dS(yi ) 4iTx2c«,Js Eg. 3. 5a Here < > denotes an ensemble average and, P|(Yi^) = ^P|(Yw^)J Eg. 3.5b t=t-r/Co„ Let p| and p 1 be ergodic random functions of time, following the assumption of Sec.3,1,0, Then by d e f i n i t i o n of ergodicity, the ensemble averages i n the above are equivalent to time averages, the l a t e r of which may be s i m p l i f i e d by the following theorem (see Appendix C f o r proof), 16 Theorem 1: P' (Xi ,t)p« (X, , f ) = P'p« (X, ,T) P| (Yi ,t)p« ( X i , f ) = - f a_ p» ( Y l ) p» ( X l ) (t)j EqwC.1 Eq.C.2 T =I+r/C0 The time variables introduced above are as follows: t i s the present time, t* i s the new time, t-=t-r/c» i s the Kirchhoff retarded time, r = t * - t i s the delayed time (assuming t'>t) and Z=t• -t=r+r/c«<, i s the delayed Kirchhoff retarded time,. Applying Theorem 1 to Eg,. 3.5 a produces the Causality Correlation Function, P'P' (x, ,r) * -x, [ n; f _a_ p'( Y l- ) p« (x,) (f) 1 dS (y; ) Eg.3,6a ^x^cm J s L bz J r=r+r/c„ We f i n d that the autocorrelation of the far f i e l d pressure i s related to the surface i n t e q r a l of the time rate of change of the cross correlation between surface pressure and far f i e l d pressure. When the Causality Correlation Function i s evaluated with the delay time (r) set equal to zero the l e f t hand side of Eq,3.6a becomes the t o t a l radiated acoustic pressure - (mean squared value) at the far f i e l d point (x^). We have. p«p« ( X l) - -x, ( n,- j _s_ p'(Yi )P' ( X i ) (r) 1 dS(y,) Eq.3.6b 4-nx2coo J s l a t J t=r/c Q O 17 Sec. 3, il. 2 The Dipole Radiation Intensity The Causality Correlation Function furnishes a unique technique f o r diagnosing noise conditions on r i g i d boundaries. The i n d e f i n i t e i n t e g r a l i n Eg,3.6a may be expressed i n d i f f e r e n t i a l form. Taking the d i f f e r e n t i a l and setting the time delay (r) to zero and di v i d i n g both sides by the d i f f e r e n t i a l area element dS (y; ) we obtain, dp'P1 ( x w y j ) - - X i n; dS UTTXC-d _ P | ( y i ) P ' ( x i ) (r) at Eg, 3. 6c x=r/c a o The l e f t hand side of the foregoing equation gives the net radiated a c o u s t i c a l pressure (mean sguared value) at the f a r f i e l d point ( X i ) that was caused by the fluctuating pressure at the surface point (y; ) . The ri g h t hand side i s proportional to the time rate of change of the cross c o r r e l a t i o n function (p|p') (f) evaluated with the Kirchhoff time delay set egual to r = r/c 0 (see Fig* 8a f o r a plot of a t y p i c a l c o r r e l a t i o n function). The 'Dipole Radiation Intensity (DRI)' at Xi of the surface point yi i s defined to be the guantity dp'p' (x; , y- )/dS* . The DRI i s uniquely defined since the surface S appearing i n Eg,3,6b i s unrestricted. The features of the DRI include a dependence upon the magnitude of the c l a s s i c a l dipole strength and a dependence upon the frequency; the dipole strength (p|) i s contained within the time derivative i n Eg,3,6c. The 1DRI d i s t r i b u t i o n 1 may be defined to be the mapping of 18 the source point to the DEI. The evaluation of t h i s d i s t r i b u t i o n amounts to a decomposition of the t o t a l f a r f i e l d pressure (mean squared value) into component pressures; each component pressure being the net radiat i o n coming from the respective source point. In the forthcoming experiments some confusion can arise in the int e r p r e t a t i o n of 'DEI d i s t r i b u t i o n s * . A circumferential p r o f i l e of the DEI (see Fig,9 a ) bears an obvious resemblance to the f a m i l i a r dipole radiation d i r e c t i v i t y (dumbell shaped) . These two polar plots are i n f a c t d i f f e r e n t . The former i s constructed for a given surface using the d i f f e r e n t i a l equation Eg,3 , 6 c ; the DEI i s evaluated while the source point i s suitably varied and the f i e l d point i s held f i x e d , giving the 'decomposition' of the mean square pressure at the f i e l d point. The radiation d i r e c t i v i t y on the other hand i s customarily obtained by measuring p'p' (x-j ) while the f i e l d point i s varied irresp e c t i v e of source point, giving the 'mapping' of the acoustical f i e l d pressure. A l t e r n a t i v e l y , within the context of The Theory of Measures 1 6, the i n t e g r a l on the r i g h t hand side. of Eg; 3.. 