AIAA-2000-1943 1 Ahuja

COMPUTATIONAL SIMULATIONS OF FORE AND AFT RADIATIONFROM DUCTED FANS*

V. Ahuja†

Combustion Research and FlowTechnology, Inc.

Dublin, PA 18917 USA

Y. Ozyoruk††

Middle East Technical UniversityAnkara, Turkey 06531

L. Long§

Pennsylvania State UniversityUniversity Park, PA 16802

ABSTRACT*

In this paper, we extend the work of Ozyoruk andLong, for predicting farfield noise from ducted fans using ahigher order coupled Euler/Navier-Stokes solver alongwith a Kirchhoff formulation, to full engineconfigurations on distributed parallel computers.Incorporating the exhaust along with the inlet raises manyissues relating to appropriate grid topologies, multigridacceleration and the unsteady numerics. These issues areaddressed in this paper and simulations pertaining to arealistic engine geometry are presented. Qualitativecomparisons are in good agreement with results from afrequency domain code, pertaining to radiation from adipole in a cylinder. Simulations relating to radiation froman engine geometry, and with and without the centerbody,with and without mass flow are also shown.

1.0 INTRODUCTION

In the past fifty years environmental awareness,increased sensitivity to noise, and protests bycommunities directly in the flight path, near airports, haveplaced huge demands on reduction of noise generated bycommercial and transport aircraft. Since the main sourceof noise is engine-related noise, the quest for quieter skieshas led to a major focus on reducing engine noise withoutsacrificing engine thrust and performance. Consequentlyengine technology has undergone a major transformation.Engines used forty years ago incorporated turbojettechnology, where the major source of sound was mixingnoise attributed to the jet exhaust mixing with thesurrounding air. Mixing noise was reduced with theadvent of low-bypass-ratio turbofan engines, since the fanwas used to augment thrust and the core and fan exhaustswere mixed internally before being vented. This has beenfurther reinforced with the introduction of the high bypassratio engines wherein a significant portion of the thrust isprovided by the fan, thereby further leading to a reductionof jet exhaust velocities. As a consequence, the emphasison noise reduction has now shifted to fan-related noise andthe acoustics of the ducted fan system. Tyler and Sofrin[1] have investigated and classified noise generatingmechanisms in axial flow systems as those due to therotating blades, the interaction between the rotor and the † Research Scientist, Member AIAA.†† Associate Professor, Department of Aeronautical Engineering,Member AIAA.§ Professor, Department of Aerospace Engineering, Associate FellowAIAA.

Copyright © 2000 by the authors. Published by AIAA with permission.

stator and the interaction of the boundary layer and inletdisturbances with the rotor blades. The fundamental tonalnoise generating mechanism is attributed to the rotatingpressure patterns or spinning modes by the rotor or the fanat the blade passage frequency or its harmonics. This is adirect consequence of blade loading and blade thickness.Also important is the interaction of rotor wakes cuttingthrough the stator blades producing noise at multiples ofthe blade passage frequency. As the fan size is increased,with increasing bypass ratio engines, rotor-statorinteractions play a more significant role as a noisegenerating mechanism. Recent advances in parallelcomputing have facilitated the simulation of entire bladerows of stators and rotors [2]. This has provided theability to capture the circumferential variation in flowfeatures such as non-uniform wakes interacting with statorvanes and the rotating modes that are responsible for noise[3].

For the purpose of design and reduction ofnoise, it becomes critical to efficiently capture the far-fieldacoustic field that is produced in the engine. However, inhigh bypass ratio engines, the BPF tones propagate insidethe engine nacelle, and aft of the stator blade row throughthe exhaust duct before being radiated out as sound. TheBPF tones radiate depending on whether the frequency isabove or below the critical value based on thecharacteristics of the noise generating mechanisms, theflow conditions in the duct and other characteristics of theduct. Furthermore, the shape of the nacelle can lead tosignificant refraction making transmission through theduct complicated. The downstream radiation is furthersupplemented by the noise due to mixing in the shearlayer at the aft end. Most of existing methods utilized inthe prediction of radiation and transmission makesimplifying assumptions that preclude capturing of thecomplex flow and wave propagation phenomena inherentin such systems, such as nonlinear effects, aerodynamicacoustic coupling and wave refraction. Eversman et al. [4]use a combination of the finite element and wave envelopetechniques with a velocity potential based formulation.Although, the method is efficient at high frequencies, itfails to account for aerodynamic-acoustic coupling andnonlinear effects that might influence the wavepropagation process. Furthermore, in the case of aftradiation a predetermined function has to be specified toaccount for the shape of the shear layer. Boundaryelement based formulations used by Myers [5], Dunn [12]have proved to be very robust and provide keen insightinto ducted fan noise. However, ???? utility is limited tosimplified flows with a uniform mean flow, where theflow does not considerably influence the acoustic field.

