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TECfJICAL MEMORANDUM 1620 ON THE CONTRIBUTION OF TURBULENT BO~NDAR(Y LAYERS TO THIE NOISE INSIDE A FUSELACE*~ by G. M. Corcos"H and HI. W. Liepmann** September, 1956 "Un~edited by the NACA (the Committee takes no responsibility for the correctness of the author 's statements. ) Aerodynamics Research Group Douglas Aircraft Company, Inc . Santa Monica, California California Institute of Technology Pasadena, California (Consultant, Aerodynamics Research Group, Douglas Aircraft Company, Inc . Santa Maonica, California) NACA TM 1420 TABLE OF COjNTENTS ABSTRACT.............................. i INTRODUCTIONI .. .. .... .. .. .. .. .. .. .. .. 1 I. THIE ACOUSTIC COUPLING OF A RANDOMLY VIBRATING PLATE WITH AIR~ AT REST .. ... ... .. .. 4 II. THE DYNAMIC BEHAVIOR OF THIE SKIN ... .. .. .. .. .. 8 a) The Mean Acceleration .. .. .. .. ... 9) b) The Length Scale A .. .. .. .. .. 12 III. THE FORCING FUNCTION .. .. ... .. ... .. .. 14 IV. SPECIAL CASES .. .... .. .. .. ... 2C 1. Convectred Turbulence ... .. ... .. .. .. 2C a) The coupling of the plate with air at rest in the case of convected turbulence .. 20 b) The response of the plate .. .. .. 21 2. The case of zero scale .. ... .. .. .. .. 23 V. SUMMARY OF RESULTS AND DISCUSSION .. .. .. .. .. .. 24 A Note on Testing ... .. .. .. .. .. 25 VI. REFERENICES .. .. .. .. ... .. ... ... .. 27 APPENDIX I: THE RANDOM RADIATION OF A PLANE SURFACE: A. Four Limiting Cases .. .. .. .. .. ... 28 B. The Noise Generated by Skin Ripples of Fixed Velocity .. .. ... .. .. 35 C. The Generation of Noise by Plate Deflection~ of Zero Scale ... .. .. .. 38 APPENDIX II: THE SIMPLIFICATION OF THE PLATE RESPONSE INTEGRAL 40 APPENDIX III: THE EVALUATION OF THIE INTEGRAL SCALE .. 42 Digilized by ille Internel Archive I i~n201wll MII1undIn~ from UniverslIv of Florida. George A. Smather Llbraries wyith support from L'rRASIS and the Sloan Foundalion hilp: wwwv\~.archlve.or g delailS oncontrlb ullonol00ulnlt NACA TM 11+20 ABSTRACT The following report deals in preliminary fashion with the transmission through a fuselage of random noise generated on the fuselage skin by a tur bulent boundary layer. The concept of attenuation is abandoned an instead the problem is formulated as a sequence of two linear couplings: theturbu lent boundary layer fluctuations excite the fuselage skin in lateral vira tions and the skin vibrations induce sound inside the fuselage. The techniques used are those required to determine the response of linear systems to random forcing functions of several variables. A certain degree of idealization has been resorted to. Thus the boundary layer is assumed locally homogeneous, the fuselage skin is assumed flat, unlined and free from axial loads and the? "cabin" air is bounded only by the vibrating plate so that only outgoing waves are considered. Some of the details of the statistical description have been simplified in order to reveal the basic features of the problem. The results, strictly applicable only to the limiting case of thin boundary layers, show that the sound pressure intensity is proportional to the square of the free stream density, the square of cabin air density and inversely proportional to the first power of the damping constant and to the second power of the plate density. The dependence on free streams velocity and boundary layer thickness cannot be given in general without a detailed knowledge of the characteristics of the pressure fluctuations in the boundary layer (in particular the frequency spectrum). For a flat spectrum the noise intensity depends on the fifth power of the velocity and the first power of the boundary layer thickness. This suggests that boundary layer removal is probably not an economical means of decreasing cabin noise. In general, the analysis presented here only reduces the determionat of cabin noise intensity to the measurement of the effect of any one of four variables (free stream velocity, boundary layer thickness, plate thickness or the characteristic velocity of propagation in the plate). The plate generates noise by vibrating in resonance over a wide! range of frequencies and increasing the damping constant is consequently an effective method of decreasing noise generation. One of the main features of the results is that the relevent quatitis upon which noise intensity depends are nondimensional numbers in which boundary layer and plate properties enter as ratios. This is taken as an indication that in testing models of structures for boundary layer noise it is not sufficient to duplicate in the model the structural characteristics of the fuselage. One must match properly the characteristics of the exciting pressure fluctuations to that of the structure. Ii TECHNICAL MFJEORAN~DUM 1420 INTRODUCT~IIO In his efforts to minimizer the noise levels for which, he is responsible, the airplan~e designer has had to pay increasing attention to a source of noise which until recently had been ignored. This is the boundary laye~r. The boundary layer will generate noise whenever it is the seat of any fluctuating phenomenon. In particular it will nurture random pressure fluctuations whenever it is turbulent. The designer's interest will, naturally center on the characteristics of that part of the boundary layer noise which has been transmitted through the fuselage skin and into the cabin rather than on that part which is radiated into the freestream. This is so because the radiation intensity is, as we shall see, a rapidly increasing function of the velocity of Uthe boundary with respect to the air and to a likely observer outside a fuelage either the relative velocity of the plane is low (as near a takeoff or landing) or the plane is considerably distant. As a consequence the practical question which has to be~ raised concerns the effects on a fuselage skcin, and on the air which it encloses, of the boundary layer pressure fluctuations acting directly on the skin.