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Numerical Simulation of Internal Flow in Aircraft Engineby Parallel Super Computer

Mitsuraasa MAKIDA*

National Aerospace LaboratoryChofu City, Tokyo, Japan, 182-0012

Tel.+81-422-47-5911, Fax.+81-422-47-7930

Abstract

A parallel numerical simulation code for the three-dimensional flow fields in an aircraft combustor, in-cluding liquid fuel spray, has been developed. Tosimulate a combustor with complicated geometry,the multi-grid method is used to divide into sub-grids and calculate on individual processor elementof a parallel super computer. And the gas-liquidphases are also separated, and they are treated alsoin parallel. And in this code, the Euler equationsare used for the fuel droplets' liquid phase assum-ing the continuous fluid, and the full Navier-Stokesequations are applied for the gas phase. Both phasesare connected through mass, momentum and energyexchange equations, and solved simultaneously. Theliquid phase has a radius distribution, and is dividedinto five groups of different initial radius, and theyare also treated as individual phases. And the vapor-ization and the combustion of fuel droplets are in-cluded in the calculation. Both phases are solved bythe finite difference method and the Harten-Yee's ex-plicit non-MUSCL modified-flux type TVD schemeis applied to convective terms and the central dif-ference scheme is to viscous terms of the gas phase.A model flow field of liquid fuel combustor, whichhas a swirler inlet region, is numerically simulatedby a parallel super computer, and some results arepresented.

Introduction

In the combustor of the aircraft engine, many de-vices are adapted to reduce the harmful exhaust gasand improve the efficiency of the combustion. For

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* Research Engineer

example, the swirler is used to promote the mix-ing and dilution air holes are devised to reduce thetemperature and suppress the NOx formation. Sothe geometry of the aircraft combustor becomes verycomplicated. On the other hand, the technologies ofthe numerical simulation are getting developed, andmany flow fields can be investigated by this method.Therefore we can now incorporate the simulationmethod to design the combustor which has compli-cated geometry. To calculate the complicated flowfield, there are some methods; for example, the un-structured grid method is getting developed and willbe very useful in the near future. But at this time,numerical methods on structured mesh have beenwidely used and are adjusted in many cases. In thisstudy, one of the methods is presented, which usesthe multi-grid method on a parallel super computer.

Besides, in aircraft engines, using liquid fuel, in-ternal flows behave as multi-phase flow and are verydifficult to be analyzed in detail only by experi-ments. To make up these difficulties, analytical re-searches by numerical simulation have been done.However, because the phenomena differ in each localarea of the combustor and total behavior is compli-cated over the whole combustor, minute simulationrequires fine calculation grid, and it takes large com-putational memory and CPU time. Therefore, thatis difficult to be treated on conventional comput-ers. We had already developed a simulation codefor an aircraft combustor with fuel spray combus-tion in two-dimensional field [1],[2],[3]. And in thisresearch, to extend this code to three-dimension, weused parallel calculation method to divide calcula-tion region and gas-liquid phases, and treated themon each processor element (PE). And we numeri-cally simulated two model flow fields of liquid fuelcombustor by a parallel super computer NWT (Nu-merical Wind Tunnel) in our laboratory.

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Gas Phase

group 5

Figure 1: Schematic of Euler method

Numerical Method

ModelingIn the numerical simulation of flow fields with

liquid spray, there are two ways of approach; theLagrangian method, which deals with the individualor representative droplets using the equation of mo-tion, and the Eulerian method, which assumes themas a continuous fluid. In the case of dense spray, thenumber of droplets is too large for conventional com-puters to simulate the problem economically, usingthe Lagrangian way. Therefore in present study, theEulerian type method is used for the liquid phaseassuming the droplet group as the continuous fluids.Further more, as shown in Fig.l, the liquid phasehas an initial radius distribution, and it is dividedinto five groups of different initial radius, and theyare treated as individual fluid. In the figure, a squarerepresents a calculation grid, and physical propertiesof each droplet in the grid amount to those of liquidphase in the grid. For example, the density of theliquid phase of group j, cij is obtained by multiply-ing mass of a single droplet (of radiusTjy, materialdensity:mj), with the number density of the droplets,as below. Physical values calculated by this way areassumed to be constant in one grid.