AIAA-2000-1943 2 Ahuja

In this paper, we propose to solve theradiation/transmission problem from first principles bynumerically solving the Euler/Navier-Stokes equations inthe vicinity of the entire engine utilizing a spatiallyhigher order time-accurate method. Our method has beenadequately proven to predict sound for the JT15D engineinlet in the past [6] and has compared favorably withsemi-analytical methods in predicting the BPF tones. Theevolution of parallel distributed computing makespermissible the simultaneous investigation of the forwardand aft radiation arcs of an engine with fine spatialresolution. An efficient implementation of the movingsurface Kirchhoff formula of Farassat and Myers [7]permits the prediction of far-field noise [6, 8, 10]. In thispaper, noise generation mechanisms are analyticallyspecified as source terms that mimic the superposition ofmultiple modes and harmonics related to the blade passagefrequency. It must be noted that our scheme is robust [8]and unlike most CAA schemes can be readily used tosolve problems involving realistic geometries and flowconditions. Furthermore, our method can be easilyincorporated in the design cycle for enhancing nacelledesign, engine ducting, analysis of cut-off modes and canbe readily extended to a comprehensive Navier-Stokessolver that could be used to analyze noise generatingmechanisms such as rotor-stator blade rows,turbulence/boundary layer interactions and shear layers.

The code used by Ozyoruk and Long [6, 8, 10]was written in connection machine Fortran (CMF), whichis very similar to Fortran 90 and High PerformanceFortran (HPF). The current code is written in Fortran 90and Message Passing Interface (MPI). The new MPI codeis very complicated compared to the CMF code. It is areal shame that HPF has not caught on, and people haveto use MPI.

In the next section, we provide details of thenumerical model including specification of the sourceterms and implementation of multigrid acceleration that iscritical to obtaining a steady state solution devoid ofnumerical noise. In section 3, we discuss results ofsimulations of duct transmission and radiation from ageneric engine and the validation of our method againstboundary element solutions for radiation of a dipole in acylinder. The paper ends with concluding remarks andrecommendations for future work.

2.0 NUMERICAL MODEL

2 . 1 Interior Equations: Navier Stokes/EulerThe 3-D time dependent Navier-Stokes/Euler

equations are solved in body fitted generalized coordinatesin the interior of the computational domain, with the non-reflecting boundary conditions solved on the outerboundaries. The time dependent partial differentialequations representing the far field conditions areexpressed in compatible form, so that the differential fluxoperator is transparent across the entire computational

domain. For the sake of completeness, the equations arewritten here in conservative generalized coordinates.

∂∂

∂

∂ξ

∂

∂η

∂

∂ζ

ˆ ˆ ˆ ˆ ˆ ˆ ˆQ

t

F F G G H Hv v v+

−( )+

−( )+

−( )= 0 (1 )

where Q is the vector of conservative state variables andF,G and H are the flux vectors. Fv, Gv and Hv are thestandard viscous fluxes.

QQ

J= (2 )

Q

u

v

w

e

=

ρρρρ

(3 )

ˆ , ˆ

ˆ

FJ

U

Uu p

Uv p

Uw p

e p U p

GJ

V

Vu p

Vv p

Vw p

e p V p

HJ

W

x

y

z

t

x

y

z

t

=++

+

+( ) −

=++

+

+( ) −

=

1 1

1

ρρ ξρ ξ

ρ ξ

ξ

ρρ ηρ η

ρ η

η

ρρ

,

WuWu p

Wv p

Ww p

e p W p

x

y

z

t

++

+

+( ) −

ζρ ζ

ρ ζ

ζ

(4 )

