Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, May 1996A9630844, AIAA Paper 96-1731

Large-eddy simulations of sound due to turbulence mixing inside an ejector

Thomas Z. DongNASA, Lewis Research Center, Cleveland, OH

Reda R. MankbadiNASA, Lewis Research Center, Cleveland, OH

AIAA and CEAS, Aeroacoustics Conference, 2nd, State College, PA, May 6-8, 1996

Large-eddy simulations of sound and flow fields generated by turbulence mixing inside an ejector nozzle are carriedout. A circular ejector configuration with a single-element primary nozzle is adopted here. The Favre-filteredNavier-Stokes equations with the Smagorinsky sub-grid scale model are solved by a high-accuracy finite differencescheme to study both the turbulence flow and the sound fields. Nonreflecting acoustics and instabitity boundaryconditions are used at the inlet as well as the exit of the ejector to eliminate the nonphysical reflections. (Author)

Page 1

96-1731

A96-30844

LARGE-EDDY SIMULATIONS OF SOUND DUE TO TURBULENCE MIXING INSIDE AN EJECTORf

Thomas Z. Dong°)NASA Lewis Research Center, MS 5-9, Cleveland, Ohio 44135

Reda R. Mankbadi*)NASA Lewis Research Center, MS 5-9, Cleveland, Ohio 44135

ABSTRACT

Large-eddy simulations of sound and flow fieldsgenerated by turbulence mixing inside an ejector noz-zle are carried out in the present paper. A circularejector configuration with a single-element primarynozzle is adopted here. The Favre-filtered Navior-Stokes equations with the Smagorinsky sub-grid scalemodel are solved by high accuracy finite differencescheme to study both the turbulence flow and thesound fields. Non-reflecting acoustics and instabilityboundary conditions are used at the inlet as well asthe exit of the ejector to eliminate the non-physicalreflections.

1. Introduction

The mixer/ejector nozzle is a new and promis-ing concept for jet noise reduction. The idea behindthis concept is to mix a large amount of ambient airwith the flow from the core engine exhaust to reducethe jet velocity and the associated noise while stillmaintaining the thrust. Many different mixer/ejectorconfigurations have been developed and studied inthe past1"2. The primary goal of the present workis to study the sound generated by the turbulencemixing and the sound interactions inside a circularejector nozzle shown in Figure 1 by direct numericalmethod. The axisymmetric computations are carriedout as an initial approach. Due to the wide range ofscales present in the high Reynolds number turbulenceflow of the present problem, a large-eddy simulationapproach is taken. The large turbulence scales arecalculated by the Favre-filtered Navior-Stokes equa-tions while the small scales are modeled by a sub-gridscale model, developed by Smagorinsky3, as a source

of dissipation. It is believed that when the flow fromthe primary nozzle is supersonic the mixing noise ismainly generated by the large scale turbulence4"7.References regarding to large-eddy simulation withsub-grid scale modeling can be found in8"12. In or-der to compute the mean flow and the noise accu-rately, the Dispersion-Relation-Preserving (DRP) fi-nite difference scheme with a selective artificial damp-ing method developed by Tarn et al 18~15 is adopted.Proper and accurate boundary condition is anotherkey factor in order to correctly compute the turbu-lence and acoustic solutions in the present paper.A detailed discussion of numerical discretization andboundary conditions with their numerical implemen-tations are included in Section 3 and 4 respectively.The large-eddy simulations and results are presentedin Section 5.

2. Governing Equations

We introduce a spatial filter

dz (1)

where K is a spatial filter function, A is the compu-tational mesh size, and D is the flow domain. Uponapplying this filter to the flow variables, the flow fieldcan be decomposed into

f = f + f (2)

where the overbar denotes the filtered field and theprime denotes the subgrid one. The Favre filter isdefined as

f=4 (3)

\ This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States.*' This work was performed while the first author held a National Research Council-(NASA Lewis Research Center) Research

Associateship. Member AIAA"I Senior Scientist, Leader, Computational Aeroacoustics. Associate Fellow AIAA

1

Assuming axisymmetry, the filtered Navior-Stokesequations in cylindrical coordinates take the form

G~

where

Q = ffv

puv - »„u - wmx -

- varrr + qr .

