applied sciences
Article A Model to Predict Acoustic Resonant Frequencies of Distributed Helmholtz Resonators on Gas Turbine Engines
Jianguo Wang * , Philip Rubini *, Qin Qin and Brian Houston
School of Engineering and Computer Science, University of Hull, Hull HU6 7RX, UK; [email protected] (Q.Q.); [email protected] (B.H.) * Correspondence: [email protected] (J.W.); [email protected] (P.R.); Tel.: +44(0)1482-465818 (P.R.)
Received: 27 February 2019; Accepted: 2 April 2019; Published: 4 April 2019
Featured Application: Passive control devices for combustion instability and combustion noise in gas turbine engines.
Abstract: Helmholtz resonators, traditionally designed as a narrow neck backed by a cavity, are widely applied to attenuate combustion instabilities in gas turbine engines. The use of multiple small holes with an equivalent open area to that of a single neck has been found to be able to significantly improve the noise damping bandwidth. This type of resonator is often referred to as “distributed Helmholtz resonator”. When multiple holes are employed, interactions between acoustic radiations from neighboring holes changes the resonance frequency of the resonator. In this work, the resonance frequencies from a series of distributed Helmholtz resonators were obtained via a series of highly resolved computational fluid dynamics simulations. A regression analysis of the resulting response surface was undertaken and validated by comparison with experimental results for a series of eighteen absorbers with geometries typically employed in gas turbine combustors. The resulting model demonstrates that the acoustic end correction length for perforations is closely related to the effective porosity of the perforated plate and will be obviously enhanced by acoustic radiation effect from the perforation area as a whole. This model is easily applicable for engineers in the design of practical distributed Helmholtz resonators.
Keywords: distributed Helmholtz resonator; acoustic radiation; hole-hole interaction effect; acoustic passive control; gas turbine
1. Introduction Helmholtz resonators are commonly employed to reduce sound pressure levels across a broad range of applications, including the built environment, industrial installations, and propulsion devices, such as gas turbines [1–3]. Combustion instability represents a significant problem in the application of low NOx emission LPC (lean premixed combustion) gas turbines. Helmholtz resonators are commonly applied to achieve attenuation effect near the instability frequencies. However, attenuation bandwidth due to Helmholtz resonance effect is often very narrow. In comparison, attenuation of noise by distributed Helmholtz resonators takes place within a much wider frequency range thanks to the stronger acoustic resistance of those smaller perforations [1–3]. The distributed Helmholtz resonator can be thought of as a special type of perforated plate absorber (PPA). The noise attenuation effect of a PPA is typically characterized by its peak damping
Appl. Sci. 2019, 9, 1419; doi:10.3390/app9071419 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1419 2 of 18 frequency and absorption bandwidth. The peak damping frequency often coincides with the resonance frequency of a PPA [4–7]: s c A fr = (1) 2π Vle p le = l + lec = l + 0.96 A0 (2) where fr stands for resonance frequency, c represents the speed of sound, V represents the volume of the resonator cavity, A is the total opening area of all perforates, le is effective length of the opening, which includes the physical plate thickness l and an acoustic radiation “end effect” length lec. A0 is the cross section area of an individual perforation. A distributed Helmholtz resonator with multiple small holes is able to significantly improve the noise damping bandwidth compared to a single neck Helmholtz absorber. On the other hand, when multiple holes are employed, additional acoustic interaction occurs between neighboring holes, thereby changing the resonance frequency of the resonator. A number of researchers, including Ingard [5], Fok [8], and Atalla [9], have demonstrated that the end correction length is affected by the interaction effect between neighboring perforations. Fok [8] tested an orifice in the center of a partition across a tube and proposed that to take into account the effect of hole interaction, the end correction length for the orifice should be corrected by a “Fok’s function”, ψ [8]:
−1 ψ = (1 − 1.4092ξ + 0.33818ξ3 + 0.06793ξ5 − 0.02287ξ6 + 0.03015ξ7 − 0.01641ξ8) (3) where ξ is the hole diameter (d) divided by hole separation distance (b). Fok’s function is later cited by other authors to consider the effect of hole interaction on acoustic radiation strength [10–14]. The acoustic effective length of the perforated plate then becomes: √ 0.96 A l = l + l = l + 0 (4) e ec ψ(ξ)
Substituting for Fok’s function gives,
p 3 5 6 7 8 le = l + 0.96 A0 1 − 1.4092ξ + 0.33818ξ + 0.06793ξ − 0.02287ξ + 0.03015ξ − 0.01641ξ
As shown in Figure1, where Fok’s function is plotted against hole diameter (d) divided by hole separation distance (b), Fok’s function is always greater than 1 and increases with increasing d/b. Fok’s function√ appears in the denominator in Equation (4), therefore, the acoustic end correction length 0.96 A0 gradually diminishes with decreasing hole separation distance due to hole interaction effect. Melling [10] provided an explanation for the diminishing end correction by a simple diagram, as illustrated in Figure2. He described the end correction effect as being the result of an “attached air mass” in the neighborhood of the perforation. Melling [10] claimed that a portion of “attached masses” for a perforation will overlap with the neighboring perforations if they are sufficiently close to each other. The overlapped portions of attached masses merge into one portion of attached mass, and as a result the average attached mass for each perforation is reduced. However, Melling [10] did not provide an explanation about the underlying physics of why the two merged “attached masses” would merge and result in a reduced total “attached mass”. Ingard [5] developed a mathematical model to describe the acoustic effect upon a single hole from the neighboring holes. He found that hole–hole interaction effect will increase the end correction magnitude. Rachevkin [12] stated that when small holes are in extreme proximity to each other, they will act as a big hole in terms of their impact on passing acoustic field. Therefore, in this situation, the overall acoustic radiation effect is not only generated by each single hole, but also the perforation region as a whole. Tayong [14,15] proposed a “geometrical tortuosity model” to account for the acoustic Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 19 frequency and absorption bandwidth. The peak damping frequency often coincides with the resonance frequency of a PPA [4–7]:
𝑐 𝐴 𝑓 = (1) 2𝜋 𝑉𝑙
𝑙 =𝑙+𝑙 =𝑙+0.96 𝐴 (2) where 𝑓 stands for resonance frequency, c represents the speed of sound, 𝑉represents the volume of the resonator cavity, A is the total opening area of all perforates, 𝑙 is effective length of the opening, which includes the physical plate thickness 𝑙 and an acoustic radiation “end effect” length
𝑙 . 𝐴 is the cross section area of an individual perforation. A distributed Helmholtz resonator with multiple small holes is able to significantly improve the noise damping bandwidth compared to a single neck Helmholtz absorber. On the other hand, when multiple holes are employed, additional acoustic interaction occurs between neighboring holes, thereby changing the resonance frequency of the resonator. A number of researchers, including Ingard [5], Fok [8], and Atalla [9], have demonstrated that the end correction length is affected by the interaction effect between neighboring perforations. Fok [8] tested an orifice in the center of a partition across a tube and proposed that to take into account theAppl. effect Sci. 2019 of, 9hole, 1419 interaction, the end correction length for the orifice should be corrected by a “Fok's3 of 18 function”, 𝜓 [8]: radiation effect of𝜓 the = perforation (1 − 1.4092𝜉 region as+ a0.33818𝜉 whole, + 0.06793𝜉 which he called− 0.02287𝜉 “heterogeneity + 0.03015𝜉 distribution effects”. (3) However, the exact definition of− parameters 0.01641𝜉 such) as tortuosity in these models require external input whereand remains 𝜉 is the difficult hole diameter for engineers (d) divided when itby comes hole separation to designing distance industrial (b). distributed resonators.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 19 simple diagram, as illustrated in Figure 2. He described the end correction effect as being the result of an “attached air mass” in the neighborhood of the perforation. Melling [10] claimed that a portion of “attached masses” for a perforation will overlap with the neighboring perforations if they are sufficiently close to each other. The overlapped portions of attached masses merge into one portion of attached mass, and as a result the average attached mass for each perforation is reduced. However, Melling [10] did not provide an explanation about the underlying physics of why the two merged “attached masses” would merge and resultFigure in 1. a Fok’sreduced function. total “attached mass”.