(1)

Liquid Phase

Gas & Liquid Phases

group 1 group 2 •group 5

Large DiameterFigure 2: Schematic of calculation region division,phase division, and group division in the liquidphase.

On the other hand, the three-dimensional non-steady full Navier-Stokes equations and species con-servation equations are used for the gaseous phase.And in order to simplify the analysis of the phenom-ena, following assumptions are considered:

1. The liquid phase is consisted of n-hexaneCsHit, and gas phase is consisted ofO2, H2O, CO^, and N2\

2. The gas phase is quasi-real such that the specificheat is represented by a polynomial function oftemperature, but the equation of state for realgas is used;

3. The bulk viscosity, Soret effect, Dufour effect,and diffusion by pressure and the effect of ex-ternal forces such as gravity are neglected;

4. Total amount of mass of mixed gas and dropletsis conserved;

5. Droplet collisions are neglected;

6. The droplet surface is smooth and spherical andis not transformed or broken-up;

7. Interactions between two phases are the dragforce, heat and mass transfer.

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Parallel Processing Method

As mentioned above, we have to calculate mixedgas phase of five species and five groups of liquidphase simultaneously, and this consumes much morecomputational memory than ordinal fluid calcula-tion. So if we want to solve them on a single com-puter, we cannot take enough calculation grid pointsbecause of the limitation of the computational mem-ory. To work out this problem, we rewrote the sim-ulation code in a parallel programming language;NWT-Fortran, which runs on the parallel super com-puter NWT (Numerical Wind Tunnel) in our labo-ratory. The NWT system has 166 PEs connected byglobal network, and each PE is a vector computerwith peek performance of 1.7G FLOPS and mainmemory of 256MB.

As shown in Fig.2, we divided computationalregion into "a" numbers of sub regions, and in eachsub region, gas and liquid phases are calculated onindividual PEs, and finally we use "2a" PEs. More-over, each PE of liquid phase includes five groups ofdifferent radius. In the future, we are going to treateach liquid group on their own PE.

Governing Equations

Governing equations are constructed follow-ing above modelings, extended to three-dimensionfrom two-dimensional equations in reference [2].Non-steady three-dimensional compressible Navier-Stokes equations coupled with continuity equationsfor each species are used for the gas phase. Whilenon-steady three-dimensional compressible Eulerequations are applied for the liquid phase. Us-ing above assumptions, the governing equations forthe gas and liquid phases are expressed as follows

Governing Equations for Gas Phase

dudt + ^ + S + H (2)

K2

Governing Equations for Liquid Phase

U =

9Uh-dt

\1

)

, _~ l (3)

CgUk2

k2 +Sklk2

(eg+Pg)uk.,

(4)

mfeai(5)

J=l

(6)Q

(7)

_

~ .In above equations, the volume fraction of dropletsfj is introduced, and substantial density of eachphases cg,cgi and c^ are expressed in terms of €j.And each phase is assumed to have partial pressurein proportion to their volume fractions.

, cLj =j=i i=l

(9)

(10)i=l

Other values; specific energy, enthalpy, heat andmass diffusions are given as follows.

e - =

l 0 (Ty )

(13)

dT

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And from the assumption, the specific heat is givenas the polynomial function of the temperature[4].

Cpi/Ri = an + a2iT + a3iT2 + a4iT3 + a5iT4 (17)

where the coefficients (iki(k = 1 ^ 5 ) are calculatedfrom the value of Cpi listed in the JANAF table.

Transport Properties

In above equations, the molecular viscosity co-efficients of each species and molecular diffusion co-efficients are calculated by the Chapman-Cowling'sequation [5] and the Eucken's empirical equation isapplied for the thermal conductivity coefficient.

>„ = 1.8829 x 10~

-0.145 -2.0

(19)

(20)

. — ' ' J T . . — F r - T - (9~\\3 — ——7,—— i -Leij — V J « 6J \ I

= 8.3280 x lO,-1 VT(23)

And from these values, /j,, K of the mixed gas arecalculated by the theory of Sutherland and Wilke[6j.

