Efregcient parallel computationsof macrows of arbitrary macruids forall regimes of Reynolds Mach

and Grashof numbersM Mulas

Department of Computational Methods for Engineering CRS4Center for Research Development and Advanced Studies in Sardinia

Area of Computational Fluid Dynamics Uta (Ca) Italy

S ChibbaroDepartment of Computational Methods for Engineering CRS4

Center for Research Development and Advanced Studies in SardiniaArea of Computational Fluid Dynamics Uta (Ca) Italy

Dipartimento di Fisica Universita di Cagliari and INFNsezione di Cagliari Cittadella Universitaria di Monserrato

Monserrato (Ca) Italy

G Delussu I Di Piazza and M TaliceDepartment of Computational Methods for Engineering CRS4

Center for Research Development and Advanced Studies in SardiniaArea of Computational Fluid Dynamics Uta (Ca) Italy

Keywords Computations Flow Fluid

Abstract This paper presents a unireged numerical method able to address a wide class of macruid macrowproblems of engineering interest Arbitrary macruids are treated specifying totally arbitrary equationsof state either in analytical form or through look-up tables The most general system of theunsteady NavierndashStokes equations is integrated with a coupled implicit preconditioned methodThe method can stand inregnite CFL number and shows the efregciency of a quasi-Newton methodindependent of the multi-block partitioning on parallel machines Computed test cases rangingfrom inviscid hydrodynamics to natural convection loops of liquid metals and to supersonicgasdynamics show a solution efregciency independent of the class of macruid macrow problem

1 IntroductionFlows of engineering interest range from high speed aerodynamics toincompressible hydrodynamics and include low speed reactive macrows and heatexchange loops in natural and forced convection just to name few examplesOn the other hand macruid dynamics applications to different macruids bring into themathematical formulation different equations of state which in turn determine

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International Journal of NumericalMethods for Heat amp Fluid FlowVol 12 No 6 2002 pp 637-657q MCB UP Limited 0961-5539

DOI 10110809615530210438337

different levels of interaction between macruid dynamics and thermodynamicsThe above range of thermo- uid dynamics situations is described in terms ofnondimensional parameters by Reynolds Mach Grashof and Prandtlnumbers If chemical equilibrium also represents a reasonable assumptionthen reactive macrows are simply treated by deregning their suitable state equationswhere heats of reaction are hidden in the mixturersquos enthalpies and speciregc heats

The desire of computational macruid dynamics (CFD) has long been thepossibility to use a unique mathematical formulation together with a uniquenumerical method to examine the variety of macruid dynamics problems all ofthem governed by the NavierndashStokes system of equations Going back to theinfancy of CFD (the 1960s and 1970s) numerical simulations for compressibleand incompressible macrows represented two classes of methods which onlyshared the name of the governing equations After the extraordinary pace ofdevelopment of CFD methods during the 1980s and 1990s in more recentyears many efforts have been made to develop unireged numerical approaches tothe solution of a wider class of macruid macrow problems For this purpose eithertypical ordfincompressibleordm methods are generalized for high speed compressiblemacrows or vice versa the efregcient use of ordfcompressibleordm algorithms is extendedto low speed macrows of incompressible macruids by means of suitablepreconditioners The two methods are now more correctly called coupled andsegregated solution algorithms

When the coupled methods are used for incompressible macrow situations themain task is to remove or tackle the singularity of the NavierndashStokes equationswhen Mach number M approaches zero (M 0) The inviscid matrix of thecoupled system of equations becomes in fact ill conditioned and though MotherNature can cope with it numerical methods break down As a resultstraightforward use of coupled methods gives severe convergence problems oreven breakdown in the presence of regions with very low Mach number

On the other hand the incompressible NavierndashStokes approximation cannotbe used whenever the macruid compressibility and the macrow conditions indicatethat kinetic and internal energies can exchange reversible pdV work which isgiven by the capability of temperature and pressure changes to determinedensity changes Somehow arbitrary values of the nondimensional parametersb DT (where b represents the macruid compressibility coefregcient at constantpressure) and the Mach number M respectively set the trade off betweenabandoning the incompressible formulation and paying a price in terms ofaccuracy of the results

Plenty of studies can be found in literature focused on the attempt togeneralize the numerical methods to a wider class of problems For low speedmacrows with variable density methods based on asymptotic expansion havebeen developed for example in Guerra and Gustafsson (1986) These methodsremove acoustic modes in the limit of M 0 avoiding the singularity of thecompressible equations Another family of methods has been proposed by

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Patnaik et al (1987) In these approaches a pressure correction equation isderived from the energy equation while density is advanced in time via acontinuity equation The method can compute aeroacoustics and nonstationarylow Mach number macrows

Segregated methods can be extended to the compressible case Thisapproach gives the best perspective in the limit M 0 where a well-knownincompressible scheme can be recovered Moreover difregculties are encounteredwhen tackling the acoustic transonic singularity which determines anonconvergence behaviour in the regime M = O(1) (see Bijl and Wesseling1996) Only the use of a regrst order upwind scheme allows the algorithm toconverge (see Bijl and Wesseling 1998) Staggered schemes are generally usedwith these methods to avoid pressure oscillations in the low Mach numberregion An exception is the work by DemirdzplusmnicAcirc et al (1993) where this approachhas been adopted with a nonstaggered scheme More information can be foundin Bijl and Wesseling (2000) and Wesseling et al (2000)

On the other side of the scientiregc community coupled algorithms have beenequipped with preconditioning methods in order to cope with eigenvaluespreading This is done by multiplying the system matrix by a preconditioningmatrix which alters the speed of the acoustic waves This makes their speed thesame order of magnitude as the speed of the entropy and shear waves ie themacruid local velocity In this way a well-conditioned system is recovered togetherwith good convergence properties

The pioneering work of Chorin (1967) introduced the method of artiregcialcompressibility and has to be considered as the oldest contribution to the regeldStarting from Chorinrsquos method Turkel (1984 1987) developed a two-parameterpreconditioning matrix where beneregts were limited to the low Mach numberregion During the 1980s the work on preconditioning techniques wascontinued by Peyret and Viviand (1985) while Merkle and co-workers (Choiand Merkle 1985) initially focused on the Euler equations and later extendedthis to include viscous effects and turbulence (Merkle 1998b Choi and Merkle1993) The procedures followed by these authors are based on preconditioningmatrices obtained by asymptotic expansions of the compressible equationsPerturbation expansions for low speed effects viscous effects and unsteadymotions are developed to ensure that the pseudo-time terms are in correctbalance with the main physical terms of the equations The preconditioningmatrix of this family can also be postulated simply providing a modiregedexpression for derivatives of the density with respect to the pressure Thispreconditioning matrix fashion was regrst announced by Venkateswaran et al(1992) for the special case of an ideal gas and then generalized to an arbitraryequation of state by Weiss et al (1995 1997) and by Merkle et al (1998)

Alternative procedures for developing preconditioning matrices have beenpursued by van Leer and co-workers (Lee 1996 Lee and van Leer 1993 vanLeer et al 1991) In general these methods make use of special variables that

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symmetrize and simplify the system matrix The preconditioned matrix isdesigned in order to obtain a condition number equal to cn =1curren 1 2 min(M 2 M 2 2) and thus cn = 1 in the limit M 0 (van Leer1991) In Lee (1996) an asymmetric preconditioning matrix is derived to removethe preconditioning sensitivity to the macrow angle due to the eigenvectorstructure The major weakness of this approach is that extension from ideal gasto a general equation of state is not straightforward

The approach followed in the present work is in line with the derivation ofWeiss and Merkle (Weiss and Smith 1995 Weiss et al 1997 Merkle andVenkateswaran 1998) for the preconditioning matrix structure The mainreason for this choice is that arbitrary macruids described by arbitrary equationsof state in analytical or tabular form can be naturally treated

2 Preconditioned governing system of equationsThe full coupled system of the unsteady NavierndashStokes equations representsthe governing equations of all macruid macrows Written in conservative formulationit is given by

shy rshy t

+shy

shy xj

(ruj) = 0 (1)

shy rui

shy t+

shyshy xj

(ruiuj + dijp 2 tij) = rgi

shy rE

shy t+

shyshy xj

(rHuj 2 uitij + qj) = r CcedilQ

where gi represent the components of the acceleration of gravity QCcedil avolumetric external power source and where the total energy E includesinternal and kinetic energy as well as the gravitational potential F = gzConstitutive relations given by Newtonrsquos and Fourierrsquos laws of viscosity andheat conductivity together with two equations of state close the mathematicalmodel

tij = mshy ui

shy xj

+shy uj

shy xi

22

3mdij

shy uk

shy xk

(2)

qj = 2 kshy T

shy xj(3)

r = r(p T ) h = h(p T ) (4)