* Two features of this problem are worth noting: To begin with, the fuselage will transmit noise only by deflecting laterally. The thickne~ss of the skin is very small when compared to the wave length in metal of audible sound waves so that, effectively there are no fluctuating pressure Gradients within the skin and hence the latter will not oscillate in lateral compression. In the second place, the turbulent pressure fluctuations in the bounary layer are random both in space an~d in time. The fluctuaions are generated locally. If they are measured simultaneously at two different points of the boundary layer, say on the skin itself, they are found to hav nro relationship with each other unless the two points are separated by a very short distance. Alike the value of the pressure fluctuation, at a given point soon loses correlation with itself.+ "Noise intensity is defined in this report as pi2 foai. It is assumed th this is the physical quantity of interest. It ha's the dimensions of an energy flux; but it is not necessarily equal to the energy flux at some point in the field, nor is it necessarily equal to the density of energy radiated away (lost) by the fuselage. "Recently two authors (Refs. 5 & 6) have suggested that the ranomes in time is not independent of the randomness in space; i~e., that the pressure fluctuations at the wall are created by the convection at a single speed of a "frozen" pattern of pressure disturbances. Some attention is paid later to this eventuality which is treated as a special case. 2 NACA TM 1420 Wle are thus led to visualize the process of transmission of boundary layer noise through the metal stein of' the fuselage as follows: A multitude of external pressure pulses push the elastic skein in and out and the skin, in turn, not unlikte a set of distributed pistons, creates inside the fuse la~se pressure waves which propagate and superimpose. This constitutes cabin noise. It is of course desirable to determine the characteristics of this nroi se. One should point out that the data for` the problem are not complete and are not likely to be so in the near future. Specifically the structure of the turbulent mechanism within the boundary layer and, in particular of the coupling between pressure fluctuations, velocity fluctuations and temperature fluctuations is not enough explained or measured to define wholly our forcing function. As a conseqqg ,ce it is not now possible to define say av:erage cabin noise intensity pi' as a function of say, free stream Mlach number, Rey~nolds number and plate characteristics. It is however possible and it is the purpose of this report to indicate the approximate functional dependence of p; on these quantities anrd thus to give similarity rules which will reduce to a minimum the amount of testing required. Hie assume at the outset that the boundryis: layer unsteady pressure field is known and that it induces small deflections in the s~in. As a consequence (a) The skinr dynamics are described by a linear equation (b) The generation of' a random pressure field inside the cabin is a linear radiation problem. Thus the mathematical techniques used are those required to obtain the response of linear sy~stems to stochastic forcing functions of several. variables. We also assume the fuselage to be a large flat plate. This assumption is not necessary but it simplifies the discussion and allows us to present mrore clearlyr th~e new features of the problem. The material in this report is presented as follows: First we study the radiation of sound from a randomly vibrating plate. It is found that the sound levels in the cabin are defined byr the intensity and the scales of the plate normal accelerations. Second, generalized Foulrier analysis is put to use in order to relate the normal acceleration of the plate to the forces exerted on it by the boundary layer flow. Third, the bolundary layer forces are defined in terms of flow character istics, and dimensional similarity is used to determine the significant parameters. Finally, the functional form of the noise intensity in the cabin is given save for an unknown function oftone nondimensional parameter. This function depends on the frequency spectrum of the pressure fluctuations in the boundary layer. Hlo measurements yielding this spectrum have been NACA TM 1620 reported to date and speculations concerninC: it would introduce in the analysis both complication and uncertainty. A summary of results is given at the end of the report. Som derivations and some of the longer arguments have been presented as appendices to the text. NACA TM 1420 I. THE ACOUSTIC COUIPLING OiF A RANDO3MLY VIBRATINGC PLATE WITHI AIR AT REST We start with Rayleigh's well known solution of the acoustic equations when the sound is generated in an otherwise unbounded stationary gas by a large~ flat plate or dise oscillating normally to its plane (Ref. I page 107). where : the static pressure p has been broken into a steady part p and a fluctuating part, pi: p = p ~j(x,y,z) + pi(x,y,z,t) V, = the normal velocityI of the plate do* = an element of plate surface area ai = the speed of sound in the air S = the vector position over which the integration is carried out =(x',o,Z') if)~ = the distance between source and field points = (x~y~ x') 2 z'2 A = the total area of the plate The subscript i refers to properties inside the cabin or on the side of the plate on which air is not flowing. The normal acceleration bVn/8t is a random fumetion of time and space and so is pi. W~e wish to evaluate the mean square of the pressure (a quantity to which sound intensity is directly proportional). W~e notice that for a sum alike, for the integral in (1))* +This discussion follows