Broadband sources of structure-borne noise for propulsors in “haystacked” turbulence

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Pergamon PII: SOO45-7!34!3(96)00261-1

& Smctures Vol. 65, No. 3, pp. 475-490, 1997 1997 Elswier Science Ltd. All tights rerrwd Printed in Great Britain 004%7949/97 $17.00 + 0.00

BROA.DBAND SOURCES OF STRUCTURE-BORNE NOISE FOR PROPULSORS IN “HAYSTACKED” TURBULENCE R. Martinez Cambridge .4coustical Associates, Inc., 200 Boston Avenue, Suite 2500, Medford, MA 02155, U.S.A. (Received 1 August 1995) Abstract-This article reviews some of the statistical and probabilistic concepts underlying a number of classical and modern studies whose objective has been to predict broadband propulsor forces due to turbulence ingestion. Part of the material is new, extending a recent “closed-form” solution for thrust to apply now to any interpretation of isotropic turbulence. The more general theory dispenses altogether with the integral sN:aleas a modeling parameter in the problem of rotor/turbulence interaction. The new formula for the net random thrust still requires only that the three-dimensional rotor analyzed have a realistically large number of blades (>5 or 6). Also reported are asymptotic results for the broadband frequency spectrum of a propulsor’s net random side force due to turbulence ingestion, as well as the torque and diametric ml>ment. The theory provides analytical and physical explanations for the details of the “haystacks” of the force frequency spectrum of a formally three-dimensional propulsor, i.e. for the size and shape of the broad humps that are both observed and rigorously computed over the blade-rate frequency and harmonics. 0 1997 Elsevier Science Ltd.

1. INTRODUCXON (a) Motivation and overview An air- or water-craft with a rotor or fanlike propulsor operating in a turbulent stream experiences a number of resultant broadband “dependent variables” that lead to either or both vibration (structure-borne noise) and sound. Among them are the propulsor’s net random thrust and side force,

which at low frequencies may induce primarily accordion- and beam-type responses, respectively, on the supporting hull or fuselage structure. These drivers clearly alsla become legitimate sources of directly radiated fluid-borne noise over that lower part of the forces’ broadband frequency spectrum where the rotor’s diameter becomes acoustically compact. Otherwise they are still relevant, though indirect, sources of fluid-borne noise via the randomly vibrating support structure in its role of sounding board. Introductory section 1 of the paper is a short review of predicnon studies of rotor/turbulence interaction. That background material appears in Subsections (b-e); its contents are as follows: (b) a brief discussion of turbulence as a random process, and of its statistical description with convective effects for a single or tandem airfoils in straight flight. (c) Same as (b), but limited to isotropic turbulence; the “tandem” airfoils, however, are now part of a three-dimensional propulsor with twisted blades that sample aerodynamically a globally correlated upwash field that becomes the physical cause of “haystacks”. (d) Further comments on (c), with an emphasis on the “standard” isotropic model for the in-line correlation

functionf(q), which serves as the basic building block in a necessarily elaborate tensor description of the flow passing through our three-dimensional propulsor. (e) Sample results of the frequency spectrum of the rotor’s random side force due to turbulence ingestion, based on a high-solidity asymptotic analysis developed elsewhere for the thrust problem, and for the standard form of f(q). This ends the “review” part of the article. Section 2 then reports new theoretical findings for a propulsor’s thrust frequency spectrum for general f(q). The new analysis shows that the standard form off(q), which has a simple exponential behavior in terms of the turbulence integral scale, may be exploited in a superposition scheme analogous to that commonly applied to gust wavenumbers to compute deterministic transient signals due, for example, to blade-vortex interaction. (b) Brief review of the probabilistic nature of the turbulence ingestion problem

Figure 1 is a schematic of a three-dimensional open propulsor turning and advancing at speeds CJand U, respectively. The turbulent field striking it is strictly random and therefore describable only in terms of its spatial and temporal statistics [l]. The sketch shows typical normal-to-blade downwash signals at two arbitrary rotor points 1 and 2. Each of the signatures depicted is one of a practically infinite number of “sample functions” [2], or “realizations” [l], also measured at that point in as many true or thought experiments. Each set of such functions is called the ensemble for that point. It is from comparing those 475

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two ensembles that one determines the joint probability of a given pair of levels for the downwash signals at points 1 and 2, for any two pair of times t, and t2. We temporarily bring the speeds C2and U to zero in order to make that probabilistic description of the aerodynamically relevant flow system more “canonical”, i.e. more in step with generic treatments of random processes that lack such convective effects. The picture will also be made simpler for now by putting points 1 and 2 along an artificial x-axis. The downwash at their positions will act normal to that connecting line, viz. along a “y-axis”. The basic mathematical operation that yields the lowest order metric for how the random downwash at point 2 is related, causally or otherwise, to that at point 1 is the correlation R”+l,

s T/2

R”+$x,

-

x2,

7)

=

lim T-w

x

-T,2

$ Uy(Xh &(X2,

t + 7). (1)

The fact that eqn (1) confines itself to describing stationary turbulence is apparent in its use of only a

single time shift 7 as the difference of our original pair of absolute times tl and t2. Homogeneity [l], which is the spatial version of stationarity, is similarly embodied in the difference x, - x2. Similar differences for the other two coordinates are implied: yl - y2 and zI - ~2. The “1” time integral in eqn (1) exists or converges if the random downwash signals are ergodic, by definition [3, for the energy interpretation made here for the “static” limit of the broadband side force, compare p. 215’s discussion after eqn (6.4.24)]. The restrictions that R”vUy be stationary and ergodic could be used to generate an alternate form of eqn (1)‘s right side in terms of direct probability concepts. That new right side would then be in the form of a double integral with r+(x), t) and 14,.(x2,t + T) as running independent variables weighted by the joint probability distribution B+“~(x, - x2, 7) which, as indicated, contains xi - x2 and 7 as parameters. Such a probability interpretation of eqn (1) is not only “physically” insightful, but is also more general than eqn (1)‘s present right side because, unlike it, it could describe nonstationary processes. Nevertheless, B”~“~ and the corresponding probability distributions for airfoil loading, radiated pressures, etc. will not be invoked again here explicitly because all of our discussion will focus on stationary processes, and will opt for eqn (1)‘s less cumbersome nomenclature.

8, =

Fig. 1. Propfan from the mid 1980s used to define the present nomenclature. The nondimensional tip radius is I. Points 1 and 2 lie along the quarter-chord curves of any two of the rotor’s B blades. Distance “q” separates points 1 and 2. The sketch shows a temporally digitized sample function of random incident upwash at each of these two typical rotor points.

Broadband sources of structure-borne noise When the turbulence is not only spatially homogeneous, but also isotropic, the correlation of normal-to-plane velocities R”++ takes on a special name [4, For g’s definition in terms off, see eqn (3-9); for the definition offin terms of an integral scale, see the unnumbered equation in the middle of p. 211. Note also from Hinze’s p. 6 that his “1” is our z and vice-versa. For the general isotropic tensor at the core of the three-dimensional rotor analyses in Refs. 9-12, see Hinze’s eqn (S-11) on p. 1861 “g(]x, - x2], T)“, which in turn is connected to the correlation of in-line velocities ‘f(]x, - x2], t)” by g(q) = [1 + (q/2) a/ aq]f(q) with q = Ix, - x2(. Traditional models of isotropic turbulence relegatef’s r dependence to that of a subparameter within the “integral scale ,4(r)“. Furthermore, most if not all of the practical studies of turbulence/airfoil “interaction” that the author is aware of, including his own, regard n as a r-invariant constant despite thse arguably unjust demands placed then on ergodicity [5-161 n’s dependence on r is supposedly weak, being related to viscous dissipation rates. Another modeling choice that is often made for convenience, but which cannot be fully reconciled with basic principles, is that of separation of variables to account for anisotropy in a homogeneous flow [17]. All correlations are then made to be proportional to the factor

exp (

1x1 - x21 A,

IYI - Yzl A,

la - ,721 - A >.