In the framework of equations (1)ndash (4) compressibilty effects come into playwhenever the conditions occur If density changes are related to temperature

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changes both buoyancy and the hydrostatic pressure gradient are allowed toplay their role on the other side high macruid speeds compared to the speed ofsound generate pressure changes which in turn determine density changesThis picture focuses more on the terms of the equations than on standard macruidclassiregcations ideal gasdynamic and liquid metal buoyant macrows for instanceshare the same thermodynamic pdV work and the reversible exchange betweeninternal and kinetic energy (namely the compressibility) as the key feature ofthe macruid macrow

Preconditiong the system (1) is achieved multiplying the unsteady terms bya suitable matrix P 2 1 or in an equivalent way the macruxes and the source termsby its inverse P System (1) can be written in compact vector form displaying

the inviscid and viscous macrux vectors components ~F and ~G and the source termarray S

shy Q

shy t+ P

shy Fx

shy x+

shy Fy

shy y+

shy Fz

shy z+ P

shy Gx

shy x+

shy Gy

shy y+

shy Gz

shy z= P S (5)

where the preconditioning matrix is given by P = MM2 1m M represents the

Jacobian matrix of the vector of conservative variables Q(rrV rE ) with

respect to the vector of the viscous-primitive variables Qv(p ~V T) Mm

represents a modireged version of M No modiregcation brings back the originalnon-preconditioned system (P = I) Matrices M and M 2 1 are given in theAppendix

The matrix M (equation (28)) contains arbitrary thermodynamics in terms ofderivatives of density and enthalpy with respect to pressure and temperature(rp rT hp hT) while the matrix Mm contains ordfmodiregedordm thermodynamics interms of rm

p To keep the condition number of O(1) rescaling of thecharacteristic speeds is obtained with the following choice of rm

p

rmp =

1

V 2p

2rT

rhT

(6)

where Vp a local preconditioning velocity plays a crucial role as it should be aslow as possible but not smaller than any local transport velocity for stabilityconsiderations

V p = min max unDx

aDx

Dp

r c (7)

Here a represents the thermal diffusivity The criterion based on the local macruidspeed u is dominant in turbulent macrows at high Reynolds numbers The velocitybased on the local pressure gradient prevents vanishing Vp at stagnationregions The two criteria based on diffusion velocities depend strongly on thegrid stretching inside the boundary layers where the macrow is viscous or heat

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641

conduction dominates Where the velocity Vp based on the maximumcharacteristic speed grows higher than the local speed of sound then V p = cand the preconditioning is locally and automatically switched off where themacrow is supersonic (rm

p = g curren c2 = rp and Mm = M) This is particularly usefulin all test cases where stagnation and supersonic regions are both present

3 Arbitrary macruidIn the above described mathematical model the choice of the working macruid istotally arbitrary Any chosen macruid can be deregned by two equations of state ofthe form of equation (4) and via the matrix M which requires the derivatives ofdensity and enthalpy with respect to pressure and temperature For any puresubstance the required derivatives are given by

rp = rp(p T) = rx (8)

rT = rT (p T ) = 2 rb

hT = hT (p T ) = cp

hp = hp(p T ) =1 2 bT

r

where the two compressibility coefregcients b and x respectively at constantpressure and at constant temperature have a straightforward analyticalderegnition for both ideal gases and liquids Strict incompressibility conditionscan be obtained providing b x 0 as input data

Moreover density enthalpy and their derivatives can be given through look-up tables for an arbitrary set of values of pressure and temperature Thisoption is useful for treating for instance reactive mixtures of gases in chemicalequilibrium where the necessary data can be obtained by a number of chemicalequilibrium codes available to the scientiregc community

4 Numerical schemeEquations are integrated with a cell-centred regnite-volume method on block-structured meshes Convective inviscid macruxes are computed by a second orderRoersquos scheme (Roe 1981 Hirsch 1990) based on the decomposition of the Eulerequations in waves so that proper upwinding can be applied to each wavedepending on the sign of the corresponding wave speed This implies aneigenvector decomposition of the matrix of the Euler system The regrst orderRoersquos scheme numerical macrux vector is given by

HFF126

642

Fi+ 1curren 2 =

F i + F i+ 1

22 1

2 [P 2 1RpjLpjLp] i+1 curren 2(Qi+ 1 2 Qi) (9)

made of a central part and a matrix dissipation part computed at the cellinterface denoted by (i + 1curren 2) Rp and Lp represent the matrices of right andleft eigenvectors of

Dp = PD = Pshy Fx

shy Qnx +

shy Fy

shy Qny +

shy F z

shy Qnz (10)

and jLpj represents the diagonal matrix whose elements ap are the absolutevalues jlpj of the eigenvalues of the system matrix Dp Second order accuracyis achieved (Hirsch 1990) for a thorough review computing the elements ap as

ap = jlpj + C(R + ) 2 C(R 2 ) (11)

with

R + =l+ (Qi 2 Qi 2 1)

l+ (Qi+ 1 2 Qi)R 2 =

l 2 (Qi+ 2 2 Qi+1)

l 2 (Qi+ 1 2 Qi)(12)

where the function C represents an appropriate limiter which assuresmonotonicity of the solution at discontinuities and at local minima andmaxima Setting C = 0 in Equation (11) the regrst order Roersquos scheme isrecovered

The expression of the eigenvectors in terms of conservative variables israther complicated Easier algebra and programming can be achievedtransforming equation (9) in the equivalent expression for the primitivevariables Qv

Fi+1curren 2 =

F i + F i+ 1

22 1

2 [MmRvp jLpjLv

p ] i+ 1curren 2(Qvi+ 1 2 Q

vi ) (13)

Euler matrices and eigenvectors in conservative and primitive variables arerelated by similarity transformations Preconditioned eigenvectors are given inAppendix A for a general equation of state Viscous macruxes and source termsare calculated with standard cell-centred regnite-volume techniques and they areboth second order accurate

5 Implicit solution algorithmOnce in regnite-volume and semidiscrete form system (5) becomes

Oshy Q

shy t= 2 P RES 2 MM

2 1m RES (14)

where RES represents the vector of residuals for the conservative variables andO the cell volume Updating is done in terms of the viscous primitive variables

Efregcient parallelcomputations of

macruid macrows

643

Qv namely pressure temperature and the velocity components This is done bymultiplying the above system by M 2 1 to the left

Oshy Qv

shy t= 2 M 2 1

m RES (15)

Systems (14) and (15) are equivalent If an implicit numerical scheme is used todiscretize the time derivative then

OQ

newv 2 Q

oldv

Dt O

DQv

Dt= 2 M

2 1m RESnew (16)

After linearization about the old time level

ODQv

Dt= 2 M

2 1m RESold +

shy RES

shy Qv

old

DQv (17)

Re-arranging multiplying by Mm to the left and dropping superscripts

MmODt

+shy RES

shy QvDQv = 2 RES (18)

In equation (18) DQv and RES are column vectors of dimension given by thetotal number N of cells where each element is an array of regve components Theunsteady term matrix on the left-hand side is a block diagonal matrix (with5 pound5 blocks) Jacobians of the residuals with respect to the variables aretypically constructed by using a two-point stencil scheme at each interface Asa result the second term within brackets is a sparse N poundN matrix with non-zero elements only on seven diagonals due to the grid structure

Denoting the block element of the seven diagonals by A BI BJ BK CI CJ andCK then the solution algorithm can be written as

MmODt

+ A DQv(i j k) + BIDQv(i 2 1 j k) + BJDQv(i j 2 1 k)

+ BKDQv(i j k 2 1) + CIDQv(i + 1 j k) + CJDQv(i j + 1 k)

+ CKDQv(i j k + 1) = 2 RES( i j k) (19)

At each time step the linearization leads to the linear system (19) which is to besolved by an iterative method When unconditional stability is achievableequation (19) becomes a Newton method which is able to deliver quadraticconvergence It has to be noticed that though not explicitly apparent fromequation (18) preconditioning enters the linear system through the inviscidnumerical macrux (13) in addition to the unsteady term People who make no useof characteristic-based schemes need little modiregcations of their codes

HFF126

644

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

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647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

Patnaik et al (1987) In these approaches a pressure correction equation isderived from the energy equation while density is advanced in time via acontinuity equation The method can compute aeroacoustics and nonstationarylow Mach number macrows

Segregated methods can be extended to the compressible case Thisapproach gives the best perspective in the limit M 0 where a well-knownincompressible scheme can be recovered Moreover difregculties are encounteredwhen tackling the acoustic transonic singularity which determines anonconvergence behaviour in the regime M = O(1) (see Bijl and Wesseling1996) Only the use of a regrst order upwind scheme allows the algorithm toconverge (see Bijl and Wesseling 1998) Staggered schemes are generally usedwith these methods to avoid pressure oscillations in the low Mach numberregion An exception is the work by DemirdzplusmnicAcirc et al (1993) where this approachhas been adopted with a nonstaggered scheme More information can be foundin Bijl and Wesseling (2000) and Wesseling et al (2000)

