[American Institute of Aeronautics and Astronautics 3rd AIAA/CEAS Aeroacoustics Conference -...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA-97-1672-CP

THE INFLUENCE OF GRAZING FLOW ON THERAYLEIGH CONDUCTIVIITY OF AN APERTURE OF

ARBITRARY SHAPE t

Sheryl M. Grace * and Michael S. Howe §and Kelly P. HoranAerospace and Mechanical Engineering Department

Boston University110 Cummington St., Boston, MA 02215

ABSTRACTThis research investigates the effect of geometry

on the unsteady motion induced in a wall apertureby a pressure perturbation in the presence of graz-ing flow. For certain combinations of perturbationfrequency, aperture size and flow speed, the aper-ture shear layer motion will augment the perturba-tion energy. Combining the three parameters intoa Strouhal number, we see that for certain rangesof Strouhal number the coupling of the aperturedynamics and the mean flow leads to the produc-tion of sound and vibration. We examine this cou-pling for both one-sided and two-sided grazing flowpast the aperture and consider several symmetricaperture geometries which include the circle, square,cross, triangle, rectangle, and one denoted as thecrown. In this study we use the Rayleigh conduc-tivity to analyze the Strouhal number dependenceof the different aperture and flow configurations fora given time harmonic applied pressure disturbance.The model is valid in the low Mach number limit,and uses linearized theory. Using this model, weshow that aperture geometry does not affect theStrouhal number range in which the aperture dy-namics add energy to the flow perturbation. Simi-larly, the Strouhal number which approximately cor-responds to self-oscillations of the shear layer re-mains relatively unchanged when the geometry ofthe aperture changes. Finally, we verify the reci-

' Copyright ©1997 by Sheryl Grace. Published by theAmerican Institute of Aeronautics and Astronautics, Inc.with permission.

'Assistant professor, Member AIAA.'Professor, Member AIAA.'Research Assistant.

procity relation in the sense that the orientation ofthe aperture in the streamwise direction does notaffect the aperture shear layer behavior.

INTRODUCTIONExperimental research has shown that unsteady

flow past wall apertures and cavities can create nar-row band acoustic tones [1, 2]. Rossiter [1] was thefirst to attribute the tones to a feedback mecha-nism involving shed vorticity from the leading edgeconvecting across the aperture and interacting withthe trailing edge to produce a pressure disturbancewhich travels back across the aperture. The toneswhich are produced correspond to continuous fre-quency ranges or Strouhal numbers in which theshear layer across the aperture exhibits unstable growth.A proper understanding of this phenomenon is im-portant because of the wide range of mechanical sys-tems in which this flow situation is present. Typi-cal examples include, perforated baffles in cross-flowheat exchangers, depressions in submarine and shiphulls, computer boards with closely spaced chip car-riers, aircraft control surfaces and fuselage openings,and flow-through resonators of automobile mufflers.

In this research we consider the low Mach num-ber limit for the flow past a symmetric aperture offinite extent and study the response of the apertureshear layer to a disturbance such as incident soundor the pressure field of a large scale flow inhomo-geneity. The mathematical model we use for thisproblem was developed in [2, 3]. It is an incompress-ible, inviscid, linearized, frequency-domain model inwhich the aperture shear layer is represented by avortex sheet. Details of the model are provided in

671American Institute of Aeronautics and Astronautics


the second section of this paper.The vortex sheet model was applied in [4] to in-

vestigate grazing flow over a slot of infinite span.For one-sided grazing flow, it was shown that en-ergy of an applied, time-harmonic pressure force isdissipated by the production of vorticity in the slotexcept for Strouhal numbers e = ̂ between 1.59and 3.49 (L is one half of the distance from the mostupstream point of the aperture to the most down-stream point, and U+ is the speed of the flow). Inthis interval energy is extracted from the mean flowand supplied to the imposed pressure field. This fre-quency range is centered approximately on the min-imum frequency of self-sustained oscillations of theshear layer in the slot. When the mean velocitiesare the same on both sides of the slot, energy is ex-tracted from the mean flow whenever the Strouhalnumber lies within one of the intervals 2.4 + nir <jj|- < 3.7 + nir, where n is an integer greater than 0(here U+ is the larger of the two flow velocities).