Quantities ,4,, /iv and _4, are integral scales in the three Cartesian directions. The obvious objection to the above is that as a generalization from isotropic to anisotropic turbulence it is fundamentally ad hoc, because it does nolt contain the former as a special case--not even when A, = A, = A,. As an extension that would. one could consider instead ~tJ4- yJ2 A;

+

(z, ~

A:

z#

’

For A, = A, = A,, this in fact becomes what we shall here call J(q): the “standard”, or Liepmann ‘x(q)” mathematical model of the general inline correlation function f(q) defined above in terms of g(q). One considers next, as a precursor to reintroducing the two effective freestreams U and n into Fig. l’s complicated three-dlimensional flow picture, a single rectilinear freestream U, that convects the random downwash pattern zr, in the direction from point 2 to 1. For the new correlation R+*J to match a specific value chosen from the “static” iJ, = 0 relief map in xl - x2, r as just discussed, it is evident that the old temporal shift t must be corrected by subtracting from it the deterministic relative retarded time

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(x, - x*)/VT. One could instead leave the meaning of z alone and correct the spatial gap xl - x2, which now effectively becomes xl - x2 - Uxr. This second approach, which we will adopt, changes the left side of eqn (1) to P+$x, - x2 - U,T, t). The y1 - yz, ZI - z2 implied arguments remain the same. The corresponding functional arguments for isotropic turbulence now become lx, - xz - UXri]and the practically constant A. These comments regarding the effects of U, on the probabilistic description of downwash signals become aerodynamically significant when one interprets points 1 and 2 as belonging either to two distinct chordwise stations of a single airfoil or, more generally, to two or more airfoils flying in tandem. If one next considers incident turbulence that is homogeneous far upstream of the finite airfoil group, the effect of the airfoils on the statistically described flow impinging them will be to distort it, and to thereby render it spatially inhomogeneous: R”“J’s XI - xZ - U,r, y, - y2, z, - z2 functional arguments would not apply then. That distortion could be calculated oftline if it were due only to the deflecting effect of each airfoil’s steady loading [18 eqn (2.30), 191. The source of the remainder of the distortion becomes clear if one makes it the only one present by considering, from the beginning, the airfoil or airfoils at zero angle of attack with respect to U,. The distorted path of the turbulence is then clearly due only to the unsteady field of fluid particle velocities generated there by the time-varying airfoil-bound vorticity [20, this excellent piece of work treats the deterministic version of the situation discussed here]. That unsteady loading in turn comes then solely from the need to cancel the downwash induced by the turbulence passing by. The acoustic and “hydrodynamic” parts of that airfoil-induced field of fluid particle velocities becomes indistinguishable over those temporal parts of the turbulence/airfoil interaction when the chords in the blade row become aeroacoustically noncompact. Nearfield sound would then be an “0(l)” contributor to turbulence distortion in the absence of the steady airfoil loads that would otherwise tend to dominate that effect. It is important to keep in mind that a state of noncompactness could come about simply from letting U, approach the local speed of sound, even if the broadband spectrum of the airfoil/turbulence interaction had a relatively lowfrequency content [21, eqn (2.32)]. A few additional remarks regarding statistical concepts: (1) the turbulence could clearly remain stationary throughout the distortion process, at least if its statistical description were kept Eulerian rather than switched to Lagrangian; (2) even if the distortion effect were not taken into account in the calculation of random airloads, i.e. even if the turbulence stayed “frozen” as it engulfed the lifting system, the resulting random acoustic field due to that unsteady random lift would clearly not be homogeneous: the airfoils’

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directivity of broadband noise due to an arbitrary arrangement of localized random loads [22, eqn (3.80)]. (c) Review of the three-dimensional rotor turbulence ingestion problem

One now restores the turning and translating speeds R and U that had been temporarily removed from Fig. 1. The effective Cartesian correlation distance q = Ix, - x1 - U,ri] in the above planar example is now replaced by its three-dimensional counterpart in cylindrical coordinates,

4 =

rf+ri - 2m

cos[Bl(r,)

-

of the strip model in the calculation of airloads: the three-dimensional breakdown of those components should discriminate against in-plane flows that are not perpendicular to the blades’ leading edges near points 1 and 2 (Ref. 11, Fig. 1 of II]). As for the “haystacks”, or the broad humps that develop over the blade rate frequency and harmonics even when the flow is fundamentally random (see Fig. 2, to be discussed in greater detail under Section (e)), they come about both theoretically and experimentally from the fact that there is always a nonzero joint probability of finding some nonzero

P(r,) + 27r(m, - m,)/B - n7] + (U7/Rt)2

The propulsor’s radius R, has been used to nondimensionalize q as well as points 1 and 2’s radial variables r, and r2. Each of the counters m1 and m2 takes on the values 1,2, . . . , B, corresponding to the discrete in-plane angular positions of points 1 and 2 along the quarter-chord curves of the rotor’s B blades. P(r,) and P(r,) are the local in-plane sweep, or skew angles at points 1 and 2. Quantity (U7/R,)* in eqn (2) denotes the square of the component of q normal to the propulsor’s plane of rotation: the rotor in the picture has no rake, so that what would have been (z, - z2 - U7/R,)* more generally has now collapsed to just (U7/R,)* [,‘z” runs along the rotor’s axis in Fig. 11. The rotor sees an ingested flow that traces out a continuum of helical paths of advance ratios J(r) = nU/[Q(rR,)] that differ from root to tip. Here J(r) becomes J,/r, where Jt is the tip advance ratio as usually defined, rrU/RR,, corresponding to r = 1 for the dimensionless radial variable. Figure l’s use of quarter-chord lifting curves from root to tip implies a chordwise compact nautical case, rather than a typically noncompact aeronautical one. The fact that that curve is drawn all the way to the blades’ tips implies the use of strip theory throughout each span, as a fast though approximate way to obtain random sectional blade loads from a knowledge of the local incident random flow [e.g. Ref. [5]; more on strip theory below, under subsection

(2)

of levels of turbulence downwash over the rotor as a whole [6-161. Mathematically the haystacks are embodied in the 2n(m, - ml)/B - Qr factor of q in eqn (2). Moreover, within that factor, R7 may be singled out as the true cause of those combination

broad humps. Theoretical models that have omitted n in this context have effectively decorrelated the airfoil sections around the rotor, and have accordingly missed predicting the spectrum’s haystacks [S, 23,241. The effect of q’s m/R, term is to reduce the stacking feature by contributing an inherently unsteady inflow. Viz., UzjR, contributes a “rain-on-the-rotor” type of term that is totally absent from the more familiar experience of a propulsor cutting through a nonuniform, but steady inflow. That type of inflow, of course, produces a line frequency spectrum of tones for all of the problem’s dependent variables, e.g. for the net propulsor forces and for their acoustic field.

dB

(41. The planar example concocted earlier concluded that the pertinent downwash correlation for homogeneous isotropic flow striking two or more thin airfoils flying in tandem was the canonical function g(q) with dimensional q given by Ix, - x2 - U,ri]. Now, for the unplanar case of a three-dimensional rotor with blades that are twisted from root to tip, the corresponding downwash correlation function must be computed from the full-fledged tensor for isotropic turbulence [4, !9-161. The components of that tensor are preferably expressed, for consistency’s sake, in the rotor’s global cylindrical coordinates. Also for consistency, it is best to invoke only those component distances within that tensor that acknowledge the use

i

;

;

;

u/an Fig. 2. Side force frequency spectra (10 log&G(o)]) for differing values of the turbulence integral scale A. Corresponding spectra for thrust would be higher by roughly 10 log,&r*/~), i.e. about 10 dB for an advance ratio J, of unity. These high-l asymptotic results agree quantitatively with similar side force calculations by Jiang er al. [9] using a rigorous numerical approach.

