1、NAS.i Contractor Report 3966 Jet Shielding of Jet Noise John C. Simonich, Roy K. Amiet, and Robert H. Schlinker CONTRACT NAS1-16689 APRIL 1986 i:i, Y t r RC:SE.C;-: :E: a from the axis and the observer is at a distance R a from the axis. The z component of the distance between the source and observe
2、r is ZOo Summary of the Approach - The analytical approach used in the present report and in the studies of Gerhold (refs. 17, 18) employs a standard formulation used by previous authors (e.g., Mani (ref. 23) in the study of shear layer refraction. The method begins with the wave equation for the Pr
3、ovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-velocity potential inside and outside the shielding jet. Since the jet flow field conditions in the z direction are constant the wave equation can be easily Fourier transformed. This simplifies the analys
4、is by eliminating the z variable until the inverse transform is performed. Assuming a sinusoidal dependence for the acoustic source and identifying a separation of variables results in a Helmholtz equation in terms of the variables R, e and w. The source field in these variables is given by Equation
5、 8, below. This source field impinges on the shear layer leading to transmitted and reflected waves given by Equations 10 and 11. The unknown amplitudes of the transmitted and reflected waves are determined by matching pressure and fluid displacement at the jet boundary. A Fourier inversion is then
6、performed on kz to obtain the final acoustic pressure result. To avoid performing the inversion numerically, the observer is assumed to be located in the far field allowing the integral to be approximated analyti cally. This approximation is based on the method of stationary phase which is commonly
7、used to make such far-field approximations. The details of the analysis are given below. General Expression for the Far Field Sound - The wave equation for the velocity potential outside and inside the jet flow field is J2 4 + ( Q a) Outside ci : + 8 (x - X 0) 8 (y) 8 (z) 0 (1) b) Inside 2 . M 2 cp
8、w2 cp V c:p - 2 I W C Z - M Zi + CZ : 0 I I (2) The velocity potential is related to the pressure field by the linearized momentum equation p: - Po where ko = w/co and a time dependence above three equations. (Gerhold uses are complex conjugates of the results Ocp Ot exp (iwt) has exp (-iwt) so foun
9、d here.) (3) been assumed in the that many of his results If the velocity potentials in Equations 1 and 2 are decomposed into their spatial Fourier components 21 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-22 CD-cp = J cP (kZ)eikzZ dkz -CD these
10、equations then become a) Outside b) Inside where -(722 + K02) 4 = QO 21TPO 8 (x-Xo) 8 (y) 2 2“ (72 + K I) cP : 0 2 02 02 72 = o X2 + Oy2 k,= ( I + Mk z) Ko = .j k2 _ k 2 o z K= /k2 _k2 I V I Z (4) (5) (6 ) (7) It should be noted that kz in Equation 7 employs a plus sign in the defini tion of kl wher
11、eas Gerhold uses a minus sign. This difference in the defini tion of the Fourier transform results in kz in Equation 26 being negative while Gerhold defines it to be positive. The solution of these equations can be written as a combination of 1nC1-dent, transmitted and reflected waves. The source gi
12、ven in Equation 1 produces an incident wave which can be represented as n= where Q Em COS m 8 m=O Em = 1m: 0 :2 m;tO (2) J m (Ko R) H m (Ko Ro) (2) Hm (Ko R) Jm (Ko Ro) Q-1Qo Ro: Xo 87TPO R Ro (8 ) (9) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
13、Here Ro is the distance of the source from the jet axis and R the distance of the observer from the jet axis; i.e., the field point. (See Equation 7.3.18 of reference 36). The reflected and transmitted waves can now be written as R ro = L m=o (2) Am Hm (KoR) cos m e (IO) “ ro CPr = mo 8m Jm (K ,R) c
14、os m 8 (I 1) Boundary Conditions - The constants Am and Bm in Equation 10 and 11 are determined by matching boundary conditions at the jet interface R = a. First, the pressure must be continuous at R = a. From Equations 3 and 4, pressure is related to by p= - ip,C, k, 4 R RO From Equations 9-11, Po
15、Co ko 1 m 0 a m cos m e J m (K 0 0 I H ,2) ( Ko R 0 I + Am H.i2) (K 0 0 I COS m eli ro =p, c, k, L 8m Jm (K,o) cos me m= 0 (13) Because e is a variable, this equation must match on a term by term basis, or (2) (2) a Em JmKoO) Hm (KoRo) + Am Hm (Ko 0) = C, kl p! Co ko Po 8m Jm (K, 0) (4) Also, there
16、must be continuity of shear layer displacement at R = a. Note that this condition is not the same as continuity of radial velocity, an error that appeared in analyses prior to the works of Miles (ref. 14) and Ribner (ref. 15) (The continuity of displacement and velocity are not the same because ther
17、e is a zero order velocity jump across the shear layer). Satisfy ing continuity of displacement for the interface requires determining the 23 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-speed at which the disturbance travels along the jet shear l
18、ayer. The Fourier component, exp (i wt + i kz z), of the potential function is a wave which travels with velocity given by V : - w/k z Z (5) In a coordinate system moving with the wave this wave component remains steady in time. In this coordinate system the fluid velocity inside the jet 1S Vi: V+ w
19、/kz (6) while the fluid velocity outside the jet is Vo = w/kz on Continuity of shear layer displacement is insured by equating the slopes of the flow vectors across the shear layer, i.e., the perturbation radial velocities divided by Vi or Vo respectively. Thus, from Equations 9-11. ,(2) (2) PI C I
20、k I , a Em Jm (Ko 0 l Hm (KoRo + Am Hm (Kool: Bm T k J m (K I 0) (8) Po Co 0 where 2 2 ko Po Co K, T= k ,2 PI C 12 Ko 09 ) Equations 14 and 18 are two equations for the two unknowns Am and Bm Elimination of Bm gives (2) I , _ a Em Hm (Ko Ro) - TJm (K, 0) Jm (Koa) + Jm (Koa) Jm (Kia) Am - (2) (2), -Hm (Koa) Jm(K, 0) + Hm (Koa) Jm (K,a) T (20) 24 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