On the other side of the scientiregc community coupled algorithms have beenequipped with preconditioning methods in order to cope with eigenvaluespreading This is done by multiplying the system matrix by a preconditioningmatrix which alters the speed of the acoustic waves This makes their speed thesame order of magnitude as the speed of the entropy and shear waves ie themacruid local velocity In this way a well-conditioned system is recovered togetherwith good convergence properties

The pioneering work of Chorin (1967) introduced the method of artiregcialcompressibility and has to be considered as the oldest contribution to the regeldStarting from Chorinrsquos method Turkel (1984 1987) developed a two-parameterpreconditioning matrix where beneregts were limited to the low Mach numberregion During the 1980s the work on preconditioning techniques wascontinued by Peyret and Viviand (1985) while Merkle and co-workers (Choiand Merkle 1985) initially focused on the Euler equations and later extendedthis to include viscous effects and turbulence (Merkle 1998b Choi and Merkle1993) The procedures followed by these authors are based on preconditioningmatrices obtained by asymptotic expansions of the compressible equationsPerturbation expansions for low speed effects viscous effects and unsteadymotions are developed to ensure that the pseudo-time terms are in correctbalance with the main physical terms of the equations The preconditioningmatrix of this family can also be postulated simply providing a modiregedexpression for derivatives of the density with respect to the pressure Thispreconditioning matrix fashion was regrst announced by Venkateswaran et al(1992) for the special case of an ideal gas and then generalized to an arbitraryequation of state by Weiss et al (1995 1997) and by Merkle et al (1998)

Alternative procedures for developing preconditioning matrices have beenpursued by van Leer and co-workers (Lee 1996 Lee and van Leer 1993 vanLeer et al 1991) In general these methods make use of special variables that

Efregcient parallelcomputations of

macruid macrows

639

symmetrize and simplify the system matrix The preconditioned matrix isdesigned in order to obtain a condition number equal to cn =1curren 1 2 min(M 2 M 2 2) and thus cn = 1 in the limit M 0 (van Leer1991) In Lee (1996) an asymmetric preconditioning matrix is derived to removethe preconditioning sensitivity to the macrow angle due to the eigenvectorstructure The major weakness of this approach is that extension from ideal gasto a general equation of state is not straightforward

The approach followed in the present work is in line with the derivation ofWeiss and Merkle (Weiss and Smith 1995 Weiss et al 1997 Merkle andVenkateswaran 1998) for the preconditioning matrix structure The mainreason for this choice is that arbitrary macruids described by arbitrary equationsof state in analytical or tabular form can be naturally treated

2 Preconditioned governing system of equationsThe full coupled system of the unsteady NavierndashStokes equations representsthe governing equations of all macruid macrows Written in conservative formulationit is given by

shy rshy t

+shy

shy xj

(ruj) = 0 (1)

shy rui

shy t+

shyshy xj

(ruiuj + dijp 2 tij) = rgi

shy rE

shy t+

shyshy xj

(rHuj 2 uitij + qj) = r CcedilQ

where gi represent the components of the acceleration of gravity QCcedil avolumetric external power source and where the total energy E includesinternal and kinetic energy as well as the gravitational potential F = gzConstitutive relations given by Newtonrsquos and Fourierrsquos laws of viscosity andheat conductivity together with two equations of state close the mathematicalmodel

tij = mshy ui

shy xj

+shy uj

shy xi

22

3mdij

shy uk

shy xk

(2)

qj = 2 kshy T

shy xj(3)

r = r(p T ) h = h(p T ) (4)

In the framework of equations (1)ndash (4) compressibilty effects come into playwhenever the conditions occur If density changes are related to temperature

HFF126

640

changes both buoyancy and the hydrostatic pressure gradient are allowed toplay their role on the other side high macruid speeds compared to the speed ofsound generate pressure changes which in turn determine density changesThis picture focuses more on the terms of the equations than on standard macruidclassiregcations ideal gasdynamic and liquid metal buoyant macrows for instanceshare the same thermodynamic pdV work and the reversible exchange betweeninternal and kinetic energy (namely the compressibility) as the key feature ofthe macruid macrow

Preconditiong the system (1) is achieved multiplying the unsteady terms bya suitable matrix P 2 1 or in an equivalent way the macruxes and the source termsby its inverse P System (1) can be written in compact vector form displaying

the inviscid and viscous macrux vectors components ~F and ~G and the source termarray S

shy Q

shy t+ P

shy Fx

shy x+

shy Fy

shy y+

shy Fz

shy z+ P

shy Gx

shy x+

shy Gy

shy y+

shy Gz

shy z= P S (5)

where the preconditioning matrix is given by P = MM2 1m M represents the

Jacobian matrix of the vector of conservative variables Q(rrV rE ) with

respect to the vector of the viscous-primitive variables Qv(p ~V T) Mm

represents a modireged version of M No modiregcation brings back the originalnon-preconditioned system (P = I) Matrices M and M 2 1 are given in theAppendix

The matrix M (equation (28)) contains arbitrary thermodynamics in terms ofderivatives of density and enthalpy with respect to pressure and temperature(rp rT hp hT) while the matrix Mm contains ordfmodiregedordm thermodynamics interms of rm

p To keep the condition number of O(1) rescaling of thecharacteristic speeds is obtained with the following choice of rm

p

rmp =

1

V 2p

2rT

rhT

(6)

where Vp a local preconditioning velocity plays a crucial role as it should be aslow as possible but not smaller than any local transport velocity for stabilityconsiderations

V p = min max unDx

aDx

Dp

r c (7)

Here a represents the thermal diffusivity The criterion based on the local macruidspeed u is dominant in turbulent macrows at high Reynolds numbers The velocitybased on the local pressure gradient prevents vanishing Vp at stagnationregions The two criteria based on diffusion velocities depend strongly on thegrid stretching inside the boundary layers where the macrow is viscous or heat

Efregcient parallelcomputations of

macruid macrows

641

conduction dominates Where the velocity Vp based on the maximumcharacteristic speed grows higher than the local speed of sound then V p = cand the preconditioning is locally and automatically switched off where themacrow is supersonic (rm

p = g curren c2 = rp and Mm = M) This is particularly usefulin all test cases where stagnation and supersonic regions are both present

3 Arbitrary macruidIn the above described mathematical model the choice of the working macruid istotally arbitrary Any chosen macruid can be deregned by two equations of state ofthe form of equation (4) and via the matrix M which requires the derivatives ofdensity and enthalpy with respect to pressure and temperature For any puresubstance the required derivatives are given by

rp = rp(p T) = rx (8)

rT = rT (p T ) = 2 rb

hT = hT (p T ) = cp

hp = hp(p T ) =1 2 bT

r

where the two compressibility coefregcients b and x respectively at constantpressure and at constant temperature have a straightforward analyticalderegnition for both ideal gases and liquids Strict incompressibility conditionscan be obtained providing b x 0 as input data

Moreover density enthalpy and their derivatives can be given through look-up tables for an arbitrary set of values of pressure and temperature Thisoption is useful for treating for instance reactive mixtures of gases in chemicalequilibrium where the necessary data can be obtained by a number of chemicalequilibrium codes available to the scientiregc community

4 Numerical schemeEquations are integrated with a cell-centred regnite-volume method on block-structured meshes Convective inviscid macruxes are computed by a second orderRoersquos scheme (Roe 1981 Hirsch 1990) based on the decomposition of the Eulerequations in waves so that proper upwinding can be applied to each wavedepending on the sign of the corresponding wave speed This implies aneigenvector decomposition of the matrix of the Euler system The regrst orderRoersquos scheme numerical macrux vector is given by

HFF126

642

Fi+ 1curren 2 =

F i + F i+ 1

22 1

2 [P 2 1RpjLpjLp] i+1 curren 2(Qi+ 1 2 Qi) (9)

made of a central part and a matrix dissipation part computed at the cellinterface denoted by (i + 1curren 2) Rp and Lp represent the matrices of right andleft eigenvectors of

Dp = PD = Pshy Fx

shy Qnx +

shy Fy

shy Qny +

shy F z

shy Qnz (10)

and jLpj represents the diagonal matrix whose elements ap are the absolutevalues jlpj of the eigenvalues of the system matrix Dp Second order accuracyis achieved (Hirsch 1990) for a thorough review computing the elements ap as

ap = jlpj + C(R + ) 2 C(R 2 ) (11)

with

R + =l+ (Qi 2 Qi 2 1)

l+ (Qi+ 1 2 Qi)R 2 =

l 2 (Qi+ 2 2 Qi+1)

l 2 (Qi+ 1 2 Qi)(12)

where the function C represents an appropriate limiter which assuresmonotonicity of the solution at discontinuities and at local minima andmaxima Setting C = 0 in Equation (11) the regrst order Roersquos scheme isrecovered

