode much is larger than theacoustic and thermal components. This is consistent with observations in other flows,including free jets Unnikrishnan & Gaitonde (2016) as well as hypersonic boundary layertransition Unnikrishnan & Gaitonde (2019). In fact, at St L = 0 .
76, the streamwisecomponent ( ˆ B x ) resembles the primary response mode of streamwise velocity in FIG. 4,consistent with streamwise velocity fluctuations being most prominent in the LES.While smaller in magnitude than the hydrodynamic content, the acoustic and thermalcomponents resemble each other, but with a phase shift of π . They are substantiallydifferent from any of the forcing or response modes from the resolvent analysis, whichhave clearly defined structures that extend in the interior of the cavity. The y componentsare inclined in the general direction of the mean shear; this is opposite to that observedfor ˆ B y . The inclined pattern of the acoustic mode resembles the radiated acoustic fieldidentified from shorter cavities, such as when L/D = 2 (Colonius 2001) in which intensestructures occur at the trailing edge with an angle of approximately 145 degrees fromthe streamwise direction.When the frequency is increased, St L = 4 .
54, the hydrodynamic component exhibitssmaller structures over shear layer regions, which are damped near the downstream wall.Unlike the results of St L = 0 .
76, where ˆ B x dominated, here ˆ B x and ˆ B y are similarin amplitude. Similar conclusions may be made in the acoustic and thermal patterns.The acoustic structure arises near the leading edge and progressively decays towards the2 Q. Liu & D. Gaitonde ˆ B x ˆ ψ Ax ˆ ψ Tx St L . .
76 -0.2 0 0.2ˆ B x -0.02 0 0.02ˆ ψ Ax , ˆ ψ Tx Figure 7.
Streamwise component of hydrodynamic, acoustic and thermal of primary forcingmodes for β = π at M ∞ = 0 . St L = 1 . St L = 2 . B x ˆ ψ Ax ˆ ψ Tx ˆ B y ˆ ψ Ay ˆ ψ Ty ˆ B x ˆ ψ Ax ˆ ψ Tx ˆ B y ˆ ψ Ay ˆ ψ Ty Figure 8.
Hydrodynamic, acoustic and thermal components of the primary response modesfor β = 2 π at M ∞ = 1 . trailing edge. This observation is consistent with the sensitivity of the initial shear layerleading to the cavity displaying more sensitivity to higher frequency Shaw (1998). Thedistinct acoustic structures between the lower and higher-frequency cases indicate thedifferent roles of forcing in the flow-acoustic feedback loop. A key conclusion thus is thatthe role of the trailing edge impingement becomes weaker for the high-frequency forcing.The corresponding forcing modes also show the hydrodynamic component dominatingthe forcing mode at M ∞ = 0 .
6. FIG. 7 shows the FT forcing mode at β = π and St L = 0 .
76 and 4 .
54. The hydrodynamic component pertains to the structure magnitudewhich is 10 times larger than the other two components. As the frequency increases, theprominent hydrodynamic structures become smaller and move upstream. At St L = 4 . π .Results for M ∞ = 1 . β = 2 π , somequalitative observations remain valid from the M ∞ = 0 . B x is the most dominant component and is generally contained in the region of themean shear layer. Similarly, the number of amplified structures increases with frequency.Key differences pertinent to control response sensitivity at the two Mach numbers, tobe discussed later, also become apparent. Despite the relatively smaller gap betweenthe natural and resolvent-derived frequencies at M ∞ = 1 .
4, the acoustic component ofthe response shows much greater sensitivity than at M ∞ = 0 .
6. In fact, at the lowerfrequency, the regular organized structures inside the cavity are not as well defined andthe response is relatively weak. However, the organized structures are very evident at St L = 2 .
43 in wavepacket-like form, which have vertical lobes over much of the cavity,before aligning in the mean shear direction near the trailing edge. Similarly, unlike at coustic response of turbulent cavity flow ˆ B x ˆ ψ Ax ˆ ψ Tx St L . .