Broadband sources of structure-borne noise It is just as important to state where the haystacks do not come from. They are not an aerodynamic “cascade” effect, i.e. they are not the result of unsteady aerodynamic interference, or multiple scattering, for the airfoil sections at a particular radial station around the rotor. One may determine that fact thought-experimentally by shutting off q’s R7 factor while computing the rotor’s unsteady loads, due now only to IJz/R,, using the deterministically improved unsteady lifting kernel that would come from expanding the matrix system to include the mutual influences of all of the airfoil sections around the rotor. That expanded meaning of the basic influence function would still not bring Ll into the picture, except as a static parameter in the definition of the total local freestream at radial station r: [u2 + (C&R,)*]‘/*.The otherwise nonappearance of R would make the frequency spectrum of all of the problem’s dependent variables monotonic, or stackless. Needless to say that this does not mean that the haystacks calculated for a globally correlated flow should not differ in level for cascase and noncascade lifting models. Obviously they should and would. (d) Review of a few spedjic rotor theories This is brief supplement to Ref. 1l’s Introductions, which hopefully already do a fair job of summarizing the history of the theoretical treatment of these rotor/turbulence problems. The reader should also be aware of important contributions by Refs [27-321. It is also important to point out that turbulence ingestion tends to be a significant source of rotor broadband noise only at relatively low frequencies up to the second or third overtone of blade rate. Flow separation noise of the “self’ kind normally takes over after that. The reader is referred to Blake [6], and to Gershfeld [33] for the latest on measurements of those higher-frequency mechanisms. One of the first studies with a direct bearing on Refs [l&12] and on the new work reported below was Sevik’s [5], which was unusual in that it kept its analysis in the spatial, rather than the wavenumber, domain. His ground-breaking treatment of a propulsor ingesting turbulence did not account for global correlations and thus understandably could not predict his accompanying measurements of haystacks. Blake [6], working with the same data, attempted to explain its humps theoretically by connecting a series of local Cartesian airload results around a quasi three-dimensional rotor. That rather clever construction of a globally correlated inflow appears, however, to overestimate the size of the haystacks beginning with the first overtone at w/Bf2 = 2. The cau,se of the overshoot is probably the difficulty which Ellake’s approach might have with combining two sets of intermediate results derived from two different coordinate systems (Cartesian and cylindrical). It is possible that that melding misses some of the huge amount of self cancellation hidden

419

in the “true” solution computed through a consistent use of the rotor’s “natural” system of cylindrical coordinates. That the problem of three-dimensional global correlations is mostly a self canceling one becomes analytically evident then [lo]. Jiang et al. [9] and Novak [ 161 arrived at similar cylindrical-coordinate formulations independently for the “exact”, or nonasymptotic version of the problem. References [ 11, 121 later showed that the high degree of cancellation is one of several interesting results brought out by a turning-point asymptotic analysis based only on the requirement that the rotor’s number of blades B be “large”. That analysis operated on the tensor of isotropic turbulence velocities as seen by the three-dimensional propulsor’s radially divergent, twisted blades. Another finding from that expansion was the degree to which the random loading is effectively pushed to the blade tips when B is high. The physical explanation of that effective concentration is as follows: the probabilistic, blade-to-blade relative unsteadiness along that outermost rotor ring is, compared to those of inboard sections, the last to disappear as the rising number of blades B takes the rotor to its effectively steady limit of an “actuator disk” [ 121.The inboard sections may be well globally correlated, but they are also essentially statistically steady due to the crowding of the airfoil sections there. The thrust response comes largely from the outer blade sections for broadband frequencies near and past blade-passage. This interpretation paradoxically attributes the haystacks to a required degree of poor blade-to-blade correlation which, however, had better not be identically zero. References [l 1, 121did not steer their high-B asymptotic analysis to the blade tips (if for no other reason, because it relied on strip aerodynamics-more on this below), but instead deduced as a by-product that the near-tip sections in fact are the overwhelming contributors to a propulsor’s aerodynamic response to turbulence ingestion, for essentially any nonzero frequency and for B as low as 6. Further “macro” evidence of this effective kinematic concentration of turbulence at the blade tips came from the theory’s computed value of the nondimensional moment arm r.(m) defined as the square root of the ratio of frequency spectra of the diametric moment and thrust. A typical value of r.(m) for B = 6 turns out to be 0.85, unity being the tips themselves. Finally, this geometric interpretation of the effective concentration of an incident flow, at the tips of a propulsor that samples it with many blades, should clearly apply as well to the more familiar tonal case, where the inflow is nonuniform but steady in its own reference frame [34, 351. A few words are probably in order by now regarding the term “strip theory” which, again, is the deterministic “transfer function” device whereby

480

R. Martinez

correlated gust upwash signals over a three-dimensional system of airfoils are commonly turned into sectional lifts. Strip theory represents at least a twofold approximation: the first is to make a wing, or blade, effectively infinite on both sides of the control point of interest-not unlike the locally applied “batlIes” used to surround the Fresnel-zone points of a Kirchhoff scatterer in geometric acoustics [43]. The second approximation traditionally has been to compute the load response assuming a spatial spectrum for the incident disturbance that contains only wavenumbers parallel to the wing’s leading edge [22]. The strip approximation is not frequency-limited in the chordwise sense: one may develop the local two-dimensional solution to the deterministic gust ingestion problem for an arbitrary state of chordwise noncompactness (numerically, if need be, cf. for example, Dunn and Farassat [36] and Adamczyk [37, the airfoil problem is attacked exactly regardless of noncompactness, on recognizing that a flat plate is a separable ellipse with a vanishing minor axis]). The acoustically compact version of the strip hypothesis becomes the frequently used Sears function [5-161, valid exactly for an incompressible medium. Strip theory in the highly noncompact regime normally relies on an analytical decoupling of load signals from the leading and trailing edges of the wing section to which it is applied [14, 21, 25, 381. The strip model, however, is wavenumber or frequency-limited in the spanwise sense. For an incompressible medium, the strip hypothesis breaks down within the hydrodynamic zone of action of the wing’s, or blade’s, tip. For a compressible one, strip aerodynamics becomes invalid roughly within an acoustic wavelength wing’s side edge or tip. Strip theory for incompressible and compressible media assumes tacitly that the spanwise wavenumber content of the generally three-dimensional incident downwash field is confined to the low end. As a compressible-regime deterministic example for which strip aerodynamics should not be expected to perform well, due solely this last restriction, one may consider an unswept wing of infinite span and constant chord chopping a solenoidal disturbance that is localized in the spanwise direction, e.g. a line vortex crossing the flight plane, but whose axis remains perpendicular to the latter. The strip hypothesis would impress the hydrodynamic spanswise behavior of the vortex onto the spanwise loading, whose exact solution, however, would clearly contain the acoustic wavenumber. That missing wavenumber would, in fact, dominate the load signal at points along the span that are far from that where the wing and vortex come together at the instant of intersection [15]. In summary, one must admit that the outboard concentration of correlated flows and sectional lifts uncovered in Refs [ 10-121 tends to subvert those analyses’ use of strip aerodynamics to calculate a rotor’s net random thrust: again, because strip theory