The expression of the eigenvectors in terms of conservative variables israther complicated Easier algebra and programming can be achievedtransforming equation (9) in the equivalent expression for the primitivevariables Qv

Fi+1curren 2 =

F i + F i+ 1

22 1

2 [MmRvp jLpjLv

p ] i+ 1curren 2(Qvi+ 1 2 Q

vi ) (13)

Euler matrices and eigenvectors in conservative and primitive variables arerelated by similarity transformations Preconditioned eigenvectors are given inAppendix A for a general equation of state Viscous macruxes and source termsare calculated with standard cell-centred regnite-volume techniques and they areboth second order accurate

5 Implicit solution algorithmOnce in regnite-volume and semidiscrete form system (5) becomes

Oshy Q

shy t= 2 P RES 2 MM

2 1m RES (14)

where RES represents the vector of residuals for the conservative variables andO the cell volume Updating is done in terms of the viscous primitive variables

Efregcient parallelcomputations of

macruid macrows

643

Qv namely pressure temperature and the velocity components This is done bymultiplying the above system by M 2 1 to the left

Oshy Qv

shy t= 2 M 2 1

m RES (15)

Systems (14) and (15) are equivalent If an implicit numerical scheme is used todiscretize the time derivative then

OQ

newv 2 Q

oldv

Dt O

DQv

Dt= 2 M

2 1m RESnew (16)

After linearization about the old time level

ODQv

Dt= 2 M

2 1m RESold +

shy RES

shy Qv

old

DQv (17)

Re-arranging multiplying by Mm to the left and dropping superscripts

MmODt

+shy RES

shy QvDQv = 2 RES (18)

In equation (18) DQv and RES are column vectors of dimension given by thetotal number N of cells where each element is an array of regve components Theunsteady term matrix on the left-hand side is a block diagonal matrix (with5 pound5 blocks) Jacobians of the residuals with respect to the variables aretypically constructed by using a two-point stencil scheme at each interface Asa result the second term within brackets is a sparse N poundN matrix with non-zero elements only on seven diagonals due to the grid structure

Denoting the block element of the seven diagonals by A BI BJ BK CI CJ andCK then the solution algorithm can be written as

MmODt

+ A DQv(i j k) + BIDQv(i 2 1 j k) + BJDQv(i j 2 1 k)

+ BKDQv(i j k 2 1) + CIDQv(i + 1 j k) + CJDQv(i j + 1 k)

+ CKDQv(i j k + 1) = 2 RES( i j k) (19)

At each time step the linearization leads to the linear system (19) which is to besolved by an iterative method When unconditional stability is achievableequation (19) becomes a Newton method which is able to deliver quadraticconvergence It has to be noticed that though not explicitly apparent fromequation (18) preconditioning enters the linear system through the inviscidnumerical macrux (13) in addition to the unsteady term People who make no useof characteristic-based schemes need little modiregcations of their codes

HFF126

644

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

symmetrize and simplify the system matrix The preconditioned matrix isdesigned in order to obtain a condition number equal to cn =1curren 1 2 min(M 2 M 2 2) and thus cn = 1 in the limit M 0 (van Leer1991) In Lee (1996) an asymmetric preconditioning matrix is derived to removethe preconditioning sensitivity to the macrow angle due to the eigenvectorstructure The major weakness of this approach is that extension from ideal gasto a general equation of state is not straightforward

The approach followed in the present work is in line with the derivation ofWeiss and Merkle (Weiss and Smith 1995 Weiss et al 1997 Merkle andVenkateswaran 1998) for the preconditioning matrix structure The mainreason for this choice is that arbitrary macruids described by arbitrary equationsof state in analytical or tabular form can be naturally treated

2 Preconditioned governing system of equationsThe full coupled system of the unsteady NavierndashStokes equations representsthe governing equations of all macruid macrows Written in conservative formulationit is given by

shy rshy t

+shy

shy xj

(ruj) = 0 (1)

shy rui

shy t+

shyshy xj

(ruiuj + dijp 2 tij) = rgi

shy rE

shy t+

shyshy xj

(rHuj 2 uitij + qj) = r CcedilQ

where gi represent the components of the acceleration of gravity QCcedil avolumetric external power source and where the total energy E includesinternal and kinetic energy as well as the gravitational potential F = gzConstitutive relations given by Newtonrsquos and Fourierrsquos laws of viscosity andheat conductivity together with two equations of state close the mathematicalmodel

tij = mshy ui

shy xj

+shy uj

shy xi

22

3mdij

shy uk

shy xk

(2)

qj = 2 kshy T

shy xj(3)

r = r(p T ) h = h(p T ) (4)

In the framework of equations (1)ndash (4) compressibilty effects come into playwhenever the conditions occur If density changes are related to temperature

HFF126

640

changes both buoyancy and the hydrostatic pressure gradient are allowed toplay their role on the other side high macruid speeds compared to the speed ofsound generate pressure changes which in turn determine density changesThis picture focuses more on the terms of the equations than on standard macruidclassiregcations ideal gasdynamic and liquid metal buoyant macrows for instanceshare the same thermodynamic pdV work and the reversible exchange betweeninternal and kinetic energy (namely the compressibility) as the key feature ofthe macruid macrow

Preconditiong the system (1) is achieved multiplying the unsteady terms bya suitable matrix P 2 1 or in an equivalent way the macruxes and the source termsby its inverse P System (1) can be written in compact vector form displaying

the inviscid and viscous macrux vectors components ~F and ~G and the source termarray S

shy Q

shy t+ P

shy Fx

shy x+

shy Fy

shy y+

shy Fz

shy z+ P

shy Gx

shy x+

shy Gy

shy y+

shy Gz

shy z= P S (5)

where the preconditioning matrix is given by P = MM2 1m M represents the

Jacobian matrix of the vector of conservative variables Q(rrV rE ) with

respect to the vector of the viscous-primitive variables Qv(p ~V T) Mm

represents a modireged version of M No modiregcation brings back the originalnon-preconditioned system (P = I) Matrices M and M 2 1 are given in theAppendix

The matrix M (equation (28)) contains arbitrary thermodynamics in terms ofderivatives of density and enthalpy with respect to pressure and temperature(rp rT hp hT) while the matrix Mm contains ordfmodiregedordm thermodynamics interms of rm

p To keep the condition number of O(1) rescaling of thecharacteristic speeds is obtained with the following choice of rm

p

rmp =

1

V 2p

2rT

rhT

(6)

where Vp a local preconditioning velocity plays a crucial role as it should be aslow as possible but not smaller than any local transport velocity for stabilityconsiderations

V p = min max unDx

aDx

Dp

r c (7)

Here a represents the thermal diffusivity The criterion based on the local macruidspeed u is dominant in turbulent macrows at high Reynolds numbers The velocitybased on the local pressure gradient prevents vanishing Vp at stagnationregions The two criteria based on diffusion velocities depend strongly on thegrid stretching inside the boundary layers where the macrow is viscous or heat

Efregcient parallelcomputations of

macruid macrows

641

conduction dominates Where the velocity Vp based on the maximumcharacteristic speed grows higher than the local speed of sound then V p = cand the preconditioning is locally and automatically switched off where themacrow is supersonic (rm

p = g curren c2 = rp and Mm = M) This is particularly usefulin all test cases where stagnation and supersonic regions are both present

3 Arbitrary macruidIn the above described mathematical model the choice of the working macruid istotally arbitrary Any chosen macruid can be deregned by two equations of state ofthe form of equation (4) and via the matrix M which requires the derivatives ofdensity and enthalpy with respect to pressure and temperature For any puresubstance the required derivatives are given by

rp = rp(p T) = rx (8)

rT = rT (p T ) = 2 rb

hT = hT (p T ) = cp

hp = hp(p T ) =1 2 bT

r

where the two compressibility coefregcients b and x respectively at constantpressure and at constant temperature have a straightforward analyticalderegnition for both ideal gases and liquids Strict incompressibility conditionscan be obtained providing b x 0 as input data

Moreover density enthalpy and their derivatives can be given through look-up tables for an arbitrary set of values of pressure and temperature Thisoption is useful for treating for instance reactive mixtures of gases in chemicalequilibrium where the necessary data can be obtained by a number of chemicalequilibrium codes available to the scientiregc community

4 Numerical schemeEquations are integrated with a cell-centred regnite-volume method on block-structured meshes Convective inviscid macruxes are computed by a second orderRoersquos scheme (Roe 1981 Hirsch 1990) based on the decomposition of the Eulerequations in waves so that proper upwinding can be applied to each wavedepending on the sign of the corresponding wave speed This implies aneigenvector decomposition of the matrix of the Euler system The regrst orderRoersquos scheme numerical macrux vector is given by