33 -0.2 0 0.2ˆ B x -0.02 0 0.02ˆ ψ Ax , ˆ ψ Tx Figure 9.
Streamwise component of hydrodynamic, acoustic and thermal of primary forcingmodes for β = 2 π at M ∞ = 1 . M ∞ = 0 .
6, where acoustic and thermal features matched each other, at M ∞ = 1 . β = 2 π and St L = 1 .
33 and 2.43 at M ∞ = 1 .
4. The acoustic and thermal components are oriented directly with the shearlayer and no substantial structures appear inside the cavity. The typical length scalereduces but the dominant areas remain unchanged. This manifests a significantly differentbehavior than the subsonic flow case M ∞ = 0 .
6. The isolated forcing modes trigger theresponse modes around shear layer region which may related to the observation fromWilliams et al. (2007). Supersonic flows may be more likely to operate in the linearregime than subsonic flows since there less overlap inside the cavity and shear layer.These differences in FT components due to compressibility and frequency now facilitatean analysis of the impact of each localized forcing component, in the context of flowcontrol. 3.4.
Fluid-thermal response to localized-componentwise forcing
A control strategy may be designed based on the information of the response outputand forcing input. The primary forcing modes show significant structures near the leadingedge of the cavity for both subsonic (FIG. 4) and supersonic cases (FIG. 5). As such,forcing around the leading edge of the cavity or on the front wall is a natural choice,with the potential to significantly modify the behavior of shear layer, as is the case inmost studies to date (Cattafesta et al. − . (cid:54) x/D (cid:54) (cid:54) y/D (cid:54) .
2) and (ii) the front wall (0 (cid:54) x/D (cid:54) . − . (cid:54) y/D (cid:54) M in equation (2.3). The forcing contains five components expressed as ˆ f =[ˆ ρ f , ˆ u f , ˆ v f , ˆ w f , ˆ T f ] in equation (2.1), of which the spanwise velocity component forcingˆ w f is neglected. In the coupling matrix M , the elements at the location of interest areset to the chosen component forcing, which localizes the input term M ˆ f . The matrix M thus serves two purposes, spatial restriction and imposition of a componentwise forcingfilter. The results of the resolvent analysis and Doak’s momentum potential theory arediscussed in the following to identify the influence of different types of forcing at thesetwo locations on energy amplification and response structures. Emphasis is placed on thehydrodynamic and acoustic components which are of interest for cavity flow control.3.4.1. Subsonic cavity flow
FIG. 10 shows energy amplification results for leading-edge and front-wall forcing at M ∞ = 0 .
6. Considering the former location first, streamwise velocity forcing results4
Q. Liu & D. Gaitonde
Figure 10.
Primary gain σ for (a) leading-edge forcing and (b) front-wall forcing with β = π at M ∞ = 0 . β St L ( a ) ( b ) G ( β, St L ) ( β, St L ) = ( π, . β, St L ) = ( π, . || ˆ B || Figure 11. (a) Ratio G ( β, St L ) for leading-edge velocity forcing over frequencies and spanwisewavenumbers at M ∞ = 0 .
6. The black dashed line and red dash-dotted line label the peaksof G ( β, St L ) and σ , respectively. (b) Spatial distribution of hydrodynamic || ˆ B || (contourplot) and acoustic || ˆ ψ A x || (black dashed lines) structures at ( β, St L ) = ( π, .
76) (top) and( β, St L ) = ( π, .
0) (bottom). The locations of their G ( β, St L ) are indicated in green and magentadots on plot (a). in the strongest energy amplification, following by the ˆ v f , ˆ ρ f and ˆ T f forcing in thatorder. The localization of ˆ u f forcing greatly reduces the energy amplification, to aboutone third that with the global forcing shown in figure 3(a). Nonetheless, compared tothe global forcing structures in FIG. 4, the much smaller localized forcing at a positionwhere an actuator may actually be placed, attains substantial energy amplification. Moreimportantly, the actual frequency value where the peak response is observed, St L =4 .