The contravariant velocity components are written as

U u v w

V u v w

W u v w

x y z t

x y z t

x y z t

= + + +

= + + +

= + + +

ξ ξ ξ ξ

η η η η

ζ ζ ζ ζ

(5 )

and pressure is computed from the conservative variablesby the relation.

p eu v w

= −( ) −+ +( )

γ ρ12

2 2 2

(6 )

The spatial derivatives are computed using afourth order stencil in each coordinate direction. Time

AIAA-2000-1943 3 Ahuja

integration is performed using a four stage Runge-Kuttamethod with a blend of Jameson’s [9] second and fourthorder type artificial dissipation, for damping the spurioushigh wavenumber components. The scheme is stable forCFL numbers less than 2 2 . However, for wavepropagation type problems CFL numbers of less than 1are preferred. This is mainly because the amplificationfactor is close to 1 for the entire range of wavenumbercomponents at CFL numbers less than 1. However, thephase error can be significant for the higher wavenumbercomponents and it limits the resolution of the scheme to12 cells per wavelength.

2 . 2 Multigrid Convergence AccelerationSince the emphasis in our approach is to utilize

the full Euler/Navier-stokes equations, all problemsregarding sound radiation presuppose an established steadystate flow solution. Once, a steady state solution hasbeen obtained, the sources responsible for generation ofsound are turned on and the wave propagation process issuperposed on the steady state solution. This places avery stringent requirement of a smooth steady statesolution devoid of spurious waves. Unfortunately, thevery characteristics of low dispersion and dissipation, thatmake the scheme suitable to simulation of wavepropagation processes, hinder convergence to steady state.The convergence problem is alleviated in our methodthrough the use of multigrid acceleration, [10] wherein thelow frequency numerical modes are aliased as highfrequency modes on successive grids of varyingcoarseness. This is particularly attractive in acousticproblems with a mean flow, because of the highresolution nature of the grid that is needed to resolveacoustic frequencies of interest. However, in the case ofcomplex geometries, especially those with multiple cornerpoints, trailing edges etc., the coarse grids need to begenerated carefully, in order to retain these definingfeatures of the geometries that are critical to the flow.This has been illustrated in figures (1a) and (1b) where thefine base grid and the coarsest level grid are shownretaining critical features of the geometry. For example,the trailing edge, beginning of the centerbody, the aft endof the nozzle, and the corner points at the junction of theaxis and the farfield are maintained in the coarse grids asconstraining parameters in much the same manner as theyappear in the finest grid. In our implementation we haveused Jameson’s full approximation scheme [11] with a 3level V-cycle formulation. Convergence of the solutionon each successive coarse grid is driven by the residual onthe finer grid with the help of a forcing function thatreflects the difference between the convective ρ anddissipative fluxes D at the two grid levels.

P W F Q D Q

F Q D Q

h hh

h h h h

h h h h

22

2 20

2 20

= −

− −( ) ( )( ( ) ( )

( ( ) ( )(7 )

Here the subscripts h stand for the fine grid and2h for the next level of coarse grids and the superscriptrepresents the initial Runge Kutta stage. The integrationprocess is performed on the next grid level as follows:

Q T Qh h h20

20( ) ( )=

Q Q J t F Q D Q P

Q Q J t F Q D Q P

Q Q

h h h h h h

h h h h h h

h h

21

20

2 20

20

2

22

20

2 21

20

2

23

20

1

41

3

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

= − ( ) ( ) − ( ) +

= − ( ) ( ) − ( ) +

=

∆

∆

−− ( ) ( ) − ( ) +

= − ( ) ( ) − ( ) +

( ) ( )

( ) ( ) ( ) ( )

1

31

3

2 22

20

2

24

20

2 23

20

2

J t F Q D Q P

Q Q J t F Q D Q P

h h h h

h h h h h h

∆

∆

(8 )

The procedure is continued until the coarsest mesh isreached. A correction to the solution is then performed onthe coarsest mesh

∆Q Q Qvh vh vh= −( ) ( )4 0 (9 )

And the accumulated correction is passed backsuccessively until the finest is reached.