Fok’s function is later cited by other authors to consider the effect of hole interaction on acoustic radiation strength [10–14]. The acoustic effective length of the perforated plate then becomes:
0.96 𝐴 𝑙 =𝑙+𝑙 =𝑙+ (4) 𝜓(𝜉) Substituting for Fok’s function gives, 𝑙 =𝑙+0.96 𝐴 (1 − 1.4092𝜉 + 0.33818𝜉 + 0.06793𝜉 − 0.02287𝜉 + 0.03015𝜉 − 0.01641𝜉 ) As shown in Figure 1, where Fok’s function is plotted against hole diameter (d) divided by hole separation distance (b), Fok’s function is always greater than 1 and increases with increasing d/b. Fok’s function appears in the denominator in Equation (4), therefore, the acoustic end correction length 0.96 𝐴 gradually diminishes with decreasing hole separation distance due to hole interaction effect. Melling [10] provided an explanation for the diminishing end correction by a
Appl. Sci. 2019, 9, x; doi: FOR PEER REVIEW www.mdpi.com/journal/applsci
Figure 2. Illustration of the Hole–Hole interaction effect, redrawn, based upon Melling [10]. Figure 2. Illustration of the Hole–Hole interaction effect, redrawn, based upon Melling [10]. In this paper, the acoustic end correction effect for distributed Helmholtz resonators or PPA’s, Ingard [5] developed a mathematical model to describe the acoustic effect upon a single hole in which holes are in proximity with each other, is conveniently proposed to be a combination of two from the neighboring holes. He found that hole–hole interaction effect will increase the end correction radiation effects—the acoustic radiation effect by each single orifice and the acoustic radiation effect by magnitude. Rachevkin [12] stated that when small holes are in extreme proximity to each other, they the overall perforation area. This paper will also propose an easily applicable model for the prediction will act as a big hole in terms of their impact on passing acoustic field. Therefore, in this situation, the of resonance frequencies of resonators. This resulting model will be of significant value in the design of overall acoustic radiation effect is not only generated by each single hole, but also the perforation practical distributed Helmholtz absorbers, especially those distributed Helmholtz resonators applied region as a whole. Tayong [14,15] proposed a “geometrical tortuosity model” to account for the on gas turbine engines to control combustion instabilities. acoustic radiation effect of the perforation region as a whole, which he called “heterogeneity 2.distribution Materials effects”. and Methods However, the exact definition of parameters such as tortuosity in these models require external input and remains difficult for engineers when it comes to designing industrial 2.1.distributed Numerical resonators. Simulation InIn orderthis paper, to investigate the acoustic acoustic end propagation correction effect through for small distributed holes, a Helmholtz time-accurate, resonators three dimensional, or PPA’s, numericalin which holes solution are wasin proximity obtained with for the each propagation other, is conveniently of a white noise proposed signal in to an be impedance a combination tube, of in two the absenceradiation of effects—the any mean flow acoustic or bias radiation flow, under effect theby assumptioneach single orifice that the and acoustic the acoustic perturbations radiation may effect be consideredby the overall to remain perforation in the area. laminar This flow paper regime. will Thealso propagationpropose an ofeasily an acoustic applicable plane model wave for in the an impedanceprediction of tube, resonance an incident frequencies normal of to resonators. a perforated This plate, resulting as show model in Figure will3 be, may of significant be represented value as in the design of practical distributed Helmholtz absorbers, especially those distributed Helmholtz resonators applied on gas turbine engines to control combustion instabilities.
2. Materials and Methods
2.1. Numerical Simulation In order to investigate acoustic propagation through small holes, a time-accurate, three dimensional, numerical solution was obtained for the propagation of a white noise signal in an impedance tube, in the absence of any mean flow or bias flow, under the assumption that the acoustic perturbations may be considered to remain in the laminar flow regime. The propagation of an acoustic plane wave in an impedance tube, an incident normal to a perforated plate, as show in Figure
Appl. Sci. 2019, 9, x; doi: FOR PEER REVIEW www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1419 4 of 18 an unsteady compressible laminar flow. Simulation of such flow regimes does not require significant modeling approximations, and therefore, in the context of this application, may be considered to be a direct numerical solution to the governing Navier-Stokes equations, provided satisfactory temporal Appl.and Sci. spatial 2019, resolution9, x FOR PEER have REVIEW been demonstrated. 5 of 19
FigureFigure 3. 3. AnAn impedance impedance tube tube configur configureded with with two two microphones. microphones.
Computational Fluid Dynamics (CFD) is based upon the numerical solution of the underlying governing equations for fluid flow under specified boundary conditions. The method has been successfully applied by a number of authors to study acoustic radiation effect for multi-perforated plate absorbers [16–19]. The simulated acoustic source applied in this work was4 a× whitex noise signal ranging between 100 Hz and 1000 Hz with a 100 dB overall sound pressure level. The governing equations may be expressed, for brevity alone, in axisymmetric cylindrical polar coordinates as: 4 × y ∂ρ ∂(ρu ) 1 ∂(ρru ) + x + r = 0 (5) ∂t ∂x r ∂r
∂(ρu ) 1 ∂(ρru u ) 1 ∂(ρru u ) ∂τ ∂p x + x x + r x = rx − + S (6) ∂t r ∂x r ∂r ∂r ∂x x
∂(ρur) 1 ∂(ρruxur) 1 ∂(ρrurur) ∂τxr ∂p + + = − + Sr (7) Plate 2 ∂t r ∂x Plater 5 ∂r ∂x ∂r Plate 7 ∂(ρh) 1 ∂(ρru h) 1 ∂(ρru h) 1 ∂ ∂h 1 ∂ ∂h + x + r = rΓ + rΓ + S (8) ∂t r ∂x r ∂r r ∂x ∂x r ∂r ∂r h where x, r are the axial and radial directions of the cylindrical coordinate system and ux, ur are the axial and radial velocities, t represents time, τrx is the axial direction viscous stress due to flow velocity gradient in radial directions. Sx, Sr are momentum source terms; h is the enthalpy and Sh the source term of the energy equation. The ideal gas equation of state, p = ρRT, is employed to determine the density of the compressible fluid, where p, ρ, T, R represent the absolute pressure, density, temperature of the medium, and the gas constant, respectively. The discretized spatial terms were approximated by a second order biased upwind scheme, which provides both the numerical accuracy of the second order upwind scheme and the convergence robustness of the first order scheme [20]. The discretized temporal terms were approximated by a Plate 24 Plate 23 bounded secondPlate order 25 accurate, implicit scheme, to robustly resolve the rapidly fluctuating pressure signal [20]. All simulations were undertaken with the double precision version of ANSYS FLUENT 17.2. A general non-reflectingFigure boundary 4. An example condition of proposedsix perforated by Poinsotplate configurations. et. al. [21,22 ] is applied to represent anechoic boundaries in CFD simulations. A two-microphoneTable transfer 1. Geometric function features was employed of the plates for resolved acoustic by analysis CFD. [23]. The normalized specific acoustic impedance of the plate absorber is defined by Kuttruff [4] as: Diameter of the Case Diameter of the Effective Impedance Tubes and x/d y/d No. = + Hole,= d +(mm) − Porosity, 𝝈𝒆𝒇𝒇 Perforated Plates (mm)z r ix (1 R)/(1 R) (9) 1 90 2 1.2 1.2 0.545 where r is the2 normalized specific90 acoustic resistance, 2 which1.5 is a measure1.5 of the0.349 resistance that the perforated plate3 presents to the90 acoustic flow, x is 2 the normalized2 specific2 acoustic0.196 reactance, which describes the phase4 difference90 between the driving 2pressure difference2.5 and2.5 the resultant0.126 orifice velocity. 