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

AIAA Meeting Papers on Disc, May 1996A9630881, NAG1-1367, AIAA Paper 96-1771

Progress in time-domain calculations of ducted fan noise - Multigridacceleration of a high-resolution CAA scheme

Yusuf OzyorukPennsylvania State Univ. University Park

Lyle N. LongPennsylvania State Univ. University Park

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

The numerical simulation of an aeroacoustics problem using the full time-dependent Euler or Navier-Stokes equationsrequires both the mean and unsteady flow fields. Sometimes the mean flow is equivalent to the steady state flow andused to start the unsteady solution process. In this case, both the steady and unsteady flow fields have to be obtainedthrough the same numerical method, so that the restart process is smooth and no spurious waves are generated.However, due to its low dissipation and consequently slow convergence, a high-resolution computational aeroacousticsscheme is not suitable for computing the steady flow field. In this paper, such a scheme is accelerated to convergence,without altering its residual, by using a full approximation storage multigrid method for the prediction of ducted fannoise. It is demonstrated that significant convergence improvements are obtained using the multigrid method, making itpossible to attain steady state solutions on extremely fine meshes designed for high-frequency radiation problems.Far-field noise results for a JT15D inlet are presented and compared with data at realistic flight configurations.(Author)

Page 1

AIAA-96-177196-1771

A96-30881

Progress in Time-Domain Calculations of Ducted Fan Noise:Multigrid Acceleration of a High-Resolution CAA Scheme

Yusuf Ozyortik* and Lyle N. LongtDepartment of Aerospace Engineering

The Pennsylvania State University, University Park, PA 16802

AbstractThe numerical simulation of an aeroacoustics problem

using the full, time-dependent Euler or Navier-Stokesequations requires both the mean and unsteady flowfields. Sometimes the mean flow is equivalent to thesteady state flow and used to start the unsteady solu-tion process. In this case, both the steady and unsteadyflow fields have to be obtained through the same numer-ical method, so that the restart process is smooth andno spurious waves are generated. However, due to itslow dissipation and consequently slow convergence, ahigh-resolution computational aeroacoustics scheme isnot suitable for computing the steady flow field. Inthis paper, such a scheme is accelerated to convergence,without altering its residual, by using a full approxi-mation storage multigrid method for the prediction ofducted fan noise. It is demonstrated that significantconvergence improvements axe obtained using the multi-grid method, making it possible to attain steady statesolutions on extremely fine meshes designed for high-frequency radiation problems. Far-field noise results fora JT15D inlet are presented and compared with data atrealistic flight configurations.

1 IntroductionAs more powerful computers are being introduced, it isbecoming possible to include more physics hi the nu-merical simulations of aeroacoustics problems by solv-ing the full Euler or Navier-Stokes equations.1'2 In thisapproach, however, the acoustic field is obtained by tak-ing the difference between the instantaneous and meanfields. If the transient part of the instantaneous pressureis associated only with acoustic waves, the mean fieldwill be^equivalent to a solution that will be obtainedby applying tune-invariant boundary conditions in thenear-field (e.g., steady solid wall boundary conditions).

'Post-Doctoral Scholar, Member, AIAAt Associate Professor, Senior Member, AIAA°© 1996 by Ozyoruk and Long. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

This solution is often referred to as the steady state so-lution. An example case where the transient flow fieldis driven only by acoustic waves is a turbofan inlet atsteady flight.

As part of an ongoing research effort to predict ductedfan noise, a hybrid, nonlinear, time-domain code hasbeen developed.1'3'4 This code solves the 3-D Eulerequations to determine the near-field acoustics and usesa moving surface Kirchhoff method5 to predict the far-field noise. The flow solver uses an explicit, spatiallyand temporally fourth-order accurate, finite difference,Runge-Kutta (R-K) time-integration scheme. Due tothe approach taken, the steady state solution is ob-tained first and then the time-accurate inlet fan-faceconditions (acoustic sources) are turned on, and thesolution is advanced in a time-accurate manner. Thisprocedure inherently requires the same scheme, namelyfourth-order central finite difference, for evaluating theresiduals of the governing equations in both the steadyand tune-accurate calculations so that starting from thesteady solution is smooth and no spurious waves aregenerated.