closely the arguments set forth in ref. 3 and the background of information on statiat~ical methods can be found in ref. 2. NACA TM 1420 and since aX = arif a does no depend on timet, we~ can write No teexpression is the correlation (i.e. the time average of the product) of the normal acceleration of the plate surface at two points 81 n 2adttw different times(t Ryan) and (t 9:)&}. If the two points vibrate completely independently from each other, this expression is zero and if the two points are brought topothLEr (Sywc therefore cl r2) the orrelation function is simply(ids/Db)l Now we assume that the average properties of the plate motion are the same anywhere on its surface and at any time (we assume statistical homogeneity and stationarity). Then the expression above for the correlation becomes merely a function of the distance ( 5\) between the two points and of('LRa We call this then IAA Now we can evaluate pi2 (Y) under a variety of assumptions for for the plate area A and for the distance Y between the plate and the observer. We will consistently hold the view that the normal acceleration at most points of the plate surface are not correlated ( ~= 0) and that two points of the plate, in order to show appreciable correlation, must be a small fraction of the total plate size away from each other. We define as h thema distance over which~bd~ is strongly correlated (i.e. g (10#dj)~ and call it the integral (length) scale*. w The integral scale is given, say in the xdirection by We should note that The integral in (4) is not finite if the plate area A is infinite The length scale h plays no role in the geometry of the problem. The integral is a function of Y, the distance to the plate, and the plate dimensions only. For instance if the plate is circular and of radius R + This case is treated more rigorously in Appendix IA NACA TM 1420 Our hypothesis can then be expressed as where R is the average linear dimension of the plate. A representative case+ Suppose that the observer distance Y to the plate is such that and that no appreciable phase difference can arise at Y between two strongly correlated signals (i.e., between sound pulses originating within jL of each other). Then, if we examined equation (3) ve see that the inner integral contributes very little except when the point 3t is approximately within a distance ), of $ Then, approximately 'Ls and 1 P(bVag Thus the inner integral is approximately and equation (3) can be rewritten I;' "~ 47\:' dsl h' (dv,)nl dr NACA TM 1420 It is apparent from this result that the distance Y from the plate to the observer is measured in terms of plate diameters, and not in terms of average correlation lengths or wave lengths )L This result holds for all the cases considered (see Appendix I) and depends only on the assumption that at a given time the plate vibrations are largely un correlated or incoherent." to define a strength. Equation (4) indicates that in order to evaluate the pressure intensity for the case considered above, one must first determine (bS /3)0 the mean square normal acceleration of the plate and \ the length scale for the plate deflections. Other cases (i.e. cases for which either the plate deflects differently or the observer moves closer to it) are treated in Appendix IA. For some of them the time scale or mean period NON is required as well as L  We rewrite Equation (4) where la (pV is a function of the plate geometry and of the distance between the observer and the plate. It is defined from equation (4) as Notice that if the integral scale Is not the same in the x' and in the z' direction we may simply substitute in equation (6) hrr' and h S/ for h\ + This result holds, as Appendix IB shows, even when the pressure is generated by the travel through air at rest of a "randomly bumpy plate" i.e., when the timewise and spacewise variations of upwash are not independent. The latter example is therefore quite distinct from the flow of an infinite (periodically) wavy wall for which the only characteristic length is the wave length. NACA TM 1420 II. THIE DYNAMIC BEHAVIOR OF THE SKIN The response of the bare fuselage skin to the random pressure field of' the boundary layer is given, in the absence of axial loads, by a plate equation. For a flat plate this equation is where x and : are the coordinates along the plate surface (x being the free stream direction), y the deflection of the plate at a point, E the modulus of elasticity of the plate material, e" its density, hL Poisson's ratio, 2h the thickness of the plate, # a damping constant which has dimensions (1/T). Damping may be present because energy is absorbed either within the skin or by the air." For air limping P0 /' gy) ~ Hlotice that to allow for air damping is to provide for a feedback in the coupling between the plate and the air at rest. On the other hand we exclude feedback between the plate and the boundary layer. In other terms we are not considering the possibility that the plate vibrations are large enough to induce timedependent pressure gradients of the same order of magnitude as our forcing function. Such a feedback would amount to panel flutter. It cannot be handled by the present method. r(x,z,t) is the random forcefLunit mass exerted by the pressure fluctuations on the plate surface. It is characterized by a power spectral density. F(kl,k2,0;) which is a continuous function of the wave numbers kl (in the x direction), k2 (in the z, direction) and of the frequency co. The coefficienrt E/30(1f2) has the dimensions of a velocity squared and it is defined as c2. *The skin construction may be such that the damping it causes is primarily viscous or primarily flexural. In the latter case it seems more appropriate to write with Ribner (ref. 5) where fig is the flexural damping constant due to the plate and/Q the damping due to the energy