Second Computational Aeroacoustics (CAA) Workshop on ...

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+ 1- j K(of)=_[p,c, a 2 (ki+ocM, )2 W_(o_)-P2c_(k2+aM2) 2 W2(00]. (14) TM?',L In deriving K(o0, we have used the condition of the continuity of the acoustic pressure and acoustic displacement at the mean flow mismatch at r = a, for x >0. K(a) is factorized into two, one is analytic in the upper half plane (+) and the other in the lower half plane (') as K(a)= K÷ (o_). K (o0. These factors will be included in the final solutions with arguments representing physical quantities. For the present problem, the factorization is obtained not in closed forms but in integral representations a s follows: 1____, LogeK(a) do_ ' Log
where f_,(O) = (-i) ' J=(vm, )p24 (1-v2 cosO')2 _ TM(1-M22) '/2 .{K÷(_rl(O)).[k+ e _r/(0)].t( k_ 1+ M z 7"/(0) = kz(c°s0'- M2 ) 1- M_ Here the modified coordinates R' and 0' are defined as (2I) R'= I r24 1-Mff X2 ' • sin0. m' k2asinO' /, +,I/
Page 1 and 2: NASA Conference Publication 3352 <s
Page 3 and 4: .Y NASA Conference Publication 3352
Page 5 and 6: PREFACE Thisvolumecontainstheprocee
Page 7 and 8: ORGANIZING COMMITTEE This workshop
Page 9 and 10: CONTENTS Preface ..................
Page 11 and 12: Large-Eddy Simulation of a High Rey
Page 13 and 14: Problem 1 Benchmark Problems Catego
Page 15 and 16: Problem 4 This is the same as Probl
Page 17 and 18: , / circilar duct I incoming sound
Page 19 and 20: Computational __ D
Page 21 and 22: where ANALYTICAL SOLUTIONS OF THE C
Page 23 and 24: The Greens function for (18)- (19)
Page 25 and 26: where b = In2/w 2. Boundary conditi
Page 27 and 28: SCATTERING OF SOUND BY A SPHERE: CA
Page 29 and 30: where h O) (z) is the spherical Han
Page 31 and 32: RADIATION OF SOUND FROM A POINT SOU
Page 33 and 34: ef. 6. Theincident pressureemitted
Page 35 and 36: 1 -g (Y3) lim .... - _ (X 3) - _ _
Page 37 and 38: 7. , , . Martinez, R., "Liner Dissi
Page 39 and 40: EXACT-SOLUTIONS FOR SOUND RADIATION
Page 41: n=l Here R,.m is the conversion coe
Page 45 and 46: ,i Log e K(ot) da, with b_+< q_+< b
Page 47 and 48: As a approachesy, onereadilyobtains
Page 49 and 50: It follows thenthat, as7/(0)approac
Page 51: Ira(a) J . :::--::::::::_'.-.::, a=
Page 54 and 55: n -3 -2 -1 0 +l 72 -6 -5 -4 -3 -2 -
Page 57 and 58: Application of the Discontinuous Ga
Page 59 and 60: eference1. In addition, the allowed
Page 61 and 62: c_ = i. $/Igl, and fl = j-_/l_l. Ea
Page 63 and 64: (a). Pressurecontour at t = 10. P 0
Page 65 and 66: i 0.01 : " o.o - _;" .:y I -0.01 _
Page 67 and 68: ...... r=5.5 --r=7.5 8e-06_ A /_ l
Page 69 and 70: COMPUTATION OF ACOUSTIC SCATTERING
Page 71 and 72: five-stage Runge-Kutta scheme (Hu e
Page 73 and 74: processing.Theunit processingtimewa
Page 75 and 76: lo ! 51- o' -1o 10 0 -10 I , •
Page 77 and 78: 0 ] ' ' ' f i , , , -10 -5 0 5 10 1
Page 79 and 80: 0. _5 i 5,0000E-6 O.O000EO -5.O000E
Page 81 and 82: o 13 soLvT ON Acoustic S ,ERIN PROB
Page 83 and 84: Gaussgrid, Xj+I/2 , mapped onto [0,
Page 85 and 86: The coefficients,a x and aY determi
Page 87 and 88: -4 -2 0 2 4 Figure 2: Subdomain dec
Page 89 and 90: four around the cylinder. The PML e
Page 91 and 92: Application of Dispersion-Relation-
Page 93 and 94:
where damping factor v = eAr × A9
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O x 10-l° 4, , , , , .... i 3.5 1_
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DEVELOPMENT OF COMPACT WAVE SOLVERS
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supportedthat C3N is long-time stab
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Figure 2 shows intersectionsof hori
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0.08 0.06 0.04 0.02 -0.02 -0.04 0 C
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OV 10p --+---=0 0t 7- 00 Op 1 c)(rU
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parallel to the radial direction, s
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¢-_ 4e--10 3e--10 2e--10 le--10 0.