However, there are several difficulties associated withusing a high-order explicit scheme for steady state cal-culations. The fourth-order scheme is really designedfor tune-accurate problems with low dissipation and dis-persion. In other words, low-frequency modes are notdamped out. Moreover, typical Met radiation problemsinvolve high blade passing frequencies (BPF). Hence,very fine meshes are usually needed, resulting in verysmall time steps and consequently slow convergence.Also, ducted fan noise predictions are often made at lowfree stream Mach numbers as associated with landing ortake-off conditions, which are considered more criticalin terms of community noise regulations. The disparityof the eigenvalues in these cases is a degrading factor hiachieving sufficiently rapid steady state solutions.

There are several convergence acceleration techniquesapplied to explicit time-marching schemes, such asresidual smoothing, and multigrid methods. Residualsmoothing is most effective when applied implicitly and,therefore, not appropriate for the current approach.The most suitable technique for the current situation

is Jameson's full approximation storage (FAS) method.6This multigrid method has been widely used in the CFDcommunity7"9 for pseudo-time marching of the time-dependent Euler/Navier-Stokes equations. However, tothe authors' knowledge, this use has remained limitedonly to second-order schemes (finite volume or finitedifference). Several other high-order implementationsappeared only together with implicit (matrix inversion)schemes.10'11 In this paper, the FAS method is appliedretaining the fourth-order spatial accuracy of our com-putational aeroacoustics (CAA) scheme.

This application is described below and several re-sults showing the convergence improvements for steadystate flow calculations around an airfoil and the JT15Dengine inlet12 are presented as well as some far-field fannoise results at realistic conditions.

2 Hybrid Ducted Fan RadiationCode

The hybrid code solves the 3-D Euler equations on a3-D body fitted coordinate system (structured meshes)and passes the near-field acoustic pressure to a Kirch-hoff method based on the formulation of Farassat andMyers5 to predict the far-field sound. The governingequations are solved in a relatively small domain usingnonreflecting boundary conditions based on the workof Bayliss and Turkel13 and Tarn and Webb.14 An or-thogonal mesh system is created through a sequenceof conformal mappings3'15 and the governing equationsare formulated in cylindrical coordinates to effectivelytreat the grid singularity at the centerline. Fourth-order accurate, cell-centered finite differencing and four-stage, noncompact R-K time integration are performedto advance the solution. Jameson type artificial dis-sipation16'17 is used to suppress spurious waves. Theacoustic source is formed using the eigensolutions of thecylindrical duct problem and the rotor-stator interac-tion theory of Tyler and Sofrin.18 The Euler solverand the Kirchhoff method are coupled such that assoon as the Euler solution becomes available, the Kirch-hoff surface integrations are performed in a recursivemanner to predict the far-field noise. All calculationsare carried out on parallel computers using the dataparallel paradigm. Ozyoruk and Long4 describes thefourth-order flow solver with emphasis on the hybridcode's parallel aspects. Ozyoriik and Long1 discussesthe acoustic source model and the Kirchhoff couplingissues for engine noise predictions.

3 Multigrid MethodThrough a multigrid method, low frequency errors ofthe fine grid solution axe transferred onto a sequence of

coarse grids and are smoothed by updating the solutionthere. Then the corrections to the solution obtained onthe coarse grids are interpolated back to the fine gridwith its own low frequency errors having been aliasedinto high frequency errors which can now be dampedout on the fine mesh.

This procedure is applied in a systematic way to pushthe low frequency errors quickly out of the domain.Hence, a significant convergence improvement is gainedbasically for two reasons.8 First, the number of opera-tions required for each iteration to update the solution isreduced significantly on coarse grids, clearly due to thefewer grid points. Second, the time step sizes on coarsegrids are larger than on the fine grid. In other words,the effect of the boundary conditions that drive the flowfield is felt more rapidly throughout the domain.

In multigrid methods the grids of varying coarsenessare usually obtained by simply deleting every other gridline of the next finer grid. The fine grid size (numberof grid points) is chosen such that the grid lines repre-senting the boundaries are retained in this process.

3.1 FAS SchemeThe semi-discretized Euler equations and the far-fieldboundary conditions3'4 can all be written as

(1)

where Q is the vector of dependent solution variables(conservative state variables for the interior points, per-turbations of primitive variables for the outer bound-aries), J is the Jacobian of the coordinate transfor-mation, .F(Q) represents the collection of the spatialderivatives and £>(Q) represents artificial dissipation.Thus, [^"(Q) - £>(Q)] is the residual. Equation 1 isintegrated using the classical (noncompact) four-stageR-K scheme, resulting in fourth-order time accuracy.