Scott extended this work on the slot by consid-ering a circular aperture in the presence of grazingflow [2, 3]. It was shown that one sided flow overa circular aperture is unstable for Strouhal numbersbetween 2.0 and 3.6. When the flow is the sameon both sides, the behavior becomes periodic in fre-quency when the Strouhal number exceeds about3.5, and perturbation energy is alternately augmentedand attenuated by the coupling with the mean flow.

An approximate treatment of tapered apertureswas discussed in [5], with particular attention totrapezoidal apertures, the extreme limiting cases be-ing apertures of rectangular and triangular cross-sections. Using the averaging method the resultsshowed that as the taper was increased the range ofStrouhal numbers for which the aperture was sup-plying energy to the perturbation also increased. Itwas also stated, that due to reciprocity, the trape-zoid could be oriented either with its longer side asthe leading edge or with the shorter side as the lead-ing edge and both aperture configurations would be-have the same.

In this research we show that aperture shape doesnot significantly alter the range of Strouhal frequen-cies for which the shear layer provides energy to theflow perturbation. We also show that the Strouhalnumber which corresponds to self-oscillation is notsignificantly changed by aperture geometry. Finally,we validate the reciprocity relation discussed in [5].

FORMULATIONNominally steady flow past an aperture with dif-

ferent flow speeds over each face is unstable. For anaperture of infinite extent, the growth of an arbi-trarily small perturbation tends to be dominated bythe appearance of interfacial Kelvin-Helmholtz waveswhose wavenumbers correspond to the most unsta-ble waves of the mean shear layer [7]. When theaperture area is finite, however, the shear layer in-stability depends critically on feedback which occursbetween the aperture's leading and trailing edges.This feedback loop consists of the shedding of vor-ticity from the leading edge, the convection of thevorticity across the aperture, the production of apressure pulse when the vorticity interacts with thetrailing edge of the aperture and the traversing ofthat pressure pulse back across the aperture to theleading edge. This feedback will either reinforce orinhibit vortex shedding depending on the phasingof the feedback signal relative to that of naturalshedding from the leading edge. This view of themotion is precisely that advanced many years agoby Rossiter [1] from his experimental investigationof tones and buffeting occurring during high speedflow over a wall cavity. As a corollary, one mightconclude that when the flows over both faces of theaperture are equal (so that there is no mean shear inthe aperture mouth), the motion becomes stable. Inpractice, however, finite wall thickness and the tur-bulent wall boundary layers will produce unsteadyshedding of vorticity from the aperture leading edge.Hence, an instability may arise, but it will be muchweaker than when the flows over both faces differ.

In order to completely quantify this qualitativepicture of disturbed motion of the shear layer it isnecessary to solve the fully viscous, nonlinear equa-tions of fluid motion. This is the case, for example,if one wants to determine the amplitude of a possibleself-sustaining, quasi-periodic mode of the aperturemotion. If, however, it is required only to calcu-late the frequencies of these oscillations (which corre-spond to the frequencies that dominate the produc-tion of sound and vibration by the flow), it is suffi-cient in a first approximation to consider a linearizedrepresentation of the shear layer motion. This resultstems from the domination of the feedback cycle bythe convection velocity of vorticity across the mouthof the aperture, which is found experimentally to beessentially independent of the amplitude of the mo-tion [8, 9, 10, 11]

The present study is restricted to low Mach num-ber flows, for which it is permissible to regard theunsteady motion in the aperture as incompressible.Furthermore, comparisons between experimental re-sults and simple approximate schemes [2, 5] indicate



that the motion may be regarded as inviscid exceptinsofar as its influence is incorporated by means ofthe standard application of the Kutta condition atthe leading edge of the aperture.

The stability properties of the aperture shearlayer can be examined in this inviscid approxima-tion by considering the linearized motion induced byan externally applied, time harmonic pressure force.This force initiates the production of unsteady vor-ticity at the aperture leading edge. In the invis-cid approximation a vortex sheet spans the aperturein the undisturbed state, and additional vorticityis shed under the influence of the applied pressure.The stability of the motion can be analyzed by con-sidering the equation governing the displacement ofthe vortex sheet from its planar form, for which theinstability frequencies coincide with the eigenvaluesof this equation.