becomes inapplicable near the blade tips, where the true three-dimensional unsteady blade loading must “relieve” hydrodynamically down to zero for the nondeterministic incompressible cases treated there. The new asymptotic model has nonetheless performed well, both with respect to rigorous numerical calculations of haystacks that have also relied on strip analyses, and against modem experimental data (which unfortunately was not publicly available at the time of this writing; the reader is referred to the authors of Ref. [9] for qualitative information regarding the match between Refs [9-161’s theories and measurements). (e) Example calculations of side-force broadband frequency spectra obtained from adapting Ref. [12]‘s asymptotic theory for thrust References [l 1, 121 based their B>> 1 asymptotic analyses for a propulsor’s frequency spectrum of thrust for ingested isotropic turbulence of the “Liepmann type”. All of the tensor manipulations of the turbulent flow sampled by the quarter-chord curves of the propulsor’s three-dimensional twisted blades were carried out there for the standard form J(q) = exp( -qR,/A) of the basic inline correlation f(q). References [ 11, 121reported their final results for a dimensionless frequency spectrum of thrust obtained from normalizing its original un-scaled formula by the constant p’U-‘R:(u/U)*, which has units of force squared per frequency, as it should (u/U is the ratio of the turbulence’s r.m.s. fluctuation velocity and the vehicle’s forward speed). The 7/w Fourier transform pairs used to derive Refs [ 11, 121’s results for thrust used one-sided cosines instead of two-sided exponentials. We shall do the same here in presenting the final formula for the side force, but in the plots shown below shall revert to the more common definition of the spectrum via exponentials: we shall introduce a factor of l/2 and accordingly lower the correlated-force predictions obtained from applying eqn (3a), given next, by 3 dB. That final asymptotic expression corresponding to the frequency spectrum of the propulsor’s resultant side force P(w) is [12]

+ZB(vB-1)

(

&-v

>I

drr’G(r) KvB - 0’ + (v)‘l’ ’

@a)

Broadband sources of structure-borne noise

481

the total behavior for S2(w-+a) is (w/BQ)-‘: Fig. 2’s curves decay by 12 dB per frequency doubling at their high end. G(r)‘s chordwise noncompact version (3b) would instead have supplied al/w2 factor [25], for an overall high-frequency decay then of (w/BQ)-~ for the force’s frequency spectrum. As in Refs [ 11, 121for thrust, the power of 2 on S*(w) Reference [ 121 describes in detail how similar is purely symbolic; it serves as a reminder of the high-B solutions may be obtained for the net random correlated character of the dependent variable in torque and diametric moment. We shall limit question. r is short for on/U, and G(r) contains the ourselves here to listing the changes necessary to product of the square of the magnitude of the write down those expressions based on Ref. 11’s appropriate aerodlynamic transfer function and formula for the frequency spectrum of net propulsor advance-ratio parameters due to blade twist [Ref. 11, thrust P(w). Those dimensionless versions of the eqn (15)]. Equation (3a) allows its radial integrations torque and moment will imply the spectrum’s overall to be cut off at an effective tip radius &rrlR, to make normalizing constant to be now p2u)R:(u/U)2, which up for an otherwise lack of spanwise tip relief. Rh/Rt contains a factor of Rt beyond the similar constant is the ratio of hub to tip radii. used to nondimensionalize the thrust and side-force Figure 2 shows sample evaluations of eqn (3a) to spectra. bring out the effect of the turbulence’s normalized To generate the diametric moment MT(w) solution: integral scale A/R,. The blade chords are taken to the simply introduce an additional r2 factor in the r chordwise compact. G(r) will therefore invoke the integrands of Ref. [ 111’seqns (17a,b). For the torque Sears function, which will depend on the local value I12(w): just divide Ref. [111’s eqns (17a,b) by the of the reduced frequency ob(r)/[U2 + (fiR,r)2]“2. The constant n2/5;?. RI-normalized spanwise halfchord distribution b(r)/ R, will be constant and equal to 0.15. R,,/R, = l/5 and 2. ASYMPTOTIC SOLUTION FOR THE HAYSTACKED &a/R, = 1 (no tip relief). The number of blades B is BROADBAND THRUST FOR NONSTANDARD FORMS 6. The rotor’s tip advance ratio J, is unity, and that OF THE IN-LINE CORRELATION FUNCTION f(q) value together with the formula J(r) = KU/ (a) Preliminaries @R,r) = J,/r fixes the blades’ twist distribution from root to tip assuming negligible angles of attack. Our fs(q) = exp( - qR,/A) “standard” form of f(q) is only one of many semi-empirical interpretations of Figure 2’s abscissa displays frequency w normalized by the blade rate BR. The corresponding tonal the solution of the equation that rules the dynamics of decay of isotropic turbulence [4, 391. A different spectrum for a steady, but spatially nonuniform wake striking the propulsor would have been confined to viewpoint would come from the inversion of the the lines w/Bn = 1, 2,. . Here, for the random “Von K&rm$n wavenumber spectrum” [4,26]. Each version of the problem, w/BR = 1 instead becomes such form off(q) supposedly does a better job than the rough center far the broadband spectrum’s first the rest in fitting the measured data over a different and main haystack subrange of the connecting distance q. Lesser humps follow at multiples of the bladeOur purpose here in generalizing Refs [ 11, 121’s analytical expressions for the haystacked thrust passage frequency. Smaller “eddies” A/R, generate spectrum will be twofold: (1) to remove from the higher levels for the higher frequencies of the theory whatever artifacts have entered it through its spectrum, as expected, and lower levels for the lowest tentative choice of_&(q) (e.g. the spurious effects of frequencies that contain the spectrum’s “banded energy” near the static limit [3]. Figure 2 indicates fs(q)‘s cusp at q = 0); and (2) to free the analysis those w/BR = 0 values for its A/R, = 0.1,0.5 curves. altogether of the concept of an integral scale A/R,, which will cease to be part of the new final expression. Respectively, they are - 28 dB and + 1 dB, which are That new solution will be “philosophically” more determined by the fi.rst term on eqn (3a)‘s right-hand powerful than its predecessor because its list of side. Since r = wA,iU = 0 at w = 0, the dependence of that first term with respect to the integral scale arbitrary parameters will have been shortened by one. This process of generalizingf(q) will turn out to be ratio is (A/RJ’, a feature now brought out by the new asymptotic theory. That first term on the right side of one of analytic continuation in the complex A/R, eqn (3a) may be interpreted as the “v = 0” part of the plane. All of Refs. [l 1, 121’s asymptotic work will come to be regarded as no more than an intermediate solution. Equation (3a) also yields the spectrum’s calculation whose actual goal was to expose the high-frequency asymptote, though with slightly more complex-singular structure of the rotor thrust as a work: one ignores eqn (3a)‘s inner two-term 1 sum, function of A/R,. References [ 11, 121’s choice of and replaces the v sum with an integral which in turn fS(q) = exp( - qR,/A) is ultimately what made that may be. evaluated in closed form. That intermediate exposure possible, and yet fs(q) will in the end come result varies as (w,‘BR)-‘. The chordwise-compact version of G(r) relevant to the present case across as just a bridging device. Its effective replacement will be the fuzzier general requirement contributes a l/w factor at high frequencies, so that with

Y=&$~(~)2+1($).

482

R. Martinez

that arbitrary forms off be a reasonably monotonic function of q. The new analysis will be simpler than it might have been in one respect: it will focus on the v # 0, or sum part of Ref. [111’s thrust expression (Ref. [111’s eqn (17a), whose v sum is similar to that given above for the side force upon setting eqn (3a)‘s 1 inner counter to zero). The neglected “v = 0” part would again be important only for the lowest of frequencies and thus does not contribute to the haystacks, which are solidly contained within the terms to be treated below. Yet another reason for omitting the v = 0 part of our starting solution is that physically it contributes minimally to the propulsor’s directly radiated broadband noise. Sound becomes proportional to the time derivative of unsteady thrust over the extremely low frequencies where the v = 0 neglected term contributes largely to Ref. [111’s eqn (17a). In the present context of the frequency spectrum of correlated random thrust, the corresponding acoustic field would be proportional to the product of Ref. [ 111’seqn (17a)‘s and a new factor of wr-which under most conditions would neutralize eqn (17a)‘s v = 0 term.