56 is the same, suggesting the promising potential from a practical standpoint forsubstituting the localized forcing for global forcing.The energy amplification of streamwise velocity forcing ˆ u f substantially reduces as theforcing location is changed from the leading edge to the front wall of the cavity, as shownin FIG. 10(b). This change highlights the importance of leading-edge forcing in the energyamplification mechanism. However, one key difference from leading edge forcing is thatthe streamwise and vertical velocity forcing trigger comparable energy amplification, withthe frequencies at which the peaks occur being slightly lower ( St L = 4 .
08 as opposedto 4 .
56 for the leading edge forcing case). The energy amplification for both densityand temperature forcing cases remain much lower than the corresponding velocity-wiseforcing. Overall, these differences reflect the fact that the strongest impact of the forcinglocation on energy amplification occurs with ˆ u f rather than on any of the other variables.The relative sensitivity of the acoustic versus hydrodynamic response may be assessed coustic response of turbulent cavity flow G ( β, St L ) = σ ( β, St L ) || ˆ ψ A x ( β, St L ) |||| ˆ B ( β, St L ) || , (3.3)where || · || denotes the magnitude of FT components and σ ( β, St L ) is the primaryenergy amplification. The magnitude || ˆ ψ A x || combines the acoustic components in x and y direction. As a representative sample, the quantification of G ( β, St L ) over spanwisewavenumber β and frequency, for the case of leading edge imposition of ˆ u f perturbationsis shown in FIG. 11(a). The black line traces the frequency where the ratio G ( β, St L )is maximum for each spanwise wave number, while the red line indicates the frequencywhere the overall energy amplification is highest. The peak corresponding to the largestmodification of the acoustic component G ( β, St L ) occurs at ( π, . St L = 4 . || ˆ B || and || ˆ ψ A x || at two frequencies. The top figureis that associated with the natural frequency St L = 0 .
76 (marked with a green dot inframe (a)) while the bottom figure corresponds to the frequency St L = 5 .
0, related to the G ( β, St L ) peak (magenta dot). In each, || ˆ B || are represented by the flooded contours,while || ˆ ψ A x || are displayed with dashed contours. For St L = 0 .
76, the hydrodynamicfeatures span approximately the region of the shear layer from the leading edge tothe trailing edge. However, acoustic structures also show a presence within the cavity,as well as on the side of the flow near the trailing edge. The peak associated with G ( β, St L ) is connected to overlapping acoustic and hydrodynamic structures. At thishigher frequency, the strongest hydrodynamic and acoustic structures move forwardcloser to the leading edge. This overlap is a feature of the location where the peak isobserved in G ( β, St L ) i.e., overlapping acoustic and hydrodynamic response componentsresult in larger modifications to the acoustic components. The exchange of energy betweenhydrodynamic and acoustic components underlying this result follow a complex dynamicsthat may be described by the dynamics of the total fluctuating enthalpy Jenvey (1989);Prasad et al. (2021). The analysis is beyond the scope of the current effort; results in thecontext of jets have been presented by Unnikrishnan & Gaitonde (2019).To illustrate the results with another type of forcing, we consider the case of ˆ T f excitation on the front-wall, which displays different local acoustic structures. The G ( β, St L ) contours and corresponding structures of || ˆ B || and || ˆ ψ A x || are shown in FIG. 12.The G ( β, St L ) peak related frequency is generally higher than the frequency related tothe σ peak, which suggests a strong effect on the acoustic response may not align withthe energy amplification at the two frequencies of interest. Examining FIG. 12(b), at St L = 0 .