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[3L- 0.06 0.04 0.02 0.00 --0.02 ...
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over ahalf-plane.Theequationsusedar
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At the computational boundaries, fl
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esolution of the grid causing the s
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0.08 q'- I ' "'I ..... T--F --[ I I
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"o_ 0.05 0.04 0.03 0.02 0.01 -0.01
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Forbothproblems(1& 2 of Category1),
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0 -3 -6 -9 -12 -15 -18 -21 -24 -27
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0.07 0.06 0.05 0.04 0.03 0.02 0.01
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•sy" _'J ApplicationAbsorb,n..oun
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First, the grid pointswill be overc
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2.3.1 Results of Problem 1 2.3 Nume
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Ov Op 0--t-+ Or = 0 (7.2) Op Ou 0;,
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To absorbthe out-goingwavesat thefa
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4.2 Inflow condition At the inflow,
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and u-velocity contours. In the vel
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mtel r BPouMLary Condition Figure 1
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0°0005 0.0004 0.0003 0.0002 0.0001
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5 0 -5 Figure 7a. 0 -5 Figure 7b. i
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008 006 0,04 i 0.02 O,., 0.0 13, .0
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i / Figure 13. Pressure contours at
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100 i0 -I 10 -z i0"3 10 .4 lO-s Fig
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100 i0 "] i0 "_ e_ i0 "s 10 -4 10 -
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4.0 3,0 2.5 2.0 0.0 4.0 _-._-,_ 3.0
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5.e-07 4.5e-07 4.e-07 3.5e-07 ¢_¢
Page 163:
5.e-07 | 4.5e-07 I 4.e-07 [ _ 2.5e-
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1 + 3 + (a) (b) Figure 1" Basic sec
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Unfortunately the resulting schemem
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Figure 3: A typical (but rather coa
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! -10.0 -3.3 3.3 10.0 Figure 5: Pre
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0.07 0.06 0.05 0.04 0,03 0.02 0.01
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CONCLUSIONS _Te have presented a pr
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The methodis basedon a least-square
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For the linear wave problem of Cate
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FIG.1 PROBLEM 2 OF CATEGORY 1 - ACO
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0 0 0 0 0 0 0 0 ! (e'£'],)d 172 -4
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"Z3 c_ E < Q "13 ii i {3. E < Fig.
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200 -100 FIG. 7 TIME HISTORY OF AN
Page 191 and 192:
ABSTRACT ADEQUATE BOUNDARY CONDITIO
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}Ve note that each v (k) is determi
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FORMULATION OF THE PROBLEM AND THE
Page 197 and 198:
where APPENDfX We considerthe follo
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0.06 0.01 -0.04 0.06 0.01 -0.04 m w
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i 0.05 - -0.00 - \ 0.05 - -0.00 - -
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ditions. The formulation and implem
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J ! s J 18D 4D 5D Figure 1. <strong
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10 -10 rm 3.0 2.5 2.0 1.5 1.0 0.5 .
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3. CATEGORY 2, PROBLEM 2 The axysim
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off the axis. Such a change often r
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In the space" below, we will concen
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3.5. Numerical Results 1 I 1 Three
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p(x) 2.0 1.5 1.0 0.5 0.0 2.0 1.5 1,
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p(x) 1.5 1.0 0.5 0.0 1.5 1.0 0.5 0.
Page 222 and 223:
For the case w " _s_, there are thr
Page 224 and 225:
where E is an (M + 1) x 5 matrix an
Page 226 and 227:
The v-velocity component in the out
Page 228 and 229:
Figure 17 shows the calculated pres
Page 230 and 231:
10-6 4.0 3.0 1.0 0.0 0.0 1.0 2.0 3.