HFF126

642

Fi+ 1curren 2 =

F i + F i+ 1

22 1

2 [P 2 1RpjLpjLp] i+1 curren 2(Qi+ 1 2 Qi) (9)

made of a central part and a matrix dissipation part computed at the cellinterface denoted by (i + 1curren 2) Rp and Lp represent the matrices of right andleft eigenvectors of

Dp = PD = Pshy Fx

shy Qnx +

shy Fy

shy Qny +

shy F z

shy Qnz (10)

and jLpj represents the diagonal matrix whose elements ap are the absolutevalues jlpj of the eigenvalues of the system matrix Dp Second order accuracyis achieved (Hirsch 1990) for a thorough review computing the elements ap as

ap = jlpj + C(R + ) 2 C(R 2 ) (11)

with

R + =l+ (Qi 2 Qi 2 1)

l+ (Qi+ 1 2 Qi)R 2 =

l 2 (Qi+ 2 2 Qi+1)

l 2 (Qi+ 1 2 Qi)(12)

where the function C represents an appropriate limiter which assuresmonotonicity of the solution at discontinuities and at local minima andmaxima Setting C = 0 in Equation (11) the regrst order Roersquos scheme isrecovered

The expression of the eigenvectors in terms of conservative variables israther complicated Easier algebra and programming can be achievedtransforming equation (9) in the equivalent expression for the primitivevariables Qv

Fi+1curren 2 =

F i + F i+ 1

22 1

2 [MmRvp jLpjLv

p ] i+ 1curren 2(Qvi+ 1 2 Q

vi ) (13)

Euler matrices and eigenvectors in conservative and primitive variables arerelated by similarity transformations Preconditioned eigenvectors are given inAppendix A for a general equation of state Viscous macruxes and source termsare calculated with standard cell-centred regnite-volume techniques and they areboth second order accurate

5 Implicit solution algorithmOnce in regnite-volume and semidiscrete form system (5) becomes

Oshy Q

shy t= 2 P RES 2 MM

2 1m RES (14)

where RES represents the vector of residuals for the conservative variables andO the cell volume Updating is done in terms of the viscous primitive variables

Efregcient parallelcomputations of

macruid macrows

643

Qv namely pressure temperature and the velocity components This is done bymultiplying the above system by M 2 1 to the left

Oshy Qv

shy t= 2 M 2 1

m RES (15)

Systems (14) and (15) are equivalent If an implicit numerical scheme is used todiscretize the time derivative then

OQ

newv 2 Q

oldv

Dt O

DQv

Dt= 2 M

2 1m RESnew (16)

After linearization about the old time level

ODQv

Dt= 2 M

2 1m RESold +

shy RES

shy Qv

old

DQv (17)

Re-arranging multiplying by Mm to the left and dropping superscripts

MmODt

+shy RES

shy QvDQv = 2 RES (18)

In equation (18) DQv and RES are column vectors of dimension given by thetotal number N of cells where each element is an array of regve components Theunsteady term matrix on the left-hand side is a block diagonal matrix (with5 pound5 blocks) Jacobians of the residuals with respect to the variables aretypically constructed by using a two-point stencil scheme at each interface Asa result the second term within brackets is a sparse N poundN matrix with non-zero elements only on seven diagonals due to the grid structure

Denoting the block element of the seven diagonals by A BI BJ BK CI CJ andCK then the solution algorithm can be written as

MmODt

+ A DQv(i j k) + BIDQv(i 2 1 j k) + BJDQv(i j 2 1 k)

+ BKDQv(i j k 2 1) + CIDQv(i + 1 j k) + CJDQv(i j + 1 k)

+ CKDQv(i j k + 1) = 2 RES( i j k) (19)

At each time step the linearization leads to the linear system (19) which is to besolved by an iterative method When unconditional stability is achievableequation (19) becomes a Newton method which is able to deliver quadraticconvergence It has to be noticed that though not explicitly apparent fromequation (18) preconditioning enters the linear system through the inviscidnumerical macrux (13) in addition to the unsteady term People who make no useof characteristic-based schemes need little modiregcations of their codes

HFF126

644

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

changes both buoyancy and the hydrostatic pressure gradient are allowed toplay their role on the other side high macruid speeds compared to the speed ofsound generate pressure changes which in turn determine density changesThis picture focuses more on the terms of the equations than on standard macruidclassiregcations ideal gasdynamic and liquid metal buoyant macrows for instanceshare the same thermodynamic pdV work and the reversible exchange betweeninternal and kinetic energy (namely the compressibility) as the key feature ofthe macruid macrow

Preconditiong the system (1) is achieved multiplying the unsteady terms bya suitable matrix P 2 1 or in an equivalent way the macruxes and the source termsby its inverse P System (1) can be written in compact vector form displaying

the inviscid and viscous macrux vectors components ~F and ~G and the source termarray S

shy Q

shy t+ P

shy Fx

shy x+

shy Fy

shy y+

shy Fz

shy z+ P

shy Gx

shy x+

shy Gy

shy y+

shy Gz

shy z= P S (5)

where the preconditioning matrix is given by P = MM2 1m M represents the

Jacobian matrix of the vector of conservative variables Q(rrV rE ) with

respect to the vector of the viscous-primitive variables Qv(p ~V T) Mm

represents a modireged version of M No modiregcation brings back the originalnon-preconditioned system (P = I) Matrices M and M 2 1 are given in theAppendix

The matrix M (equation (28)) contains arbitrary thermodynamics in terms ofderivatives of density and enthalpy with respect to pressure and temperature(rp rT hp hT) while the matrix Mm contains ordfmodiregedordm thermodynamics interms of rm

p To keep the condition number of O(1) rescaling of thecharacteristic speeds is obtained with the following choice of rm

p

rmp =

1

V 2p

2rT

rhT

(6)

where Vp a local preconditioning velocity plays a crucial role as it should be aslow as possible but not smaller than any local transport velocity for stabilityconsiderations

V p = min max unDx

aDx

Dp

r c (7)

Here a represents the thermal diffusivity The criterion based on the local macruidspeed u is dominant in turbulent macrows at high Reynolds numbers The velocitybased on the local pressure gradient prevents vanishing Vp at stagnationregions The two criteria based on diffusion velocities depend strongly on thegrid stretching inside the boundary layers where the macrow is viscous or heat

Efregcient parallelcomputations of

macruid macrows

641

conduction dominates Where the velocity Vp based on the maximumcharacteristic speed grows higher than the local speed of sound then V p = cand the preconditioning is locally and automatically switched off where themacrow is supersonic (rm

p = g curren c2 = rp and Mm = M) This is particularly usefulin all test cases where stagnation and supersonic regions are both present

3 Arbitrary macruidIn the above described mathematical model the choice of the working macruid istotally arbitrary Any chosen macruid can be deregned by two equations of state ofthe form of equation (4) and via the matrix M which requires the derivatives ofdensity and enthalpy with respect to pressure and temperature For any puresubstance the required derivatives are given by

rp = rp(p T) = rx (8)

rT = rT (p T ) = 2 rb

hT = hT (p T ) = cp

hp = hp(p T ) =1 2 bT

r

where the two compressibility coefregcients b and x respectively at constantpressure and at constant temperature have a straightforward analyticalderegnition for both ideal gases and liquids Strict incompressibility conditionscan be obtained providing b x 0 as input data

Moreover density enthalpy and their derivatives can be given through look-up tables for an arbitrary set of values of pressure and temperature Thisoption is useful for treating for instance reactive mixtures of gases in chemicalequilibrium where the necessary data can be obtained by a number of chemicalequilibrium codes available to the scientiregc community

4 Numerical schemeEquations are integrated with a cell-centred regnite-volume method on block-structured meshes Convective inviscid macruxes are computed by a second orderRoersquos scheme (Roe 1981 Hirsch 1990) based on the decomposition of the Eulerequations in waves so that proper upwinding can be applied to each wavedepending on the sign of the corresponding wave speed This implies aneigenvector decomposition of the matrix of the Euler system The regrst orderRoersquos scheme numerical macrux vector is given by

HFF126

642

Fi+ 1curren 2 =

F i + F i+ 1

22 1

2 [P 2 1RpjLpjLp] i+1 curren 2(Qi+ 1 2 Qi) (9)

made of a central part and a matrix dissipation part computed at the cellinterface denoted by (i + 1curren 2) Rp and Lp represent the matrices of right andleft eigenvectors of

