Prediction of Acoustic Loads on a Launch Vehicle: Non- Unique Source Allocation Method

Transcription

1 Prediction of Acoustic Loads on a Launch Vehicle: Non- Unique Source Allocation Method Mir Md. Maruf Morshed 1 Jubail University College, Jubail Industrial City, Jubail, 31961, Kingdom of Saudi Arabia Colin H. Hansen 2 The University of Adelaide, Adelaide, SA, 5005, Australia and Anthony C. Zander 3 The University of Adelaide, Adelaide, SA, 5005, Australia Severe acoustics loading, which occurs during the flight mission of launch vehicles, is often responsible for damage to the payload inside the fairing compartment. The prediction of the acoustic loads on the payload fairing generated by large propulsion devices requires the use of analytical and numerical methods to identify the acoustic sources and the spectrum of the acoustic loads. Work presented in this manuscript investigates the nature of the external acoustic pressure excitations on the fairing in the low frequency range from 50 Hz to 400 Hz, during the lift-off of a launch vehicle. The acoustic pressure excitation acting on a Representative Small Launch Vehicle Fairing (RSLVF) was estimated from the complex acoustic field generated by the rocket engine exhaust gases. The estimation procedure involved the use of a non-unique source allocation technique which considered acoustic sources along the rocket engine exhaust flow. Numerical and analytical results for the acoustic loads on the fairing agree well. = coefficient matrix Nomenclature = cylinder radius, m = speed of sound in the exhaust flow, m/s 1 Assistant Professor, Department of Mechanical Engineering, Jubail University College, Jubail Industrial City 31961, Kingdom of Saudi Arabia. 2 Professor, School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia. 3 Associate Professor, School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia. 1

2 = speed of sound in air, m/s = frequency band number = diagonal matrix ( ) = solid angle, rad = nozzle exit diameter, m = directivity index, db = frequency, Hz = frequency bandwidth, Hz = coefficient matrix () = Hankel function or Bessel function of the third kind of m th order and argument, z = identity matrix () = Bessel function of the first kind of m th order and argument, z = wavenumber, rad/m = overall acoustic power level, db, =,, sound power level for each segment, db = sound pressure level for each segment in each frequency band, db,,, = sound pressure level on the circumference of the vehicle, db,!",,, = overall sound pressure level, db # = total number of terms $ = term number % = total number of point sources & = source number, and number of nozzles = field point or observation point ',( = ' ),( * = acoustic pressure, Pa incident sound pressure, Pa 2

3 ' = scattered sound pressure, Pa ' +, - = total sound pressure at the surface of a cylinder, Pa = spatially dependent factor. = source strength, m 3 /s. = projection point, and integration point on the boundary / = distance between the integration point and observation point, and resultant oblique distance, m 0 = radial distance, and distance of the source from the nozzle to the centre of each segment, m 1 = exhaust velocity, m/s 2, = acoustic power of each source for each frequency band, Watt 2() = sound power per Hz, Watt 2(0) = sound power as a function of distance between the nozzle exit and the source, Watt 2!" = overall acoustic power, Watt 3 = X component of the cylindrical coordinate axes 4, = core length, m 4 5 = axial distance of the observation point on the vehicle from the nozzle exit, m 4 6 = vertical distance from the nozzle exit of the sources along the flow axis, m 4 7 = distance of the source from the vehicle axis along the flow axis, m 4 = length of segment, m 8 = Y component of the cylindrical coordinate axes 9 = Z component of the cylindrical coordinate axes = elevation height, m : = angle from the horizontal axis of the noise radiation, deg ; = azimuthal angle, deg < = incline angle between the line joining the observation point to the source location and the normal of the vehicle axis, deg 3