Jameson's FAS multigrid time-advancing algorithm,however, uses his and his co-workers compact R-Kscheme,16 which is given as

= Qn,Q« = Qn

Qn+l _

where the superscript n is the time step, Ai is the timeincrement from one time step to the next, and a =[1/4,1/3,1/2,1].

Here we adopt this scheme directly for the steadystate calculations, since any time-advancing methodwill not alter the steady state solution so long as theresidual evaluation scheme remains the same. In otherwords, at the steady state dQ/dt = 0 leaving the fieldgoverned by :F(Q) - D(Q) = 0.

It is useful to introduce the following notation indiscussing Jameson's FAS multigrid scheme below. Atransfer (restriction) operator for the state variables isdenoted by T, for the residuals by W, and an interpola-tion (prolongation) operator is denoted by X. All theseoperators take on subscripts and superscripts that indi-cate the mesh spacing of the two grids that are involvedin the data link, whose direction is understood to bealways toward the grid of the superscript. These gridsare assumed to be given by the sequence, from the meshof the desired resolution (ft) to the coarsest,

(3)Gh, G^h, — > Gr/,/2,

where a subscript indicates the mesh spacing of the grid,and r and v are integers that are given by integer powersof 2.

First the solution on mesh G^h is initialized throughthe restriction of the data from mesh Gh as

= 72*0*. (4)where Q/i is the current value of the solution variableon mesh G/,. Then a residual forcing function "P^h isestablished such that the solution on grid G^k is drivenby the residual calculated on grid G/, using the solutionQ/,. This is achieved by setting

(5)

This function is added to the residuals of the R-Kscheme, which become

_ /i(Qh)) since the other terms willcancel. Thus the solution on grid G-ih. will be drivenby the residual transferred from grid G/,. It should benoted that Jacobian and the tune step in the above R-K scheme are associated with grid G^h, where the timestep (local) will be significantly larger, consequently afaster propagation of the signals (errors) and an un-proved convergence rate will result.

The above procedure is continued until the coarsestmesh (Gvh) is reached, where the correction to the so-lution will be given by

= o(4) - o(0) (7)Then, when going back from a coarse grid to the nextfiner mesh, the accumulated correction is given, on anintermediate grid Grh/2, by

- 0$) (8)

where Qrh/a is the solution on mesh Gr^/2 after the R-K time stepping on grid Grh/2 and before the transferfrom grid Grh, and Q^% ig tne ^na^ value of Qr/,/2resulting from both the correction calculated in the timestep on grid Grh/2 and the correction transferred fromgrid Grh-

Finally the updated solution on the fine grid is givenby

Qn+l = (9)

The restriction and prolongation operators used togenerate the data connections between the grids for thesecond-order finite difference and finite volume multi-grid methods have been established fairly well.6'9 Forexample, in finite difference methods, direct injection ofthe data from a fine grid to the next coarser grid is onepractical and extensively used method9 (Figure 1). Infinite volume methods the transfer of the data is real-ized through a volume weighted average of four cells onthe fine grid which make up one cell of the next coarsergrid. This way the flow state variables are conserved.6It is very natural to take the volume weighted averageof the data hi finite volume methods because of the factthat the data is assumed to reside at a cell center, whichon the next coarser grid will be affected mostly by thelargest of those four constituent cells.

FINE MESH COARSE MESHjj _ C

X X

X X1- » -t

X X

X X

t — >' —— <

D 9 15X X

X X

«* x°

X Xa b

) —— • ———— 4

? ^

) «

t fl

S ———— (

X

•J X

j —————————— j

X

) ———————— «

a bj —————————— «

• Mesh nodes (Finite difference)® Injected points from fine to coarse mesh (Finite difference)x Cell centers (Rnite volume)a Data transfer, averaging (Rnite volume)—-Grid lines deleted to obtain next coarser mesh

Figure 1: Data transfer from fine mesh to the coarsemesh.

As indicated, the current algorithm uses a fourth-order accurate cell-centered finite difference method,wherein no grid points coincide with the boundaries ofthe domain. Only in this way does the mesh structureof our current method resemble that of the classical fi-nite volume method. The weighted cell centers hi ourmethod, which are the grid points of its finite differencescheme, are not the geometrical cell centers. Instead,they are obtained via high-order interpolations, or themesh formed by these points is directly produced us-ing the grid generator described in Ozyoruk3 such thatno grid points exist at the solid boundaries. This is

important in terms of the smoothness of the geometri-cal derivatives over the stencils of the finite differencescheme.