One then notes that the new denominator integrand in eqn (5) may be rewritten as

1

of the r

1

iR,/A + k,(r) - iRJA - k,(r)

’

(7)

and so it follows that eqn (5) now becomes

27r’ R, p n 0

IVI2 1: P(w) = J:

,,z., [V’ +2v(& - V)]

(b) Extension of P(w) to generaZf(q) The Iv] 2 1 part of Ref. [Ill’s high-B solution for p(o), based on fs(q) as the standard building-block function. is

1 ’ k,(r)

dr l(ve;:$$q’

32n3 R,

=J:

0

7

1 iR,/A - k,(r)

’

(8)

(4)

Putting eqn (3b) for y into the denominator of the above’s r integrand generates the following alternate form of eqn (4):

Id 3 1: F(o)

1 iR,jA + k,(r)

m I?*,=;,

The next step is to digress temporarily by defining the spatial Fourier transform pair for the general “anisotropic” form of f(q): &,(q, OS,&), whose new arguments are the usual spherical-coordinates angles. Their “s” subscript denotes spatial domain. Their spectral counterparts on F&k, Ok,&) contain a “k” subscript. The transform pair is

=[v2+2v(+)]

(5) . l’.

f ,(l$.$

where k,(r) is given by

(6)

x

db J0

sin4,~fsdq, OS,4.1 e*‘l.

(9b)

Broadband sources of structure-borne noise For the isotropic special case,f..i, becomesf(q) and the above collapse to

f(q) = 27r

continues to hold on Ct. With eqn (11a) and (12), eqn ( 1Oa) becomes

f(4) = i

’ d& sin 4;

483

s

mdkk*F(k) 5 c,,,(-i)” [ * :ym] m-0 0

s

. F(k) e-Y~as+~, (lOa)

dr e@‘t-keinhc,(13)

C,

1 Ok) = 2%

a (I dqq sin W(q).

(lob)

s

Equation (lOa) is the result of the usual rotation of the running vector k’, whose new angle & is now measured away from the direction of the spatial vector q. Equation (lob) has carried out the corresponding manipulation in the spatial domain with respect to an intermediate meridional angle #J: that has been integrated out analytically from the right side of the expression. An analogous integration over q& is clearly immediately possible in eqn (1Oa) because F depends on k only. We deliberately avoid that operation and instead bring in the following well-known identity [40, p. 159, eqn (1), which holds exactly. Note that Sommerfeld’s Z, functions become .Z. in the modern nomenclature]

e-WC+

=

f

t,(

cosm&J,(kq),

_i)m

m=ll

where [1 + (- l)“]/[l - m*] is to be replaced by zero form= 1. Equation (13) could be regarded as one of a number of initially acceptable spectral representations of the general form of f(q). That for our purposes eqn (13) turns out to be the most preferred is evident from the fact that it is in the form of a superposition of exp( - qRJA) which results with the nondimensional wavenumber R,/A being replaced by 6)

iIm{

t

l!-

(1 la)

@I

where

1 form=0 ‘m’= 2form> 1’

1

(*lb)

Next one takes the average of the contour representations for the Hankel functions ZQ, II in eqn (9.1.25) of Ref. [41] to arrive at

Mkq) =

-&

s dc

CC

CC - k@ahC

(12) Fig. 3. (a) Analytic continuation of R,/A to k sinh r with r

where C, is as indicated in Fig. 3a here. Of special note is that Cc is such that Ret is always slightly negative, and that on the two semi-infinite legs of C, one that -sinh e = has - sinh( f ire + Ret) =: - sinh( - Ret), where this last argument of the hyperbolic sine is positive. The reliance of Ref. [I 11’s analysis on the exponential decay of the standard function exp(-qR,/A) thus

complex. The r plane describes the trajectory of the root r = r+ that falls inside C( for I@), and of the root r- for Z,( - /J) that similarly lies within Cc. The beginning point of each root path corresponds to B = 1 and the last to z!I= co. The range 1 < 1 < co determines the limits of integration in the final k integral in eqn (19). (b) Implied analytic continuation of the p(r) parameter beyond the positive real line. The indicated cuts make eqn (A3)‘s roots unambiguous. The figure shows the required approaches of the 1 < /I < co real line segment for Z&3) (right side of the picture), and for Z<(-/3) (left side). Zc(--8) demands the circumvention of the cut at ZV= - 1.

R. Martinez

484

the new nondimensional running wavenumber k sinh 5. It follows that if one analytically continues eqn (8) to Rt/A values along the contour C,, according to the transformation R,/A-+k sinh 5, one immediately arrives at the needed generalization to arbitrary forms of f(q) for Ref. [Ill’s asymptotic theory of broadband rotor thrust. So, upon defining for convenience a new variable j?(r) as

B(r) = k/k(r),

(a)

(14) (b)

one finds the new form of the Iv1> 1 part of P(w)

p1(o)= Jf =

[v~+2v(+l)]

f

y=_-co

Oc dkk2F(k) f

E,(m=O

5II

.

dl sinh 5 em{ s cc

Fig. 4. (a) Theoretical models of the inhne isotropic-flow correlation function f(q): (1) fs(q) = exp( - qRt/A) is the “standard”, or Liepmann form used throughout Refs [lfk12]; and (2) the remaining curves are for the proposed Gaussian model fG(q) that introduces “n” as an additional free parameter. The ordinate of the plot is linear. (b) 20 log,, of the ratio in eqn (29, in a sense the spectral version of Fig. 4a with the standard result acting again as a normalizing function. The segment of the abscissa that is of significance to a given rotor case comes from eqn (26) with the radial variable r spanning the normalized blade length

1)”[l:‘;?“]

1 isinh<+fi-’

&m/R, - Rb/Rt.

-

1 i sinh 5 - 8-l

’

(1% A)

from which one finds that (Appendix

The extra sinh integrand in eqn plicative constant additional sinh 5 identity

< term in the numerator of the 5 (15) comes from the R,/A multioutside the ,4 sum in eqn (8). This factor is removed by the algebraic 1

+isinhl-j?-’ sinh 5

sinh 5

i sinh 5 + B-’ - i sinh < - /?-’

1 = ifi-’ isinhc+B-’

ik&) =- k

= - 4n: Im{log[l

- /P(r)]}.

(17)

The p(r) has been cut as shown in Fig. 3b. The shape of those cuts, and the fact that k and k”(r) in

1 +isinhr-/?-I

their definition of /3(r) are both positive and real, leads to

Im{log[l - P(r)l) = WwAl - B(rN

1 i sinh 5 + B-’

(16)

OforO
(lga) (lgb)

Broadband sources of structure-borne noise

485

-20

n z

-25

I

I

1

0.5

I

I

I

1

I

I

I

I

1

2.5

2

1.5 O/B0

Fig. 5. 10 loglo[Wo)]* (1~13 1 part) for the standard and Gaussian isotropic-flow models off(q).