4 = = constant > = phase angle, rad? = angular frequency, = density of air in the exhaust flow, kg/m = equilibrium density of the fluid, kg/m 3 Γ = Surface boundary of the exterior boundarys I. Introduction For many years researchers have been trying to develop a suitable analytical methodology for the prediction of acoustic loads on launch vehicles, based on different sub-scale or full-scale tests of supersonic and subsonic engine exhaust jet flows. For example, Mayes et al. 1 conducted near-field and far-field noise surveys on straight horizontal solid fuel rocket engines for a range of nozzle exit pressures, although this arrangement excludes the influence of the launch pad. The measurements were presented for a 1,500-pound-thrust engine and for several 5,000-pound-thrust engines for which the nozzle exit flow conditions varied widely. From the near-field measurements in the frequency range from 5 Hz to 2.5 khz; the apparent source of noise was found to be approximately 20 diameters downstream of the nozzle exit. The far-field measurements showed that the maximum overall noise levels for all engines occur between 30 to 45 from the axis of the flow. Tedrick 2 conducted acoustical measurements of clustered and single-nozzle rocket engines, and discussed the characteristics of the near-field and far-field noise fields. Experimental results for various types of rocket engines, including exhaust flow configurations, and general discussions can be found in a wide range of literature, including Lassiter and Heitkotter 3, Cole et al. 4, Eldred et al. 5 and Morgan et al. 6. Based on several experimental models, Morgan et al. 6, Franken et al. 7 and Potter and Crocker 8 proposed an extensive empirical prediction methodology to calculate the near-field and far-field noise produced by jet and rocket flows, including the deflected flow from various types of exhaust deflectors. Mel'nikov et al. 9 and Dumnov et al. 10 investigated empirical techniques to define the maximum acoustic loads on the fairing on the basis of flight and experimental model data. In their work, the interaction of the jet flow with the launch pad was extensively studied to quantify the noise sources using empirical techniques. Using experimental and theoretical studies as a basis, Dementjev et al. 11 developed a semi-empirical technique to determine the noise due to the interaction of supersonic high temperature jets with different shaped 4

5 deflectors. All the aforementioned empirical techniques require a considerable amount of experimental data and this is one of their major shortcomings. The standard empirical methodologies and recommended implementation practices are outlined in NASA-SP , a space vehicle design criteria document, which is used in the current work to determine the acoustic loading on the launch vehicle fairing. Two different source allocation techniques have been proposed in NASA-SP , based on experimental data for chemical rockets and generalized sound spectrum levels. The first technique (unique source allocation method) assumes a unique source location along the exhaust flow in each frequency band, whereas the second technique (non-unique source allocation method) assumes that the noise in each frequency band is generated throughout the exhaust flow. Although both techniques predict the acoustic loads at a specified point on the launch vehicle, the effects of the scattering of the sound waves from the launch vehicle surface are completely omitted from the analysis using either technique. Morshed et al. 13 extended the unique source allocation method for scattering effects and developed analytical and numerical tools to examine the acoustic loads on the surface of a Representative Small Launch Vehicle Fairing (RSLVF). However, the extended technique considering the scattering effects did not implement the non-unique source allocation method, and this implementation is the focus of the investigation reported here which is an extension of the previous work 13. The intention is to provide the inputs needed to evaluate noise and vibration control treatments for launch vehicles. The non-unique source allocation method was used here to allocate sources along the engine exhaust flow for each of three one-third octave bands corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz, assuming that all the power in the one-third octave band is concentrated at a single frequency, and the method was extended, using the analytical and numerical tools developed by the authors, to examine the acoustic loads at the surface of the RSLVF. In order to keep the quantity of figures and results for a reasonable amount, only the aforementioned onethird octave bands were chosen for analysis in this manuscript, however a similar approach can be used for the other one-third octave bands. The results reported here and obtained using the non-unique source allocation method are compared with the results obtained using the unique source allocation method presented in previous work 13. The non-unique source allocation method was used here as it provides improved accuracy over NASA-SP and other previous work This is a result of the use of the process of combining the new analytical and numerical methods to describe the sound pressure as a result of the reflecting surface of the launcher, which permits convergence of the external 5