—<Tg0

(4)

(5)

(6)

(7)

(8)L 0 J

Here u and v are the velocity in x and r directionsrespectively,

&ij = &ij + 1*j (9)

and9k = 9k (10)

The turbulence sub-gird stress terms TV,- and heat fluxterms *t, according to Smagorinsky8 and others*-10,are mq deled as

and

(11)

(12)

where ey are the Favre filtered strain rate tensors, C,is the specific heat at constant volume and Pt is theturbulence Prandtl number. The effective turbulenceviscosity vt is modeled by

where the filter width A/ is given by

A, = (ABAr)*

(13)

(14)

and C, is the so-called Smagorinsky's constant. Inthe present work, these turbulence parameters arechosen to be C, — 0.1 and Pt = 1.0 according toreferences8"12.

3. Numerical Discretization

With the DRP finite difference scheme andthe selective artificial damping method the governingequations in (4) are discretixed as:

(n)

j=-S

1 ~(«)

»m

/*«

r/,m T ̂ ,m

,o(»).d*J* (15)

_ n(*0 (16)3=0

The last two terms in (15) are the artificial dampingterms where /i0 is the artificial viscosity. All the abovecoefficients can be found in references18"16.

4. Boundary Conditions

4.1. Wall Boundary ConditionOn the solid surface of the primary nozzle and

ejector, only the slip boundary condition V • n^au =0 is imposed. The numerical treatment of solid wallboundary conditions derived by Tarn and Dong16 forhigh order finite difference schemes with the minimumnumber of ghost values is used here to maintain thehigh order accuracy of the interior scheme. Interiordifferencing technique is used to avoid introducing un-necessary ghost values. The coefficients of the Interiordifferencing are derived in a way consistent with theinterior DRP scheme.

In the present paper the boundary condition onthe surface of the acoustic liner or soft wall is notimplemented, namely all the wall surfaces are solid.

4.2. Outflow Boundary ConditionTo let the large structure turbulence or the insta-

bility waves in the mixing layer as well as the acousticwaves leave at the exit of the ejector without generat-ing noticeable reflections, the outflow boundary con-ditions for instability waves derived by Tarn et al19 is

imposed there. The boundary equations can be writ-ten as

(17)tt-ev-f

Lp-plwhere the overbared quantities are the time indepen-dent mean quantities and x is in the axial directionof the jet. The constants Co and C\ are obtainedfrom the dispersion relation of the instability waves.In the present computation only the first order deriva-tive terms are used. Co is then the phase speed of theinstability wave. Assuming the mixing layer is formedby two layers of fluids moving at speeds of u\ and «j(itj, > 1*2). Co is approximated by «a -t- 0.7(«i — «j) .

4.3. Inflow Boundary ConditionsSince the flow is supersonic at the inlet of the

primary nozzle, the flow variables there are specifiedas

"Piui0

Pi

exp(i(kx-ui))} (18)

where the first part of the sum is the uniform jetflow condition inside the primary nozzle and the sec-ond term is the internal noise source with ampli-tude e. This internal source term in the presentsimulations is used to excite the instability waves orthe large structure turbulence in the mixing layer.(/>(/«•),«(»«•), »(*«•),p(/cr))r in the source term is theeigenvector of duct acoustics in r-direction which in-volves Bessel functions for circular ducts. For a givenangular frequency u and a wave number in r-directionK the wave number in r-direction Jb can be computedfrom the dispersion relation of the duct acoustics

<19>

where Mj is the jet Mach number. Details about ductacoustics can be found in18

At the inlet of the by-pass channel where flow issubsonic, the first order absorbing boundary conditiondeveloped by Engquist and Majda20 is used as follows:

d_at + (*-»)£

P-P1u-nV-9p-pl

= 0 (20)

where the overbar denotes the time independent meanquantities and 5 is the speed of sound in the mean flow.