∆Q Q Q I Q Qrh rhnew

rh rhrh

rhnew

rh/ / //

2 2 22 0= − = −( )( ) (10)

Q Q Q Qhn

hnew

h h+ = = +1 ∆ (11)

Restriction of data Tnh, onto coarser grids is accomplishedthrough direct injection, whereas prolongation Irh

rh / 2 is

performed by bilinear interpolation.Needless to say, including the multigrid

procedure in the solution algorithm has resulted in betteroverall accuracy for the steady state solution, and in turnfor the acoustic pressure levels. Multigrid acceleration hasalso resulted in tremendous savings in computationalresources, as is seen in Figure 2, which is a convergencecomparison for single grid and multigrid convergencecharacteristics for a converging-diverging nozzle typeflow, that is indicative of wide range of flow regimes interms of Mach Number. The computational savings dueto acceleration in convergence of the multigrid solution faroutweigh the added expense of the three-level iterativemultigrid method over a single level computation.

2 . 3 Kirchhoff Formulation:Limitations on computational resources preclude

the extension of the computational domain into the far-field. The acoustic response in the near-field is bridged bymeans of the Kirchhoff formula [15] of Farassat andMyers [7] to the far-field. This formulation of theKirchhoff integration surface is for arbitrarily moving and

AIAA-2000-1943 4 Ahuja

deforming surfaces and is coupled with our Euler-NavierStokes solver.

On simpler geometries, the Kirchhoff surface canbe conveniently constructed along constant surfacesdefined by ξ, η or ζ grid lines, since the surfaceparameters, such as normal vector and surface area areknown apriori. In more complicated geometries,combinations of grid lines can be utilized to construct aclosed surface, with the given exception that this surfacewill now have piecewise discontinuous normal vectors andsurface areas. In such cases, the individual surfaceelements are tagged and an additional data structure ofpointers is stored.

For a nondeforming Kirchhoff surface that is inrectilinear motion, the Kirchhoff formula for the acousticpressure ′p at the observer location x and observer time ttakes the following form:

41 1

1 22π

τ′( ) =

−+ ′

−∫∫p x tE

R M

p E

R MdSs

R R, [

( ) ( )]

*

(12)

where

R = R , R = x - y = M R/Rv v v v v v

τ( ) ⋅, MR , (13)

and

E n p M n M p

M n

c M

M n

c

p

R

1

1

= − ⋅∇ ′ + ⋅( ) ⋅∇ ′( )+

− ⋅−( ) − ⋅

′

∞ ∞

ˆ ˆ

cos ˆ ˆ

v v

v vϑ ∂

∂τ

(14)

E

M

MM n

R2

2

21

1= −

−( )− ⋅cos ˆϑv

(15)

The Kirchhoff surface and the observer are seen to travel at–M∞, and the elapsed time for a signal to travel from aKirchhoff surface element to the observer will always bethe same, thereby, ostensibly simplifying the numericalcoupling between the flow solver and the Kirchhoffmethod. Thus, the emission time for a Kirchhoff surfaceelement e is given by

τe et R c∗∞= − / (16)

where

Rx y M R

e =− −( ) +∞1 1

2β(17)

with β2 21= − ∞M and

R x y x y x y= −( ) + −( ) + −( )[ ]1 12 2

2 22

3 32β (18)

in which x1 and y1 are the projected components of thevectors x and y on the Mach number vector M ,respectively, at the observer time. The components x2 ,x3 and y2 , y3 are the transverse components to x1 andy1, respectively.

The Kirchhoff surface integrations are preformedassuming that the integrand in Eqn. 12 is constant over aKirchhoff surface element, and the time is discretized inthe same manner as in the flow solver. The acousticpressure required by Eqn. 12 in the vicinity of theKirchhoff surface is obtained from the flow solver bysubtracting the mean pressure from the instantaneouspressure. In most cases the time increment of the flowsolver, ∆t, from one time step to the next is sufficientlysmall to calculate the time derivative of the acousticpressure on the Kirchhoff surface using a first-orderbackward difference. Then,

∂∂τ τ

′ = ′ − ′ −p p p

te e

nen

*

* * 1

∆(19)

where the superscript n* is the time level such that, whenmultiplied by ∆ t, it gives a discrete time that is theclosest to the emission time τe

∗ . This is only anapproximation provided that ∆ t is sufficiently small.Hence we do not perform interpolations for the exactemission time data. The gradient of the acoustic pressureis calculated using the same finite difference stencils as theflow solver. At this point the observer time is consideredthe same as the flow solver. Hence, for the discreteacoustic pressure at the observer point and time, we canwrite

′[ ] = ∑ − ( )[ ]{ }=

∞p x n t P n R c t See

NE

e e, int / ∆ ∆ ∆1

4 1π(20)

where the operator int( ) rounds the real argument to theclosest integer. The time level n R ce− ( )( )∞int / in Eqn.