6 pressure field calculations at the surface of the launcher. The current work uses the Boundary Element Method (BEM) for numerical calculations to predict the acoustic load on the RSLVF and to assess the validity of the analytical model. The BEM software used here was written in MATLAB and is known as Open BEM, an open source code mainly developed by the Acoustic Laboratory, Technical University of Denmark 14. The BEM was implemented for the three dimensional geometry of the RSLVF considering the effects due to the diffraction of sound waves from the ends, and the results are compared with the results obtained analytically for an infinite cylinder of constant diameter. II. Theory A. Analytical Modeling The theoretical description of the three-dimensional acoustic loading on a cylinder due to a point source has been extensively discussed previously by Morshed et al. 13,15. This work was used as a basis for the current study and was extended to apply to a number of point sources along a line as shown in Fig. 3. Following the previous theoretical work 13,15, the following assumptions were made: (i) the object is an idealized long cylinder, since the geometry of a launch vehicle fairing is cylindrical, (ii) there will be very little effect on the external sound pressure field due to the diffraction of sound waves from the ends of the cylinder and thus this effect is neglected in the following analysis, and (iii) the cylinder wall is considered to be acoustically hard so that all of the scattered waves proceed outward from the surface. Consider a point source located on the X axis which produces spherical waves that impinge obliquely on the surface of the cylinder as shown in Fig. 1. The sound waves travel a distance R from the source and impinge obliquely on the surface at point P of the cylinder. Without repeating the lengthy mathematical derivations 13,15 for this particular arrangement of acoustic loading on the cylinder, it can be shown that the total external sound pressure at the surface of the cylinder is the superposition of the incident wave pressure p i and scattered wave pressure p s as follows 13,15 : ( ) ( ) t i s p ( R, k, a, z, φ, Q, t) = p R, k, a, z, φ, Q, t + p R, k, a, z, φ, Q, t a s s s N ( ) M 1 m iγm = p R, z, t εmi cos( mφ ) Jm( ka) ie sin γm Hm( ka), n= 1 m= 0 (1) 6

7 where the first summation on the right hand side is for N point sources along a line and the second summation is for M terms required in the series calculation. The quantity ε m = 1 if m = 0 and 2 if m > 0. The spatially dependent factor, pʹ (R, z, t), is a function of distance between the point source and observation point at the surface of the cylinder, and can be determined as Qs i ( kr ωt ) p ( R, z, t) = iωρo e. (2) 4π R Fig. 1: Geometry of obliquely incident waves for three-dimensional cylindrical coordinate axes. B. Numerical Formulation The numerical formulation technique using the boundary element method (BEM) for the exterior boundary problem has been extensively discussed by Morshed et al. 13,15. This technique has been used and extended in the current work to include a number of point sources along a line. For the exterior problem where the observation point is located on the boundary Γ, the Kirchhoff-Helmohltz (boundary) integral solution for the time harmonic sound pressure p can be determined as C D 'C D=F GHCI,J D L GKCJD 'C.D MΓC.D+4P' ) C D. (3) Eq. (3) can be reduced to a matrix formulation as STU(V=U(VST+4P( *, (4) 7

8 where [C] is equal to 2 πin (I N is an N N identity matrix of the N calculation points on the N nodes) for the surface formulation, and [H] is an H H global matrix of the H calculation points on the H elements. Here both [C] and [H] are known but the complex vector U(V is unknown. Therefore to find U(V, Eq. (4) reduces to STU(V= 4P( *, (5) where [A] = [H C]. The distances between the m integration points Q m (x, y, z), and n source points P n (X, Y, Z), need to be determined for the incident sound pressure from each source. Hence, the right side of the Eq. (5) may be calculated as Q N i p = i e n= 1 4π R s i( kr ωt) ωρ ο, (6) where /=X. $ 2 (3,8,9) & 2 (4,Z.) and N is number of point sources. C. Non-Unique Source Allocation Methodology The non-unique source allocation method is more complicated than the unique source allocation method described in the previous study by Morshed et al. 13. The non-unique source allocation method recognizes that the rocket noise in each frequency band is generated throughout the exhaust flow, and a unique, single equivalent point source location along the flow axis is not assumed to exist as in the unique source allocation method. The nonunique source allocation methodology is based on experimentally measured data and normalized results, and is described in detail in NASA-SP Predictions can be made for sources located in the near-field as well as in the far-field along the exhaust flow. In this section, the acoustic loading on the RSLVF, calculated using the nonunique source allocation technique, was observed to be dominated by the sources closest to the vehicle, with the contribution from other sources being negligible. For the prediction of the acoustic loading on the vehicle, it is necessary to distribute the sources relative to the core length along the exhaust flow. The rocket flow can be divided into a number of segments and the overall acoustic power for each segment can then be calculated using the estimated sound power per unit core length and the length of each segment using 12 8