Dp = PD = Pshy Fx

shy Qnx +

shy Fy

shy Qny +

shy F z

shy Qnz (10)

and jLpj represents the diagonal matrix whose elements ap are the absolutevalues jlpj of the eigenvalues of the system matrix Dp Second order accuracyis achieved (Hirsch 1990) for a thorough review computing the elements ap as

ap = jlpj + C(R + ) 2 C(R 2 ) (11)

with

R + =l+ (Qi 2 Qi 2 1)

l+ (Qi+ 1 2 Qi)R 2 =

l 2 (Qi+ 2 2 Qi+1)

l 2 (Qi+ 1 2 Qi)(12)

where the function C represents an appropriate limiter which assuresmonotonicity of the solution at discontinuities and at local minima andmaxima Setting C = 0 in Equation (11) the regrst order Roersquos scheme isrecovered

The expression of the eigenvectors in terms of conservative variables israther complicated Easier algebra and programming can be achievedtransforming equation (9) in the equivalent expression for the primitivevariables Qv

Fi+1curren 2 =

F i + F i+ 1

22 1

2 [MmRvp jLpjLv

p ] i+ 1curren 2(Qvi+ 1 2 Q

vi ) (13)

Euler matrices and eigenvectors in conservative and primitive variables arerelated by similarity transformations Preconditioned eigenvectors are given inAppendix A for a general equation of state Viscous macruxes and source termsare calculated with standard cell-centred regnite-volume techniques and they areboth second order accurate

5 Implicit solution algorithmOnce in regnite-volume and semidiscrete form system (5) becomes

Oshy Q

shy t= 2 P RES 2 MM

2 1m RES (14)

where RES represents the vector of residuals for the conservative variables andO the cell volume Updating is done in terms of the viscous primitive variables

Efregcient parallelcomputations of

macruid macrows

643

Qv namely pressure temperature and the velocity components This is done bymultiplying the above system by M 2 1 to the left

Oshy Qv

shy t= 2 M 2 1

m RES (15)

Systems (14) and (15) are equivalent If an implicit numerical scheme is used todiscretize the time derivative then

OQ

newv 2 Q

oldv

Dt O

DQv

Dt= 2 M

2 1m RESnew (16)

After linearization about the old time level

ODQv

Dt= 2 M

2 1m RESold +

shy RES

shy Qv

old

DQv (17)

Re-arranging multiplying by Mm to the left and dropping superscripts

MmODt

+shy RES

shy QvDQv = 2 RES (18)

In equation (18) DQv and RES are column vectors of dimension given by thetotal number N of cells where each element is an array of regve components Theunsteady term matrix on the left-hand side is a block diagonal matrix (with5 pound5 blocks) Jacobians of the residuals with respect to the variables aretypically constructed by using a two-point stencil scheme at each interface Asa result the second term within brackets is a sparse N poundN matrix with non-zero elements only on seven diagonals due to the grid structure

Denoting the block element of the seven diagonals by A BI BJ BK CI CJ andCK then the solution algorithm can be written as

MmODt

+ A DQv(i j k) + BIDQv(i 2 1 j k) + BJDQv(i j 2 1 k)

+ BKDQv(i j k 2 1) + CIDQv(i + 1 j k) + CJDQv(i j + 1 k)

+ CKDQv(i j k + 1) = 2 RES( i j k) (19)

At each time step the linearization leads to the linear system (19) which is to besolved by an iterative method When unconditional stability is achievableequation (19) becomes a Newton method which is able to deliver quadraticconvergence It has to be noticed that though not explicitly apparent fromequation (18) preconditioning enters the linear system through the inviscidnumerical macrux (13) in addition to the unsteady term People who make no useof characteristic-based schemes need little modiregcations of their codes

HFF126

644

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

conduction dominates Where the velocity Vp based on the maximumcharacteristic speed grows higher than the local speed of sound then V p = cand the preconditioning is locally and automatically switched off where themacrow is supersonic (rm

p = g curren c2 = rp and Mm = M) This is particularly usefulin all test cases where stagnation and supersonic regions are both present

3 Arbitrary macruidIn the above described mathematical model the choice of the working macruid istotally arbitrary Any chosen macruid can be deregned by two equations of state ofthe form of equation (4) and via the matrix M which requires the derivatives ofdensity and enthalpy with respect to pressure and temperature For any puresubstance the required derivatives are given by

rp = rp(p T) = rx (8)

rT = rT (p T ) = 2 rb

hT = hT (p T ) = cp

hp = hp(p T ) =1 2 bT

r

where the two compressibility coefregcients b and x respectively at constantpressure and at constant temperature have a straightforward analyticalderegnition for both ideal gases and liquids Strict incompressibility conditionscan be obtained providing b x 0 as input data

Moreover density enthalpy and their derivatives can be given through look-up tables for an arbitrary set of values of pressure and temperature Thisoption is useful for treating for instance reactive mixtures of gases in chemicalequilibrium where the necessary data can be obtained by a number of chemicalequilibrium codes available to the scientiregc community

4 Numerical schemeEquations are integrated with a cell-centred regnite-volume method on block-structured meshes Convective inviscid macruxes are computed by a second orderRoersquos scheme (Roe 1981 Hirsch 1990) based on the decomposition of the Eulerequations in waves so that proper upwinding can be applied to each wavedepending on the sign of the corresponding wave speed This implies aneigenvector decomposition of the matrix of the Euler system The regrst orderRoersquos scheme numerical macrux vector is given by

HFF126

642

Fi+ 1curren 2 =

F i + F i+ 1

22 1

2 [P 2 1RpjLpjLp] i+1 curren 2(Qi+ 1 2 Qi) (9)

made of a central part and a matrix dissipation part computed at the cellinterface denoted by (i + 1curren 2) Rp and Lp represent the matrices of right andleft eigenvectors of

Dp = PD = Pshy Fx

shy Qnx +

shy Fy

shy Qny +

shy F z

shy Qnz (10)

and jLpj represents the diagonal matrix whose elements ap are the absolutevalues jlpj of the eigenvalues of the system matrix Dp Second order accuracyis achieved (Hirsch 1990) for a thorough review computing the elements ap as

ap = jlpj + C(R + ) 2 C(R 2 ) (11)

with

R + =l+ (Qi 2 Qi 2 1)

l+ (Qi+ 1 2 Qi)R 2 =

l 2 (Qi+ 2 2 Qi+1)

l 2 (Qi+ 1 2 Qi)(12)

where the function C represents an appropriate limiter which assuresmonotonicity of the solution at discontinuities and at local minima andmaxima Setting C = 0 in Equation (11) the regrst order Roersquos scheme isrecovered

The expression of the eigenvectors in terms of conservative variables israther complicated Easier algebra and programming can be achievedtransforming equation (9) in the equivalent expression for the primitivevariables Qv

Fi+1curren 2 =

F i + F i+ 1

22 1

2 [MmRvp jLpjLv

p ] i+ 1curren 2(Qvi+ 1 2 Q

vi ) (13)

Euler matrices and eigenvectors in conservative and primitive variables arerelated by similarity transformations Preconditioned eigenvectors are given inAppendix A for a general equation of state Viscous macruxes and source termsare calculated with standard cell-centred regnite-volume techniques and they areboth second order accurate

5 Implicit solution algorithmOnce in regnite-volume and semidiscrete form system (5) becomes

Oshy Q

shy t= 2 P RES 2 MM

2 1m RES (14)

where RES represents the vector of residuals for the conservative variables andO the cell volume Updating is done in terms of the viscous primitive variables

Efregcient parallelcomputations of

macruid macrows

643

Qv namely pressure temperature and the velocity components This is done bymultiplying the above system by M 2 1 to the left

Oshy Qv

shy t= 2 M 2 1

m RES (15)

Systems (14) and (15) are equivalent If an implicit numerical scheme is used todiscretize the time derivative then

OQ

newv 2 Q

oldv

Dt O

DQv

Dt= 2 M

2 1m RESnew (16)

After linearization about the old time level

ODQv

Dt= 2 M

2 1m RESold +

shy RES

shy Qv

old

DQv (17)

Re-arranging multiplying by Mm to the left and dropping superscripts

MmODt

+shy RES

shy QvDQv = 2 RES (18)

In equation (18) DQv and RES are column vectors of dimension given by thetotal number N of cells where each element is an array of regve components Theunsteady term matrix on the left-hand side is a block diagonal matrix (with5 pound5 blocks) Jacobians of the residuals with respect to the variables aretypically constructed by using a two-point stencil scheme at each interface Asa result the second term within brackets is a sparse N poundN matrix with non-zero elements only on seven diagonals due to the grid structure

Denoting the block element of the seven diagonals by A BI BJ BK CI CJ andCK then the solution algorithm can be written as

MmODt

+ A DQv(i j k) + BIDQv(i 2 1 j k) + BJDQv(i j 2 1 k)

+ BKDQv(i j k 2 1) + CIDQv(i + 1 j k) + CJDQv(i j + 1 k)