5 90 2 3 3 0.087 R represents the6 reflection coefficient.90 For a PPA installed 2 at the4 end of an4 impedance0.049 tube, the acoustic 7 90 2 5 5 0.031 8 135 3 1.2 1.2 0.545 9 135 3 1.5 1.5 0.444 10 135 3 2 2 0.250 11 135 3 2.5 2.5 0.160 12 135 3 3 3 0.111 13 135 3 4 4 0.063
Appl. Sci. 2019, 9, x; doi: FOR PEER REVIEW www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1419 5 of 18 energy that is not reflected by the absorber is absorbed by the PPA and the acoustic energy absorption coefficient, therefore, becomes [4]:
4r α = 1 − |R|2 = (10) (1 + r)2 + x2
As can be seen from Equation (10), the optimum absorption effect of a PPA takes place at the frequency where acoustic reactance is zero, which also coincides with the resonance frequency of a PPA [4]. The range of geometric configurations considered is summarized in Table1. All plates have the same porosity of 0.0123 and the same thickness of 2.5 mm. Figure4 illustrates a selection of the geometric configurations considered. The ratios of hole separation distance to hole diameter (x/d, y/d) vary from 1.2, to 1.5, 2, 2.5, 3, 4, 5, where x is the hole pitch in the horizontal direction, y is the hole pitch in the vertical direction, and d is the diameter of the hole. The size of the perforations varies from 2 mm to 3 mm and 4 mm. Those plates with 2 mm, 3 mm, and 4 mm perforations are installed in impedance tubes with diameters of 90 mm, 135 mm, and 180 mm, respectively. The length of all orifices is kept consistent at 2.5 mm. All plates are backed by a 25 mm deep cylindrical air cavity, which has an identical cross-section to upstream impedance tubes. Real world perforated plates are sometimes more densely perforated in one direction than in other directions; therefore, perforated plates (plate 23, plate 24, and plate 25) with different aspect ratios were taken into consideration as well.
Table 1. Geometric features of the plates resolved by CFD.
Diameter of the Diameter of the Case No. Impedance Tubes and x/d y/d Effective Porosity, σ Hole, d (mm) eff Perforated Plates (mm) 1 90 2 1.2 1.2 0.545 2 90 2 1.5 1.5 0.349 3 90 2 2 2 0.196 4 90 2 2.5 2.5 0.126 5 90 2 3 3 0.087 6 90 2 4 4 0.049 7 90 2 5 5 0.031 8 135 3 1.2 1.2 0.545 9 135 3 1.5 1.5 0.444 10 135 3 2 2 0.250 11 135 3 2.5 2.5 0.160 12 135 3 3 3 0.111 13 135 3 4 4 0.063 14 135 3 5 5 0.040 15 180 4 1.2 1.2 0.545 16 180 4 1.5 1.5 0.444 17 180 4 2 2 0.250 18 180 4 2.5 2.5 0.160 19 180 4 3 3 0.111 20 180 4 4 4 0.063 21 180 4 5 5 0.040 22 135 3 1.5 5 0.105 23 135 3 2 5 0.079 24 135 3 3 5 0.052 25 135 3 4 5 0.039 CFD: Abbreviation for “Computational Fluid Dynamics”; x: hole separation distance in the horizontal direction as shown in Figure4; y: hole separation distance in the vertical direction as shown in Figure4; σe f f : effective porosity as defined in Equation (11). Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 19
Figure 3. An impedance tube configured with two microphones. Appl. Sci. 2019, 9, 1419 6 of 18
4 × x
4 × y
Plate 2 Plate 5 Plate 7
Plate 23 Plate 25 Plate 24
Figure 4. An example of six perforated plate configurations. Figure 4. An example of six perforated plate configurations. The porosity of a perforated plate is defined as the ratio of the total opening area to the whole surface area of the perforatedTable 1. plate, Geometricσ = featuresnA0/A pof= theA plates/Ap, resolved where n byis CFD. the number of holes, which equals to 25 for the PPA simulated in this work and 32 for those experimentally tested. Diameter of the In the aboveCase definition, A is the cross-sectionDiameter area of the of a single hole, A representsEffective the total opening Impedance0 Tubes and x/d y/d No. Hole, d (mm) Porosity, 𝝈 area of all holes, and PerforatedAp stands Plates for the(mm) total surface area of the perforated plate backed𝒆𝒇𝒇 by a cavity. In this paper, a1 new parameter,90 effective porosity, 2 is introduced,1.2 and for1.2 a parallel0.545 distributed plate, as shown in Figure2 4, it is defined90 to be: 2 1.5 1.5 0.349 3 90 2 2 2 0.196 4 90 2 2.5 2.5 0.126 σ = πd2/4xy (11) 5 90 e f f 2 3 3 0.087 6 90 2 4 4 0.049 where d is diameter7 of the hole,90 x and y are hole separation 2 distance5 in5 two directions,0.031 as shown in Figure4, πd2/48 is the opening135 area in a rectangular 3 perforation1.2 region1.2 whose area0.545 is xy. Effective porosity depicts9 the uniformity135 of hole distribution 3 in a perforated1.5 plate1.5 backed0.444 by a cavity. It is 10 135 3 2 2 0.250 equal to classical11 porosity if all135 holes are distributed 3 evenly on2.5 a perforated2.5 surface.0.160 It will be larger than the classical12 porosity if some135 or all holes are more 3 densely3 distributed3 in a0.111 local area of a noise damping surface.13 135 3 4 4 0.063 The phenomena of acoustic radiation takes place in close proximity to the individual holes in the perforatedAppl. Sci. 2019 plate., 9, x; doi: Therefore, FOR PEER thisREVIEW requires a computational grid of sufficientwww.mdpi.com/journal/applsci resolution to capture the local fluid dynamic and acoustic phenomena. The structural composition of the computational grid employed in this work is illustrated in Figure5. Block-structured meshes with a cell size of 2.6 mm were built within the air cavity and impedance tubes, as shown in Figure5a, and block-structured meshes were further refined near the orifices until the mesh size reached a minimum of 0.2 mm in the orifice, as shown in Figure5b. A further mesh with a resolution of 0.1 mm in the orifice was generated for a separate test of grid dependency. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 19 Appl. Sci. 2019, 9, 1419 7 of 18 The results show a very similar character and support the use of the 0.2 mm resolution grid as being
Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 19
The results show a very similar character and support the use of the 0.2 mm resolution grid as being an acceptable grid-independent solution for acoustic impedance.
(a) (b)
FigureFigure 5.5. GridGrid resolution resolution in in an impedancean impedance tube tube configured configured with platewith No. plate 7 absorber: No. 7 absorber: (a) grid resolution (a) grid resolutionin the test rigin the and test (b) rig grid and resolution (b) grid resolution in the hole in (Hole the ho Diameterle (Hole = Diameter 2 mm). = 2 mm).