6b can be viewed as being an i n d e f i n i t e i n t e g r a l whose domain i s the set of a l l measureable surfaces S. In t h i s case, i t can be shown that the l e f t hand side of Eg;3.'6b i s i n a sense a 'Lebesgue measure function' which also just happens to equal the far f i e l d acoustical pressure p'p' (x^). I t can be subsequently shown that the set of quantities { dp* p * (x,,y ' ) / d s | y, e S J exists and i s c a l l e d the 'dis-integration' of p'p* (x^) for the surface S. Moreover, the integrand of Eq,3 , 6 b , which we are denoting by dp'p* (xi,y;)/dS, may be shown to be unique. Incorporating the term dis- i n t e g r a t i o n into the d e f i n i t i o n f o r the DEI would y i e l d a 19 more concise d e f i n i t i o n ; the quantity p'p'(xi) would be c a l l e d the DEI, and the set of quantities { dp 1?' (xj,yi)/dS } would be called the Pis-integrated Dipole Radiation Intensity (DDRI). However t h i s change in nomenclature w i l l not be made herein. The cross c o r r e l a t i o n function P5 (Yi) P1 (*i) (£) i s more conveniently expressed as a dimensionless c o e f f i c i e n t . D e f i n i t i o n : We define the cross correlation c o e f f i c i e n t to be, C(x 1,y i) ( f ) PMYOPMX,) (?) -RMS i R M S Eq, 3, 7a r =r/c c The source and far f i e l d RHS pressures are equal to the square roots of t h e i r respective autocorrelation functions evaluated with the time difference (r) set equal to zero, Using Eq,C,1 we obtain. \\?T\" (Yi ) = J PS (t)p' ( f ) (Yi ) I = J P|P? (Y, ,r) I t «=t r=0 (x P* (t) p' ( f ) (x,) I t • =t -[ P'P' (X,,T) | T=0 Eq, 3, 7 b Eq,3.7c Each EMS pressure i s also related to a sound pressure l e v e l by. ,RMS _ p R E F antilog.o SPL/20 Eq. 3,8 where p R e F = 20 x 10-6 N/m 2« Introducinq these d e f i n i t i o n s i n t o Eq.3,6c yields the following recipe for the DRI at a point, 20 clp'p 1 ( x 1 # Y i ) -XJ n, (y,) _ a _ C ( x 1 / Y i ) ( t ) ds 4irxc» I a t Eg. 3 . 9 r=r/c< p?„ ant R E F ilog.o |SPL(y,) ^+ SPL(xQj Thus the DEI i s porportional to the slope of the normalized cross c o r r e l a t i o n function C(x^ ,y- ) ( r ) * and the antilog of the source & f a r f i e l d sound pressure l e v e l s . The slope i s evaluated with the Kirchhoff time delay set equal ; to r/c M;. Forthcoming sections outline the experimental procedure used to construct the d i s t r i b u t i o n of the DRI over the surface of the emersed cylinder model. 21 Sec,4 EXPERIMENT AL PROCEDURE Sec,4,0 Design and Fabrication of the Cylinder Probe An exploded view of the cylinder model i s shown in Fig.5a. The probe was used to detect the flow induced pressure fluctuation at a point on the surface of the cylinder model. The basic probe design consisted of a miniature microphone set i n a cavity within the midsection of the rod. The cavity was coupled to the surface by a c a p i l l a r y tube which was terminated with a pinhole. The pinhole at the surface could be moved into d i f f e r e n t positions in the flow by rotating or t r a n s l a t i n g the rod (see Fig,2a), The rod measured 1.27 cm in diameter and 17 cm i n length and was d r i l l e d out to accommodate a one guarter inch Bruel and Kjaer Condenser Microphone (Type 4136) connected to a B & K Preamplifier (Type 2618), The microphone cartridge.was sealed in the small cavity by an O-ring, The O-ring was positioned between the microphone membrane and the r e l i e f vent (see Fig.5a). The a i r - f i l l e d cavity i n front of the membrane was coupled to the surface of the cylinder model by the tiny c a p i l l a r y tube* The c a p i l l a r y tube measured .091 cm (,036 inches) i n diameter at the surface of the cylinder and was ,318 cm (,125 inches) i n lengths An a i r f i l l e d cavity that i s coupled to the atmosphere by a narrow c a p i l l a r y constitutes a Helmholtz resonator, The probe acted as a Helmholtz resonator and conseguently introduced an exaggerated response and a substantial phase lag near the Helmholtz resonant freguency. 