20 is identical to the time level n* that appears in Eqn.19. Implementation on a distributed computingenvironment is performed by letting each processoridentify that part of the Kirchhoff surface which is locatedin its computational domain. Before the time integrationprocess begins, the lag time associated with each elementis calculated and a global minimum is communicated toeach processor. This marks the starting point of theacoustic pressure storage array on all processors. Duringthe iterative process, each processor sums thecontributions of acoustic pressure from the elements in itscomputational domain. After the simulation, the acousticpressure arrays from all processors are summed up to getthe final response in the farfield.

AIAA-2000-1943 5 Ahuja

2 . 4 Source TermsThe source terms that simulate the effect of the

sound generating mechanisms involving rotor-statorinteractions are derived from infinite duct theory. Ductacoustic modes are classified as (m, µ) representing thecircumferential and radial modes. The circumferential modeis related to the number of stator vanes and rotor bladesthrough the relation

m = n B + sV s=…-2,-1,0,1,2…

The acoustic pressure in an infinite cylinder as a functionof (r, θ, x) is given as

ˆ , , ,

exp ,

p x r A J k r

i k x m

nm nm m m

x m nm

µ µ µ

µ µ

( ) = ( )+ +( )[ ]

θ

θ φ (21)

A summation over all the eigensolutions corresponding toall active modes gives us a representative source term thatcan be specified at the fan face

′ ( ) = ℜ

µ

−

µ=

∞

=−∞

∞

=

∞∑∑∑p x r t p ef nm

i nt

mn

, , , ˆθ ω

01

(22)

The axial wave number kx m, µ is given by

kM k k M k

Mx m

f f m

f,

,µ

µ=− − − −( )

−

2 2 2

2

1

1(23)

And the axial wavelength λ πa

x mk=

µ

2

,with the sign of k

determining an upstream or downstream travelling. Thecut-off ratio of a particular mode (m,µ) is given by

ξmT

f m f

nBM

r k Mµ

µ

=−1 2

(24)

where MT is the rotor tip speed.

2 . 5 Grid TopologyAmong the different grid options, the use of an

O-H grid type seems attractive for engine calculationssince it is typically used in steady state computations.However this grid typology has a singularity at the high-lite of the nacelle requiring clustering in the region (asshown in Fig. 3). Furthermore, the clustering is carriedall the way to the boundaries and it creates difficulties inresolving higher order schemes like those utilized in wavepropagation problems. Furthermore, the grids need to beorthogonal to each other, especially close to thecenterline, for the sake of geometrical conservation during

discretization of the equations. For engine inletconfigurations, grids generated (Fig. 4) by conformalmappings (Ref. 13) have provided smooth orthogonalgrids. Based on this topology, we generated grids (Fig. 5)for the entire inlet/exhaust configuration. Although, thesegrids are orthogonal throughout, they exhibit onedrawback. There is an interchange in the marchingdirection at two points as shown in Fig. 5 on the cowlsurface. These crossover points have difficulties with ahigher order scheme thereby limiting its utility for ourpurposes. In this paper we utilize a wrap around O-typegrid starting at the fan face and terminating at oneexhaust/stator plane. This type of grid traces a wavefrontand naturally coarsens close to the farfield boundary andprovides fine resolution in the region of interest.

3.0 DISCUSSION OF RESULTS

3 . 1 Radiation from a Dipole in a CylinderIn order to test our approach for entire engine

calculations and evaluate the appropriateness of the gridtopology in use, we compare our results for radiation froma dipole in a cylinder, with the boundary element methodTBIEM3D. This is a close representation of the radiationproblem from an engine. The TBIEM3D [12] code is afrequency domain code. This code can simulate radiationfrom cylindrical shaped containers (with both ends open)including forward flight effects. The noise due to the fanblades is modeled using rotating or non-rotating dipolesources. This is a linearized code with limited cowlingshape options, but is extremely useful for validation andtrade studies.