Figure6 illustrates the normalized(a) specific acoustic impedance determined(b) from the simulation for perforatedFigure 5. Grid plate resolution No.7, employing in an impedance both the 0.2tube mm configured and the 0.1with mm plate resolution No. 7 absorber: computational (a) grid grids. The resultsresolution show in the a verytest rig similar and (b character) grid resolution and support in the ho thele (Hole use of Diameter the 0.2 mm= 2 mm). resolution grid as being an acceptable grid-independent solution for acoustic impedance.
Figure 6. Normalized specific acoustic impedance of perforated plate No. 7 absorber acquired by 0.1 mm and 0.2 mm resolution computational grids in the hole.
2.2. Experimental Validation Figure 6. Normalized specific acoustic impedance of perforated plate No. 7 absorber acquired by FigureSimulation 6. Normalized of the acoustic specific signalacoustic propagatio impedancen of through perforated a distributedplate No. 7 absorber Helmholtz acquired resonator by 0.1 was 0.1 mm and 0.2 mm resolution computational grids in the hole. undertakenmm and by0.2 usingmm resolution a laminar computational flow solver, grids which in the it hole.self is free from significant physical models. Verification2.2. Experimental of the Validation solution methodology in the form of numerical errors associated with the quadrature2.2. Experimental scheme Validation and discretization can be achieved by the selection of high order numerical Simulation of the acoustic signal propagation through a distributed Helmholtz resonator was schemesSimulation and demonstrated of the acoustic through signal a testpropagatio of grid dependency.n through a distributed Helmholtz resonator was undertaken by using a laminar flow solver, which itself is free from significant physical models. undertakenIn order by to usingprovide a laminara validation flow of solver, the results which from itself the iscomputational free from significant simulations, physical experimental models. Verification of the solution methodology in the form of numerical errors associated with the quadrature Verificationdata was collected of the forsolution eighteen methodology separate plates in the in aform dedicated of numerical experimental errors test associated rig, illustrated with the in scheme and discretization can be achieved by the selection of high order numerical schemes and quadratureFigure 7 and scheme Figure 8.and discretization can be achieved by the selection of high order numerical demonstrated through a test of grid dependency. schemesThe andtest demonstratedrig (Figure 7) consiststhrough ofa testan upstreamof grid dependency. duct section, a downstream duct section, and a In order to provide a validation of the results from the computational simulations, experimental test sectionIn order installed to provide in abetween validation them, of the each results with from an internal the computational diameter of simulations, 165 mm. Four experimental opposing data was collected for eighteen separate plates in a dedicated experimental test rig, illustrated in dataloudspeakers was collected were forinstalled eighteen on theseparate upstream plates duct in .a Thededicated test rig experimental was designed test to rig,include illustrated anechoic in Figures7 and8. Figureterminations 7 and Figureto minimize 8. reflections from the change in impedance at the inlet and exit. In all test configurations,The test rig no (Figure mean flow7) consists through of thean upstreamduct or bias duct flow section, through a downstream the plates was duct applied. section, and a test section installed in between them, each with an internal diameter of 165 mm. Four opposing loudspeakers were installed on the upstream duct. The test rig was designed to include anechoic terminationsAppl. Sci. 2019, 9 ,to x; doi:minimize FOR PEER reflections REVIEW from the change in impedance at thewww.mdpi.com/journal/applsci inlet and exit. In all test configurations, no mean flow through the duct or bias flow through the plates was applied.
Appl. Sci. 2019, 9, x; doi: FOR PEER REVIEW www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 19
Appl.The Sci. resonant 2019, 9, x FOR cavity, PEER FigureREVIEW 8, was constructed from a curved rectangular test plate with a8 lengthof 19 of 0.05 m and an arc length of 0.0594 m, mounted flush to the side wall of the test section duct, with an attachedThe resonant fan-shaped cavity, cavity Figure of 8,0.051 was m constructed depth. Referring from a curvedto Figure rectangular 8, the volume test plate of the with cavity a length is: of 0.05 m and an arc length of 0.0594 m, mounted flush to the side wall of the test section duct, with 41.25° an attached𝑉 fan-shaped = 𝜋 (0.051 cavity+0.165/2+0.002 of 0.051 m) depth.− (0.165 Referring/2+0.002 to) Figure 8, the 0.05 volume=0.002 of the 𝑚 cavity is:(12) 360° 41.25° where 51 mm is𝑉 the = 𝜋 (0.051cavity depth,+0.165/2+0.002 165 mm) is− the (0.165 internal/2+0.002 diameter) of the 0.05 tube,=0.002 2 mm 𝑚 is the thickness(12) 360° of plate, and 41.25° represents the angle of the fan-shaped cavity. whereA total 51 mm of 32is the circular cavity perforations,depth, 165 mm as is exemplithe internalfied diameterin Figure of 9,the were tube, drilled 2 mm isin the the thickness x and y directions,of plate, andrespectively. 41.25° represents The diameters the angle of of these the fan-shapedorifices were cavity. 2 mm, 3 mm, or 4 mm. The thickness of A total of 32 circular perforations, as exemplified in Figure 9, were drilled in the x and y all experimentally tested plates was kept the same at 2 mm. directions, respectively. The diameters of these orifices were 2 mm, 3 mm, or 4 mm. The thickness of Detailed geometric features of the orifices are exemplified in Figure 9 and detailed in Table 2. A all experimentally tested plates was kept the same at 2 mm. signal generator and amplification was employed to generate a 100 dB white noise signal within a Detailed geometric features of the orifices are exemplified in Figure 9 and detailed in Table 2. A bandwidth of 100 Hz–1000 Hz. Two microphones were installed flush with the resonator top wall signal generator and amplification was employed to generate a 100 dB white noise signal within a and the acoustic duct wall, respectively. The resulting magnitude of the pressure response function, bandwidth of 100 Hz–1000 Hz. Two microphones were installed flush with the resonator top wall PRF,and is the defined acoustic below, duct wall,based respec upontively. the Fast The Fourier resulting Transform magnitude (FFT) of the of pressure the individual response microphone function, signals:PRF, is defined below, based upon the Fast Fourier Transform (FFT) of the individual microphone 𝐹𝐹𝑇(𝑃_𝑎𝑐𝑜𝑢𝑠𝑡𝑖𝑐_𝑚𝑖𝑐𝑟𝑜𝑝ℎ𝑜𝑛𝑒_1) signals: 𝑃𝑅𝐹 = 𝐹𝐹𝑇(𝑃_𝑎𝑐𝑜𝑢𝑠𝑡𝑖𝑐_𝑚𝑖𝑐𝑟𝑜𝑝ℎ𝑜𝑛𝑒_2)𝐹𝐹𝑇(𝑃_𝑎𝑐𝑜𝑢𝑠𝑡𝑖𝑐_𝑚𝑖𝑐𝑟𝑜𝑝ℎ𝑜𝑛𝑒_1) Appl. Sci. 2019, 9, 1419 𝑃𝑅𝐹 = 8 of 18 𝐹𝐹𝑇(𝑃_𝑎𝑐𝑜𝑢𝑠𝑡𝑖𝑐_𝑚𝑖𝑐𝑟𝑜𝑝ℎ𝑜𝑛𝑒_2)
Figure 7. Experimental Test Rig Setup. FigureFigure 7. Experimental Test Test Rig Rig Setup. Setup.