22 The resonant frequency i s approximately given by the Helmholtz formula, Here S i s the cross sectional area of the c a p i l l a r y , V i s the -jr volume of the cavity and 1 E F F i s the e f f e c t i v e tube length. Near the resonant frequency, the pressure response of the Helmholtz device rapidly increases and the phase angle abruptly increases. The response i s maximum at resonance (after which i t f a l l s off) and pressure lags f l u i d v e l o c i t y by 90°. The Helmholtz resonant freguency of the probe was found by a procedure outlined i n the next section to be about 2 k hz. By comparison, the the far f i e l d pressure was monitored with a one half inch BSK Condenser Microphone (Type 4133) which had f l a t response and introduced n e g l i g i b l e phase s h i f t up to 20 k hz, The probe was subsequently redesigned to achieve a higher resonant freguency. The aim was to make the resonant frequency of the probe higher than freguencies that were known to dominate the spectrum of the far f i e l d pressure f l u c t u a t i o n (see Sec.4,2). With t h i s reguirement s a t i s f i e d , the probe should have a f l a t response ( < ±1.5 ) db and the phase s h i f t between the probe and the far f i e l d microphone should be minimal ( < 15°), over the desired range of frequencies (the range that i s involved i n p' (x^ ,t) or P'P' ( r ) ) . These conditions ensure that the cross co r r e l a t i o n The e f f e c t i v e tube length i s s l i g h t l y greater than the geometric tube length due to i n e r t i a of the a i r beyond the ends of the tube. f Eq,4. 1 23 function p|p* (r) which i s the es s e n t i a l ingredient used to obtain the DEI, i s not distorted by extraneous resonance peaks or by phase s h i f t s introduced in either channel (see Appendix B for more d e t a i l on the phase s e n s i t i v i t y of the cross c o r r e l a t i o n function). The freguency range of the probe can be extended by optimizing the parameters given i n Eq;.4, 1 , Several changes were made to the basic probe design i n order to achieve the optimum design. Cardboard discs made with a paper punch were c a r e f u l l y inserted on the inside of the microphone grid cap, The s l o t s of the grid cap were f i l l e d in with s i l i c o n e rubber. These two modifications reduced the cavity volume thereby increasing the resonant frequency, Furthermore, the c a p i l l a r y tube was tapered, From the surface, half the length of the tube was bored out to ,091 cm (.036 inches);. The other half was bored out to > 198 cm (,078 inches), Tapering the tube resulted i n a smaller e f f e c t i v e tube length and an increase i n the resonant frequency. The resonant frequency of the re-designed one guarter inch probe was found to be about 5 k hz. Comparing Fig,7c with Fig.6a, shows that the upper frequencies i n the far f i e l d spectrum s l i g h t l y overlap the resonance frequency of the one quarter inch probe. The proximity of these frequencies creates a p o s s i b i l i t y of high frequency d i s t o r t i o n s a f f e c t i n g the cross co r r e l a t i o n function (p^p* (f) )• A second probe was designed i n an attempt to achieve a resonant frequency that was higher than the marginal 5 k hz that was measured f o r the one guarter inch prober The design of the second probe was aimed towards further reducing the cavity volume thereby increasing the Helmholtz 24 resonant freguency,. To achieve t h i s , a one eighth inch BSK Condenser Microphone Cartridge (Type 4138) was mounted in a smaller cavity in a d i f f e r e n t cylinder model, The membrane of the one eighth inch microphone was very close to the grid cap so cardboard discs could not be employed, after several attempts at optimization i t was found that the resonant freguency of the one eighth inch probe was also about 5 k hz, The actual freguency response curves for the.one guarter and one eighth inch probes were recorded along with th e i r phase response by a procedure outlined i n the next section. The response curves are shown i n Figs;,6a & b , For both probes, the freguency response was f l a t (within ±1,5 db) out to about 3 k hz with a resonant boost occurring at the upper freguency range. The phase s h i f t -e-A - -e-o was recorded near probe resonance and i t was observed to increase abruptly near