Ki

x<(24)

° 2v/2{l

Interaction TermsThe interaction between gas and liquid phases

are considered as the source term like in the Par-ticle Source in Cell (PSI-cell) model [7]. Ag and AIare the interaction coefficients of two phases by thedrag force which induces momentum and energy ex-change between them. And momentum changes inboth phases are presented as follows:

d(cgu) _

(26)-\ — C-l n jr\.l 4 \U — 11-1 <* ) — 11-1 •; 77-iJ. J J \ J I J

A"=LV;(27)

Where CoSi is the Stokes drag coefficient, and thedrag coefficient with combustion CDJC is correctedby the mass transfer number £?y, from that with-out combustion CD, which is given by Henderson'sequation [8]. Here, Bj is given by evaporation theoryas follows.

ri _ "(Joo) V 3) /OQ\Bj — ————T————— (^8)

In the same way, Bg and BI are interaction coeffi-cients of heat transfer, and energy changes are givenbelow.

de•9 _

dt

dt

,-B,

= E 7=~-————+u-(u-uh)

BH = + Utj • (u- Ulj)

In above eqautions, the Nusselt number NUJ is cal-culated by Carlson & Hoglund's equation[9], and TSis temperature at droplets surface.

Vaporization and CombustionVaporization and combustion of fuel droplets

are treated in simple ways. The evaporation of fueldroplets are assumed to be in quasi-steady state, andfollows so called "dMaw". In this process, dropletsgain their energy from the surrounding gas to evap-orate. The vaporization to the gas phase can beconsidered as the mass flux from the liquid phaseaccompanied by the momentum and energy trans-fer, where the mass flux from the liquid phase to thegas phase is given by the following equation:

(30)And for the chemical reactions in the com-

bustion, a simplified single-step reaction equation isused. For the gas phase, an empirical modified Ar-rhenius formula for the forward reaction rate kf isgiven as follows [10]:

— O2 6C02

kf = 5.7xlOnexp (- 30.0

dt

20 (31)

3-25[02]1-5

(32)

9j

(29)

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AIAA 98-0725

Where the universal gas constant R0 is 1.9862 x10~3 kcal/mole/Kiu this unit, and [CsH^] is themole fraction of CeHu.

Numerical Algorithm

The governing equations for the two-phase flowin the physical domain x — y — z are transformedinto those in the computational domain £ — t] — £and solved by the finite difference method. Inboth phases, the Harten-Yee's explicit non-MUSCLmodified-flux type TVD scheme [11] is applied toconvective terms and the central difference schemeis to viscous terms of the gas phase. This scheme hassecond-order accuracy in time and in space. Phys-ical properties at the boundaries of cells, which arenecessary to calculate the numerical flux, are givenby the generalized Roe's average. In both phases,source terms are added to the schemes explicitly.

Results of Test Case Calculations

Using the numerical methods mentioned above,we have done calculations for two cases; (A) a simplecombustor shape model consisted with 2 regions, and(B) a complicated geometry with swirler inlets andtotally consisted with 6 regions.

Case (A)

In case (A), the numerical grid system and bound-ary names are shown in Fig.3. This grid system isconsisted of two grids with 30 x 30 x 36 grid points,and grid 1 aims to make fine mesh near the fuel injec-tor. For boundary conditions, air and fuel dropletsare injected from (D with inlet velocity of 50m/sec,inlet angle of 60 degrees and swirl angle of 45 de-grees. For this case, inlet boundary condition atthe swirler is given by constant values. And air anddroplets are injected from © with inlet velocity of22.5m/sec, and inlet angle of 30 degree.

Moreover as mentioned before, liquid phase hasan initial radius distribution, and they are given byfollowing equation:

(33)

Where rim means the value of TIJQ which correspondsto maximum NIO, a is decided from rim and loadingratio of the fuel, and /3 is a parameter to control thewidth of the distribution shape. Each liquid groups'initial radius r^ and number density A . are givenas shown in Table 1, and the SMD (Sauter MeanDiameter) becomes 20//m.

312345

rij0 (iJ-m)2.474.957.429.9012.4

NiJQ (1/m-5)7.74 x 10IU

1.01 x 1011

1.07 x 1011

1.02 x 1011

8.89 x 1010

Table 1: ry0, NIJO of each group in liquid phase

Boundary (a) ~ (d) are non-slip adiabatic walls,(e) and (f) are symmetric boundary, and (g) is anoutlet-boundary. And in the region where grid 1 andgird 2 overlap, physical properties of each grid areinterpolated from another grid. Notation £. rj. C inFig.3 represent the computational domain £ — rj — C,which is transferred from the physical domain x -y-z.