51 mm 51 mm
MMicricropophhoonne e2 2
FigureFigureFigure 8. 8.The 8.The The position position position and and and shape shape shape of of of the the the distributed distributed distributed Helmholtz Helmholtz resonator resonatorresonatoron on on thethe the experimentalexperimental experimental test test rig. rig.rig. The test rig (Figure7) consists of an upstream duct section, a downstream duct section, and a test section installed in between them, each with an internal diameter of 165 mm. Four opposing loudspeakers were installed on the upstream duct. The test rig was designed to include anechoic Appl.Appl. Sci. Sci. 2019 2019, 9, x;9, x;doi: doi: FOR FOR PEER PEER REVIEW REVIEW www.mdpi.com/journal/applsciwww.mdpi.com/journal/applsci terminations to minimize reflections from the change in impedance at the inlet and exit. In all test configurations, no mean flow through the duct or bias flow through the plates was applied. The resonant cavity, Figure8, was constructed from a curved rectangular test plate with a length of 0.05 m and an arc length of 0.0594 m, mounted flush to the side wall of the test section duct, with an attached fan-shaped cavity of 0.051 m depth. Referring to Figure8, the volume of the cavity is:
◦ 2 2 41.25 3 V = π[(0.051 + 0.165/2+0.002) − (0.165/2+0.002) ] × ◦ × 0.05=0.002 m (12) 360 where 51 mm is the cavity depth, 165 mm is the internal diameter of the tube, 2 mm is the thickness of plate, and 41.25◦ represents the angle of the fan-shaped cavity. A total of 32 circular perforations, as exemplified in Figure9, were drilled in the x and y directions, respectively. The diameters of these orifices were 2 mm, 3 mm, or 4 mm. The thickness of all experimentally tested plates was kept the same at 2 mm. Appl. Sci. 2019, 9, 1419 9 of 18 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 19
FigureFigure 9. Orifice9. Orifice separation separation distances distances (mm) (mm) in in different different directions directions ofof testedtested perforated plates 1–6. 1–6.
Detailed geometricTable 2. Geometric features ofdetails the of orifices perforated are plates exemplified for experimental in Figure data9 and collection. detailed in Table2.
A signal generator and amplificationPlate was employedPlate Axial to generateHole a 100Hole dB white noiseEffective signal within a Case # Diameter Porosity bandwidth of 100 Hz–1000 Hz.Circumferential Two microphones Length were installeddistance flushdistance with the resonatorPorosity top wall and EXP (mm) 𝝈 the acoustic duct wall, respectively.Length (mm) The resulting(mm) magnitude-x (mm) of the-y (mm) pressure response function,𝝈𝒆𝒇𝒇 PRF, is defined below,1 based2.0 upon the59.4 Fast Fourier Transform50 (FFT)6.0 of the12 individual0.0338 microphone0.0436 signals: 2 2.0 59.4 50 6.0 9 0.0338 0.0582 3 2.0 59.4 50 6.0 6 0.0338 0.0873 FFT(P_acoustic_microphone_1) 4 2.0 PRF59.4= 50 4.2 12 0.0338 0.0623 5 2.0 59.4 FFT(P_acoustic50 _microphone4.2 _29 ) 0.0338 0.0831 6 2.0 59.4 50 4.2 6 0.0338 0.1247 7 3.0 59.4 50 6.0 12 0.0762 0.0982 8 Table3.0 2. Geometric details59.4 of perforated50 plates for6.0experimental 9 data0.0762 collection. 0.1309 9 3.0 59.4 50 6.0 6 0.0762 0.1963 10 3.0 Plate 59.4 Plate Axial 50 Hole 4.2 Hole 12 0.0762 0.1402 Case # Diameter 11 3.0Circumferential 59.4 Length 50Distance-x 4.2Distance-y 9 Porosity0.0762σ Effective0.1870 Porosity σ EXP (mm) eff 12 3.0 Length (mm)59.4 (mm) 50 (mm) 4.2 (mm) 6 0.0762 0.2805 113 2.04.0 59.459.4 5050 6.06.0 1212 0.03380.135 0.1745 0.0436 214 2.04.0 59.459.4 5050 6.06.0 99 0.03380.135 0.2327 0.0582 315 2.04.0 59.459.4 5050 6.06.0 66 0.03380.135 0.3491 0.0873 416 2.04.0 59.459.4 5050 4.24.2 1212 0.03380.135 0.2493 0.0623 517 2.04.0 59.459.4 5050 4.24.2 99 0.03380.135 0.3324 0.0831 618 2.04.0 59.459.4 5050 4.24.2 66 0.03380.135 0.4987 0.1247 7 3.0 59.4 50 6.0 12 0.0762 0.0982 EXP: Abbreviation for “Experiment”; x: hole separation distance in the horizontal direction as shown 8 3.0 59.4 50 6.0 9 0.0762 0.1309 9in Figure 3.0 9; y: hole separation 59.4 distance 50 in the vertical 6.0 direction 6as shown 0.0762in Figure 9; 𝝈𝒆𝒇𝒇: 0.1963effective 10porosity 3.0 as defined in 59.4 Equation (11). 50 4.2 12 0.0762 0.1402 11 3.0 59.4 50 4.2 9 0.0762 0.1870 12 3.0 59.4 50 4.2 6 0.0762 0.2805 3.13 Results 4.0 59.4 50 6.0 12 0.135 0.1745 14 4.0 59.4 50 6.0 9 0.135 0.2327 3.1.15 Acquisition 4.0 of Model 59.4 50 6.0 6 0.135 0.3491 16 4.0 59.4 50 4.2 12 0.135 0.2493 17Figure 4.0 10 displays 59.4normalized specific 50 acoustic 4.2 reactance 9 curves for 0.135 the full set of 0.3324 twenty-five 18 4.0 59.4 50 4.2 6 0.135 0.4987 plates (Table 1) for which results were obtained using CFD simulation. All four sub-figures EXP: Abbreviation for “Experiment”; x: hole separation distance in the horizontal direction as shown in Figure9; y: hole demonstrate that the acoustic reactance curves shift to lower frequencies as the distance between separation distance in the vertical direction as shown in Figure9; σe f f : effective porosity as defined in Equation (11). individual holes within a perforated plate decreases. According to Equation 1, these reductions in resonance frequencies imply that the end correction length increases with the decreasing hole separation distance. This is contrary to Fok’s function [8], which suggests that hole interaction effect reduces the overall acoustic end correction length, as given in Equation (4).
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3. Results
3.1. Acquisition of Model Figure 10 displays normalized specific acoustic reactance curves for the full set of twenty-five plates (Table1) for which results were obtained using CFD simulation. All four sub-figures demonstrate that the acoustic reactance curves shift to lower frequencies as the distance between individual holes within a perforated plate decreases. According to Equation (1), these reductions in resonance frequencies imply that the end correction length increases with the decreasing hole separation distance.
ThisAppl. is Sci. contrary 2019, 9, x FOR to Fok’s PEER REVIEW function [8], which suggests that hole interaction effect reduces the11 of overall 19 acoustic end correction length, as given in Equation (4).
Figure 10. CFD obtained resonance frequencies of perforated plate No. 1–25 absorbers (as listed in Table1). Figure 10. CFD obtained resonance frequencies of perforated plate No. 1–25 absorbers (as listed in TheTable acoustic 1). velocity magnitude in the vicinity of the perforations is visualized using the results from the CFD simulations of Plate No. 7 with distantly spaced holes, shown in Figure 11, and Plate No. 2 with closely spaced holes, shown in Figure 12. Figure 13 presents a cross-section view of the same holes. There is a clear indication that when holes are distant from each other, such as those on Plate No. 7 absorber, the oscillating masses for each orifice are so far apart that they barely affect each other. The holes radiate pressure fluctuations separately. By contrast, for plate No. 2, the acoustic velocity does not diminish so rapidly when neighboring holes are very close, due to the combined momentum of the interacting jet flows from neighboring holes. As a result, the strong oscillating mass is sustained further downstream and forms an extra film of what may be considered to be high acoustic velocity, as shown in Figure 12. This gives rise to the extra acoustic radiation effect from the perforation area as a whole to the acoustic downstream space. In this paper, this extra radiation effect is named “acoustic radiation effect due to the overall perforation area”.