resonance but was less than 3 degrees for frequencies below 4 k hz, f o r both probes, Thus i t was found that the resonant frequency and the phase c h a r a c t e r i s t i c s of the two probes were about the same, However the design of the one quarter inch probe was apparently superior to that of the one eighth inch probe. Although the one eighth inch probe was advantageously the smaller of the two probes , 1EFF. was larger in the one eighth inch probe than i t was in the one quarter inch probe. Furthermore the cavity volume i n the one eighth inch probe was probably larger than i t was i n the one guarter inch probe since the volume i n between the s l o t s of the one eighth inch microphone grid cap was not f i l l e d in with s i l i c o n e rubber, The g r i d cap could not be removed or modified without risking damage to the microphone membrane contained therein* These disadvantages 25 that are found in-the design of the one eighth inch probe l i m i t that probes performance over the one guarter inch probe. Sec; 4; 1 The Measurement of Probe Ch a r a c t e r i s t i c s : Freguency and Phase Response The frequency response of the probe was obtained by exciting the device with a constant amplitude sinusoidal sweep tone generated by a loud speaker, The tests were carried out i n the anechoic chamber at the Department of Mechanical Engineering at UBC. ft one half inch BSK Condenser Microphone (Type 4133) was positioned several excitation wavelengths down the primary lobe of a good guality loudspeaker* The cylinder model was placed beside the microphone. The loudspeaker was driven with B&K Sine Random Generator (Type 1024) connected i n series with a 50 watt power amplifier. The SPL of the excitation was held constant by a feedback loop* The SPL near the probe was fed back into the Compressor Input of the generator via the one half inch microphone. The compressor section of the generator modulated (increased or decreased) the output of the generator i f the SPL started to d r i f t . The probe response was amplified by a BSK Measuring amplifier (Type 2606). This amplified signal was connected to the y axis of a B&K Level Recorder (Type 2305); the x axis was synchronized with the sweep frequency on the generator* The recorder plotted probe response (in db) versus frequency (see Fiqs,6a & b). . 26 The phase s h i f t introduced by the Helmholtz resonator probe was measured using an accurate dual beam oscilloscope. Channel B on the C,E,0. was triggered with the generator signal (used as a reference) and channel A was triggered with the probe s i g n a l . The two traces were observed on the C.B,0. , From t h i s , the phase s h i f t -e-c, - -e-6 was recorded near resonance, for each probe (see Figs, 6a & b). Sec.4* 2 Freguency Spectra f o r the Source Point and for the Observation Point Freguency spectra of surface pressure were obtained with the cylinder model spanning the potential core region of the j e t ; the position of the o r i g i n 0 i s given by the coordinates (xf = 0, x£=0, X'=1D=3. 8x10- 2 m) in t h i s case (see Fig.2a). The pinhole was moved onto the jet axis to the point yi=(E=d/2 # 0=O<>, h=0) and the pot e n t i a l core spectrum was recorded (see Fig,7a), The pinhole was subseguently moved into the mixing layer to the point yi = (E=d/2, 0=0°, h=1,91x10-2 m) , and the mixing layer spectrum was recorded (see Fig.7b). A one half inch BSK Condenser Microphone (Type 4133) was moved to the far f i e l d point x|=(x|=0# x«=-3.00 m, x«=0) so that i t was 90° to the jet axis, and the f a r f i e l d spectrum was charted (see Fig*7c), The amplifier and loud speaker introduced n e g l i g i b l e phase s h i f t between channels A and B for the range of freguencies near probe resonancei. 