9 where the term, ( ) L 10 log xw t r W x W ( r) x = 10log + L + 10log xt t w, seg 10 W 10 WOA 10 OA the subscript seg indicates the segments of exhaust flow., (7) refers to the relative sound power per unit core length at location r and The normalized relative sound power spectrum level with modified Strouhal number, f rae Uea o, is given in NASA-SP This sound power spectrum can then be converted to a conventional one-third octave bandwidth as follows 12 : W ( f, r) U a U a L = 10log + L 10log + 10log f e o e o w, seg, b 10 w, seg b W ( r) rae rae where the term, ( ) ( ) 10 log 10 W f, r Ueao W r rae, (8), refers to the relative sound power spectrum level. The acoustic power of each source corresponding to each segment for a frequency band of interest can be determined from the acoustic power level, L w,seg,b presented in Eq. (8), as follows: W seg b L w, seg, b 10, = 10, Watts (9) where the subscript b indicates the frequency band. The strength of each source corresponding to each segment for a frequency band of interest can be determined as follows 22 : Q 4πWseg, b s, seg, b 2 k ρae =. (10) The sound pressure level on the circumference of the vehicle, including the effects due to the reflecting surface of the vehicle, for a band centred on any frequency can be expressed as L p, seg, b, 10 (,,,, φ, s, seg, b, t) 2 t φ = 10log p DI( ) 3, (db re 20 Pa) a R k a z Q + θ µ (11) where the pressure quantity, t p a, represents the total sound pressure on the vehicle due to incident and scattered 9

10 waves, and is a function of the distance R from the source to the observation point on the vehicle, wave number k, radius of the vehicle a, elevation height z, azimuthal angle ϕ and source strength Q s,seg,b, and the term DI (θ) is the directivity index. The pressure quantity t p a can be determined using Eqs. (1) and (2) for the analytical calculations and Eqs. (5) and (6) for the numerical calculations by evaluating {p}. The first summation presented in Eq. (1) and the summation presented in Eq. (6) need to include all the sources for each frequency band of interest, using logarithmic summation of L p,seg,b,ϕ for all the segments. Then the overall sound pressure level calculated by the logarithmic summation of L p,seg,b,ϕ over all segments of the exhaust flow for a frequency band of interest becomes 12 L,,, 10,OA,, 10log10 10 p seg b φ Lp,seg bφ =, (db re 20µPa). (12) All seg In Eqs. (11) and (12), the sound pressure due to scattering from the reflecting surface of the vehicle as well as the spreading due to distance, are included by the term p t a. The effect of the scattered sound field on the sound pressure on the surface of the vehicle was not considered in the previous analysis presented by NASA-SP III Results and Discussions For the analytical and numerical prediction of acoustic loading on the fairing, a Representative Small Launch Vehicle Fairing (RSLVF) of overall length 5.33 m and maximum diameter of m was used, which has been used by others in previous work 13,23,24. Figure 2 shows the surface elements and nodes of the RSLVF generated in ANSYS using quadratic eight node elements and a mesh-only element, which is MESH 200. The ANSYS.LIS files containing the descriptions of all the surface elements (726 elements) and nodes (2120 nodes) were then imported into MATLAB for the numerical BEM analysis using the codes written in MATLAB. The RSLVF was considered to be positioned at the top of the rocket during lift-off. The geometry of the launch environment used for calculating the acoustic loading on the RSLVF is shown in Fig. 3, and is the same as used in previous work 13. The parameters shown in Fig. 4 are defined in the nomenclature section. It was assumed that all the sources are situated along the exhaust flow axis, generating spherical waves which impinge obliquely on the fairing. For the current work, it was assumed that the fairing structure and the flow axis were situated at x 1 = 15D e 10