Figure 11. Acoustic velocity magnitude contour near perforated plate No. 7 absorber.
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Appl. Sci. 2019, 9, 1419 11 of 18
From the other perspective, a close hole separation distance (x/d = 1.5) in Plate No. 2 makes the “attached mass” near each individual hole slightly horizontally narrower in comparison with that observed in Plate No. 7 due to hole interaction effect, yet the “attached mass” near each individual hole is somewhat stretched in the vertical direction. However, the “attached mass” near each individual hole can be considered to not be affected to a noticeable degree by acoustic radiations due to the overall perforation area. Therefore, it may be considered that the end correction due to acoustic radiation for an individual hole is not affected by the acoustic radiation effect, due to the overall perforation area. The acoustic radiation effect from the perforation area as a whole manifests itself by forming an extra acoustic radiation effect in the vicinity of the overall perforated area. The acoustic radiation effect of a perforated plate is treated as a superimposition of two radiation effects: the “individual hole radiation effect” and the “overall perforation region radiation effect”. An acoustic signal experiences the single hole radiationFigure effect 10. CFD and theobtained perforation resonance region frequencies radiation of effectperfor simultaneously.ated plate No. 1–25 A simpleabsorbers model (as listed is proposed in in thisTable paper 1). by assuming that acoustic end correction lengths caused by these effects are additive.
Appl. Sci. 2019Figure, 9, x FOR 11. PEER Acoustic REVIEW velocity magnitude contour ne nearar perforated plate No. 7 absorber. 12 of 19
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Figure 12. Acoustic velocity magnitude contour near perforated plate No. 2 absorber. Figure 12. Acoustic velocity magnitude contour near perforated plate No. 2 absorber.
(a)
(b)
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Appl. Sci. 2019, 9, 1419 12 of 18 Figure 12. Acoustic velocity magnitude contour near perforated plate No. 2 absorber.
(a)
(b)
Appl. FigureSci. 2019 13., 9, Acousticx; doi: FOR velocity PEER REVIEW magnitude contour near an orifice with different holewww.mdpi.com/journal/applsci separation distance: (a) plate No.7 absorber: x/d = 5, y/d = 5, (b) plate No.2 absorber: x/d = 1.5, y/d = 1.5.
As was introduced earlier, the end correction length due to single hole radiation effect has been analytically derived to be [4–6]: p lsh = 0.96 A0 (13) Consider an extreme case where all perforations are just next to each other (x/d=1). The orifices nearly work as a single large hole in terms of acoustic radiations. Therefore, it is proposed that acoustic end corrections due to the overall perforation area follow a similar form with that caused by an individual hole. √ lop = C · 0.96 A (14) where A0 stands for area of an individual perforation and A is the overall opening area of all orifices. A correcting parameter C is introduced to consider the extent to which the perforation region is acting Appl. Sci. 2019, 9, 1419 13 of 18 like a single large hole. C is large when holes are close to each other and it is zero when holes are far away from each other. A simple model is proposed in this paper by assuming that acoustic end correction lengths caused by these effects are additive. p √ lec = lsh + lop = 0.96 A0 + C · 0.96 A (15) where σe f f represents the effective porosity of the perforation area; lsh is the end correction length due to acoustic radiation effect near an individual hole and lop stands for the end correction length due to the acoustic radiation effect due to the overall perforation area. C is a factor for the overall perforation area radiation effect. The factor C for all twenty-five numerically investigated PPAs was determined from Equations (1) and (15), as listed in Table3, and then plotted against effective porosities, as presented in Figure 14.
Table 3. Numerical results of resonance frequencies and end correction lengths for all simulated perforated plates.
Case No. CFD σeff fr (Hz) lec (mm) C 1 0.545 429 5.562 0.654 2 0.349 457 4.275 0.503 3 0.196 492 2.970 0.349 4 0.126 517 2.194 0.258 5 0.087 535 1.696 0.199 6 0.049 561 1.062 0.125 7 0.031 576 0.721 0.085 8 0.545 373 8.141 0.639 9 0.444 401 6.452 0.506 10 0.250 435 4.512 0.354 11 0.160 461 3.320 0.260 12 0.111 481 2.529 0.198 13 0.063 510 1.544 0.121 14 0.040 530 0.955 0.075 15 0.545 331 11.000 0.647 16 0.444 357 8.764 0.516 17 0.250 392 6.037 0.355 18 0.160 417 4.471 0.263 19 0.111 437 3.411 0.201 20 0.063 470 1.988 0.117 21 0.040 492 1.179 0.069 22 0.105 473 2.721 0.2234 23 0.079 490 2.123 0.1665 24 0.052 508 1.522 0.1194 25 0.039 521 1.136 0.0891
A model relating the overall perforation area radiation effect to the effective porosity was obtained by regression analysis, yielding: √ p lop = 0.96 A · 0.78 σe f f − 0.11 (16)
The total end correction length considering both acoustic radiation effects then becomes: √ p p lec = lsh + lop = 0.96 A0 + 0.96 A 0.78 σe f f − 0.11 (17)
The resonant frequency of a distributed Helmholtz resonator
s s v c A c A c u A fr = = = u √ √ (18) 2π Vle 2π V(l + lec) 2π t √ V(l + 0.96 A0 + 0.96 A 0.78 σe f f − 0.11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 19
Table 3. Numerical results of resonance frequencies and end correction lengths for all simulated perforated plates.
Case No. 𝝈𝒆𝒇𝒇 𝒇 (Hz) 𝒍 (mm) C CFD 𝒓 𝒆𝒄 1 0.545 429 5.562 0.654 2 0.349 457 4.275 0.503 3 0.196 492 2.970 0.349 4 0.126 517 2.194 0.258 5 0.087 535 1.696 0.199 6 0.049 561 1.062 0.125 7 0.031 576 0.721 0.085 8 0.545 373 8.141 0.639 Appl. Sci. 2019 9 , , 1419 9 0.444 401 6.452 0.506 14 of 18 10 0.250 435 4.512 0.354 11 0.160 461 3.320 0.260 where lec represents the total12 end correction 0.111 length481 of a PPA, lsh2.529represents 0.198 end correction length as a result of acoustic radiation13 effect due 0.063 to individual510 perforation,1.544and lop represents0.121 end correction length as a result of acoustic14 radiation 0.040 effect due to530 the overall0.955 perforation region.0.075 The above model 15 0.545 331 11.000 0.647 was fitted to the available data16 within an 0.444 effective porosity357 range8.764 of 3.14–55% 0.516 and an aspect ratio range of 0.4:1–1:1. 17 0.250 392 6.037 0.355 The model provides an18 overall assessment 0.160 for417 the impact4.471 of hole interaction0.263 effect on acoustic 19 0.111 437 3.411 0.201 radiation effect by introducing20 a simple 0.063 additive term470 “overall perforation1.988 area0.117 radiation effect”, under the assumption that acoustic√ 21 radiation 0.040 effect from the492 overall perforation1.179 area0.069 does not change the end correction length 0.96 A022caused by acoustic 0.105 radiation473 from each2.721 individual 0.2234 hole. 23 0.079 490 2.123 0.1665 As shown in Figure 14, the later term lop reduces to zero when σe f f ≈ 2%, which suggests the 24 0.052 508 1.522 0.1194 situations where the holes25 are far enough 0.039 away from521 each other1.136 that the perforation0.0891 area as a whole does form acoustic radiations. In these situations, holes radiate sound separately.