radiated to the air. As is shown in Appendix IB the noise intensity within the fuselage may or may not be related to the acoustic energy radiated by the plate and thu~~tcee~ aitdb h lt n hus P may or may not be NACA IM 1420 We still have to specify the spacewise boudr conditions on the plate and we are led, for the sake of simplicity, to either one of two limiting cases. In the first case, the~ forcing; funtion (the random pressure field in the boundary layer) is characterized by an integral scale so large that at a given time, a skin are~a between two stiffeners (assumed rigid) is very likely to be subjected to a pressure load of the same sig~n (see Fig. Ia). This allows us to exrpress y and f as functions of t only. Z OR X & OR X FclG C F us. Ib In the second case, the integral scale of the forcing function, is very small in comparison with the distance between two stiffeners an the behavior of the skin is in the ave~rage very much as though the supports were removed to infinity (see Fig. Ib). The real case will in general be intermediate between these two limiting examples. However, the first case seems to apply to boundary layers of excessive~ thickness: A reasonable guess for the average correlation length might be one dis placement thickness s +; for d+ to be~ larger than the spacing between stiffeners (of the order of a foot) the boundary layer thickness Al would have to be of the order of five! feet or more. This unlikely case is treted in reference (7). On the other hand, the second limiting case (Fig. Ib) would seem to provide a reasonable model for boundary layer thickness no0t exceeding one foot. This is the model discussed now. a) The Mean Acceleration According to our assumption, the average motion NACA TM 1420 10I is not sensibly affected by the presence of stiffeners. A large number of: pulses act on the sktin at a given time between two consecutive stiffeners. The random pulses may be positive orr negative and thus there will be a large number iof load rev~ersals between supports. Then the effect of the boundary conditions can be expected to become small, in the average. Consequently one can define a generalized admittance and use it in much the same way as is often done in onedimensional problems." For instance, the mean square plate displacement is given by Here the mean square of' the forcing function f' is related to the spectrumn by: l;ffis the generalized admnittance, and kl, k2 and Wt are respectively thel wave number In thle x( directiojn, the wave number in the z directions and the frequency. The determination of If)(is easy once it is realized that this expression is the square of' the Fourier transform of the fundamental solution and so can be written by inspection. Thus an average solution of (7) is: and  fr.uw~ h Bl (9) Equation (9) gives the mean square response of an unbounded plate to a random forcing function. One should notice that the plate will always exhibit resonance no matter what value the dam~ping constant /8 may have. This resonance occurs, not at a given frequencyi or at a set of discrete + See in particular reference (2). NACA TM 1420 WITH 1~= 0.1 ano +Z = A THE ADMITTANCE MAP FOR AN UNBOUNDED PATE FIGURE 2 //X C~~4 e n2 5! C'h',3;IC~2 Gi NACA TM 1420 frequencies but over the whole frequency spectrum, whenever the follow ing relationship obtains between frequencies and wave numbers:+ We can visualize the resonance condition as a crest or ridge in the wave space (see Fig. 2) which originates at the line k2 = (3/chf2 and which becomes hiigher and steeper as the wave number and the frequency increase. Thus the effective damping is a function of the exciting frequency. b) The Length Scale A Equation (6) shows that h2 is needed as well as ( VI/}t)2 Now A is a length scale. It was defined, say in the x direction as Ewa and could be termed the equivalent length of perfect correlation. There are various ways of evaluating the integral scale. Perhaps the most convenient one for our purpose is that (found for instance in Ref. 2 Eq. IIS) which is derived from the relationship between correla tion functions and spectral functions. Thus if a stationary random fune tion J(t) possesses a correlation function which is sufficiently well behaved, + H~ere resonance is defined as the maximum of the response curve 1/j(k) holdinggg) constant. The locus of maximae holding k constant is given by: These maximae correspond only for zero damping. NACA TM 14c20 one can define a spectral density function Nov for the particular casek 4 = this gives Since n, = *~~ d] (o ) we have the result that z Pe) (10 ) We~ hv already obtained a forma representation for the spectral density function of the plate. It is the integrand of Eq. (9) so we! can write, in view of (10) (11) and a similar expression for hy . **j O /t,,c 31 [nlh e z r SE'A4,s. / bVn 2. NACA TM 11620 III. THE FORCING FUNJCTION We suppose that a turbulent boundary layer develops on the skin of the airplane~ (on one side of our plate). The forces which excite the plate are the pressure fluctuationls experienced by the plate itself. We assume that all characteristics of the boundary layer are fixed once we have specified the boundary lawyer thickness b, the free stream ve~locity Ueo and density In terms of pressure fluctuations this implies that at a fixed point of the "vetted" surface of the skin, we have for the mean square pressure fluctuations ~ fo Ut and the: integral scales, i.e. are proportional to b. Also the relative contribution to pressure intensity of the various frequency bands must be a function of Ugg,5, amndtJ only so that f aC is a random function of three independent variables, x,z,t, PIis related to a threedimensional spectrum by: anc that : NACA TM 1420 Loosely speaking, this means that a characteristic frequency for pressure fluctuations is proportional to Use /, and a characteristic wave length is proportional to d Now the forcing function of Eq. (7) is a forcefunit mass so that according to our similarity hypothesis B' One can thus define a spectral function associated with the forcing function f(x,z,t): and thus S~i"~b 6lk'L (12) Here F2 is a function of K1, K2, and .11 only and these are nondimensional variable s: Jr.