76, the local intense acoustic response occurs around the forcing location, while,the hydrodynamic response structure remains over the shear layer region. When theforcing frequency increases, the peak related response structure in FIG. 12(b) resemblesthe case of ˆ u f shown in FIG. 11 at St L = 5 .
0, which has an overlapping area over theshear layer. It is noteworthy that except for the difference in the value of G ( β, St L ),that at the same high-frequency, the different forcing types have similar acoustic andhydrodynamic response structures. This observation holds for all cases examined, andindicates that the high-frequency selection mechanism is less influenced by the locationor type of forcing.6 Q. Liu & D. Gaitonde β St L ( a ) ( b ) G ( β, St L ) ( β, St L ) = ( π, . β, St L ) = ( π, . || ˆ B || Figure 12. (a) Ratio G ( β, St L ) for front-wall temperature forcing at different frequenciesand spanwise wavenumbers for M ∞ = 0 .
6. The black dashed line represents the peak relatedfrequency over spanwise wavenumbers and red dash-dotted line labels the peak of σ relatedfrequencies over spanwise wavenumbers. (b) Spatial distribution of hydrodynamic || ˆ B || (contourplot) and acoustic || ˆ ψ A x || (black dashed lines) structures at ( β, St L ) = ( π, .
76) (top) and( β, St L ) = ( π, .
0) (bottom). The locations of their G ( β, St L ) are indicated in green and magentadots on plot (a). Figure 13.
Primary gain σ for (a) leading-edge forcing and (b) front-wall forcing with β = 2 π at M ∞ = 1 . β St L ( a ) ( b ) G ( β, St L ) ( β, St L ) = ( π, . β, St L ) = ( π, . || ˆ B || Figure 14. (a) Ratio G ( β, St L ) for leading-edge velocity forcing at different frequencies andspanwise wavenumbers in the case of M ∞ = 1 .
4. The black dashed line and and red dash-dottedline represent the peaks G ( β, St L ) and σ related frequency over spanwise wavenumbers,respectively. (b) Spatial distribution of hydrodynamic || ˆ B || (contour plot) and acoustic || ˆ ψ A x || (black dashed lines) structures at ( β, St L ) = ( π, .
33) (top) and ( β, St L ) = ( π, .
35) (bottom).The locations of their G ( β, St L ) are indicated in green and magenta dots on plot (a). coustic response of turbulent cavity flow β St L ( a ) ( b ) G ( β, St L ) ( β, St L ) = ( π, . β, St L ) = ( π, . || ˆ B || || ˆ B || Figure 15. (a) Ratio of G ( β, St L ) for front-wall temperature forcing over frequencies andspanwise wavenumbers at M ∞ = 1 .
4. (b) The hydrodynamic || ˆ B || (contour plot) and acoustic || ˆ ψ A x || (black dashed lines) components for two representative cases at ( β, St L ) = ( π, .
24) and( π, . G ( β, St L )and σ related frequency over spanwise wavenumbers, respectively. The locations for tworepresentative cases at ( β, St L ) = ( π, .
24) and ( π, .
43) are indicated with green and magentadots, respectively.
Supersonic cavity flow
As noted for the subsonic case, localized forcing in the supersonic case also reducesthe energy amplification relative to the original unconstrained forcing. However, manyfeatures, including its distribution over frequency and spanwise wavenumber resemblethat of the global case (shown earlier in FIG. 3(b)). FIG. 13 shows the energy ampli-fication for leading-edge and front wall forcing cases for M ∞ = 1 .
4. As for M ∞ = 0 . u f . In contrast, the effect of the other types of forcing is negligible.Based on these similarities and differences, we discuss the distribution of G ( β, St L ) andresponse structures of leading-edge ˆ u f and front-wall ˆ T f in detail.The distribution of G ( β, St L ) indicates stronger acoustic response at lower spanwisewavenumbers. FIG. 14(a) shows the variation of G ( β, St L ) with frequency and spanwisewavenumber. As the spanwise wavenumber increases, the value of G ( β, St L ) decreases,and the peak related frequency gradually shifts to higher frequencies, as indicated bythe black dashed line in FIG. 14. The corresponding structures related to the naturalfrequency St L = 1 .