FigureFigure 14. 14.Proposed Proposed model model for for end end correction correction factor factor as as a functiona function of of effective effective porosity. porosity.
3.2.3.2. Validation Validation of theof the Model Model TheThe model, model, as describedas described earlier earlier in Equationsin Equation (17) (17) and and (18), (18), was was determined determined based based on on numerical numerical resultsresults alone. alone. In In this this section, section, thethe modelmodel is validated by by comparison comparison with with experimental experimental results results for foreighteen eighteen distributed Helmholtz resonators.resonators. Effect Effectiveive porositiesporosities ofof thesethese experimentally experimentally tested tested perforatedperforated plates plates range range from from 0.0436 0.0436 to to 0.4987, 0.4987, porosities porosities of of these these plates plates ranges ranges from from 0.0338 0.0338 to to 0.135, 0.135, andand perforation perforation aspect aspect ratios ratios range range from from 1:1 1:1 to 1:3.to 1:3. Figures 15–17 display the magnitude of pressure response functions between microphone 1 in the resonator and microphone 2 in the acoustic propagation duct. The experimental results suggest that resonanceAppl. Sci. frequencies2019, 9, x; doi: ofFOR PPAs PEER shift REVIEW to lower frequencies as the holes approachwww.mdpi.com/journal/applsci each other. This finding is in agreement with CFD investigations within a wide range of effective porosities from approximately 0.04 to 0.54. This range of 0.04–0.54 covers the effective porosity of most practical PPAs that exhibit a hole–hole acoustic interaction effect. An even lower porosity than 0.04 does not exhibit a noticeable hole–hole interaction effect. A higher effective porosity than 0.54 is not investigated because a PPA with higher effective porosity is nearly a single hole in terms of acoustic radiation effect (for example, for a plate with an effective porosity of 0.6, the ratio of hole diameter (d) to hole center distance (b) is 0.874, which means the orifice is so close that only a distance of 0.126d exists in between their edges). Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 19
Figure 15, Figure 16, and Figure 17 display the magnitude of pressure response functions between microphone 1 in the resonator and microphone 2 in the acoustic propagation duct. The experimental results suggest that resonance frequencies of PPAs shift to lower frequencies as the holes approach each other. This finding is in agreement with CFD investigations within a wide range of effective porosities from approximately 0.04 to 0.54. This range of 0.04–0.54 covers the effective porosity of most practical PPAs that exhibit a hole–hole acoustic interaction effect. An even lower porosity than 0.04 does not exhibit a noticeable hole–hole interaction effect. A higher effective porosity than 0.54 is not investigated because a PPA with higher effective porosity is nearly a single hole in terms of acoustic radiation effect (for example, for a plate with an effective porosity of 0.6, the ratio of hole diameter (d) to hole center distance (b) is 0.874, which means the orifice is so close that only a distance of 0.126d exists in between their edges). Resonance frequencies of all eighteen PPAs are determined from Figure 15, Figure 16, and Figure 17 and then listed in Table 4. Then, the experimental end correction length was calculated by employing Equation (1). End correction lengths acquired by the proposed model (Equation (17)) are similarly provided in Table 4. The experimental end correction lengths for all eighteen distributed Helmholtz resonators are plotted against the model results, as shown Figure 18. The figure shows that the proposed model resulting from the regression of the CFD simulation data successfully provides a satisfactory agreement with the experimental results. The relative errors with respect to the experimental results are provided in Table 4 as well. Both experimental results and model results predict an increase of end correction length with closer hole separation distance due to the hole–hole interaction effect. By contrast, as listed in Table 4, predictions according to Fok’s function (Equation 4) always generate
end correction lengths less than 0.96 𝐴 , which are much lower than experimental results. On the other hand, as can be seen from the error analysis listed in Table 4, the model tends to slightly underestimate the acoustic end correction length in comparison with some of the experimental data; this model underestimation is more significant for plates with a y = 12 mm, where there are holes very close to cavity walls. As a result, a noticeable hole-to-wall interaction effect will Appl. Sci.probably2019, 9, 1419contribute to the error. 15 of 18
Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 16 of 19 FigureFigure 15. Experimental15. Experimental pressure pressure response response functions for for plate plate 1 to 1 toplate plate 6. 6.
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Figure 16. Experimental pressure response functions for plate 7 to plate 12. FigureFigure 16. 16.Experimental Experimental pressure pressure response response functions functions forfor plate 7 to plate 12. 12.
FigureFigure 17. 17. Experimental pressure response functions for plate 13 to plate 18. Experimental pressure response functions for plate 13 to plate 18.
TableFigure 4. Comparison 17. Experimental of experimental pressure and response fitted model functions results for for plate end 13 correction to plate 18.lengths.
𝒍𝒆𝒄-𝑭𝒐𝒌 𝒍𝒆𝒄-EXP 𝒍𝒆𝒄-Model Model Relative Case # Table𝝈𝒆𝒇𝒇 4. Comparison𝒇 (Hz)-EXP of experimental𝒇 (Hz)-Model and fitted model results for end correction lengths. 𝒓 𝒓 (mm) (mm) (mm) Error (%)
1 0.0436 579 595 <1.7𝒍𝒆𝒄- 𝑭𝒐𝒌mm 𝒍𝒆𝒄2.45-EXP 𝒍𝒆𝒄2.21-Model Model−9.80 Relative Case # 𝝈𝒆𝒇𝒇 𝒇 (Hz)-EXP 𝒇 (Hz)-Model 2 0.0582 𝒓 573 𝒓 579 <1.7(mm) mm (mm)2.54 2.45(mm) Error−3.59 (%) 1 3 0.04360.0873 579556 595554 <1.7 mmmm 2.822.45 2.862.21 1.18−9.80 2 4 0.05820.0623 573563 579575 <1.7 mmmm 2.712.54 2.512.45 −7.04−3.59 3 5 0.08730.0831 556554 554557 <1.7 mmmm 2.862.82 2.802.86 −1.921.18 4 6 0.06230.1247 563520 575531 <1.7 mmmm 3.522.71 3.292.51 −6.41−7.04 07 0.0982 683 719 <2.55 mm 5.19 4.49 −13.5 5 0.0831 554 557 <1.7 mm 2.86 2.80 −1.92 08 0.1309 676 691 <2.55 mm 5.34 5.03 −5.79 6 0.1247 520 531 <1.7 mm 3.52 3.29 −6.41 09 0.1963 644 649 <2.55 mm 6.09 5.95 −2.34 07 0.0982 683 719 <2.55 mm 5.19 4.49 −13.5 10 0.1402 650 683 <2.55 mm 5.94 5.18 −12.8 08 0.1309 676 691 <2.55 mm 5.34 5.03 −5.79 11 0.1870 642 654 <2.55 mm 6.14 5.83 −5.10 09 0.1963 644 649 <2.55 mm 6.09 5.95 −2.34 12 0.2805 606 613 <2.55 mm 7.14 6.92 −3.02 10 0.1402 650 683 <2.55 mm 5.94 5.18 −12.8 11 0.1870 642 654 <2.55 mm 6.14 5.83 −5.10 Appl.12 Sci. 20190.2805, 9, x; doi: FOR606 PEER REVIEW 613 <2.55 mm 7.14 www.mdpi.com/journal/applsci6.92 −3.02
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Resonance frequencies of all eighteen PPAs are determined from Figures 15–17 and then listed in Table4. Then, the experimental end correction length was calculated by employing Equation (1). End correction lengths acquired by the proposed model (Equation (17)) are similarly provided in Table4.