= o/ U, In terms of these nondimensional units, Eq. (8) becomes: ~y 91 U~ Ua, NACA TM 11s20 and Eq. (9) becomes which can be written Again, F2 is a function of the3 integration variable only, so that one can write b(*l" po', U + ch (5 &4 Equation (14) yields the twonondimensional parameters upon which the plate dynamics depend. The first one, Ch /SJo 04 i the product of a speed of free stream Mach umer, ( ) and a Speed of propagation of waves in the plate thickness ratio (to~dr )ai crrs The second one, pbu plate thickness is a nondimensional damping parameter which is, alike, a function of plate and free stream properties. If we treat the equation for the integral scale (Eq. 11) in the same way, we notice that no new nondimensional parameter occurs, so that, at most SL c (15) 7 5GS T Cao4jLr(~ d~acboK ck ~=rp~'rtpr I SUao nU JL F(Kln)r(L ak 'e NACA TM 1420 We now vish to investigate the form of the functions H and L in Eas. (14) an (15) respectively. :First, we make an assumption which is not strictly necessary but which simplifies the manipulation of Eq. (12). We take the function F(KI, KtlR) to be symrmetrie in KI and Kg, which leads us to define a new wave number. and to write Thus, Eq. (13) becomes Now, the dapng circumtances it S ~ 'L~~IC~o Uao bth" paramneter 13141D is assumed small and under these can be shown (see Appendix II) that (16) The small difference between these two integrals can easily be evalaed for arbitrarily small values of pllUa even though both integral are unbounded as P*** This leads us to believe tha for low damping the main contribution to the inner integral comes from the resonansce condition Thus, if the spectral function F( L,K) is reasonably wide, i~e. ifbF bK((1 over a large range of K Eq. (16) suggests tha we write (r7) 1:JL"drr IraDIc CCIH du(hc Fl kJT) r ILC I c Ke ~r lb'0~; Uclr, Kdl NACA TM 1420 The requirement that F be flat in KC when compared to 1/X(K) is equivalent to the requirement that the average correlation distance or integral scale for the boudr layer pressure fluctuations be small compared to integral scale of the plate deflection. Translated in physical terms the simplifica tion suggested here is prompted by the following remark: If the plate has some stiffness, it makes little difference whether the forcing function is assumed to be distributed over smal distances or made of con centrated loads (see Fig. 3) FIGURE 3 Thus a satisfactory model for the problem at hand would rain drops on a metal roof. Equation (16) allows us to to get be the impact of integrate over K, blbo FlJb~"~;E~J' ' ~" 9."uf t 4 a2h~+ ~:~~ ~oC  a' h'e ~ dJL (18) (~~3~ns \ loI"n ")dn bV~ndt ~ bu n ch ~(~, NACA TM 1420 The expression can be evaluated only when F(K n) is known. It may be an increasing or adeceasig fncton olinh .In the absence of data on the spectral function F, we will not tep odfn t The function H defined by Eq. (14) can be written: c~~~h & ']I . where f,, is an unspecified function related by (18) to the bounar layer pressure spectrum. In order to determine p 2 we need to find out, in addition, wha quantities the integral scale h depends on. Here we make use of con siderations which are similar to those yielding Eq. (16;) (see Appendi III). The result is that whref is another function related to the boundary layer pressure spectrum by 1II(4). Hlow we are able to write Eq. (6) as (21) here Expression (21)gves the functional dependence of pressure intensity "inside" on boundary layer parameters for a typical case. The only quantity, not immediately available is h(b */ck). It is probable that we shall have to await experimental data to define its numerical value reliably. NACA TM 1420 IV. SPECIAL CASES 1. Convected Turbulence Two authors (ref. 5 & 6) have recently suggested that the boundary layer pressure fluctuations at any point of the fuselage skin are caused essentially by the passage over the point of a fixed (i.e. time inde pendent) pattern of pressure disturbances carried downstream at a fixed convective velocity. So far, experimental evidence in proof or disproof is lacking. However, it is interesting to incorporate this special case in the general formulation which has been presented. Both the response of the plate and the coupling of the plate with the air at rest must then be reconsidered. a) The coupling of the plate with air at rest in the case of convected turbulence If a fixed spatial pressure distribution is carried downstream on the surface of the plate, it is easy to show that the (infinite) plateY response will be of the same kind, i.e., that it will consist of ripples which are randomly distributed in space but which travel through the plate at the same convective '.elocityr as the boundary layer disturbance. The determination of the pressure field inside the fuselage is not in principle different for this case and has been carried out in Appendix I~B*t. The result is that for both subsonic moving ripples (with convection velocity U, C ELi ) and moderately supersonic ones: where I For higher supersonic speeds, the function of geometry and Mach number appearing as an integral is more complicated. The equation (I.10) above has the same form as equation (6). On the other hand there is a sharp difference in terms of energy radiated by the plate between the subsonic and the supersonic case, since no energy at all is radiated by subsonic ripples while the supersonic ones do generate some. One must, then, make "Here the presence of transversal