+ CKDQv(i j k + 1) = 2 RES( i j k) (19)

At each time step the linearization leads to the linear system (19) which is to besolved by an iterative method When unconditional stability is achievableequation (19) becomes a Newton method which is able to deliver quadraticconvergence It has to be noticed that though not explicitly apparent fromequation (18) preconditioning enters the linear system through the inviscidnumerical macrux (13) in addition to the unsteady term People who make no useof characteristic-based schemes need little modiregcations of their codes

HFF126

644

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

Fi+ 1curren 2 =

F i + F i+ 1

22 1

2 [P 2 1RpjLpjLp] i+1 curren 2(Qi+ 1 2 Qi) (9)

made of a central part and a matrix dissipation part computed at the cellinterface denoted by (i + 1curren 2) Rp and Lp represent the matrices of right andleft eigenvectors of

Dp = PD = Pshy Fx

shy Qnx +

shy Fy

shy Qny +

shy F z

shy Qnz (10)

and jLpj represents the diagonal matrix whose elements ap are the absolutevalues jlpj of the eigenvalues of the system matrix Dp Second order accuracyis achieved (Hirsch 1990) for a thorough review computing the elements ap as

ap = jlpj + C(R + ) 2 C(R 2 ) (11)

with

R + =l+ (Qi 2 Qi 2 1)

l+ (Qi+ 1 2 Qi)R 2 =

l 2 (Qi+ 2 2 Qi+1)

l 2 (Qi+ 1 2 Qi)(12)

where the function C represents an appropriate limiter which assuresmonotonicity of the solution at discontinuities and at local minima andmaxima Setting C = 0 in Equation (11) the regrst order Roersquos scheme isrecovered

The expression of the eigenvectors in terms of conservative variables israther complicated Easier algebra and programming can be achievedtransforming equation (9) in the equivalent expression for the primitivevariables Qv

Fi+1curren 2 =

F i + F i+ 1

22 1

2 [MmRvp jLpjLv

p ] i+ 1curren 2(Qvi+ 1 2 Q

vi ) (13)

Euler matrices and eigenvectors in conservative and primitive variables arerelated by similarity transformations Preconditioned eigenvectors are given inAppendix A for a general equation of state Viscous macruxes and source termsare calculated with standard cell-centred regnite-volume techniques and they areboth second order accurate

5 Implicit solution algorithmOnce in regnite-volume and semidiscrete form system (5) becomes

Oshy Q

shy t= 2 P RES 2 MM

2 1m RES (14)

where RES represents the vector of residuals for the conservative variables andO the cell volume Updating is done in terms of the viscous primitive variables

Efregcient parallelcomputations of

macruid macrows

643

Qv namely pressure temperature and the velocity components This is done bymultiplying the above system by M 2 1 to the left

Oshy Qv

shy t= 2 M 2 1

m RES (15)

Systems (14) and (15) are equivalent If an implicit numerical scheme is used todiscretize the time derivative then

OQ

newv 2 Q

oldv

Dt O

DQv

Dt= 2 M

2 1m RESnew (16)

After linearization about the old time level

ODQv

Dt= 2 M

2 1m RESold +

shy RES

shy Qv

old

DQv (17)

Re-arranging multiplying by Mm to the left and dropping superscripts

MmODt

+shy RES

shy QvDQv = 2 RES (18)

In equation (18) DQv and RES are column vectors of dimension given by thetotal number N of cells where each element is an array of regve components Theunsteady term matrix on the left-hand side is a block diagonal matrix (with5 pound5 blocks) Jacobians of the residuals with respect to the variables aretypically constructed by using a two-point stencil scheme at each interface Asa result the second term within brackets is a sparse N poundN matrix with non-zero elements only on seven diagonals due to the grid structure

Denoting the block element of the seven diagonals by A BI BJ BK CI CJ andCK then the solution algorithm can be written as

MmODt

+ A DQv(i j k) + BIDQv(i 2 1 j k) + BJDQv(i j 2 1 k)

+ BKDQv(i j k 2 1) + CIDQv(i + 1 j k) + CJDQv(i j + 1 k)

+ CKDQv(i j k + 1) = 2 RES( i j k) (19)

At each time step the linearization leads to the linear system (19) which is to besolved by an iterative method When unconditional stability is achievableequation (19) becomes a Newton method which is able to deliver quadraticconvergence It has to be noticed that though not explicitly apparent fromequation (18) preconditioning enters the linear system through the inviscidnumerical macrux (13) in addition to the unsteady term People who make no useof characteristic-based schemes need little modiregcations of their codes

HFF126

644

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

Qv namely pressure temperature and the velocity components This is done bymultiplying the above system by M 2 1 to the left

Oshy Qv

shy t= 2 M 2 1

m RES (15)

Systems (14) and (15) are equivalent If an implicit numerical scheme is used todiscretize the time derivative then

OQ

newv 2 Q

oldv

Dt O

DQv

Dt= 2 M

2 1m RESnew (16)

After linearization about the old time level

ODQv

Dt= 2 M

2 1m RESold +

shy RES

shy Qv

old

DQv (17)

Re-arranging multiplying by Mm to the left and dropping superscripts

MmODt

+shy RES

shy QvDQv = 2 RES (18)

In equation (18) DQv and RES are column vectors of dimension given by thetotal number N of cells where each element is an array of regve components Theunsteady term matrix on the left-hand side is a block diagonal matrix (with5 pound5 blocks) Jacobians of the residuals with respect to the variables aretypically constructed by using a two-point stencil scheme at each interface Asa result the second term within brackets is a sparse N poundN matrix with non-zero elements only on seven diagonals due to the grid structure

Denoting the block element of the seven diagonals by A BI BJ BK CI CJ andCK then the solution algorithm can be written as

MmODt

+ A DQv(i j k) + BIDQv(i 2 1 j k) + BJDQv(i j 2 1 k)

+ BKDQv(i j k 2 1) + CIDQv(i + 1 j k) + CJDQv(i j + 1 k)

+ CKDQv(i j k + 1) = 2 RES( i j k) (19)

At each time step the linearization leads to the linear system (19) which is to besolved by an iterative method When unconditional stability is achievableequation (19) becomes a Newton method which is able to deliver quadraticconvergence It has to be noticed that though not explicitly apparent fromequation (18) preconditioning enters the linear system through the inviscidnumerical macrux (13) in addition to the unsteady term People who make no useof characteristic-based schemes need little modiregcations of their codes

HFF126

644

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

6 Construction of Jacobian matricesThe inviscid numerical macrux at the interface between cells (i j k ) and(i + 1 j k) given by equation (13) depends on the values of the primitivevariables on both sides of the interface With the convention that the cell

surface normalDS points towards the direction of increasing index i the

contribution Fi+ 1curren 2 DS is to be added to the residual of the cell (i j k ) and

subtracted to the residual of the cell (i + 1 j k) Dropping subscripts j and k forthe sake of clarity and noting that the inviscid numerical macrux depends only onthe two states on the left and right of the cell interface it follows

RES(i) = RES(i) + DSFi+ 1curren 2(Q

vi Q

vi+ 1) (20)

RES(i + 1) = RES(i + 1) 2 DSFi+ 1curren 2(Q

vi Q

vi+ 1)

Differentiation of equation (13) with respect to Qv allows the construction of thematrix block elements A Brsquos and Crsquos collecting all contributions by sweepingthrough all cell interfaces in the three structure directions

A i = Ai + DS(H i + D1+1 curren 2) curren 2 (21)

A i+ 1 = A i+ 1 2 DS(H i+ 1 2 D1+ 1curren 2) curren 2

B i+1 = B i+ 1 2 DS(H i + D1+ 1curren 2) curren 2

C i = C i + DS(H i+ 1 2 D1+ 1curren 2) curren 2

where the Jacobian H whose analytical expression is given by shy F curren shy Qvcorresponds to the ordfcentralordm part of the numerical macrux expressionDifferentiation with respect to Qv of the second part of equation (13)representing the scheme matrix numerical dissipation is based on the locallinearization approach of Roe where the contribution D1+1curren 2 = [MmRv

p jLpjLvp]

is locally frozen and evaluated at the cell interface based on the Roersquos averagestate a function of left (i ) and right (i+1) states

The viscous macruxes of system (1) are given by

Gr = 0 (22)

Gru = txxnx + txyny + txznz

Grv = txynx + tyyny + tyznz

Grw = txznx + tyzny + tzznz

Efregcient parallelcomputations of

macruid macrows

645

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

GrE = (utxx + vtxyny + wtxznz + qx)nx + (utxy + vtyyny + wtyznz + qy)ny

+ (utxz + vtyzny + wtzznz + qz)nz

Displaying the macruxesrsquo dependence upon the primitive variables gives

Gru = m 2nx

shy x+

ny

shy y+

nz

shy zshy u + m

ny

shy xshy v + m

nz

shy xshy w (23)