11 upstream and x 2 = 5D e downstream of the vehicle respectively. A reference point was chosen at φ = 0, which is the front point on the vehicle facing the exhaust flow. Hence φ =180 is the rear point on the vehicle. For simplicity, it was assumed that the temperature along the flow axis was uniform and T = 1000 C. At that temperature the speed of sound and density in air are m/s and 0.28 kg/m 3 respectively. For the analytical and numerical analyses, Engine E was used which has the core length of x t = 1.25m and is the same as used in previous work 13.The flow axis was divided into ten segments along the flow with an assumed length of 1m for each segment. At the centre of each of the ten segments a point source completely uncorrelated with the other nine was assumed. The distances from the vehicle axis to the centre of each segment were estimated and used to determine the necessary parameters for each source. The specifications of each source, and the calculated and estimated values of all the parameters are reported in Tables A1 and A2 in the Appendix. The calculated acoustic power level for each band of interest corresponding to each segment is reported in Table A3 in the Appendix. The calculated source strength of each source corresponding to each segment is provided in Table 1. The source directivity index was assumed to be the same for radiation to all the circumferential positions on the vehicle for any specified elevation angle from the source to launch vehicle. Both the analytical and numerical approaches calculated the acoustic loading around the circumference of the RSLVF at a height of z = 2.17m from the bottom face of the RSLVF. This height was chosen because at that height there are sixty circumferential nodes which are sufficient to achieve a good numerical estimation of the sound pressure pattern for comparison with the analytical result. The descriptions of sixty circumferential nodes and their respective positions corresponding to azimuthal angles are provided in Table A4. 11

12 Fig. 2 RSLVF surface elements and nodes. The solid circles show the circumferential nodes at a height of z = 2.17m on the RSLVF. Z Observation point, P Launch Vehicle Normal to vehicle R Nozzle r Axis of flow Bucket Point source Fig. 3 Geometry of source locations relative to the vehicle and flow axes. 12

13 Table 1 Calculated source strength for Engine E for ten sources corresponding to ten segments for each 1/3 octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz. Core length, x t = 5.15m; overall acoustic power level, L w = dB; length of each segment, x = 1m; speed of sound and density in air at T = 1000 o C are m/s and 0.28 kg/m 3 respectively. Source strength, Q s, seg, b 10 6 (m 3 /s) Source Number 50 Hz Hz Hz For the analytically calculated acoustic loading on the RSLVF, using the non-unique source allocation method, the maximum diameter of m of the RSLVF was used as the infinite cylinder diameter. The surface sound pressure levels were evaluated as a function of azimuthal angle ;. The calculations were conducted considering the impacts of the scattered waves using Eqs. (1), (2) and (12), of the acoustic loading on the RSLVF from ten point sources acting as a line of point sources, as shown in Fig. 3, at each of the 1/3 octave bands corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz. Fig. 4 shows the analytically calculated overall sound pressure levels at the circumferential surface of the RSLVF at a height of z = 2.17m from the bottom face of the RSLVF. The results show that for the small chemical rocket Engine E the sound pressure level reaches around db at the front of the RSLVF facing the exhaust flow, for ten point sources located close to the vehicle. For the numerical calculations, Eqs, (5) (6) and (12) were used and the results for each band centered on respective center frequencies are shown in Figs. 5a-5f. The results show that for very low frequencies (50 Hz and 100 Hz), the highest acoustic pressures mostly occur on the surface of lower portions of the RSLVF, as shown in Figs. 5a-5d. The reasons for this are: (a) the sources are located below the RSLVF, along the exhaust flow axis; and (b) the RSLVF dimension is relatively small compared to the wavelength of the low frequency noise. As the frequency increases much of the acoustic pressure incident on the vehicle becomes scattered over the entire RSLVF since the RSLVF dimension is no longer smaller than the acoustic wavelength, as shown in Figs. 5e-5f. It appears that the overall sound pressure level reaches around db for ten individual point sources situated close to the vehicle acting as a line of point sources, along the exhaust flow, as shown in Figs. 5e-5f for the frequency of 400 Hz. The sound pressure levels on the circumferential nodes at a height of z = 2.17m of the RSLVF were calculated for each onethird octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz and are shown in Fig. 6. The results show similar pressure fluctuations to the analytical results shown in Fig. 4, which were calculated using the non- 13