For the multigrid application here we generate in eachof the f, T) and ^-directions (curvilinear coordinate di-rections) twice as dense a mesh as the fine mesh. Werefer to this new mesh as the source mesh, which nowhas grid lines on the boundaries of the domain. Thesource mesh readily provides the weighted cell centers,or the grid points of the current finite difference methodfor all the meshes used in the multigrid convergence ac-celeration process. This point is illustrated, for clarity,in only one dimension, in Figure 2. Hence, a given meshlevel is at least as smooth as the others.

1 -D Source mesh: Twice as dense as GhH — I-H-H — HH — I I I I I I I I I I I

i l l illi

i | i1 , grid point

MeshGh

Mesh G,h

Mesh G4h

Figure 2: Obtaining all levels of grid from a sourcemesh.

3.2 Restriction (transfer) operatorsIn the current approach the transfer operator T for thedependent variables is defined as

=7?hQfc = 4 (10)ce«=l

where the four cells that are involved in the summationmake up one cell of grid C?2h, as illustrated in Figure1 for a finite volume method. This simple averaging ispreferred over the volume weighted averaging of a finitevolume method due to our grid system.

However, the transfer of the residuals is performedalong the same lines as Jameson's method. We simplysum the residuals of those four constituent cells on gridGh to obtain the residual of a GZH cell. Thus for theinterior grid points we simply write

2>2fc(Q2fc) =4

ccll=l(11)

Since the time-accurate calculations use nonreflect-ing boundary conditions on the outer boundaries of thedomain, steady state calculations for the engine inletproblems are also performed using nonreflecting bound-ary conditions. These conditions are put in the sametime-dependent partial differential form as the interiorequations (see Ozyoruk3 or Ozyoruk and Long4). Thenthe multigrid method is applied to the entire system ofequations. Therefore, the residual transfer operator isdefined differently for the far-field boundary points. Anaveraging along the boundary and a consequent scalingby the coarse cell Jacobian is performed for the transferof nonreflecting boundary conditions residuals.

3.3 Prolongation (interpolation) opera-tor

The corrections are prolongated back to the next finermesh by using interpolation. For simplicity, considerthe 1-D coarse mesh and the next finer mesh shownin Figure 3. Two finer grid points fall between twocoarse grid points. The variation between two coarsegrid points is assumed linear and the data at the finergrid points are simply calculated by using the formulae

«2h+l/2 _2h -

tfc _h —

_ -rh »2h,»2h-fl _ _(—

(12)

(13)

where a superscript indicates the grid point the data isassociated with (see Figure 3). Special treatments, suchas extrapolation, are needed at or near the boundaries.

I2k+1/2 ^+1COARSE MESH

Qh ih ih+l FINE MESH

Figure 3: Grid points involved hi prolongation of coarsemesh data to fine mesh.

3.4 Cycling StrategyV-cycles with 3 mesh levels are used, although usingmore grid levels generally improves the efficiency ofmultigrid methods.6'9'19 After every restriction of thedata from a fine grid to the next coarser grid, the solidwall and the fan-face boundary conditions are appliedto prevent large jumps that might trigger large adap-tive dissipation coefficients. Similarly, these boundaryconditions are also applied after every prolongation op-eration of the data from a coarse mesh to the next finer

mesh. The V-cycles are started on the finest mesh fromthe first step of the pseudo-time advancement and con-tinued until convergence is obtained.

4 ResultsIn this section, both steady flow simulations using themultigrid method and acoustic radiation simulationsfrom a turbofan engine inlet are presented.

4.1 Steady State SolutionsIn multigrid work the RMS residual or the error is usu-ally plotted versus the number of work units so that theconvergence rates can be compared in the same normswith those given by using single meshes. This is becauseone has to perform more operations in a multigrid cy-cle than in a single mesh cycle (equivalent to one timestep or iteration for single mesh). The increase in thetotal number of operations is usually proportional top = (P+Z)?e=oZs V2*)' where p is the overhead cost andthe sum is the increase due to performing calculationson multiple grids. The overhead is usually associatedwith the restriction and prolongation operations as wellas the application of the solid wall, far-field, and thefan-face boundary conditions in the case of an engineinlet after each of the transfer operations. The totalcost increase factor P for the current code is about 1.8.Therefore, the ratio of the multigrid convergence rateto single mesh convergence rate per fine mesh iterationmust be over at least 1.8 so that one can talk aboutconvergence improvement using the multigrid method.In other words, the convergence improvement per timestep using multigrid method must not be onset by theintroduced overhead and the increase in the number ofoperations. One good measure would be the conver-gence rate given per CPU time. However, the CPU timeis usually machine dependent. Therefore, we presentthe results showing the convergence versus the numberof cycles curve.