The problem solved here consists of a dipoleplaced at the center (on the axis of symmetry). Theboundary element method utilizes an infinitely thin shellto represent the cylinder. In our approach, the shell has afinite thickness and the edges are rounded to alleviateproblems in gridding. A wrap-around O-type grid is usedwith the grid being cyclic in the primary direction. As isseen in Figure 6, the cylinder is located at –15 < x < 15and y = 15. The dipole can be represented as acombination of two monopoles with the phase shift of180 degrees or as a source term [14] in the momentumequation given by

S Ax

r= −( )

∞ρ ω

ωexp sin3

22

∆ ∆ t (25)

where ∆ω is the radius of Gaussian pulse in terms ofnumber of cells.

The source term is valid everywhere inside the cylinder andis zero outside. For these computations a wavenumber of267 radians/sec is utilized and no mass flow is specified.Figure 6 shows the instantaneous pressure contours of theTBIEM3D compared to the Euler solution. The twosolutions indicate close agreement, and the Euler solution

AIAA-2000-1943 6 Ahuja

seems to capture the interference effects behind thecylinder due to waves emanating from both the aft andfront ends.

There is a slight discrepancy in the solutions due tothe rounding of the edges. This is better seen in Figure 7that depicts the comparison of sound pressure levels. Aswe move away from the cylinder, axially, a phase shifterror starts to dominate the Euler solution. This is due topoor resolution that is an artifact of grid coarseninginherent in grid topologies of this nature.

3 . 2 Radiation from EngineOur next set of results pertains to computations

regarding a realistic ducted fan and exhaust system. Theengine contains 18 rotor blades and a set of 42 statorvanes. The operating flow regime for this engine dictatesa freestream Mach No. of 0.2 and the fan rotates at anRPM of 5210 rev/min. This engine has a highlightradius of 0.28 meters and a centerbody that variessignificantly throughout the length of the cowl andextends beyond, arcing towards the centerline as it reachesthe compressor exhaust. The rotor blades extend from–0.041m to 0.047m and the stator blades are locatedbetween 0.1478m and 0.193m. For the purpose of ourcalculations, we have taken a fixed rotor and stator planethat is obtained by arithmetical averaging. For ourpurposes, these represent the inlet and exhaust planes ofour calculations. Furthermore, the compressor exhaust iseliminated by carrying the aft end of the centerbody all theway to the axis of symmetry.

3 . 3 Radiation of the (0,0) + (0,1) Modes withNo Mass Flow and without a Centerbody

For our first calculations we removed thecenterbody, so that our system now approximates acylindrical duct with a cowl. For the purpose ofgenerating the zeroth circumferential mode at the bladepassage frequency, we assume that the number of statorvanes is equal to the number of rotor blades. The sourceconsists of a combination of the (0,0) and (0,1) modes,each having a 3kPa amplitude. From past experience [6]it has been found that this high source amplitude does notcreate any nonlinearities in the absence of mass flow.Both at the exhaust and inlet planes momentum sourceterms are specified based on the linearized momentumequations and the acoustic pressure obtained fromcylindrical duct theory.

ρ ρρ ω

ˆ ,u

k

np

nmx m

nm1( ) = ℜ ′

µ ∞

µ

∞µ (26)

ρ ρρ ω

uk

n

J k r

J k rp

nmm m m

m mnm2( ) = ℜ

′ ( )( ) ′

µ ∞

µ

∞

µ

µµ (27)

ρ ρρ ω

um

n rp

nm nm3( ) = ℜ ′

µ ∞

∞µ (28)

ρ θ ρu u inm

umn1 1

011 2 3( )( ) = ( )∑∑∑ =µ=

∞

=

∞x, r, , t ˆ , , , (29)

where the value of P′nmµ can be found from Eqn. 22.Based on cylindrical duct theory, the cut-off ratios for thetwo modes are ξmµ(0,0) = ∞ and ξ mµ (0,1) = 2.109indicating both modes to be cut-on. As is seen fromFigure 8 the waves are well resolved almost until thefarfield boundary and the interference patterns closelyresemble those obtained for the dipole in the cylinder.The sound pressure level directivity patterns also closelyresemble those obtained for the cylindrical duct with thedipole. The normalized SPL levels are shown in Figure 9and indicate multiple lobes in both the aft and the frontends. The maximum levels occur close to the inlet axis.