33 and St L = 2 .
35 for β = π are shown in FIG. 14(b) with themagnitudes of || ˆ B || and || ˆ ψ A x || as before. At St L = 1 .
33, the strong acoustic componentoccurs at the rear part of the cavity with a vertical extent expanding into the cavity.The corresponding hydrodynamic structure gradually spreads over the shear layer. Forthe case of St L = 2 .
35, both structures are more diffuse, but there is significant overlapin the shear layer region. This feature resembles the observation for subsonic cases where G ( β, St L ) peaks at higher frequencies.Besides a relatively lower magnitude of G ( β, St L ) for the case of front-wall ˆ T f , itscorresponding acoustic structures are distinct from other cases. FIG. 15 shows thedistribution of G ( β, St L ), magnitude of || ˆ B || and || ˆ ψ A x || at St L = 0 .
24 and 2 .
43, which arerepresentative cases to elaborate the unique acoustic response to front-wall temperatureforcing. Overall, the values of G ( β, St L ) are much smaller than for ˆ u forcing. Theslightly higher value of G ( β, St L ) emerges for the case with low frequency St L < . St L = 0 .
24 is shownin FIG. 15(b) (upper figure). The localized intense acoustic structures appear whereforcing is executed. However, the hydrodynamic component remains spread out over the8
Q. Liu & D. Gaitonde cavity due to the convective nature. For structures related to the G ( β, St L ) peak (lowerfigure), the dominant acoustic component occurs inside the cavity and overlaps with thehydrodynamic component. These features are common with acoustic and hydrodynamicdistributions associated with G peak at different forcing cases. The overall low magnitudeof G across frequency and spanwise wavenumber suggests the front-wall ˆ T f forcing isineffective at triggering hydrodynamic and acoustic response.
4. Concluding Remarks
We have combined resolvent analysis and Doak’s momentum potential theory toinvestigate the input-output characteristics of subsonic ( M ∞ = 0 .
6) and supersonic flows( M ∞ = 1 .
4) past an
L/D = 6 cavity at Re = 10 , i.e., thatprominent in the corresponding LES, for both subsonic and supersonic cases. Regardlessof Mach number, the acoustic component appears at the trailing edge for the casewith natural frequency. As the forcing frequency is increased, the intense regions of theacoustic mode moves upstream and decays progressively towards the trailing edge. Thus,high-frequency forcing where the resolvent analysis predicts peak amplification inducesdifferent acoustic responses compared to natural frequency forcing.The coupling matrix is used to isolate the influence on energy amplification andacoustic response of the different forcing types, such as streamwise, wall-normal veloc-ity, thermal or density perturbations, and actuator localization to accessible regions,specifically the leading edge or the front wall. For both Mach numbers, leading-edgeforcing achieves more extensive energy amplification than the front-wall case. The ratiobetween the acoustic and hydrodynamic component G ( β, St L ) indicates strong acousticmodification. The related peak frequency is higher than that associated with maximumenergy amplification. At spanwise wavenumbers and frequencies where G ( β, St L ) ismaximum, the acoustic and hydrodynamic response structures overlap in the region ofthe shear layer. Overall, the current effort provides a deeper understanding of the flow-acoustic input-output characteristics from a linear perspective. Future investigation willfocus on the identification and differentiation of linear observations in the nonlinear LEScavity flows. Acknowledgments
The authors gratefully acknowledge the support of AFOSR (FA9550-19-1-0081; Pro-gram Officer: G. Abate) and ONR (N00014-17-1-2584, Program Officer: S. Martens).The authors would like to acknowledge Dr. K. Taira from University of California, LosAngeles and Dr. Y. Sun from Syracuse University for providing time-averaged cavity flowdata.
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