27 Frequency spectra of pressure were obtained with the cylinder model spanning the mixing region of the j e t ; the.position of the or i g i n 0 i s given by the coordinates (x* = 0, x» = 0, x'=5D=19.0x10-2 m) in t h i s case (see Fig. 2b) „ The mixing region spectrum was recorded with the pinhole at the point yi=(R=d/2, 0=00, h=6) , the same orientation as was used for potential core t r i a l (see Fig;„7d). Once again the f a r f i e l d spectrum was charted with the one half inch microphone at the previously given far f i e l d point (see Fig.7e), The locations of the mixing layer and the mixing region are shown in Fig-3a. Spectral analysis of the pressure signals was performed with a B&K Band Pass F i l t e r Set (Type 1614),. The said f i l t e r set has a constant percentage bandwidth which i s 23% of the center frequency or equivalently one t h i r d of an octave* Consequently, i t was necessary to account for the e f f e c t of increasing bandwidth in the one t h i r d octave spectra i n order to obtain a more meaningful measure of high frequency energy i n the pressure signals. For each one t h i r d octave spectrum, spectrum l e v e l s associated with frequencies above the center frequency of the peak spectrum l e v e l were corrected f o r increasing bandwidth. The correction results i n a 3 db decrease i n the spectrum l e v e l * each time the freguency i s doubled from the center frequency of the peak spectrum l e v e l . These corrected l e v e l s are indicated by arrows i n Figs.7a to 7e i n c l u s i v e , The one t h i r d octave spectra were obtained in the following manner. The f l u c t u a t i n g signal from each microphone was fed into the input of a measuring amplifier. The f i l t e r set was introduced at an intermediate stage of the amplifier; the amplifier was 28 provided with Ext. F i l t e r jacks on the rear panel for t h i s purpose.. The f i l t e r e d output of the amplifier was recorded on the yi axis of the B&K Level Recorder (Type 2305) . The recorder provided a voltage pulse back to the f i l t e r set. This pulse triggered the f i l t e r set each time a t h i r d octave band had been charted. The combined e f f e c t was to plot the f i l t e r e d response in db against the center freguency of the one t h i r d octave band. Sec. 4;3 Ca l i b r a t i o n of Instruments The sound pressure l e v e l i n a i r i s defined i n units of decibels by. SPL = 10 log 1 0 f p'p 1) Eq,4,2 where p R e p= 20x10-* pascals which i s the minimum sound i n t e n s i t y perceptible by the human ear. The measuring amplifier was eguipped with a precision galvanomet er. i The scale, marked in decibels (SPL) required accurate c a l i b r a t i o n * A B&K Pistonphone (Type 4220) was used for t h i s purpose. The sig n a l from the Pistonphone generated a scale reading on the measuring amplifier that corresponded to (124, 0±, 1) db. The Pistonphone generated a high SPL. The excitation freguency was 250 hz, This combination was suited f o r c a l i b r a t i o n of the probe; a high SPL tone dominates ambient (background) level s and produces a steady scale reading; a low freguency tone 29 remains unattenuated as i t t r a v e l s down a c a p i l l a r y tube. However, a t i g h t seal was reguired between the pinhole and the Pistonphone to avoid leakage of low frequency energy* An adaptor was used that f i t t e d snugly into the Pistonphone, The end of the adaptor had a b u i l t in O-ring which formed a seal between the pinhole and the Pistonphone. Sec. 4.4 The Measurement of Plow Parameters The value of the atmospheric pressure was obtained from the airport and the temperature inside the anechoic chamber was measured with a mercury thermometer marked to an accuracy of ±.05°C. The stagnation pressure head at the jet nozzle was measured with a mercury manometer to an accuracy of ±5% on the scale reading. Sec,4,5 The Construction of the DRI Dis t r i b u t i o n The DRI i s related to the cross c o r r e l a t i o n function (PfjP' (r)) by the following recipe (see Sec.3. 1.2), dp'p' ( x 1 # Y l ) n, (yj) dS 4iT xCco ±_ C (x, ,y, ) it) | Eg. 3. 9 a-c J r =r/c«, Pf« ant i l o g l o |SPL( Y i)^+ SPL(x 1)| Here C(r) i s the dimensionless cross correlation c o e f f i c i e n t o defined to be, 30 C(xi,y-) (£) = P K T i l P ' l x i ) (r) Eq.3.7 a t =r/c 0 0 The d i s t r i b u t i o n i s constructed by evaluating Eq.3.9 at an appropriate number of source points (y^ ) with the.far f i e l d point (x^) at a fixed distance 90° to the jet axis. The parameters used to evaluate Eq.3.9 were obtained as follows: the anqle subtended by x} and n, (y, ) i s equivalent to (j> to f i r s t order (see Fiq, 2a), and factor; an anomaly i n the