Table 4. Comparison of experimental and fitted model results for end correction lengths.
Model Relative Case # σ f (Hz)−EXP f (Hz)−Model l −Fok (mm) l −EXP (mm) l −Model (mm) eff r r ec ec ec Error (%) 1 0.0436 579 595 <1.7 mm 2.45 2.21 −9.80 2 0.0582 573 579 <1.7 mm 2.54 2.45 −3.59 3 0.0873 556 554 <1.7 mm 2.82 2.86 1.18 4 0.0623 563 575 <1.7 mm 2.71 2.51 −7.04 5 0.0831 554 557 <1.7 mm 2.86 2.80 −1.92 6 0.1247 520 531 <1.7 mm 3.52 3.29 −6.41 07 0.0982 683 719 <2.55 mm 5.19 4.49 −13.5 08 0.1309 676 691 <2.55 mm 5.34 5.03 −5.79 09 0.1963 644 649 <2.55 mm 6.09 5.95 −2.34 10 0.1402 650 683 <2.55 mm 5.94 5.18 −12.8 11 0.1870 642 654 <2.55 mm 6.14 5.83 −5.10 12 0.2805 606 613 <2.55 mm 7.14 6.92 −3.02 13 0.1745 778 790 <3.4 mm 7.86 7.55 −3.88 14 0.2327 757 753 <3.4 mm 8.41 8.52 1.31 15 0.3491 719 700 <3.4 mm 9.54 10.15 6.37 16 0.2493 728 744 <3.4 mm 9.26 8.78 −5.20 17 0.3324 703 707 <3.4 mm 10.07 9.93 −1.37 18 0.4987 671 655 <3.4 mm 11.29 11.88 5.21
Appl.σ eSci. f f : Effective2019, 9, x porosity; FOR PEERfr (Hz)-EXP:REVIEW resonant frequency obtained by experimental method; fr (Hz)-Model: resonant17 of 19 frequency obtained by the proposed model as given in Equation (18); lec-Fok: acoustic end correction length obtained by13 Fok’s0.1745 function as given778 in Equation (4);790l ec -EXP: acoustic<3.4 mm end correction7.86 length from experiment;7.55 lec-Model:−3.88 predicted14 0.2327 acoustic end correction757 length by753 using the proposed<3.4 mm model as given8.41 in Equation8.52 (17). 1.31 15 0.3491 719 700 <3.4 mm 9.54 10.15 6.37 The16 experimental0.2493 end728 correction744 lengths for all<3.4 eighteen mm distributed9.26 Helmholtz8.78 resonators−5.20 are 17 0.3324 703 707 <3.4 mm 10.07 9.93 −1.37 plotted18 against0.4987 the model671 results, as shown655 Figure<3.4 18 mm. The figure11.29 shows that11.88 the proposed5.21 model resulting from the regression of the CFD simulation data successfully provides a satisfactory agreement 𝜎 : Effective porosity; 𝑓 (Hz)-EXP: resonant frequency obtained by experimental method; 𝑓 (Hz)- with the experimental results. The relative errors with respect to the experimental results are provided Model: resonant frequency obtained by the proposed model as given in Equation (18); 𝑙 -𝐹𝑜𝑘: in Table4 as well. Both experimental results and model results predict an increase of end correction acoustic end correction length obtained by Fok’s function as given in Equation (4); 𝑙 -EXP: acoustic length with closer hole separation distance due to the hole–hole interaction effect. By contrast, as listed end correction length from experiment; 𝑙 -Model: predicted acoustic end correction length by using in Tablethe4, proposed predictions√ model according as given to in Fok’s Equation function (17). (Equation 4) always generate end correction lengths less than 0.96 A0, which are much lower than experimental results.
FigureFigure 18. 18. ValidationValidation ofof the proposed model against against experimental experimental results. results. (— (—: model: model results, results, +: +:experiment experiment results). results).
4. Conclusions A model for the prediction of resonance frequencies of distributed Helmholtz resonators is proposed. The model was initially derived based on high resolution computational simulations for twenty-five distributed Helmholtz resonators. End correction length was proposed to be correlated with two acoustic radiation effects, namely, acoustic radiation effect from the individual hole and acoustic radiation from the overall perforation area. The model was then acquired by making a convenient assumption that the end correction lengths due to these two acoustic radiation effects are additive. It is noteworthy that the assumption is not strictly validated analytically, as its purpose is to make the model easy to use in practice. The proposed resonant frequency model was successfully validated by comparison with experimental data for a further eighteen distributed Helmholtz resonators. The validation exercise demonstrated that the proposed model displays very good agreement with the experimental results for resonance frequencies for distributed Helmholtz resonators within a wide range of effective porosities (0.04–0.54), porosities (0.0123–0.135), and perforation aspect ratios (1:1–1:3). A new term, “overall perforation area acoustic radiation effect”, is proposed to account for the acoustic radiations from the perforated area as a whole. This resulting model is easily applicable for engineers and is of significant value in the design of practical distributed Helmholtz resonators, such as those distributed Helmholtz resonators applied on gas turbine engines to control combustion instabilities.
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On the other hand, as can be seen from the error analysis listed in Table4, the model tends to slightly underestimate the acoustic end correction length in comparison with some of the experimental data; this model underestimation is more significant for plates with a y = 12 mm, where there are holes very close to cavity walls. As a result, a noticeable hole-to-wall interaction effect will probably contribute to the error.
4. Conclusions A model for the prediction of resonance frequencies of distributed Helmholtz resonators is proposed. The model was initially derived based on high resolution computational simulations for twenty-five distributed Helmholtz resonators. End correction length was proposed to be correlated with two acoustic radiation effects, namely, acoustic radiation effect from the individual hole and acoustic radiation from the overall perforation area. The model was then acquired by making a convenient assumption that the end correction lengths due to these two acoustic radiation effects are additive. It is noteworthy that the assumption is not strictly validated analytically, as its purpose is to make the model easy to use in practice. The proposed resonant frequency model was successfully validated by comparison with experimental data for a further eighteen distributed Helmholtz resonators. The validation exercise demonstrated that the proposed model displays very good agreement with the experimental results for resonance frequencies for distributed Helmholtz resonators within a wide range of effective porosities (0.04–0.54), porosities (0.0123–0.135), and perforation aspect ratios (1:1–1:3). A new term, “overall perforation area acoustic radiation effect”, is proposed to account for the acoustic radiations from the perforated area as a whole. This resulting model is easily applicable for engineers and is of significant value in the design of practical distributed Helmholtz resonators, such as those distributed Helmholtz resonators applied on gas turbine engines to control combustion instabilities.
Author Contributions: Conceptualization, J.W. and P.R.; methodology, J.W.; software, J.W. and Q.Q.; validation, J.W., P.R., and Q.Q.; formal analysis, J.W.; investigation, J.W.; resources, P.R., B.H., and Q.Q.; data curation, J.W. and B.H.; writing—original draft preparation, J.W.; writing—review and editing, J.W., Q.Q., and P.R.; visualization, J.W.; supervision, P.R.; project administration, P.R. Funding: This research received no external funding. Acknowledgments: The University of Hull and China Scholarship Council are greatly acknowledged for the financial support in making this research possible. Conflicts of Interest: The authors declare no conflict of interest.
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