The right sides of eqns (18a,b) and the definition of p(r) in eqn (14) cause the k integral in eqn (15) to have k,(r) as its lower limit rather than zero. Substitution of results of eqns (16) and (18a,b) into eqn (15) gives

fs(q) = exp( - q&/A), and which via eqn (lob) the wavenumber spectrum

(214 It follows Fl(k,(r)) -= kv(r)

. r dkkF(k), *k,(r)

where the [k;‘(r)d/dk,(r)]* applied to yield

/VI2 1: z-$0)=-~

w

operation may finally be

“Z, [v2+2v(zi

- v)]

The prime on F’(k,(r)) denotes differentiation with respect to k,(r). The benchmark check for eqn (20) is eqn (5) which, again, corresponds to

has

that 4 -2

0

1 Rt 1 [k:(r) + (R,/.4)2]3’

(21b)

which when substituted into eqn (20) reproduces eqn (5). The new solution in eqn (20) is structurally as simple as eqn (5), the less general theory from which it came: the extension brings no additional computational or analytical penalties. A/R, is obviously no longer a parameter, since it has been superposed away. The asymptotic theory continues to be physically revealing, though in a different way from Ref. [ 11J’s standard-fresults: the new eqn (20) isolates the kv(r) “wavenumber” defined in eqn (6) as the only one of significance in the turbulence sampling process for a high-b rotor. And eqn (20) suggests that the rotor’s random thrust depends only on the first derivative of the spectrum of the in-line correlation f(q) evaluated at that wavenumber. Lastly, this generalizing analysis applies just as it is to the torque and diametric moment solutions discussed above for fs(q); and with only minor obvious modifications to the side force in eqn (3a).

R. Martinez

486

The mathematical operations leading to eqn (20) justify themselves a posteriori: the F(k, (r)) integrand must exist for all r between the hub (r = Rh/RI) and the actual (r = 1) or effective tip (r = R.a/R,). The new theory is now also available to accept directly as input empirical forms off(q) through the derivative of their F(k) transforms. We stated earlier that one obvious application of the new theory would be to compare the effect on thrust of “slight” differences in existing or possible future models of f(q). The illustrative calculations that follow consider instead gross differences amongst two postulated forms off(q). The first will model, viz the standard Liepmann be fs(q) = exp( - qR,/A). The second will be Gaussian in shape, with spatial and spectral forms given, respectively, by

hb)=exp(-$(:)J Fo(k)=g, (22aN where in eqn (22b) the compound short for

parameter

A is

The latter is usually small, but nonzero, so long as the blade chords around the propulsor do not lie on the plane of rotation, i.e. so long as they have some pitch into the freestream U). The rotor parameters in Fig. 5 are 6(r)/R, = 0.18 (halfchord “b” is again taken as constant from root to tip), %/R, = 1, Rh/Rt = l/5, B = 6, A/R, = l/10, and a blade twist distribution J(r) = J,/r from root to tip again determined by a value of unity for the advance ratio Jr. The R,/A-normalized version of k,(r) is

where the compound parameter BnA/U is formed from the product of more primitive quantities, namely nB A/Rt. J;‘. Its value for the specific case of interest is 1.88. At the haystack frequencies w/B0 = v, the wavenumber ratio in eqn (26) ranges from about 0.6~ at r = RR/R1 = 1 to 3v at r = Rh/R, = 0.2, e.g. putting v = 1, it follows that at the blade-passage frequency under the main, or first haystack 0.6 < k’ - ‘(r) < 3. &IA

Equations (22a,b) and (23) have introduced a new parameter “n” to control the width of the proposed Gaussian. That they have done so, and moreover that they continue to rely on A/R,, is clearly for purposes of presentation only: eqn (20) invokes neither “n” nor A/R,. The new Gaussian fG(q) intersects, by construction, the standard formfs(q) at the value q* of the correlation distance q given by q* = d/R,.

(24)

The expected differences in thrust frequency spectra will obviously be proportional to the ratio F;;(k)

K(k)=

Jc64ns’*(1 +

k2A2/Rf)exp

{ - k*(;Rd’}.

(25) Figure 4a,b plots h(q), fo(q) and eqn (25) for Gaussian cases with n = 1,2,3. Figure 4a’s abscissa q has been renormalized by the integral scale A/R,, and Fig. 4b’s correspondingly becomes k divided by the flow’s “characteristic” wavenumber R,/A. Figure 5 shows associated predictions for the part of the total thrust frequency spectrum, which derives from the correlation of turbulent inflow components normal to the rotor disk (Ref. [12], from which this figure comes, chose to display its results in this piecemeal fashion in order to draw more detailed interpretations. Another figure there shows the contribution to the thrust of the cross-correlation of in-plane and normal-to-disk flow components.

This becomes the relevant range of k”(r)/(R,/A) to be “pulled out” of Fig. 4b’s abscissa. The plotted ratio of spectra from eqn (25) is much greater than unity over most of that segment for n = 3. The main haystack of the thrust frequency spectrum in Fig. 5’s n = 3 curve accordingly lies well above that of the standard curve. For n = 2 in Fig. 4b, the same 0.6 c k/(Rt/A) < 3 segment encompasses values of eqn (25) that are sometimes greater than unity and sometimes less. The thrust frequency spectrum for the n = 2 Gaussian fG(q) in Fig. 5 correspondingly fluctuates relative to the standard case for frequencies near o/BS2 = 1. The two thrust curves intersect at two points, roughly at o/BR = 0.6 and at w/BR = 1.25. For n = 1, finally, Fig. 4b shows that over 0.6 < k/(R,/A) < 3 eqn (25)‘s ratio of wavenumber spectra is a small fraction and so, Fig. 5’s n = 1 curve lies far below that of the standard case for wIBR z 1. 3. CONCLUSIONS

The frequency spectra of the net random thrust, side force, etc. acting at the hub of a propulsor cutting through homogeneous isotropic turbulence depend on the derivative of the wavenumber spectrum of the inline correlation function f(q). The argument of that differentiated spectrum is the single special wavenumber from eqn (6). The new asymptotic solutions for those force spectra are now available as sources of structureborne noise [43], and under conditions of global

Broadband sources of structure-borne acoustic radiated,

compactness, single-dipole

also as sources of directly broadband

noise.

Acknowledgements-lduch

of the new work reported here was performed for the: Office of Naval Research, with James Fein and Dr Peter Majumdar as project managers. The author is again indebted to Dr C.-W. Jiang of NSWC/ Carderock for a number of stimulating conversations on the subject. REFERENCES Batchelor, The Theory of Homogeneous Turbulence, Chap. 1. Cambridge University Press, Cambridge (19561. S. Crand&l,~Random Vibration, Vol. I, 1st Edn, Chap. 2. MIT Press. Cambridae. MA (1958). H. Tennekes’ and J. < ‘Lumley, A’ First Course in Turbulence, Chap. 6. MIT Press, Cambridge, MA (1972). J. 0. Hinze, Turbulence. McGraw-Hill, New York (1959). M. Sevik, Sound radiation from a subsonic rotor subjected to turbulence. NASA SP 304, pp. 493-511 (1971). W. K. Blake, Aerohydroacoustics for Ships. DTRC Monograph Series, pp. 1181-1192 (1984) [also published as Mechanics of Flow-Induced Sound and Vibration, Academic Press, New York (1986)]. G. F. Homicz and A. R. George, Broadband and discrete frequency radiation from subsonic rotors. J.

1. G. 2. 3. 4. 5. 6.

7.

K.

Sound Vibr. 36, 151-177 (1974).

8. 9.

10. 11.

12.

13. 14. 15.

16. 17. 18.