3 . 4 Radiation of the (0,0) + (0,1) Modes withNo Mass Flow and with a Centerbody

In the next set of calculations we include thecenterbody in our inlet/exhaust system. Again, we assumeparity in the number of stator vanes and rotor blades inorder to generate the zeroth circumferential mode at theblade passage frequency. The (0,0) and (0,1) modes areexcited at an amplitude of 3kPa and the boundaryconditions are imposed in much the same manner as theprevious case. However, in this case Bessel functions ofthe second order are utilized to evaluate the acousticpressure source terms since the exhaust and inlet planesare now representative of annular ducts. As aconsequence, the cut-off ratios for the exhaust plane areξmµ(0,0) = 1.3 and ξmµ(0,1) = 0.65 and for the inlet planeare ξmµ(0,0) = 1.62 and ξmµ(0,1) = 0.83. Both the (0,1)modes are now cut-off and this explains the rapiddampening of waves as they travel outwards. As is seenin Figure 10 this effect is more pronounced at the aft end,where the cut-off ratios are lower compared to the frontend.

3 . 5 Radiation with Mass Flow and CenterbodyIn this section, we present results with mass

flow. The steady state solution was obtained with massflow at the inlet face being 17.8 kg/sec and the mass flowspecified at the state of exhaust phase is increased by 10%(20.05kg/sec.). This simulates the effect of an engine andthe velocity at the backend is matched to that observedexperimentally. Mach number contours of the steady statesolution are shown in Figure 11. Multigrid accelerationwas used, although convergence dropped only four ordersof magnitude. This may be due to the unsteadinessinherent in the shear layer at the aft end. A source termdefining a plane wave excitation at the blade passagefrequency was turned on at the stator and rotor planes andthe acoustic pressures are shown in Figure 12. A distinctwas pattern is seen both at the fore and the aft end.

AIAA-2000-1943 7 Ahuja

Further investigation is being carried out to improve thecharacteristic boundary conditions in the near fields asacoustic pressure levels indicate reflections in the domain.

4 . 0 SUMMARY AND FUTURE WORK

In this paper we have extended the hybrid, timedomain ducted for noise prediction capability developed forengine inlets by Ozyoruk and Long to entire engineconfigurations. An MPI based parallel version of the codesuitable to distributed computing platforms has beendeveloped. The code solves the 3-D nonlinearEuler/Navier-Stokes equations and is coupled to aKirchhoff method for farfield predictions. Theimplementation of multigrid is essential or convergenceacceleration and accuracy purposes. Comparisons withfrequency domain codes show good agreement for radiationfrom a dipole in a cylinder. Furthermore, simulations ofradiation from a full engine configurations have beendemonstrated. The results show qualitative trends thatclosely approximate the patterns obtained for one dipole inthe cylinder that are considered representative of engineradiation patterns.

Most simulations involved the zerothcircumferential mode. Currently, analysis of the spinningmodes are underway. In particular, the sixthcircumferential mode is being investigated for the engineconfiguration shown in this paper. Efforts are alsounderway to use a soft wall boundary condition tosimulate the effect of engine liners.

5.0 ACKNOWLEDGEMENTS

The authors wish to acknowledge the technicalsupport and insightful comments provided by Rumsey,R.T. Biedron and F. Farassat of the NASA LangleyResearch Center. The lead author would like to thank Ms.Katherine Young for her help in preparing the paper.

6.0 REFERENCES

[1] Tyler, J.M., and Sofrin, T.G., “Axial FlowCompressor Noise Studies,” SAE transactions,Vol.70, 1962, pp 309-332.

[2] Ahuja, V., Deshpande, M., Feng, J., Wang, L., andMerkle, C.L., “Practical Applications of ParallelProcessing in Computational Fluid Dynamics,”Proceedings of 13th AIAA CFD Conference,Snowmass Village, CO, June 29-July 2, 1997.