Calculated results for case (A) are shown on Fig.4and Fig.5. In Fig.4, velocity vectors of the gas phaseon a computational plane of £ = 5, and stream-lines originated from this plane are shown. Fromthe streamlines we can see that; a recirculation zoneexists inside the fuel spray injected from inlet (2),and outside of that zone, complicated flows are gen-erated in tangential direction induced by interactionwith the swirling flow from inlet ©. Existence of therecirculation zone have also appeared in the calcula-tion of two-dimensional spray[2], [3]. In Fig.5, pro-jections of streamlines of the liquid group 1 and 5 tox-y plane are shown. As mentioned before, these twogroups have different droplet diameter; group 1 hassmaller diameter and group 5 has larger one. Thesestreamlines show that, small droplets behave alongthe gas phase while large droplets have inertia to flyin the direction of initial injection. This feature isalso consistent with two-dimensional case[3].

To see the flow field in more detail, streamlines of(1) gas phase and (2) liquid phase group 1, projectedto the x-z plane, are shown in Fig.6. It is seen thatliquid fuel injected from inlet (D is involved in theflow generated from the swirler, and comes to regionnear the wall. Besides, liquid fuel from the inlet ®goes in the direction of the injection to form liquidfilms, and in that region, gas phase flow is separatedand a recirculation zone is generated. And in Fig.7,profiles of (1) density of liquid phase group 1 and (2)mass transportation to the gas phase (evaporation)are shown. From these figures, the fuel liquid seemsto spread as mentioned above, and the evaporationoccurs in the near region of the inlet.

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(d)

(e)

(1) Projection to y-z plane (2) Whole view

Figure 3: Calculation grid system and boundary names in case (A).

(1) Whole view (2) Projection to x-z plane

Figure 4: Velocity vectors and streamlines of gas phase in case (A).

>-/: / ?&38$K8&&$ '" ^ n ">-r--/ W$9F3l£^^%^11 TlS^^ i in /IlLLQ^^PKSTT

(1) group 1 (2) group 5

Figure 5: Streamlines of liquid phase in case (A).

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(1) gas phase (2) liquid phase (j=l)

Figure 6: Streamlines projected to the x-z plane in case (A).

Case (B)

/•-i \ j . (2) mass transportation

Figure 7: Liquid phase density and mass transport in case (A).arbitrary number of sub-grid to one PE, and real al-location map of the sub-grids to PEs in case (B) areshown in Tabel 2. In this table, PG and PL meanPE which is used for gas phase and liquid phase, and6 PEs are used in total.

The total view of the computational region in case(B) is shown in Fig.8. In this case, as mentionedbefore, in order to get exact inlet condition fromthe swirler, the swirler region is also integrated inthe calculation region. And the total system is con-sisted with 6 regions; 4 swirler vane regions, 1 regionnear the injector, and 1 main chamber region. Eachgrid system and boundary names are shown in Fig. 9.As shown in this figure, grid 1^4 has 10 x 10 x 12grid points, and grid 5 and grid 6 has 20 x 20 x 40grid points. It usually happens, like in this case,that some small sub-regions have to be involved inthe calculation, but cannot have same grid points ofthe main region, because if we adopt too many gridpoints to these small regions, the time step of thecalculation is suppressed by the small separation ofgrid points. We could treated each sub-grid on in-dividual PE in spite of grid size, and use 12 PEs intotal, but this is not efficient in the view of memoryusage of the computer. So in this study, we allocate

Swirler grid

zone

PE-

Z1 Z2 Z3 Z4 Z5 Z6

(15x15x20)x4 20x20x4120x20x41

Table 2: Allocation map of grid system to PEs incase (B).

As shown in Fig.9, velocity is given at © bound-ary, and after calculation in the swirler vane re-gion, inlet velocity profile from the swierler's out-let boundary (2) is connected to grid 5 through (3)

7

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boundary. Values at boundary (4) and (5) are inter-polated as in case (A), and other boundaries are alsothe same as in case (A).