Page 233 and 234:
7'/ L/3
Page 235 and 236:
w E W C w s S [.;sing this ceil-cen
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neighbors are involved. Both of the
Page 239 and 240:
3e- 10 2.5e-10 ._ 2e-10 e-, • 1.5
Page 241:
Future work involves the inclttsiot
Page 244 and 245:
NONLINEAR, HYBRID CODE The hybrid d
Page 246 and 247:
z agree better now although there a
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Figure 1: Snapshot of the acoustic
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-11 -12 -13 --% oT .9o -14 -15 Cat
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THREE-DIMENSIONAL CALCULATIONS OF A
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the IBM SP2. The computational doma
Page 257 and 258:
intersects the center of the sphere
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ON COMPUTATIONS OF DUCT ACOUSTICS W
Page 261 and 262:
It can be easily seen that duct aco
Page 263 and 264:
the duct diameter D for problem 1 a
Page 265 and 266:
To fix "this problem, a multi-domai
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60.0 , _ , , ,. _- f , 40.0 j 20.0
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0.004 [-- ' r .... , - , ," . ....
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w .................... 0..IJ'_l._ .
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{u} B= P 0 U M_v { M_p+w } ooP D= 0
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aseline case, as it employed a very
Page 278 and 279:
the incoming wavesolution from the
Page 280 and 281:
P(z) 1.0 Pressure Envelope ((o=7.2)
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v.rti_'al ¢li_tlllb;tl_('v. ;t11,1
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,l,>mairlOD. the,_4,_'r_ll_[ mt_'_L
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This is r,',',,g1_iz_',l a,,__zl ,q
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i ..J 2o i ¢.5 04 -2 0 fj,o_ DDD \
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2 8 ]or o 0,0 02 0.I 0.6 \ Circumfe
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Numerical Algorithm Equations (1) c
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The results are divided into differ
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0.012 0.01 0.008 0.006 0.004 0.002
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2 ... - -_ Patternrepeats I -- [ --
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-- o26 -7/ COMPUTATION OF SOUND GEN
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2.0 1.0i CI (Im.)o.o i ; -1.0 -2.0
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120 100 SPL, dB (re: 20 _Pa) 8O L
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REFERENCES 1. Lighthill, M. J., "On
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Perform numerical simulations to es
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[ RANS - Dirichlet B.C [ NNNNe , Pe
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.? _i, _ 80 .... 6O O0 02 04 0.6 0.
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A VISCOUS/ACOUSTIC SPLITTING TECHNI
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Ov' _" v o%' u'v' , OV v' OV Uv' Vu
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2 2 2 2 +L_ °_ _[_-_)J--T[_+d _) T
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corrector method was chosen. It is
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where # = p', u',v',p'. At the oute
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Finally, the latter portions of the
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p' o oo_ 9'3 t3 O3 75 7O 65 Nearfie
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LARGE-EDDY SIMULATION OF A HIGH REY
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In the implementation of the modeli
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to Cs = C 1/2 = 0.1. These LES gave
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a) c) ! • t r 2 i " i Figure 2. C
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t))
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A COMPARATIVE STUDY OF LOW DISPERSI
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Naturally, a left upwinded formula
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where F and G are flux vectors, and
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temporal integration of the semi-di
Page 349 and 350:
vent reflected numerical waves from
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0.5 0.4 0.3 0.2 0.1 -0.1 0.6 0.4 0.
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13- Q- 0.2 0.15 0.1 0.050 [ -0.05 -
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1 e-06 9e-07 8e-07 7e-07 6e-07 5e-0
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133 cL cL 0.008 0.006 0.004 0.002 -
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0.0014 0.0012 0.001 0.0008 0.0006 P
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OVERVIEW OF COMPUTED RESULTS Christ
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10 -10 13.0 _i,i_ Illlllll, H _l,,l
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0.05 0.00 -0.05 0.05 0.00 -0.05 0.0
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0,05 0.00 -0.05 0.05 p(t) 0.00 -0.0
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SOLUTION COMPARISONS. CATEGORY 1: P
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_-12 8 Cat 2, Prob 1 " [_ Myers (BE
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p2=D(8) o.010 0.009 0.008 O.O07 0.0
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_:D(O) 0,12 0.11 0.10 0.09 0.08 0.0
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i° o. $ £ ,./ q 0.00 I i 1 1 0.20
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er .< o
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a- < (J O O ¢q o O O d J J o 0.00
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SOLUTION COMPARISONS: CATEGORY 4 Ja
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presenceof thewind tunnelwalls. The
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i == Slrul and Pylon Fan-OGV Pressu
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z >, O I.O O O O O (:3 O Some of no
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