R. Mani, Noise due to interaction of inlet turbulence with isolated stators and rotors. J. Sound Vibr. 17, 251-260 (1971). C.-W. Jiang, M. (Chang and Y.-N. Liu, The effect of turbulence ingestion on propeller broadband forces. David Taylor Research Center Report SHD-1355 (1991). R. Martinez and E;. Weissman, Spatial-domain analysis of the thrust on a propeller cutting through isotropic turbulence. CAA Rept. U-1894-358.47 (1990). R. Martinez, Asymptotic theory of broadband rotor thrust, Part I: manipulations of flow probabilities for a high number of blades; Part II: analysis of the right frequency shift of the maximum response. J. appl. Mech. 63, 136148 (1996). R. Martinez, Closed-form solutions for the random forces on a three-dimensional propulsor in a turbulent stream. CAA Reports. U-2124-386 and U-2224-386, prepared for the Office of Naval Research (1993/1994). R. Martinez, Aeroacoustic diffraction and dissipation by a short propeller cowl in subsonic flight. NASA CR 190801 (1993). R. Martinez and Q. E. Widnall, Unified aerodynamic acoustic theory for a thin rectangular wing encountering a gust. AIAA J. lB, 636-645 (1980). R.-Martinez, Lifting-surface theory for propfan vortices striking a downstream wing. J. Aircraft 26. 629-633 (1989): specifically. compare the shapes” of Figs. 3 and 4. S. Novak, Thrust spectrum of a rotating propeller in a turbulent wake. ATT and Bell Laboratory Report 46211910701-OlTM (1991). C. S. Ventres, M. A. Theobald and W. D. Mark, Turbofan Noise G(rneration; Vol. 1, Analysis. NASA CR-167952 (1982). M. E. Goldstein, Unsteady vertical and entropic distortions of potential flows around obstacles. J. Fluid Mech. 89, 433-468 (1978).

19. H. M. Atassi, The Sears problem for a lifting airfoil revisited-new results. J. Fluid Mech. 141, 109-122 (1984). 20. I. Lindblad, The effect on acoustic radiation of mutual interaction between a line vortex and an airfoil. MIT Master Thesis. MIT, Cambridge, MA (1983).

noise

481

21. M. T. Landahl, Unsteady Transonic Flow, p. 30. Pergamon, Oxford (1961). 22. M. E. Goldstein, Aeroacoustics. McGraw-Hill, New York (1976). 23. S. A. L. Glegg, Broadband noise from ducted propfans. AIAA Paper 93-4402, 15th Aeroacoustics Conf., Long Beach, CA (1993). 24. R. K. Amiet, Noise produced by turbulent flow into a propeller or helicopter rotor. AZAA J. 15, 307-308 (1977). 25. R. K.‘Amiet, High-frequency airfoil theory for subsonic flow. AIAA J. 14. 10761082 (1976). 26. R. K. Amiet, Acoustic radiation from an airfoil in a turbulent stream. J. Sound Vibr. 41, 407420 (1975). 27. S. Breit and A. L. Dickinson, A broadband noise analysis code for propellers and appendages. BBN Report. 7298 (1990). 28. N. A. Brown, Aspects of the noise of propellers operating in turbulent flows. In: Flow Noise Modeling, Measurement, and Control. NCA-Vol. 15/FED-168 (1993). 29. D. B. Hanson, Unified aeroacoustics analysis for high speed turboprop aerodynamics and noise, Vol. IDevelopment of Theory of Blade Loading, Wakes and Noise. NASA CR 4329 (1991).

30. M. S. Howe, The influence of surface rounding on trailing edge noise. J. Sound Vibr. 126, 503-523 (1988). 31. E. J. Kerschen and P. R. Gliebe, Noise caused bv the interaction of a rotor with anisotropic turbulence. AIAA J. 19, 717-723 (1981).

32. D. E. Thompson, Propeller time-dependent forces due to nonuniform flow. ARL TM 76-48 (1976). 33. J. Gershfeld, Ph.D. thesis for Catholic University (in preparation). 34. D. G. Crighton and A. B. Parry, Asymptotic theory of propeller -noise, Part II: supersonic _single-rotaiion propeller. AZAA J. 29, 2031-2037 (1991). 35. E. Envia, Asymptotic theory of supersonic propeller noise. AIAA J. 32. 239-246 (1994). 36. M. H. Dunn and F. Farassat, State of the art of high speed propeller noise prediction-a multidisciplinary approach and comparison with measured data. AIAA Paper 90-3934 (1990). 37. J. J. Adamczyk and R. S. Brand, Scattering of sound by an aerofoil of finite span in a compressible stream. J. Sound Vibr. 25, 139-156 (1972).

38. J. J. Adamczyk, The passage of an infinite swept airfoil through an oblique gust. NASA CR 2395 (1974). 39. G. Treviiio, Isotropic analysis of grid turbulence. fnt. J. Engng Sci. 27, 1463-1471 (1989). 40. A. Sommerfeld, Partial Differential Equations in Physics, p. 159. Academic Press, New York (1949). 41. M. Abramowitz and I. Stegun, Handbook of Mathematical Physics. Dover, New York (1968). 42. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, p. 366. Academic Press, New York (1980). 43. M. C. Junger and D. Feit, Sound, Structures, and Their Interaction. MIT Press, Cambridge, MA (1986). APPENDIX A PROOF OF EQUATION (17) (1) Statement of the problem The alleged identity in eqn (17) is $,c-(-t)-[w]

J‘, d5 e”(

1 isinh<+p-l+isinhr-8-r

1

= - 4n Im{log[l - /I*(r)]}.

(Al)

488

R. Martinez

The fundamental analytic continuation step in the overall analysis has clearly been the passage from eqn (8) to eqn (15), which has introduced a contour Ce in the calculation of propulsor thrust due to the chopping of general isotropic turbulence. But the original contour that defined A(kq) in eqn (12) did not have to be as precise then as it now has to be in eqn (Al). That is because the algebraic part of eqn (Al)% r integrand is clearly more complicated than that of eqn (12). Equation (12)‘s algebraic integrand is just unity, whereas eqn (Al)‘s now contains poles in the complex 5 plane. The well-posed global kinematics of turbulence “sampling” by the rotor will make up for those inherited ambiguities of Cc. The additional physical arguments contributed by this chopping process will be barely restrictive. Our main guiding light will simply be the requirement that the left side of eqn (Al), which is the result of that sampling, yield a real quantity. And since the [I + (- I)“]/(1 - m’) factor in eqn (Al)% sum is nonzero only for even MS, so that the factor (- iy” is already real, it follows that the final result of the 5 integral itself must be real: Ci must somehow make this possible in eqn (Al). The tentative shape for Cc will be as shown in Fig. 3a, with the associated cuts in the /3 plane indicated in Fig. 3b. Figure 3b thus represents a second necessary analytical construction process in the whole theory, this time with respect to a wavenumber variable /3(r) = k/k,(r), rather than with respect to the complex flow parameter R,/A = k sinh 5. The cut plane /l will have the effect of automatically confining the logarithm in the final result on eqn (Al)‘s right side to its principal-value sheet: since Fig. 3b states that /I - 1 = e-‘*(I - B) generally, so that 1 - b = eln(p - 1) for real /l greater than 1. The purely imaginary vertical leg of C, in Fig. 3a is to be regarded as the limiting form of a similar integration segment slightly shifted to the left, along which Ret is “ -c”, an arbitrarily small number [see further comments just before eqn (13) in the main text]. These stipulations are enough to “fix” the 5 and /l planes, and thereby eliminate all of the earlier-cited uncertainties regarding both planes in the context of eqn (Al). The direct analysis of eqn (Al)‘s left side may now proceed along two self-suggesting steps: (1) the calculation of the 5 integral, and (2) the closed-form evaluation of the m sum with its contents so derived. The important thing to note is that both sets of operations may be carried out without in any way invoking the specific form of the function F(k) in eqn (15). (2) Calculation of the 5 integral One begins by choosing the imaginary part of 8, which plays the role of parameter in the-l; integral, as negative. This is convenient because ultimately the region in B of physical interest will be limited to thepositiveieal half line 0 c p < co, which in Fig. 3b may be approached from the lower half plane without running into branch cuts. The development will treat the first of the two integrand factors in eqn (Al), i.e., will define the integral B as

Then one will determine from this generally complex fuzlction the result corresponding to Z&3)for /3 eventually The problem in eqn (A2) with Cc as in Fig. 3a is similar to the familiar one posed by eqns (3-29) or (3-34) of Ref. [42]. One sets z = e( and finds that the complex-valued zeros z = z* of i sinh r + 8-l = 0 occur at Z* =

i~-‘*J~=

i-IJS’B

1

.