bulkheads will change the picture because of multiple reflections of the ripple. **cThis problem can also be viewed as a steady (randomly bumpy) wing problem from the standpoint of a stationary observer. MIACA TM 1420 a distinction between the results in terms of pressure intensity (the quantity of practical interest) and in terms of energy radiation. This distinction stems from the fact that (as is pointed out on page 7) the acoustical field investigated is truly a near field. b) The response of the plate According to the convective hypothesis, time is not an independent variable once the convective velocity [f, is fixed. Translated in tenns of the spectral density T'1~(c,kl,k2) of the pressure fluctuations, this means that I (A) ,hlak2)is zero, except whenP = U, g, or in non dimensional form, when n.s rum We rewrite equation (12) for this special case. 4 3,' u, QS kL 'SS~ k,, a,, ,> Here 6 [ KsL~ is the Dirac delta function of the variableR.. Then the plate response becomes bVl~n \~t at 1 99~ u~ QL~' (23) he re Now if we assume as before that F is symmetric in K1 and K2 and substitute ~= .kh;l KI a26 7(~11 K,~~n~61" Uao ~Uo~ R+ F(~,,HIJL)b~*~ II a, d~cl ace, ~ao[' klL(22) urf JL F~ ~ K,,~~ dn<~ dKL q 6 YI 4p~ ~r LII\ r ~~hZ r~trK~ r. ~r h~ L ~~, i~~ r.~ crJ~I; Fl K,,~z~ UL~bp) NACA TM 1420 we~ finally get Here and ~bzUf d' BZ E a~ k+ c)p ItrCosCa Tr(Beosesuslde c (24) 3 = *,/ U, T.(aeo581~) F(K,,~, U, with KIL K+ = CoS is indicated by the following dimensional argument. and Th length scale X The mean correlation length or integral scale is a weighted average of all wave lengths, so that dimensionally I Since resonance dominates the plate response, = is given from the plate response equation (equation 23) byr the resonance condition h A similar reasoning would have yielded, in the nonconvective case, C~(h/,,5) VE instead of eq. (20). Combining (24) and (25) according to (I.10) we notice that we can still write as in equation (21). (25) h'L~ p 4:, 0^ (24 Here 61Is a weak function of the kach number as seen from (I.10). NAC TM 16420 2. The case of zero scale Under some circumstarnces it is possible that the space average of th plate motion vanishes, i.e.: This does not ean tha th norma accelerations at two neighboring points show no correlation, but that the correlation function becomes negative J as indcated in Fig. (4) and in suc a way that its space integral vanishes. We can then consider the! normal accelerations as dipoles rather tha sources and we are led to a slightly di fferent radiation problem. Appendix IC shows, however, that if one defines a length 2 ba such that FIGURE 4 The results are again ide:ntical in forn with those of equation (6). B~ere? hL can be viewed as the mean moment arm of deflection moments. A~lternately one can redefine the integral scale as ,== CCl o (27) where c is a constant. Equation (27) can thus be used to define the! integral scale in any event. NACA IN 1420 V. SUMMARY OF RESULTS AND DISCUSSION Appendix I discusses in addition to the cases mentioned in the text a few examples which provide different limiting conditions. Thus the observer is brought close to the plate (Y(( 7\ ). A short time scale is considered etc..... The common feature of all these analyses is that the resulting mean noise intensity can always be represented, say by equation (26). We shall therefore retain this equation: as ~ r~st Here Pi" Q)i a; 96 6 Il~op 6 zh r" v MI UI c general statement that we can make at the present time. =mean square noise intensity inside =air density inside =speed of sound inside = air density in the free stream =plate density = free stream velocity =boundary layer thickness =plate thickness =viscous damping constant (of units 1/time) =perpendicular distance between observer and fuselage =geometry of the plate = Mach number Ula convectivee velocity of turbulence pattern =characteristic velocity in the plate NCACA TM 1420 25 For all but high supersonic velocities, the dependence of on Na~ is quite small and can be disretgarded. The function@~ (Y,g), a quantity which does not depend on the dynamics of the problem but only on its geometry should be modified to take into account the fact that the fuselage is a cylinder and not a large flat plate. Thne form of the function 5 cannot be given here both because no information is yet available on boundary layer pressure spectra an because S depends too critically on the tyipe of model assumed. How ever, if is measured while any one of the four vaiable defining S (6,Uap or b) is varied, then the functional forn of thte noise intensity inside a fuselage can be determined. Thus the main contribution of the analysis is to diminish the extent of the testing required. One of the conclusions which can be drawn frau the foregoing equation is that unless the boundary layer pressure spectrum is a ve sharp function of frequency (which would makre S very sensitive tolUIICht ) it is not practical to decrease cabin noise by boundary layer suction: Since the noise intensity is a weak function of boundary layer thickness, decreasing appreciably cabin noise would involve the remloval of a prohibitive amount of air. Another conclusion is that increasing the damping is a very effec tive way of limiting the production of noise of all frequencies, since the structure transmits sounds essentially by resonance. The analysis which has been presented deliberately omitted some of the features of the problem which would influence the results an intro duce new parameters. For instance, the fuselage of commrcial airplanes is usually subjected to an axrial. tension as well as other loadis. In addition the skin is curved. To account for these featurs of the prob lem one would introduce further terms in the differential equation describing the plate and one could treat it in much the same way as has been done