The linear stability characteristics of the shearflow over an aperture govern the properties of theRayleigh conductivity, denoted as KR(U), which isthe reciprocal of the aperture impedance [12]. Fortime harmonic pressures p£e~twt and pje~""* ap-plied above and below the aperture respectively, theRayleigh conductivity is defined by the relation

(1)

where Q is the volume flux at the aperture, p+ — p~is the applied pressure difference across the aperture,w is the radian frequency, and p0 is the fluid density[12].

The pressure difference produces a small ampli-tude displacement £ of the vortex sheet from itsundisturbed form. Thus, in the linearized approxi-mation and in terms of the coordinate system shownin Figure 1, the aperture volume flux is given by

/

OO ,.00 Q

\ -i(u + iu+j—•oo ./-oo °yi,00- ,00 g

= I I -i(u+iU -^—J-oo J-oo °yi

/

OO ,00

/ C(»i,i/-oo J — oo

, 1/2,0)^2/2

(2)

since £ vanishes on the wall.The eigenvalues of the equation governing the

aperture shear layer displacement are identical to thepoles of the Rayleigh conductivity function. Hence,

the real parts of the poles of KR(U) in the com-plex frequency plane coincide with the linear insta-bility frequencies of the aperture motion [13], anddetermine the Strouhal numbers of a possible self-sustaining oscillation and the frequencies of the nar-row band tones radiated into the flow. In the ab-sence of mean flow KR is a real valued quantity thatdepends only on the shape and size of the aperture.

To calculate the conductivity according to (1) wemust first calculated the displacement C of the vortexsheet. To do this we make use of the procedure usedby Scott [2, 3] for a circular aperture. The incom-pressible motion on either side of the sheet dependson the solution of the Laplace equation

= 0 (3)

for the velocity potential $. The unsteady pressureinduced by the motion of the vortex sheet is thenobtained from the linearized Euler equation in theform

f\

p~ = +ipo(u -HI/-— -)ax\

--)^+, x 3 > 0 (4)

x3<0. (5)

The condition that the pressure is continuous acrossthe vortex sheet accordingly implies that

Ox if\

OX i-— )*-, 0:3 = 0 (6)

The solution of Laplace's equation (3) is obtainedby application of Green's Theorem using a Green'sfunction whose normal derivatives vanish on zg = 0and 2/3 = 0, i.e.,

b±fy\ — G(x\y)dyidy2, y = (ylt j/2, 0)

(7)

y1 = (yi,!/2,-i/3).By substituting the relation



leading edge "2shed vorticity

u-Figure 1: Grazing flow past a wall aperture.

into (7), and integrating by parts, we find that equa-tion (6) can be cast in the form

Po

L- (8)

where the integration is restricted to the apertureopening. To simplify this, we introduce the followingnondimensionalization and rescaling where lengthsare nondimensionalized by the half length of theaperture in the streamwise direction and velocitiesby C/+ which is the larger flow velocity:

X,Y,Z = x/L,y/L,{/L

Ur = U~/U+

Z' =- Po

(9)(10)

(11)

(12)

where e is the Strouhal number. Then, by integrat-ing (8) with respect to the second order differentialoperator on the right hand side, we obtain

Z'(yi,y a ,0).\X-Y\

= 1 + a(X2)eiflXl + /3(X2)eic*Xl (13)

Here a and /? are unknown functions of X% whichmust be determined to ensure that the Kutta con-dition is satisfied at the leading edge of the aper-ture [2, 3]; £1 and £2 are the nondimensional Kelvin-Helmholtz wave numbers for a velocity discontinuity[7],

U+ £2 = U+ - iU~ (14)

Note that e\ = e^ = e when U+ = U~. In this casethe right hand side of Equation (13) can be reducedto

The integral equation (13) is solved by means ofa standard quadrature scheme which is discussed inthe next section.

NUMERICAL IMPLEMENTATIONIn this research we solve the two-dimensional gov-

erning integral equation (13) numerically in order toobtain the Rayleigh conductivity for various sym-metric aperture shapes. The equation is discretizedby covering the aperture with a cartesian grid asshown by the schematic in Figure 2. On each gridcell Z'(Y\, YI, 0) is assumed to be constant, but Green'sfunction is integrated analytically. The Kutta condi-tion is imposed by requiring that C = 0 (which is the



flow direction

quadraturegrid

aperture boundary

grid cells where £ = 0.