[3] Rumsey, C.L., Biedron, R.T., Farassat, F. andSpence, P.L., “Ducted Fan Engine AcousticPredictions Using a Navier Stokes Code,” Journal ofSound and Vibration, Vol. 213, No. 4, June 1998, pp643-664.

[4] Everson, W., “A Finite Element Formulation for AftFan Duct Radiation,” AIAA Paper No. 97-1648.

[5] Myers, M.K., “Boundary Integral Formulations forDucted Fan Radiation Calculations,” CEAS-AIAAPaper No. 95-076, June 1995.

[6] Ozyoruk, Y., and Long, L.N., “Computation ofSound Radiating from Engine Inlets,” AIAA Journal,Vol.34, No. 5, May 1996.

[7] Farassat, F. amd Myers, M.K., “Extension ofKirchoff’s Formula for Radiation from MovingSurfaces,” Journal of Sound and Vibration, Vol. 123,June 1988, pp451-460.

[8] Ozyoruk, Y., and Long, L.N., “A New EfficientAlgorithm for Computational Aeroacoustics onMassively Parallel Computers,” Journal ofComputational Physics, Vol. 125, No. 1, pp. 135-149, April, 1996.

[9] Jameson, A., Schmidt, W., and Turkel, E.,“Multigrid Algorithms for Compressible FlowCalculations. In Lecture Notes in Mathematics,Number 1228 in Multigrid Methods II” Proceedingsof the 2nd European Conference on MultigridMethods, October 1985, pp. 66-201, Spring Valley,1986.

[10] Ozyoruk, Y., and Long, L.N., “MultigridAcceleration of a High Resolution AeroacousticScheme,” AIAA Journal, Vol. 35, No. 3, pp. 428-433, March, 1997.

[11] Jameson, A., “Numerical Solutions of the EulerEquations by Finite Scheme methods using Runge-Kutta Time-stepping Schemes,” AIAA Paper No. 81-1259, 1981.

[12] Dunn, M. H., “TBIEM3D A Computer program forPredicting Ducted Fan Engine Noise,” Version 1.1,NASA CR-97-206232, Sept., 1997.

[13] Ives, D.c. and Menor, W.A., “Grid Generation forInlet-Centerbody Configurations using Conformalmapping and Stretching,” AIAA Paper No. 81-0097,1981.

[14] Pierce, A.D., “Linear Acoustics,” Encyclopedia ofApplied Physics, Vol. 1, pp. 137-182, 1991.

[15] Lyrintzis, A.S., “The Use of Kirchoff’s Method inComputational Aeroacoustics,” FED-Vol. 147,Computational Aero and Hydro Acoustics.

AIAA-2000-1943 8 Ahuja

Fig. 1a. Fig. 1b.Fig. 1. Fine and coarsest grids of the multigrid cycle.

Fig. 2. Convergence comparisons with and withoutmultigrid for a converging-diverging nozzle.

Fig. 3. H-grid used in steady state computations of anengine inlet.

Fig 4. Conformal mapping based grid used for solvingradiation problems from inlet. Fig. 5. Grid for entire engine-like geometry extending the

topology of conformal mapping based grids.

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Fig. 6. Radiation patterns of acoustic pressure for a dipole in a cylinder.The top figure is TBIEM3D solution. The bottom figure is Euler calculation.

Fig. 7. SPL contours for a dipole in a cylinder.The top figure is TBIEM3D solution. The bottom figure is Euler calculation.

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Fig. 8. Radiation from engine without centerbody for (0,0 =(0,1) modes and no flow.Patterns range from –300Pa to 300Pa. A section of the grid is shown in the bottom half of the figure.

Fig. 9. SPL levels for radiation from engine without centerbody with (0,0) = (0,1) modes.

Fig. 10. Radiation from engine with centerbody with (0,0) + (0,1) modes and no flow. Acoustic pressure patterns rangefrom –300Pa to 300Pa. The bottom half showing grid used for the calculation.

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Fig. 11. Mach number contours for the steady state solution of engine with centerbody.

Fig. 12. Acoustic pressure patterns from engine with mean flow and plane wave excitation.Levels range from –400Pa to 400Pa.