5. Simulation and Results

5.1. Physical and Computational SetupsThe computation domain of the present simula-

tions is shown in Figure 2. Both primary nozzle andejector are circular with diameters D and 3D respec-tively. The length of the nozzle is one D while thelength of the ejector shroud is 6D. The total tem-perature of the primary nozzle flow and the by-passchannel flow are assumed to be the same. The com-putation starts with two uniform initial conditions inwhich the Mach number of the jet Mj is chosen to be1.5 and the Mach number of the co-flow MI, is chosento be 0.4. The jet from the primary nozzle is assumedto be perfectly expanded, namely the static pressureof the jet PJ is equal to the one of the by-pass flow piin the initial conditions.

The flow variables are non-dimensionaliied bythe following characteristic scales:

length scale =D (primary nozzle diameter)velocity scale =Uj (jet speed )

time scale =D/Uj (21)density scale =PJ (jet density)

pressure scale =PJU*

In the present simulation the Reynolds number Re =2i£z£ is equal to 1000, the Piandtl number Pr = c-£-

Py *is 0.72 where Cp is the specific heat at constant pres-sure and k is the coefficient of heat conductivity, andthe dimensionless frequency Strouhal number St =gg- is taken to be 0.2. The corresponding period T*w«jis equal to 5 characteristic time units. In the sourcefunction in equation (18), only the first duct mode inr-direction and the faster downstream moving modeis taken. So the source acoustic wave is uniform in rdirection, e is taken to be 0.025.

The mesh size is chosen to be A« = A*1 =D/33. The dimensionless ratio of Ai/Az is chosento be 0.0125 to ensure the numerical stability and lownumerical dissipation. The dimensionless artificial vis-cosity in equations (9) is taken to be /*„ = 6 X 10~8 tosuppress the high frequency spurious solutions. Thischoice of /*„ has very small effect on acoustic waves

with wave length longer than 6&x as indicated inreference14

5.2. Results and Analysis.The simulation is carried out on IBM RS/6000

workstations. After the transient has left the com-putational domain, a periodic stage is reached. Themean flow solution is then computed by averagingthe solutions over two periods (2T). The root-mean-squared pressure fluctuation Prm is computed by tak-ing the maximum of \p—pme<m \ over a period (T). Thewhole procedure takes about 8 hours of CPU timewith 3/4 of it time driving the transient out of thecomputation domain.

The Mach number distribution of the mean solu-tion is plotted in Figure 3. It is observed that a mixinglayer has developed and its thickness is growing down-stream. The growth or the spreading of the mixinglayer is also shown by the u-velocity profile at variousdownstream locations plotted in Figure 4. Figure 5shows the n-velocity profile along the inlet boundary.A non-uniform entrainment of ambient air in radialdirection is observed with the strongest entrainmentoccurs near the surface of the primary nozzle. Thesefigures have shown qualitatively correct trends of themean solutions. It is also noticed that a special wavepattern is formed inside the jet due to the interactionbetween the source acoustic waves and the instabil-ity waves in the shear layer. Even though that thejet from the primary nozzle is set to be perfectly ex-panded, the low frequency source acoustic wave hasactually generated a shock wave during its expansionstage and an expansion wave during its compressionstage at the exit of the primary nozzle. Weak shockstructures are, therefore, formed and trapped insidethe jet. Because of this periodic switch from shock toexpansion wave at the nozzle exit, the shock structureinside the jet is not as regular as those observed in anunder- or over-expanded supersonic jet. The strengthof this shock structure decays when it moves down-stream as it is seen in Figure 3. This structure can alsobe seen in the root-mean-squared pressure and axialmomentum fluctuation fields Prm* and purm, plottedin Figure 6 and 7. Figure 8 shows the root-mean-squared pressure fluctuation distributions at severalradial locations. It can be seen that the amplitude ofthe huge structure turbulence inside the mixing layergrows downstream. The wriggles on the Prm, right