here. The general methods which have been used are adaptable in addition to the study of a germane problem, the fatigue of panels which are buffeted by a turbulent boundary layer. A NOTE ON TESTING The discussion of the various limiting solutions makes it clear that for the transmission of boundary layer noise through a structure~, the ratio of outside (boundary layer) noise to inside (cabin) noise is in general a function of boundary layer as well as structural character istics. This is to say, first, that an attenuation coefficient cannt be defined by testing the structure alone with a standard noise source. Thus accurate testing requires at the outset that th model be tested NACA TM 1420 for transmission of a noise similar to boundary layer noise. The main property of such a noise, as we have seen is that it must be random in space as well as in, tim, whch precludes the use of one or a few concentrated sources as noise generators. The only proper substitutes for boundary layer pressure fluctuations are forcing functions whose effects on a fuselage are local.* The impact of water drops for instance might be found adequate simulation. Further, similarity in testing requires the matching of' parameters which are ratios of plate and forcing funtion properties. For instance if the forcing function used in the test is a turbulent boundary layer, similarityr parameters are: *This is not true of jet noise which is Generated away from the fuselage. NaACA TM 1420 VI. REFERIEICES 1. Lord Rayleigh: The Theory of Sound. Dover Publications, New York. Volume II (1945). 2. Liepmann, H. W.: Aspects of the Turbuence Problem (Part I) ZAMP, Volumet III (1952) pp 321342. 3. Liepman, H.W.: On the Application of Statistical. Concepts to the Buffeting Problem. Journal of the Aeronautical Sciences 19 (1952) pp 793800. 4. Timoshenko 8. and Young, D. A.: Advanced Dynamics. Mc~ratwKll, New Yorkt (1948). 5. Ritne~r, H. S.: Boundar~yLayer Induced Noise in the Interior of Aircraft. UTIA Report No. 37 (April 1956). 6. Kraichnaun, R. H.: Iloise Transmission from Boundary Laye~r Pressure Fluctuations. To be published. 7. Co~rcos G. H. and Liepmuann, H. W.: On the Transmission Through a Fuselalle Wlall of Boundary Layer Noise. Douglas Report Noe. SM19570. (1955) MACA TM 1420 AiPPENDIX I PHE RANDOM RADIATION OF A PLANE SUIRFACE: A. FOUR LIMITING CASES In order to determine the couling between. fuselage vibrations and cabin air one has to choose a model for the correlation 7 1 between the normal accelerations at two different points of the plate. The model which was discussed and for which equation (4) was made plausible is predicated upon two conditions: A. That the observer is distant enough so that a large number of plate elements vibrating independently contribute sound in comparable amounts, i.e. Here as before, h is the integral (length) scale for the plate normal accelerations and Y is the perpendicular distance between the observer and the plate. B. That the time scale of the phenomenon is large enough so that the differences in phase (introduced by the. unequal distance from the point at Y to the? various points of a plate element of length PL) are unimportant, i.e. ai is the speed of sound in the fuselage air, and 8 is the integral (time) scale for the phenomenon: Then one can choose a s;imrple model for the :orrelation function 1 ' where a is the delta function. The norma accelerations are assumed perfectly correlated within a length Lh and not at al for distances NACA TS( 1c20 greater than PL Thlen rr ZI Rt 4 Ad/CIROPI~ay& FICUTRE 5 Under these conditions the~ noise at the microphone is contributed primarily from a single plate element which in the average vibrates in phase. The evaluation of this contribution is particularly simple. We can vaite, very nearly andupon integrating which is equation (4). This case,h 4))\ 44 4)X 1.<$ ,;Ls)4 coragendsi to the following conditions. The passenger (or the mderophone) is far from the plate (in terms of PL ), the boundary layer is thick and the(3 airpane velocities low. One may well wondr about cases for which these conditions do not aply. Whle it apears difficult to answer such a qury with Genrerality it is possible to consider other limiting cases. For instance let us assume that condition 2 still applies but that our observer is extremely close to the plate. This would correspond to the following physical case: A thin fuselage skin, a thick boundary layer, a low airplane velocity and we are3 measuring neise by placing a imicroprhone very close to the skin and insulating it on all sides except the side which faces the skin. Then X7\pY }\44 CL'shSM O IRCA TM 1420 (I.2) and If, for the sake of definiteness, we assume the element circular, then and since '} ) Y it is permissible to write The pressure intensity is therefore given as Thus Eq. (6) applies for the very close as well as for the very far field when phase effects are not important ( A Regi * Now assume that we carry on the same experiment but that the boundary layer is thin and that the velocity of the airplane is high so that the exciting frequencies are high. Let us assume in addition that the skin is thick, so that y (4~ h : < C /A i FUSELAGE~ BOUym9RY LAYER FIGURE 6 NACA TM 1420 N~ow the time scale of the plate motion is short and phase effects are prevalent. We define a simple time history in analogy to the space description of Eq. (I.1) The microphone still receives signals effectively only from one plate element an al points within that element vibrate in phase but the pressure pulses originating from that element do not arrive at the microphone in the sam time. Then: ......... 6( I 3) 4Is6 q*** St* at, rs0. 