14 unique source allocation method. The small differences in pressure magnitude are a result of diffraction around the ends of the cylinder which is taken into account in the BEM analysis but not in the analytical model. The BEM analysis allows determination of the sound pressure near the ends of the cylinder by considering a quarter-point technique (r 2/3 ) for particle velocity near the edge of the cylinder, where r is the distance from the edge. This approach approximates the condition that the particle velocity tends to infinity as r tends to zero, as explained in detail by Juhl 14. The analytical and numerical results also indicate that for each frequency band of interest, the sound pressure amplitude varies relatively smoothly at the front face of the RSLVF facing the exhaust flow and varies aggressively at the back of the RSLVF, due to the reinforcement and cancellation that occurs between the diffracted waves passing around each side of the RSLVF. The sound pressure fluctuation increases as the frequency band of interest increases because the amount of interference of the two diffracted waves passing around the two sides of the RSLVF increases. It is noticeable that the non-unique source allocation method results in a smaller acoustic pressure magnitude compared with the results presented in Figs. 7 and 8 (Morshed et al. 13 ), using the unique source allocation method. The reason for this is that for each frequency band of interest, the source strength of each source calculated using the non-unique source allocation method was less than the source strength of a single equivalent point source calculated using the unique source allocation method (see Table 2 for source strengths calculated using unique source allocation method, in Morshed et al. 13 ). 14

15 Fig. 4 Analytically calculated overall sound pressure level at the RSLVF surface, using the non-unique source allocation method, for ten individual point sources along the exhaust flow, for each one-third octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz. [x 1 = 15D e, x 2 = 5D e and reference pressure 20µPa] (a) Front face, 50 Hz. (b) Rear face, 50 Hz. 15

16 (c) Front face, 100 Hz. (d) Rear face, 100 Hz. (e) Front face, 400 Hz. (f) Rear face, 400 Hz. Fig. 5 Numerically calculated sound pressure excitation at the surface of the RSLVF at each one-third octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz, using the non-unique source allocation method. [x 1 = 15D e, x 2 = 5D e and reference pressure 20µPa] 16

17 Fig. 6 Numerically calculated sound pressure levels as a function of RSLVF circumferential location for each one-third octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz, using the nonunique source allocation method. [ z = 2.17m, x 1 = 15D e, x 2 = 5D e and reference pressure 20µPa] Fig. 7 Analytically calculated sound pressure levels at the RSLVF surface, for an equivalent single point source for each one-third octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz, using the unique source allocation method. Results taken from Morshed et al. 13.[z = 2.17m, x 1 = 15D e, x 2 = 5D e and reference pressure 20µPa] 17

18 Fig. 8 Numerically calculated sound pressure levels as a function of RSLVF circumferential location, for each one-third octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz, using the unique source allocation method. Results taken from Morshed et al. 13. [z = 2.17m, x 1 = 15D e, x 2 = 5D e and reference pressure 20µPa] Table 2 Calculated source strength for Engine E for each 1/3 octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz, using the unique source allocation method. Overall acoustic power level, L w = dB; speed of sound and density in air at T = 1000 o C are m/s and 0.28 kg/m 3 respectively. Source Strength, Frequency, Hz Qs,b 10 7 (m 3 /s) IV. Conclusions The acoustic noise generated on a launch vehicle fairing during launch has been estimated for a particular chemical rocket Engine E, which has the Mach number of 4.07 and the core length of 5.15m. The engine exhaust noise has been assumed to originate along the exhaust flow, perpendicular to the vehicle axis, and has been modelled using the non-unique source allocation method to determine the acoustic pressure loading at the surface of the Representative Small Launch Vehicle Fairing (RSLVF) at each one-third octave band corresponding to centre 18