4.1.1 RAE 2822 Airfoil

First, we present results for the steady state flow aroundthe RAE 2822 airfoil at a free stream Mach number ofMOO = 0.725 and an angle of attack of a = 2.92°. Thesolution was performed on 3 different size domains usingthe Euler equations together with 2 different far-fieldboundary conditions sets, namely the Riemann typeboundary conditions20 and the Bayliss-Turkel type non-reflecting boundary conditions.13 The fine meshes of themultigrid runs all included 192 x 96 x 1 cells (includingthe ghost cells). Due to their C topology, these mesheshad high aspect ratio cells in the far-field, especially inrelatively large domains.

Figure 4 shows the convergence histories of the sin-gle grid and multigrid runs. The convergence improve-ments by the multigrid application are clear. For thesecases the Riemann type boundary conditions were used.It is evident that the domain size (indicated by 5c, lOcand 25c in the figure; c = chord length) does not havemuch effect on the single-grid convergence of the solu-tion. However, the multigrid convergence is influencedby the domain size. This is mainly due to the strongstretching of the grid points in the far-field region onthe relatively large domains.

0

-2

c- -4

JL -8at2 -10

-12

-14

RAE2822, M_=0.725, o=2.92°

1Qc 25c'"fe'tSpO^

^X^vwv-^Single grid

domain size 5c

1000 2000 3000 4000 5000 6000Number of cycles

Figure 4: Convergence histories of the Euler runson varying-size domains. RAE2822 airfoil, MOO =0.725, a = 2.92°.

Figure 5 illustrates the Mach contours about the air-foil. Although the mesh was relatively coarse in thechord-wise direction over the airfoil, the shock was cap-tured very well, as evident from the figure. Figure 6indicates that the numerical pressure distribution overthe airfoil is highly affected by the computational do-main size. The differences between the Euler solutionsand the experiment21 observed in this figure are mainlydue the viscous effects plus more importantly the windtunnel effects that the experimental measurements in-volve. Therefore, pressure coefficient comparisons areusually performed at modified angles of attack match-ing the experimental lift coefficient (see, for example,Coakley22). However, this figure is significant to vali-date the multigrid solutions. They agree perfectly withthe single grid solutions. Figure 7 indicates a very in-teresting point. It is clear that, as the domain size is in-creased, the use of the Riemann type far-field boundaryconditions results in increasingly over-predicted pres-sure distributions, while the use of the Bayliss-Turkeltype boundary conditions results in decreasingly over-predicted solutions as compared to the experiment.21However, both type boundary conditions yield converg-ing solutions as the domain size is increased further andfurther.

Mach contours

X

o

1.5

1.0

0.5L

0.0

-0.5

-1.0

RAE2822, M_=0.725, o=2.92°

0.0 0.2 0.4 0.6X/C

0.8 1.0

Figure 5: Mach contours around the RAE2822 airfoil.Moo = 0.725, a = 2.92°.

RAE2822, M_=0.725, = 0.725, a = 2.92°.

4.1.2 JT15D inlet

The low Mach number cases presented in this sec-tion pertain to the JT15D inlet geometry12 (withouta centerbody) at zero angle of attack. All mesh levelswere generated using conformal mapping.3 The far-fieldboundaries were placed as close as only 1.5-2.0 inlet di-ameters from the inlet lip. At the fan, 1-D characteristicbased nonreflecting boundary conditions were used.3'23

Figure 8 compares the convergence histories (densityresidual) of the single mesh and multigrid runs for afree stream Mach number of 0.204 and a mass flow rate(MFR) of 14kg/s. The fine mesh for this case had192 x 32 x 1 cells with aspect ratios varying from nearly1 to 5. Clearly there is a significant improvement hiconvergence per time step using the multigrid method.It is extremely important to drive the numerical errorsto very low levels so that they do not contaminate the

Figure 7: Effects of the far-field boundary condi-tions on the Euler solution around the RAE2822 air-foil. (Ri=Rfemann type BCs; BT=Bayliss-TurkeI typeboundary conditions). MOO = 0.725, a = 2.92°.

acoustic solutions. Therefore, in noise prediction calcu-lations, the steady state flow residual is usually driven10-13 orders of magnitude down from its initial value.It has been experienced that the single grid runs do notusually drive the residual down to these levels at a con-stant rate. Sometimes the solution does not convergeat all when these levels are aimed.