inside the mixing layer (r/D = 0.5) are caused by theinterference of the weak shock structure inside the jetmentioned above. It is also noticed that the directinfluence of the weak shock structure vanishes outsidethe mixing layer. The power spectrum of the axialvelocity inside the mixing layer (r/D = 0.5) com-puted at various downstream locations is plotted inFigure 9. The spectrum is obtained by taking dis-crete Fourier transform of the time derivative of axialvelocity «. So the mean is automatically dropped out.It can be seen that the harmonics of the principle fre-quency are generated due to the non-linearity. Figure10. shows the instantaneous p — pmetm distribution atnine stages of a period in the region above the mixinglayer (r/D > 0.5). It again shows that the Kelvin-Helmhotz instability waves generated near the lip ofthe nozzle inside the mixing layer are amplified whilethey travel downstream. The speed of the K-H insta-bility waves CK-B agrees very well with the approx-imation Co in equation (IT). It also shows that anup-stream traveling acoustic wave is generated whichpropagates through the by-pass channel against themean flow. This acoustic wave is believed to be gen-erated by the interaction of instability waves and theweak shock structures inside the jet, namely it is theshock associated acoustic wave. No noticeable reflec-tions from the ejector exit or from the by-pass channelinlet are observed from Figure 10. This indicates theaccuracy of the chosen non-reflecting boundary con-ditions.

Conclusion

A large-eddy simulation of turbulence mixing in-side an ejector is studied in the present paper. Thesuccess of this large-eddy simulation has provide a toolfor studying the high speed, high Reynolds numberturbulence flow and the generated noise. The primaryresults show that the simulation has captured the cor-rect features of the turbulence miring and the acous-tics. The high order DRP finite difference schemeand the acoustics and instability wave boundary con-ditions implemented in the present simulation haveplayed an important role in getting the correct so-lutions. Due to the lack of experimental or analyt-ical data to compare with, the assessment can onlybe made qualitatively. Future investigation is needed

and could include the quantitative solution validation,extension of the computation domain the free field tocompute the fat field jet acoustics and higher Reynoldsnumber computations.

References

1 Lord, W.K.; Jones, C.W.; Head, V.L.; Krejsa,E.A.: "Mixor ejector nossle for jet noise suppre-rion", AIAA paper 90-1909, July 1995, Orlando.

2 Tfflman, T.G., Paterson, R.W., Press Jr., W.M.;"Supersonic Nozzle Mixor Ejector", J. of Propul-sion and Power, Vol. 8, No. 2, March-April1992, pp. 513-519.

8 Smagorinsky, J.: "General Circulation experi-ments with the primitive equations, I The basicexperiment", Monthly Weather Review, Vol. 91,pp. 99-164, 1963

4 Mankbadi, Reda R.: Transition, Turbu-lence, and Noise", Kluwer Academic Publishers,Boston, 1994.

5 Mankbadi, Reda R.; Hayder, M.Ethesham;Povinelli, Louis A.: "Structure of supersonic jetflow and its radiated sound", AIAA J., Vol.32,No.5, May 1994.

e Mankbadi, Reda R.; Shih, S.H.; Hixon, R.;Povinelli, L.A.: "Direct computation of soundradiation by jet flow using large-scale equa-tions", AIAA paper 95-0680, Jan. 1995, Reno.

7 Tarn, C.K.W: "Jet noise generated by large-scalecoherent motion", Chap. 6 of Aeroaconstics ofFlight Vehicles: Theory and Practice, NASARP-1258, 1991.

8 Erlebacher, G., Hussaini, M.Y., Speziale, C.G.;"Toward the large-eddy simulation of compress-ible turbulent flows", J. Fluid Mech., (1992) Vol.238, pp. 155-185.

9 Zang, T.A., Dahlbnrg, R.B., Dahlburg, J.P.;"Direct and large-eddy simulations of three-dimensional compressible Navior-Stokes turbu-lence", Phys. Fluids A 4 (1), January, 1992.

10 Moin, P., Squires, K., Cabot, W., Lee, S.; "A dy-namic subgrid-scale model for compressible tur-bulence and scalar transport", Phys. Fluids A 3(11), November 1991.