Figure 2: Grid used in quadrature of integral equa-tion with the locations for the application of the Kuttacondition marked.

same as imposing Z1 = 0) on the first two grid cellsin each grid row of constant Y%; this is equivalent torequiring that the displacement of the vortex sheetand its streamwise derivative vanish at the leadingedge. The first two Z' values in each grid row maythen be replaced by the unknown functions ^(Yz)and f3(Y-2) which are constant for each grid row (ofconstant Y"2)-

When Z', a and /3 have been determined using acomplex matrix solver, the scaled conductivity, KR,is calculated using the following simple integrationtechnique

- Po -"//.Z'dS

. = 1.7 = 1

RESULTSWe have considered numerous symmetric aper-

ture shapes, and have calculated the scaled Rayleighconductivity (KR = 2L(F — iA) for both one-sidedand two-sided equal grazing flow with time harmonicapplied pressures p+ = P/e~'wi ,p~ = 0. The shapeswe consider include the circle, square, cross, twosizes of forward triangles (upstream pointing) andbackward triangles (downstream pointing), crown,and rectangle as shown in Figure 3.

The cross-sectional geometries we have studiedwere chosen for several reasons. We use the circleand tapered apertures as a benchmarking tool forour numerical code. The cross-shaped aperture wassuggested by Chanaud [14]. The crown was testedin hopes that it would modify the vortex shedding

from the leading edge of the aperture and cause asignificant shift in the self-oscillation frequency.

For flow past just one face of the aperture, thecomponents of the scaled conductivity, F and A, areplotted vs. the Strouhal number, in Figures 4 and 5.As was noted earlier, it is the real part of the polesof the conductivity function which correspond to theself-oscillation frequencies. These approximately co-incide with the minima in A. And the regions whereA < 0 indicate regions where the aperture shearlayer supplies energy to the applied pressure pertur-bation. Hence, these are the regions of instability forthe aperture shear layer. As the Strouhal number in-creases the magnitude of the square root singularityin £ near the trailing edge of the aperture increases(for the case of one-sided flow past the aperture).This makes the numerical evaluation of £ and thesubsequent integration inaccurate past a Strouhalnumber of about 5. Therefore we only show valuesfor the conductivity corresponding to Strouhal num-bers of 0 to 5.

For the one-sided flow case, there is one contin-uous region of Strouhal numbers in which A < 0;and, the location of the minimum in this regionvaries only slightly for the different aperture geome-tries. We also observe that the jagged leading edgeof the crown aperture does not affect the shed vor-ticity enough to greatly change the dependence ofthe shear layer motion on Strouhal number. Finally,we observe that the conductivity is independent ofthe orientation of the triangular-shaped apertures(upstream pointing or downstream pointing). Thissupports the reciprocity relation discussed in [5].

All of these points are made more clearly whenthe data is plotted simultaneously. To make a faircomparison on one single plot, it is necessary to nor-malize the scaled conductivity by the square rootof the aperture area rather than 2 times the halflength in the streamwise direction. Figure 6 showsthe comparison of I<R/\fA vs. Strouhal number forthe circle, square, cross, crown and rectangle. Fig-ure 7 shows the comparison for the large and smallforward and backward facing triangles. Some nu-merical error is present for the small forward trian-gle at the higher Strouhal numbers. This is a resultof the difficulty in capturing the leading edge Kuttacondition using a rectangular grid to discretize anaperture with such a sharp point. By refining thegrid, this error is minimized.

We then considered equal flow on both faces ofthe apertures. For the two-sided grazing flow, theproblem is numerically stable and we can calculate£ and its integral for very high Strouhal numbers.



We choose here to use Strouhal numbers from 0 to15. Figures 8 and 9 show T and A for all of the aper-ture shapes. Again the similarity between the resultsis clear. The minima of A all occur at roughly thesame values of c, as shown more clearly in the renor-malized simultaneous graphs in Figures 10 and 11.One difference however, is that for the case of thelarge triangular apertures, A first becomes negativeat about e = 10.3 as compared to e = 2.5 for theother geometries. More research is needed to fullyunderstand this behavior. Comparing the results forthe forward and backward triangles as shown in Fig-ure 11 again we see that reciprocity is affirmed.