0

-2

T3to -6

-12

Convergence history, JT15D inletM_=0.204,0=0°, MFR=14 kg/s

single mesh (192x32x1)

3 levels of mesh, V-cycles

2 4 6 8Number of iterations (x103)

10

Figure 8: Convergence history of the steady state prob-lem on the fine mesh (192 x 32 x 1). JT15D inlet,Moo = 0.204, a = 0°, MFR = Ukg/s.

-Through multigrid iterations the residual is expelledmore quickly out of the domain as mentioned earlier.This is evident from the mass flow ratio histories ofthe single grid and multigrid runs, as shown in Fig-ure 9. The mass flow ratio with the multigrid methodcame very near the specified mass flow ratio and sta-bilized within only 250 time steps while this took ap-

proximately 1700 steps for the single grid. This be-havior of multigrid methods is crucial in quick aerody-namic design analyses.19 The 1-D characteristic fan-face boundary condition implementation described inOzyoriik3 yielded an extremely accurate mass flow ra-tio at convergence. The computed mass flow ratio is0.594 as compared to the specified 0.596.

0 r Convergence history, JT15D inletM_=0.192, cc=0°, MFR=22 kg/s

0.75

•B0.702.20.65COCO

iO.60

0.55

Mass flow convergence, JT15D inletM_=0.204, a=0°, MFR=14 kg/s

single mesh (192x32x1)3 levels of mesh, V-cycles

1 2Number of iterations (x103)

Figure 9: Convergence in terms of the mass flow ratioon the fine mesh (192 x 32 x 1). JT15D inlet, MOO =0.204, a = 0°, MFR = 14 kg/s.

In Figure 10 the convergence histories for MOO =0.192 and MFR - 22 kg/s are presented. The finemesh for this case had 384 x 96 x 1 cells with aspect ra-tios ranging from nearly 1 to 5. This mesh was designedto resolve waves at up to 4.8 KHz in the near-field. Itis extremely difficult to attain a steady state flow fieldin reasonable number of time steps on this land of finemeshes using the fourth-order accurate algorithm, as in-dicated by the single grid convergence curve. However,using the FAS multigrid technique a significant improve-ment was observed. It is evident by comparing Figures8 and 10 that the improvement tends to increase as themesh is further refined. This can be attributed to theincrease in the transferred error bandwidth during therestriction and prolongation operations.

Figure 11 illustrates the histories of the mass flow ra-tios. Again the mass flow ratio for the multigrid casesettled very quickly compared to the single grid case.The final mass flow ratio at convergence is 0.935, while0.938 was specified as the operating condition. The er-ror is well below 1%.

The flow field Mach number contours for this case atconvergence are shown in Figure 12. The flow accel-erates in the throat region due to the high mass flowrate. The Mach number in this region reaches 0.27.This means that the wavelength of the acoustic waveshi the upstream direction is shortened by a factor of0.73 (= 1.0 — M). This is a disadvantage in the ducted

¥-2-

A-4-§-

cc,

jf-e

-83 levels of mesh, V-cycles

2 4 6 8 10 12Number of iterations (x103)

Figure 10: Convergence history of the steady state prob-lem on the fine mesh (384 x 96 x 1). JT15D inlet,Moo = 0.192, a = 0°, MFR = 22 kg/s.

1.00

_o"§0.90

CO

jo 0.80

0.70

Mass flow convergence, JT15D inletM_=0.192, 0=0°, MFR=22 kg/s

Single mesh (384x96x1)3 levels of mesh, V-cycles

1 2 3Number of iterations (x103)

Figure 11: Convergence in terms of the mass flow ratioon the fine mesh (384 x 96 x 1). JT15D inlet, M^ =0.192, a = 0°, MFR = 22 kg/s.

fan noise computations. Notice that the contour lineshi this figure are extremely smooth across the centerlineshowing no signs of a singularity problem. This can befurther observed when contours of the ratio of the lo-cal total enthalpy to the free stream total enthalpy areexamined in Figure 13. The ratio is extremely closeto unity everywhere except near the solid wall wherethe effect of artificial dissipation is observed. This ef-fect, however, is only slight. These results are extremelyimportant in terms of showing that the method can ac-curately predict steady inlet flow fields.

4.2 Inlet Radiation CasesThe ultimate test of the current code is carried out inthis section. Far-field noise of a spinning mode of theJT15D engine is predicted at actual flight conditions.