11 Edison, T.M.; "Numerical simulation of turbu-lent Rayleigh-Bernard problem using numerical

sug-grid modelling", Journal of Fluid Mechanics,Vol.158, pp 245-268, 1985.

12 Deaidroff, J.W.; "Numerical Study of Three-dimensional turbulent channel flow at largeReynolds number", Journal of Fluid Mechanics,Vol. 41, pp 453-480, 1970.

18 Tarn, C.K.W.; Webb, J.C.: "Dispersion-Relation-Preserving finite difference schemes forcomputational acoustics", J. Comput. Phys.,Vol. 107, Aug. 1993, pp. 262-281.

14 Tarn, C.K.W.; Webb, J.C; and Dong, T.Z.: "Astudy of the short wave components in compu-tational acoustics," J. Comput. Acoustics, Vol1, 1993, pp. 1-30.

15 Tarn, C.K.W; Shen, H.; "Direct Computation ofnon-linear acoustic pulse using high order finitedifference scheme", AIAA paper 93-4325, Oct.1993.

16 Tarn, C.K.W.; Dong, T.Z., "Wall boundary con-ditions for high-order finite difference schemesin computational aeroacoustics," Theoret. Com-put. Fluid Dynamics, Vol. 6, 1994, pp. 303-322.

17 Tarn, C.K.W., Auriault, L.; "Time-domainboundary condition for acoustically treated sur-faces", FED-Vol. 219, Computational Acoustics,ASME 1995.

18 Eversman, Walter: "Theoretical models for ductacoustic propagation and radiation," NASA Ref-erence Publication 1258, Vol. 2., 101, (1991).

19 Tarn, C.K.W.; Mankbadi, Reda R.; ... : "Out-flow boundary conditions for instability and dis-persive waves", In preparation.

20 Engquist, B. and Majda, A., "Absorbing bound-ary conditions for the numerical simulation ofwaves", Math. Computation, Vol. 31, July 1977,pp. 629-651.

Ejector Shroud

Secondary Row V '

Primary Nozzle —— * ~^ —— > Elector Exit

^—— ~~^~^-^——————————— 1

r

By-pass inlet

PrimaryNozzle inlet

tjecior exit: * •—••—••• 6D --.-\........*

'\Ejector Shroud ; \

1.5D *

A Dl2 iCneter Line

Figure 1. Ejector with a single-element primary nozzle Figure 2. Computation Domain

r/D

0,0,

0,0 1.0 2.0 3.0 4,0 5,0 6,0 7.0

Downstream distance x/D

Figure 3. Mach number distribution of the mean flow

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.0

Figure 4. u-velocity profiles at various downstreamlocations.

0.50.30 0.38

Figure 5. u-velocity profile along the inletboundary of the by-pass channel.

0.020

0.015 -

0.010

0.005

0.000

r/D

0,0 1.0 2.0 3.0 4,0

Downstream Distance x/D

Figure 6. Root-mean-squared pressure fluctuation distribution

6.0 7,0

r/D

2.0 3.0 4,0 5.0 6,0 7.0

Downstream distance x/D

Figure 7. Root-mean-squared axial momentum fluctuation distribution.

0.0 1.0 2.0 3.0 4.0 5.0 6.0

x/D = 0.27: Circlex/D = 2.18: Squarex/D = 4.09: Diamondx/D = 6.0 : Star

Figure 8. Root-mean-squared pressure fluctuationdistribution at various radial locations.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.1Strouhal number St

Figure 9. Power spectrum of axial velocity at variousdownstream locations in the mixing layer r/D=0.5.

r/D

t=TO

t=TO+T/8

HO+T/4

t=TO+3T/8

t=TO+T/2

t=TO+5T/8

t=TO+3T/4

t=TO+7T/8

t=TO+T

Oil___LOI

2.0 3,0 4.0 5.0

Downstream distance x/D

6.0 7.0

Figure 10. Instantaneous pressure fluctuation at 9 stages of a period T in the region right above theshear layer (r/D > 0.5)

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.Fig. 3

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.Fig. 6

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.Fig. 7

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.Fig. 10