Calculated results for case (B) are shown on Fig. 10and Fig.ll. In Fig.10, streamlines are shown for(1) gas phase and (2) liquid phase group 5. Theselines are generated from different position; (D and(D are from the fuel inlet on a computational planeof C = 20, (2) and © are likely from C = 15, and ©is from inside of the swirler vane. Because the planeof £ = 15 includes exit of the swirler, and C = 20does not, streamlines from them behaves somewhatdifferently, and each line shows existence of recircu-lation zones. From the comparison of (1) and (2),it can be seen that liquid phase has inertia to gostraight, but in the downstream, it is effected by thegas phase and caught in the recirculation zone. Andat that zone, from Fig.ll, it is seen that evaporatedfuel gas has high concentration.

Concluding remarks

We have made a parallel numerical simula-tion code for three-dimensional flow fields in an air-craft combustor, including fuel spray, and simulatedmodel flow fields, which have swirler inlet regions.From the presented results, following phenomenacan be seen:

1. A recirculation zone is generated inside the fuelspray field, and the flow field around that re-gion is effected by the swirling flow and becomescomplicated.

2. Small droplets are likely to behave along thegas phase while large droplets have large inertiato go straight, so the distribution profiles aredifferent in the spray field.

And in this study, we have done relatively small sizesimulation using 6 PEs, but we can extend the com-putational region only by adding PEs and calculatethem in parallel, without being bothered by the lim-itation of computational memory and time.

Nomenclature

A, B interact coefficientc densityCD drag coefficientCp specific heate specific energyh specific enthalpyh° specific enthalpy in steady statekf forward reaction rateL latent heat of vaporization/ property of liquid phasem mass flow by vaporizationmi material density of dropletsrhki mass diffusion of In number of gas species (n = 5)m number of liquid groups (m — 5)N number density of dropletsP pressurePr Prandtl numberq heat diffusionr\ radius of a dropletR universal gas constantt timeT temperatureu velocityYi mass fraction of ith speciesx, y, z physical coordinate£, r/, C calculation coordinateT viscous shear stressK thermal conductivityH dynamic viscosityp material densitye volume fraction of dropletsu> mass rate of production of ith species

Subscriptsg gas phasei property of i-th gas species

(l:Csff14, 2:O2, 3:H2O, 4:CO2, 5:N2)j jth group of liquid phaseI liquid phases property at the droplet surfaceco property at infinity0 initial condition

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References[1] M.Makida, A.K.Hayashi, "Numerical Analysis

on Fuel Spray Jet", Proc. of the 18th Int.Sympo. on Space and Tech. and ScL, pp.709(1992).

[2] A.K.Hayashi, M.Makida, T.Fujiwara, "SprayJet Combustion Using yhe Group CombustionTheory", Proc. of IUTAM (1994).

[3] M.Makida, A.K.Hayashi, T.Fujiwara, "Numer-ical Simulation of Fuel Droplets' Behavior inSpray Combustion", J. of the Japan Soc. forAeronautical and Space ScL, Vol.43, No.498,pp.390 (1995).

[4] W. C. Gardiner, Jr : Combustion Chemistry,Springer-Verlag, pp.485.

[5] Chapman,S. and Cowling,T.G., "The Mathe-matical Theory of Non-Uniform Gases", (1952).

[6] C. R. Wilke,"A Viscosity Equation for GasMixtures", J. of Chemi. Phys., vol.18 (1950),pp.517.

[7] C.T,Crowe,"The Particle-Source-In Cell (PSI-CELL) Model for Gas-Droplet Flows", J. FluidsEngi. p.325, (1977).

[8] Henderson,C.B., "Drag Coefficients of spheresin continuum and rarefied flows", AIAA J.,Vol.14, pp.707, (1976).

[9] Carlson, D. J. and Hoglund, R. F., "Particledrag and heat transfer in rocket nozzles", AIAAJ., Vol.2 (1964), pp.1980-1984.

[10] W.Bartok, "Fossil Fuel Combustion", A Wiley-interscience Pub..

[11] H. C. Yee, "Upwind and Symmetric Shock-Capturing Schemes", NASA Technical Memo-randum 89464.

Figure 8: Total view of calculation grid system in case (B).

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grid 6(20x20x40)

gridS(20x20x40)

grid I~grid4(10x10x12)

Figure 9: Calculation grid system and boundary name in case (B).

(1) gas phase (2) liquid phase 5

Figure 10: Streamlines of gas and liquid phase in case (B).

Figure 11: Fuel gas concentration in case (B).

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