A final interesting point can be made for theequal two-sided flow case. The overall trends in theconductivity directly reflect the general elements ofthe geometry of the aperture. For some of the aper-tures, the magnitude of the maxima and minimain the real and imaginary parts of the conductiv-ity remain the same value indefinitely, e.g. square,and rectangle. However, for those apertures, whoselength from leading edge to trailing edge continu-ously decreases with span, the values of the maximaand minima decrease with e, e.g. circle, and tri-angles. For the cross-shaped aperture, where thereare two distinct lengths in the streamwise direction,there are two distinct values of the peaks in the con-ductivity which continue repeating indefinitely. Thecrown-shaped aperture is a mix of all of these. Thereare two main lengths which are reflected in the con-ductivity, but since there is a gradual change be-tween the two lengths, the values of the correspond-ing maxima and minima decrease with epsilon.

In a frequency domain calculation we cannot cal-culate the magnitude of the instabilities, only theassociated frequencies, but it is surmised that thesevarying values in the maxima and minima may givea qualitative description of the magnitude of the in-stabilities. The validity of this statement is left forfuture investigations.

CONCLUSIONSGrazing flow past a wall aperture with the addi-

tion of a time harmonic applied pressure perturba-tion creates unsteady motion in the aperture shearlayer. The shear layer behavior can both augmentand attenuate the energy in the flow perturbationdepending upon the coupling between the motionand the mean flow. Of particular interest are theStrouhal number ranges which correspond to aug-mentation of the perturbation energy since it is thesefrequencies which dominate the production of soundand vibration. Here the Strouhal number is based

on the frequency of the applied pressure disturbance,the half length of the aperture in the streamwise di-rection and the velocity of the flow past the aperture.

The aperture dynamics are reflected in the Rayleighconductivity insofar as the real parts of the com-plex poles of this function correspond to the self-oscillating frequencies. Studying the Rayleigh con-ductivity for solely real frequencies, still allows usto predict the various Strouhal regions and to esti-mate the self-oscillating frequencies. This is done bystudying the imaginary part of the Rayleigh conduc-tivity. When it is less than zero, the aperture shearlayer motion serves to enhance the applied pressuredisturbance. The minima of these regions occur ap-proximately at the Strouhal numbers which corre-spond to self-oscillation.

We study finite apertures of various shapes andconsider the cases when there is flow past only oneface of the aperture and when there is equal flowon both faces of the aperture. The shapes discussedin this report include the circle, square, cross, largeand small forward triangles, large and small back-ward triangles, crown and rectangle. We show thatthe range of Strouhal numbers associated with theperturbation energy enhancement and in particularthe Strouhal number that is associated with self-oscillation of the aperture shear layer remain rela-tively unchanged as the shape of the aperture changes(for both the one-sided and two-sided flow cases).Furthermore using the forward and backward fac-ing triangles we verify the reciprocity relation thatstates that the streamwise orientation of the aper-ture is immaterial. We also show that a qualitativeassessment of the geometry of the aperture can bemade from the general trends in the Rayleigh con-ductivity for the two-sided flow case. The distinctnumber of values taken by the minima or maximaindicates the number of length scales present in theaperture geometry in the streamwise direction. Forexample, the Rayleigh conductivity for the cross-shaped aperture shows two distinct minima values.

ACKNOWLEDGMENTSThe authors would like to thank Kadin Tseng,

consultant at the Boston University Scientific Com-puting and Visualization facility, for his assistantwith the numerical implementation.

REFERENCES

[1] J. E. Rossiter, (1964), "Wind tunnel exper-iments of the flow over rectangular cavitiesat subsonic and transonic speeds," Aero. Res.



flow direction

2L

2L i

a , , ......... ,

Figure 3: Aperture cross-sections studied, a) circle, b) square, c) cross, d) forward pointing triangle, e) backwardpointing triangle, f) smaller forward pointing triangle, g) smaller backward pointing triangle, h) crown, i) rectanglewith aspect ratio 3.

Counc. RkM, no. 3438.

[2] M. S. Howe, M. I. Scott and S. R. Sipcic, (1996),"The Influence of Tangential Mean Flow on theRayleigh Conductivity of an Aperture," i Pro-ceedings of the Royal Society of London A452,2303 - 2317.