4~r 4t ~ICt A' is simply Equation (I.3) is evaluated by noticing that: Here f; are the real roots of C( ) = 03 which are included in the interval between and and b. W~e only have one root, namely rl = r2. If we choose to integrate, say, with respect to s2 first we get (assumning again that the element is circular bIr A'Wed. 4, I v.l r 9.& #./t~'+Y; 4 (I.5) and according to (I.4) the inner internal yields: NACA TM 1420 so that The time scale appears explicitly in the answer. For the unbounded plate however it is simply proportional to f /Uao just as the time scale f or the boundary layer pressure fluctuations. Finally we may consider a physical case for which phase effects are important and for which the microphone has been placed a large distance away from the plate. i.e.: )19 <( A /4*, j r 4 Y FL./SELAGE SA/JV MICROPHONESE~ FIGURE 7 NAA M1420 Now the contribution from each subelement of vibrating plate is still in the average independent from that of the next one. However, there are in addition cancellations from. withiin one element just as in the previous case. This will happen if the boundary lawyer is thin, the airplane velocity is high LI small) an the observer is far from the fuselage wall. In ordr to evaluae thiis limiting case we first specify the tim behavior of the correlation function: wert write 3 (P,h e,ejJd(qp~(Lth, e..e) (1[.0 so that 4A and we integrate first with resptect to .SL Using the same techniques as in the previous exape, we ge~t: Nov ve assume that Integrating with respect to 02, 91 1 successively: ata Je(o 'ho a NACA TM 1420 For a circular plate of radius R, this would give 9;' 1TT i~g (~,w O) h( o) In general, and defining a function. of the plate geome~try and of the distance Y only, we have 10;" ~icx~; (Is) "( b\rtbtt c" le d ld~ ~(%lx~~ NA~CA TM 1420 APPED~IX I B. TH NOISE GEEAE BY SKIN RIPPLES OF FIXED VELOCIT: If the turbulence pattern is frozen, as discussed in section IV1 ripples will travel through the (infinite) skin at a fixed convective velocity. Then the~ correlation f~unction.'t mut be written differenty: where UI is the speed of propagation of the rippe ( turbulence convective speed) and therefore where now ~lr J ~,'Z ~ Etl''L~t ~~ The inner integral is of' the form which can be written as in part A. The expression I l~s) hz ~; 'IbJ1I dLllldrt: r \~e CI 1 dre ~' skz~l~,~,~ NACA TM 1420 has either one or two real roots depending as 41,41 or Cfl respectively. For M ( 4 Sthe only real root is and thus the inner integral yields Il J~' so that: equation (4) for low to the root X, w 5s X"p;.t err2~ (I.lOa) notice that equation (I.10a) above tends to convective speeds. For NI ( (I.9) has, in addition root Given by Another .~ 5 (M:CI)PI C L Llrl plZ~ I ICt is easy to show that this root exists for al values of xl. In order to simplify the integration let us assume slightly supersonic conditions; i.e. let us write where f. ~e ~ Then '12 ~ C~tlL) lfl*4~ ~)~ (Ca) NACA TM 1420 and the inner integral = I so that for the supersonic case: I a (*Ml, j Ms (I.10b) A result which is save for a constant coefficient the same as I.10a. i 'LIL ~i M,[r,+ U;xl') *~L I 4 v at] NACA TM 11420 APPER3DIX I C. THE GENERATIONI OF NOISE BY PLATE DEFLECTION'S OF ZERO SCALE If the space average of the correlation function is zero and if the plate vibrations are isotropic in x' and z' one can define a new length scale as a moment arm: ( s is a fixed point) alternatively one can define a modified integral scale h"; = I 1 ( 7, ) 1 Here h''=: C ~CI where c is a constant. Then, one can idealize provided It follows that and integrating with respect to h'29c~ e~~t 51\ ~, bp, bet Ilt 9~ dp, hi Now, unless a or b = 0 t "I 16 PILP'.)"P the correlation function as = (~6V~~~2 P('.~qb'(LP~ Jb (1) 5()b~~J.t6t NACA M 1)+20 and thus Is,4ip. keplcy b 9((II~d9~ rPI 1; ILs. 4sro) I Idj S~'bP ,,,~~ 3 rt~ zi~ rc'Sq; h')S(a6F~)~ Leae h'~4L~!YTtz. \t 51 IOi~' If there is a (time) microscale 0) (OrC P~ d~ ILu ~7;5_ Z~C h'"Pr"rb~rcr'c4;" 4" \0"/ Here the plate has been assumed circular and R is its diame~ter. ;2 j* "'9[6~ )LZ t~Fl APPETH)IX I~I THIE SIMPLIFICATION OF THlE PLATE RESPONSE INTEGRAL (Equation 16) We consider the approximation equation NACA TM 1420 (16) The righithand side is clearly un~bounded as the damping constant /3 L O since its value is explicitly proportional to 1/;a (see for instance EQ.(18)). On the other hand the difference between the left and the righthiand integrals is finite for (3 = lj. To show this we write Then the leftiland side becomes fojr (9 = 3 suchitd a I) (II1) and the rightha~nd side becomes; for 9 = 0 as C on CL (II2) ( ~t1)2 f (~_()2 (~+l)e I so tha the difference D between expressions (Ill) and (II2) is anCi+~scr dc ho & 0.. (II4) on L~ ada == KAM NACA TM 1420 41 This e~xprssion is finite and of course independent of fS so that we can concude that the lefthand integral of (16) is unbounded fo /i = O. Further, it is clear that D is a regular function of/C3 so tha th ratio of th lef'than~d side to the righthand side of Eq. (16) can be made ar bitraily close to unity, by choosing arbitrarily small /3. If a corree tion is desired a numerical check indicates that Eq. (IIA) gives a good approximation to the error made even with moderately large damping. NIACA TM 1420 AP~PEIJDIX III THE EVALUATION OF THE INTEORAL SCALE Our starting point is Eq. (ll). In terms of nondimensional variables it becomes We now simplify the denominator by writing successively Then we define (III1) (III2) RJLK~Z Equation (III2) can now be written and if we replace K.2 b;, its value at resonance," namely + The justification for that step is identical to that advanced in Appendix II. Ch .. (fl As h4.) 12 J2. F , NACA TM 14210 We can write > V. * rr p~l~s ,, 8 a'h' ~To P6 (III3) If we compare (III]) to (183) we Get immediately F( )~da cks U, 9. (II14) NACA LangLey Field Va. X. e *I 9'K i 9,'R., T(~l go v. LI d,,l ( R(JShiL.JL)dh I" '' .L {Ch!\ X,~~~ e (ii " I 0 0 C. 97 L0 "~. 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