(A3)

One of the 5 solutions corresponding to these z values will fall inside and the other outside the contour Cc in Fig. 3a

for any given unrestricted value of 8-r. But the multivaluedness of the right side of eqn (A3) itself with respect to B-l, or /I, causes these two roots to become interchangable with each circuit around the points fi = 1, - 1; so that the individuality of eqn (A3)‘s two possible values for the purpose of further analysis becomes meaningless unless it is established through cuts such as those in Fig. 3b. Those cuts in /.I make the identity m= -iJF* hold over the whole cut ~9plane, and the phase of J--b* - 1 will be automatically limited throughout by the inequality -_IC

argm

<

C O=-Irn&Ki

G 0.

(A4)

It follows also that arg(m) = 0 for @along Fig. 3b’s positive or negative real axes. The c* solutions corresponding to z* are

(A54 The magnitude of the complex quantity within parentheses in eqn (A5a) determines the sign of Ret*, which must be negative if the root in question is to be inside Cc in Fig. 3a. So from

WW it now follows that for the (+) version of eqn (ASb) the quantity Irnm in the numerator diminishes the factor 1 + Im(m), while the (-) solution increases it above 1. The (+) solution in eqn (A5a) therefore falls within the Cc contour and the (-) solution does not. Finally then

=

-

&+

2ni 8

. z (I smh C + 8-‘)I( - C+

_ zrr

emlOsl~~~’ +

JTT)

= coshpog(i/l-’ + Jl

- /3*)]

Now fi may be analytically continued from its temporary position. in the lower half plane to the positive real -axis in Fig. 3b, where it belongs for the nhvsical nroblem. the analysis of the-second r- integral in eqn (Al)‘s left-hand side, i.e. of

s

U-8)

dt em<

= c,isinht-b-”

follows similarly: instead of eqn (A3) the roots equation now is z* = -q-

*

$q

= +.

(AT)

There are two ways to proceed: (1) a straightforward one where the previous analysis is repeated, but is now based on the (-) solution of eqn (A7). since that solution and the

Broadband sources of structure-borne given in eqn (A3) produces the lbination in the numerator of eqn (A5b). (2) A second, more interesting way that is really a shortcut but that requires care: to simply switch b for -/3 in the final result in eqn (A6), which holds for B real and positive, but which was developed assuming at the beginning that the imaginary part of j? was negative. The application of method 2, which comes from adapting eqn (A6), must therefore start with the initial temporary assumption that gave that solution; viz., with Im b < 0. Figure 3b shows that the exchange /?+ -B and the limiting process are immediately applicable to eqn (A6) for B real and positive in the range 0 < /? < 1, because the landing point does not have tas cross a branch cut then. So one gets that

noise

489

In summary, from the final equalities of eqn (A8) and (AlO) it follows that

r,(D) + M-8)

=

1 sCCdr e’“’( i sinh 5 + 8-j 1 + i sinh l - 8-I

= 2Re{F

1

(JGJQY}. (All)

2n(-i)/?(-i)“+’

0 < p < 1: Z,(-/?) =

This result holds for all real /I. It is algebraically singular for B = k 1 though integrable there. Assuming consistently that the imaginary part of B was initially negative guaranteed the existence of all intermediate quantities. That assumption obviously served as a bridging step. The right side of eqn (A9) produces, as expected, a quantity that is second order in /3 for vanishing /?: the B factor within parentheses approaches then z -b/2, so that Zc(fi+0) + 4(-B+O) N #I”+‘Re(-i)“+‘. And since Re( -i)“+ ’ is nonzero beginning with m = 1 rather than m = 0, it follows that &(& + Zc(-b-0) u /P.

$7

(3) Evaluation of equation (A 1)‘s m sum The next step is to recall the canonical formula [42] where * denotes complex conjugate. The same blind use of the switch /?+ -p does not work for 1 < B < co because, again as shown on Fig. 3b, in order to reach the negative real axis one must account for the change in the sign of d/‘s’ across the cut springing from fi =-1. i.e., on-must not only let /?+-B, but also put Jjc_ l__Jpz-z 1. The original 1,(b) solution in ;;in)(A6) for fi real an’d > 1 is (since m = i&??

dx cos nx =-+(+j-): o 1 +acosx J+

(A12) For the present purposes, its left side becomes

dxcosnx s _I 1 +acosx (A9)

Following now the prescription just deduced to obtain :ia;a) from this (/I-+-S.--+-m), gives

Ial< 1.

_(-1)” 2a

’ dxcosnx so a-‘-cosx

_X(-1)n *& cosnx a so 2xa-‘-cosx’

(A13)

from which it follows that the quantity *(+M)

1
2xi(-B)(-i)“+’ x/3=

x

generates the cosine Fourier-series coefficients of u-l - cos x, i.e. that at least for (al < 1 the following holds:

(

-i&yl>-

~O~“cOs.x{~(~~}=y_,_lcosx.

_F

-iy-1 2-l

(

m >

(A14) The G factor is still given by the prescription in eqn (1 lb). Figure 3b has analytically continued the quantity within curly brackets in eqn (A14) beyond the segment Ial < 1, with our physically meaningful variable fi playing the role of the

R. Martinez

490

meaningless generic “a” in eqn (A14). One identifies now eqn (Al) as claiming that

1 isinhr+B-“isinh5-B-’

1

= - 4n Im{log[l - /P(r)]}.

= - 2xi

(Al5)

Then substituting for 1,(/I) from eqn (A6), and applying eqn (Al2), leads to

’ d& sin 4; f

(Al61

- cos 4;.

-Los 4L

-

8-~

+Los 4;

I .

tA18)

t,,,( - i)” cos m&J

m=O

0

- 2ni

B_,

Equation (Al 1) has established that the C, integral in eqn (Al 8) is real, so that if the left side of eqn (Al 3) is to produce a real quantity, as claimed by the right side of that expression, the & integrand over n/2 < 4; < x must become the complex conjugate of the & integrand in 0 < 4; < n/2. The sin 4; factor in eqn (Al5)‘s left side is already the conjugate of itself with respect to the two 4; subdomains (degenerately so, since sin r$Land sin@ - 4;) (= sin r#$) are obviously real). The rest of the 4; integrand appears in eqn (Al8), whose right side now confirms in a simple way the left side’s also becoming the conjugate of itself for n/2 < & < x vs 0 < 4; < x/2. One can therefore express eqn (Al5)‘s left side, with the aid of eqn (Al8) as

s =p

I

s cc

1 ‘d5em’ isinh<+/I-‘+isinht-B-I

1

{

=Rr[ -4ni

r

dcX{8_,s~~s

$; - s,$i~s

=4rc Im

12 d[log(/V-’ - cos 4;)

4L}:

And similarly, from eqn (A8),

2 Em(-i)mCOSM&

m-0

Is’0

dt emi sisinht-B-l

I

+ log(B-1 + cos &)I 1 = - 4n Im{log(l - /P)}.

(A19)

Finally, substituting [1 + (- l)“]/( 1 - m’) for m # 1 (and 0 for m = 1) for the elementary integral x cos[m(n - &)I

=p

And so,

2si 2xi - cos(n - &) = p-1 + cos qfd

(A17)

that appears on the left side of eqn (Al9), proves eqn (Al). Since from Fig. 3b it is evident that (1 - fi) = ei$9 - 1) for /l> 1 and real, it follows that Im{log(l - B’)} = Im{log(l - /I)} = n as noted on eqn (18b) in the main text.