[3] M. I. Scott, (1995), "The Rayleigh Conductiv-ity of A Circular Aperture in The Presence ofA Grazing Flow," Master's Thesis, Boston Uni-versity.

[4] M. S. Howe, (1981), "On the Theory of Un-steady Shearing Flow Over a Slot," Philosophi-cal Transactions of the Royal Society of London,Vol. A-303.

[5] M. S. Howe, (1996), "Influence of Cross-Sectional Shape on the Conductivity of a WallAperture in Mean Flow," Boston UniversityTechnical Report, No. AM-96-034, Nov.

[6] M. S. Howe, (1996), "Influence of Wall Thick-ness on Rayleigh Conductivity and Flow-Induced Aperture Tones," Boston UniversityTechnical Report, No. AM-96-022, Sept. J. Flu-ids and Structures (in press)

[7] Sir H. Lamb, (1995), Hydrodynamics, 6th Ed.,Cambridge University Press, New York, pg.373.

[8] A. Powell, (1961), "On the edgetone," Journalof the Acoustical Society of America, Vol. 33,pp.395 - 409.

[9] D. K. Holger, T. A. Wilson, and G. S. Beavers,(1977), "Fluid mechanics of the edgetone",Journal of the Acoustical Society of America,Vol62, pp. 1116- 1128.

[10] D. Rockwell, (1983), "Oscillations of ImpingingShear Layers," AIAA Journal, Vol. 21, pp. 645-664.

[11] W. K. Blake, and A. Powell, (1986), "The De-velopment of Contemporary Views of Flow-toneGeneration", pp. 247 - 325 of Recent Advancesin Aeroacoustics, (edited by A. Krothapali andC. A. Smith). Springer.

[12] J. W. S. Rayleigh, (1945), The Theory of Sound,Vol. 2, Dover, pg. 173.

[13] M. S. Howe, (1996), "Edge, cavity and aperturetones at very low Mach numbers," Journal ofFluid Mechanics Vol. 330, 61 - 84.

[14] R. C. Chanaud, (1994), "Effects of geometryon the resonance frequency of Helmholtz res-onators," Journal of Sound and Vibration, Vol.178. no. 3, pp. 337-348.



-30.5 1 1.5 2 2.5 3 3.5 4 4.5 5

coL/U+0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0)L/U+

Figure 4: The Rayleigh conductivity for one-sided grazing flow past aperture with shapes circle, square, cross,forward triangle, backward triangle, small forward triangle, small backward triangle, and crown.



-30 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

COL/U+

Figure 5: The Rayleigh conductivity for one-sided grazing flow past the rectangular aperture.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Forward largeBackward largeForward smallBackward small

Figure 6: Real part (top) and imaginary part (bottom)of the Rayleigh conductivity normalized by the squareroot of the area for circle, square, cross, crown andrectangle apertures with flow on one side.

2 2.5 3 35 4 4.5 5eoL/U+

Figure 7: Real part (top) and imaginary part (bottom)of the Rayleigh conductivity normalized by the squareroot of the area for the large and small, forward andbackward facing triangle apertures with flow on oneside.



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15coL/U+

-iL54

backward [tdangle

small IforWaifd jtriaiigl

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15o)L/U+

Figure 8: The Rayleigh conductivity for equal two-sided grazing flow past apertures with shapes: circle, square,cross, forward triangle, backward triangle, small forward triangle, small backward triangle and crown.



-1.50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

OOL/U+

Figure 9: The Rayleigh conductivity for equal two-sided grazing flow past the rectangular aperture.

-»- Circle- Square->— Cross

10 11 12 13 14 15 __ Crown

-a- Rectangle

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15coL/U+

Figure 10: Real part (top) and imaginary part (bot-tom) of the -Rayleigh conductivity normalized by thesquare root of the area for circle, square, cross, crownand rectangle apertures with equal flow on both faces.

Forward largeBackward largeForward small

• Backward small

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15coL/U+

Figure 11: Real part (top) and imaginary part (bot-tom) of the Rayleigh conductivity normalized by thesquare root of the area for the large and small, forwardand backward facing triangle apertures with equal flowon both faces.


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