Mach

Muitigrid, V-cycles

0.99860.9985

Muitigrid, V-cycies

Figure 12: Mach contours at the steady state. JT15Dinlet, MTO = 0.192, a = 0°, MFR = 22 kg/s.

Figure 13: Total enthalpy contours at the steady state.JT15D inlet, M«, = 0.192, a = 0°,MFR = 22 kg/s.

The JT15D engine configuration has an array of 41 rodsplaced in front of the 28-blade rotor, which producesmode (13,0) interaction tones at the BPF.12'24 Theflight Mach number MOO is 0.204 and the inlet ductcarries a mass flow of 17.297 kg/s at an engine speedof 8120 revolutions per minute (RPM). There is noangle of attack. The (13,0) mode at these conditionshas a BPF of 3789.3 Hz and cut-off ratio of fi3,0 =1.27. This frequency corresponds to the dimensionlessfrequency parameter (27rl?PF)x(r//c/) = 18.65, whererf and Cf are the duct radius and the speed of sound atthe fan stage, respectively.

The steady state part of the problem was solved onan axisymmetric, 384 x 96 x 1-cell mesh using the multi-grid technique. This mesh is the same as the one usedin the previous section and is shown in Figure 14. Thesteady state solution was then spread out onto one ofthe 13 periodic 3-D grids to start the time-accurate partof the solution, using periodic boundary conditions inthe circumferential direction. One indeed does not needto solve this problem on a fully 360° mesh, since asingle spinning mode generates periodic pressure pat-terns in the circumferential direction. The (13,0) modegenerates periodic patterns at every 360°/13 degrees.Therefore, the problem was solved on a 384 x 96 x 16mesh, having 16 cells per circumferential lobe. Thetime-accurate R-K iterations were carried out using atime step size At = 0.25/(384 x BPF), and the Kirch-hoff integrations were performed at every 16th step ofthe R-K iterations.

Figure 15 shows the location of the Kirchhoff surfaceand the steady state pressure contours for this case. Asnapshot of the acoustic pressure contours in the verti-cal plane of the inlet is shown in Figure 16, where also

shown are the RMS acoustic pressure contours. The di-rectivity trend of the radiating (13,0) mode is evidentfrom this figure. The far-field sound pressure levels ascalculated for 24 observer points at a distance of 30.48 mat every 3° are shown in Figurg 17 along with the ex-perimental data24 and the finite element-wave envelope(FE-WE) solution of Eversman et a/.24

Figure 18 shows the far-field sound pressure level(SPL) for the conditions, MOO = 0.200, MFR =19.275 kg/s and BPF = 4360 Hz. The frequency pa-rameter for this case is k/r/ = 21.48. For this case atime step size of At = 0.4/(384 x BPF) was used andthe Kirchhoff integrations were performed at every 12thR-K iteration.

The comparison of the current simulations with theexperimental data reveals very good agreement in gen-eral, although the predictions have the SPL peaks atsomewhat higher angles. Similar differences are ob-served between the FE-WE solutions and experiment,as well. However, the current simulations have about a2-degree better prediction of the peak SPL angles thanthe FE-WE method. The reason for the differences be-tween the experiment and the simulations is not clear.At this point we can only speculate about the differ-ences, such as that the source model used is only ap-proximate and does not exactly represent the real acous-tic source, namely the interactions of the rods with therotor. Nonetheless the source has been modeled reason-ably well.

5 ConclusionsConvergence acceleration is essential in the current ap-proach of the ducted fan noise prediction method solv-

JT15D mesh (384x96x1) JT15D at M_=0.204, MFR=17.297 kg/s

Every 3rd grid line shown

Figure 14: The mesh system and two typical Kirchhoffsurfaces around the JT15D inlet.

ing the full, time-dependent Euler equations togetherwith nonreflecting boundary conditions. For this pur-pose, a multigrid convergence acceleration techniquethat retains the high-order accuracy of a typical, ex-plicit, high-resolution CAA algorithm has been devel-oped and implemented. Particularly, Jameson's fullapproximation storage method has been utilized. Themethod has been applied successfully both to the time-dependent Euler and nonreflecting boundary conditionsequations. Although only 3 mesh levels with V-cyclesare used, the improved convergence characteristics ofthe hybrid code with the multigrid method make it pos-sible to attain steady state flows at very low Mach num-bers on very fine grids. Example calculations showingthe use of the method have been carried out for bothexternal and internal flow problems. Far-field noise sim-ulations have been performed for an actual turbofan en-gine inlet and comparisons with flight data and othernumerical results have indicated very good agreement.