19 frequencies of 50Hz, 100Hz and 400Hz, respectively. The non-unique source allocation method assumes that the exhaust flow can be represented by a number of point sources distributed through the flow for each frequency band of interest. Using this method, ten individual point sources were arbitrarily positioned close to the vehicle. The analytical and numerical results showed that the overall sound pressure level reaches around db and db, respectively, for the band centred on frequency of 400 Hz, for ten individual point sources situated close to the vehicle acting as a line of point sources along the exhaust flow. The numerical results produced similar pressure fluctuations, at each band of interest, to the fluctuations in the analytical results. There were small differences in the pressure magnitudes between the results obtained analytically and numerically because of diffraction around the edges of the Representative Small Launch Vehicle Fairing (RSLVF) which was taken into account in the Boundary Element Method (BEM) analysis in contrast to the results calculated analytically. Therefore, the pressure magnitudes of the results obtained analytically were slightly different to the results obtained using the Boundary Element Method (BEM). Despite this little difference, it can be seen that for non-unique source allocation technique the results obtained using the developed BEM and analytical tools agree well. These can be the necessary inputs to evaluate noise and vibration control treatments for launch vehicles It has been observed that the non-unique source allocation method results in a smaller acoustic pressure magnitude compared with the unique source allocation method. The reason for this is, for each frequency band of interest, the strength of each source calculated using the non-unique source allocation method was less compared with the strength of a single equivalent point source calculated using the unique source allocation method. It is intended to extend the present work in the future by including the effects of diffraction around the ends of the cylinder in the analytical model. This will enable the development of a full analytical model of acoustic loading on finite length cylinders, representative of geometries such as cylindrical launch vehicles. Appendix Table A1 Details of the ten sources for Engine E. Core length, x t = 5.15m; overall acoustic power level of the rocket engine, L w = dB; length of each segment, x = 1m. [Data estimated from Figure 12 in NASA-SP ] Source Number Estimated Source Distance, x 3 (m) Source Distance From the Nozzle, r (m) Distance per unit Core Length, x 3 / xt Elevation Angle, β (Degrees) Directivity Angle, θ (Degrees) Estimated Directivity Index, DI (db) Estimated Sound Power per Unit Core Length, xtw ( r) 10log10 WOA (db) Calculated Acoustic Power per Segment, L w, seg (db) 19

20 Table A2 Estimated relative sound power level for Engine E for ten sources for each 1/3 octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz. Core length, x t = 5.15m; overall acoustic power level of the rocket engine, L w = dB; speed of sound in the exhaust flow, a e = m/s; length of each segment, x = 1m. [Data estimated from Figure 13 of NASA-SP ] Source Number W ( f, r) U Estimated relative sound power spectrum level, ea 10 log o 10, db W ( r) rae Hz Hz Hz Table A3 Calculated acoustic power level for Engine E for ten sources corresponding to ten segments for each 1/3 octave band corresponding to centre frequencies of 50 Hz, 100 Hz and 400 Hz. Core length, x t = 5.15m; overall acoustic power level, L w = dB; speed of sound in the exhaust flow, a e = m/s; length of each segment, x = 1m. Calculated acoustic power level in each band for each segment, L w, seg, b (db) Source Number 50 Hz Hz Hz Table A4 Descriptions of sixty circumferential nodes at a height of z = 2.17 m from the bottom face of the RSLVF. Node Azimuthal Node Node Azimuthal Node Node no. angle,ϕ order no. angle, ϕ order no. Node order Azimuthal angle, ϕ (Degrees) (Degrees) (Degrees)

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