Acknowledgment

This work was supported by the NASA Langley Re-search Center, under the grant NAG-1-1367.

References[1] Ozyoriik, Y., and Long, L. N. Computation of

Steady p, N/m2A 103624.61

103189.31102753.80102318.30101882.80101447.30101011.7910057629100140.789970528

Figure 15: Steady state pressure around the JT15D in-let and the location of the Kirchhoff surface. MOO =0.204, MFR = 17.297 kg/s, BPF = 3789.3 Hz.

sound radiating from engine inlets. AIAA Jour-nal, May 1996.

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JT15D at M.=0.204, MFR=17.297 kg/s(13,0) mode,_BPF=3789.3 Hz

100

Figure 16: Instantaneous and RMS acoustic pressurecontours in the vertical plane. (13,0) mode, JT15Dinlet, MOO = 0.204, MFR = 17.297 kg/s, BPF =3789.3 Hz.

[9] Chima, R. V., Turkel, E., and Schaffer, S. Com-parison of three explicit multigrid methods for theEuler and Navier-Stokes equations. AIAA Paper87-0602, 1987.

[10] Agarwal, R. K. Unigrid and multigrid algorithmsfor the solution of coupled, partial-differentialequations using fourth-order-accurate compact dif-ferencing. MDRL 81-35, 1981.

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[12] Preisser, J. S., Silcox, R. J., Eversman, W., andParret, A. V. A flight study of tone radiation pat-terns generated by inlet rods in a small turbofanengine. AIAA Paper 84-0499, 1984.

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[14] Tarn, C. K. W., and Webb, J. C. Dispersion-relation-preserving finite difference schemes forcomputational physics. Journal of ComputationalPhysics, 107, pp. 262-281, 1993.

[15] Ives, D. C., and Menor, W. A. Grid generationfor inlet-centerbody configurations using conformalmapping and stretching. AIAA Paper 81-0097,1981.

CDTJ

90

Ii£0.TJ

OCO

80

70

ExperimentFinite element/wave envelope

. (Eversman et al. [24])- Current simulation

20 30 40 50 60 70 80 90Angle from inlet axis, Deg.

Figure 17: Far-field sound pressure level of the(13,0) mode. JT15D inlet, Mx = 0.204, MFR -17.297 kg/s, BPF = 3789.3 ffz.

100

CDT3

I90

80

a.TJo

CO

60

ExperimentFinite element/wave envelope

(Eversman et al. [24])Current simulation

20 30 40 50 60 70 80 90Angle from inlet axis, Deg.

Figure 18: Far-field sound pressure level of the(13,0) mode. JT15D inlet, M^ = 0.2, MFR =19.295 kg/s, BPF = 4360 tfz.

[16] Jameson, A., Schmidt, W., and Turkel, E. Nu-merical solutions of the Euler equations by finitevolume methods using Runge-Kutta tune-steppingschemes. AIAA Paper 81-1259, 1981.

[17] Swanson, R. C., and Turkel, E. Artificial dissi-pation and central difference schemes for the Eulerand Navier-Stokes equations. AIAA Paper 87-1107,1987.

[18] Tyler, J. M., and Sofrin, T. G. Axial flow compres-sor noise studies. SAE Transactions, 70, pp. 309-332, 1962.

[19] Jameson, A. Successes and challenges in computa-tional aerodynamics. AIAA Paper 87-1184, 1987.

[20] Long, L. N., Khan, M. M. S., and Sharp, H. T.Massively parallel three-dimensional Euler/Navier-

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Stokes method. AIAA Journal, 29(5), pp. 657-666, May 1991.

[21] Cook, P. H., McDonald, M. A., and Firmin, F. C.Airfoil RAE 2822 pressure distribution and bound-ary layer measurements. Technical report, AGARDAR 138, 1979.

[22] Coakley, T. Numerical simulation of viscous tran-sonic airfoil flows. AIAA Paper 87-0416, 1987.

[23] Giles, M. B. Nonreflecting boundary conditionsfor Euler equation calculations. AIAA Journal,28(12), pp. 2050-2058, December 1990.

[24] Eversman, W., Parret, A. V., Preisser, J. S., andSilcox, R. J. Contributions to the finite elementsolution of the fan noise radiation problem. Trans-actions of the ASME, 107, pp. 216-223, 1985.

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