Grv = mnx

shy x+ 2

ny

shy y+

nz

shy zshy v + m

nx

shy yshy u + m

nz

shy yshy w

Grw = mnx

shy x+

ny

shy y+ 2

nz

shy zshy w + m

nx

shy zshy u + m

ny

shy zshy v

GrE = knx

shy x+

ny

shy y+

nz

shy zshy T + uGru + vGrv + wGrw

where the superscript means frozen interface values Differentiations ofequation (23) with respect to u v w and T give the viscous macruxes contributionsto the block Jacobians

A i = A i + G (24)

A i+ 1 = A i+1 + G

B i+ 1 = B i+ 1 2 G

C i = C i 2 G

where the 5 pound5 matrix G is regnally given by (all elements evaluated at cellinterface)

G =DS 2

O

0 0 0 0 0

0 m (1 + n2x) mnxny mnxnz 0

0 mnxny m (1 + n2y ) mnynz 0

0 mnxnz mnynz m (1 + n2z ) 0

0 m (u + V nnx) m (v + V nny) m (w + V nnz) k

(25)

where O respresents the cell average volume at the interface and DS the

HFF126

646

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

interface area It can be noted that the diagonal elements are positive deregniteNear solid boundaries viscous contributions to Jacobians increase the stabilityof the solution algorithm

7 Parallel redndashblack relaxation schemeAt each linearization step equation (19) is solved with an iterative redndashblackrelaxation scheme The redndashblack scheme can be seen as the inherentlyparallel version of the GaussndashSeidel scheme rather than choosing a sweepdirection regrst all red cells then all black cells are treated in a chessboardpattern The red loop reads as

DQn+ 1v ( i j k) = 2 Mm

ODt

+ A2 1

[RES(i j k) + BIDQnv(i 2 1 j k)

+ BJDQnv(i j 2 1 k) + BKDQ

nv(i j k 2 1) + CIDQ

nv(i

+ 1 j k) + CJDQnv(i j + 1 k) + CKDQ

nv(i j k + 1)] (26)

Before the red-cell loop inversion of the block diagonal elements (MmO curren Dt +A) is carried out for all blocks In a multi-block code the red loop must be madefor all blocks in parallel (no matter whether in a single processor or in a parallelmachine) Then for all blocks the black-cell loop is given by

DQn+ 1v (i j k) = 2 Mm

ODt

+ A

2 1

[RES(i j k) + BIDQn+ 1v ( i 2 1 j k)

+ BJDQn+ 1v (i j 2 1 k) + BKDQ

n+ 1v (i j k 2 1)

+ CIDQn+ 1v (i + 1 j k) + CJDQ

n+ 1v (i j + 1 k)

+ CKDQn+ 1v (i j k + 1)] (27)

where the contributions at the right-hand side are all at the new iteration levelAt each iteration step n in between the red and the black-cell loop boundaryupdates (DQvs) are exchanged between adjacent blocks The solution algorithmperformance is totally independent of the number of blocks and can be easilyported to parallel machines implementing MPI communications with no lossof convergence rate

8 Turbulence modelThe Spalart amp Allmaras turbulence model (Spalart 1994) is solved with animplicit algorithm decoupled from the main regve equations The solutionalgorithm is the same as the one just described All Jacobian 5 pound5 block

Efregcient parallelcomputations of

macruid macrows

647

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

matrices for A B and C become scalar values Special care must be taken forthe Jacobian contribution coming from the source and sink terms in theequation The procedure described in detail in the cited reference is capable ofassuring stability of the implicit time integration method

9 ResultsFour test cases covering a broad variety of applications are presented in thefollowing They include inviscid and turbulent test cases incompressiblehydrodynamics liquid metal natural convection loops and supersonic nozzlegasdynamics All cases were run with inregnite CFL number starting from trivialuniform conditions At each linearization step the redndashblack relaxationscheme was iterated till convergence This was achieved choosing a number ofiteration steps about one and a half times the number of grid nodes normal tothe solid walls The resulting quasi-Newton method showed excellent efregciencyand convergence history independent of the block mesh partition

91 Low speed turbulent macrow over a backward facing stepThe work of Le et al (1997) (DNS simulation) is used for comparison TheReynolds number is 5100 based on the step height h The computationaldomain starts at xcurren h = 10 upstream of the step location and ends at xcurren h = 20downstream The resulting two-block mesh is made of 48 pound40 and 64 pound80cells A severe mesh stretching is provided close to the solid boundaries with avalue of the non-dimensional grid spacing y

+ of the order unity The DNSsimulation determines a re-attachment point at a distance of xcurren h = 628 fromthe step compared to xcurren h = 575 of the present Spalart amp Allmaras simulation

Figure 1 shows the convergence history of streamwise momentum andturbulence model equations Figure 2 shows velocity and Reynolds stressproregles at four different locations downstream of the step The discrepancy atstation xcurren h = 6 is affected by the earlier re-attachment of the present versusthe DNS simulation

92 Buoyancy-driven annular loopA natural convection macrow of a liquid metal (the eutectic Li17-Pb83 with Pr =32 pound102 2) is driven by a volumetric heat addition supplied in a sector of thedomain at a rate CcedilQ = 84 kWcurren kg The walls are kept at a constant temperatureof 300 K The resulting Grashof number is Gr = 2 pound106 The internal radius is102 2 m while the external radius 2 pound102 2 m The grid has 120 cells in thepoloidal direction and 40 cells in the radial direction (Di Piazza 2000 Di Piazzaand Mulas 2001)

Figure 3 shows the convence history This is the only case where thesimulation does not reach machine accuracy As a matter of fact an explicit runalso showed the same stagnating convergence Figure 4 shows the temperatureregeld The macrow circulation is clockwise The uniformly heated sector is visiblein the reggure

HFF126

648

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

93 Inviscid hydrodynamic macrow over a ship hullA six-block OndashH type mesh is used to simulate the inviscid macrow over a S60

type ship hull The available towing tank measurements are provided by

Longo (1996) The hull surface is described by 226 nodes in streamwise

direction and 21 nodes in crosswise direction with about a total of 150000 cells

Figure 1Backward facing step

convergence

Figure 2Backward facing stepvelocity and Reynolds

stress proregles

Efregcient parallelcomputations of

macruid macrows

649

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

No effect of free surface is taken into account so that measurement conditionsfor low Froude numbers (Mulas and Talice 1999) are approached

Figure 5 shows the convergence history Figures 6 and 7 show the computednon dimensional lateral force and z-moment coefregcients compared tomeasurements for conditions with Fr = 010

Figure 3Buoyancy driven loopconvergence

Figure 4Buoyancy driven looptemperature regeld

HFF126

650

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

94 High enthalpy hydrogen rocket nozzle macrowThe last case deals with a new concept of rocket propulsion from an idea ofPhysics Nobel prize Carlo Rubbia (Augelli et al 1999) capable of speciregcimpulses of the order of 2000 s A hydrogen macrow enters the stagnation chamberfrom a distributed inlet at about 1500 K As it macrows through the chambertoward the convergent-divergent nozzle its temperature increases up to about10000 K due to direct energy conversion of the kinetic energy of the regssion

Figure 5Ship hull test case

convergence

Figure 6Ship hull test case non

dimensional forcecoefregcient vs drift angle

Efregcient parallelcomputations of

macruid macrows

651

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

fragments originated from a thin layer of Americium 242 (smeared over thechamber walls and undergoing critical nuclear reaction) to hydrogen internalenergy This special case of volumetric heat addition was modelled with aMonte Carlo method coupled to the CFD code (Mulas 2000 Leonardi et al2001)

This is an extremely stiff problem for three reasons regrstly because thewhole spectrum of Mach number is present (inlet velocities are of the order offew centimeters per second and exit velocities reach 20000 ms ) then becausethe speed of sound of hydrogen at 10000 K is about 10000 ms withcorresponding Mach number in the stagnation chamber of the order of 10 2 6regnally the transonic region at the nozzle throat represents another source ofstiffness Due to the operating temperature range hydrogen equations of stateare produced with a chemical equilibrium code in the form of look-up tables Asalready mentioned the output of the chemical equilibrium code takes intoaccount the hydrogen dissociation and ionization processes occurring attemperature intervals depending on the pressure level Radiative andabsorption effects due to high temperature were also taken into account Theimplicit preconditioned method with inregnite CFL number allows an efregcientmacrow calculation Preconditioning automatically switches off as the macrow locallyreaches sonic conditions Figure 8 shows the grey-scale temperature regeld and afew streamlines Also shown the domain block partition

10 Concluding remarksA unireged numerical preconditioned method for the solution of macruid macrowproblems belonging to a wide class of applications of engineering interest has

Figure 7Ship hull test case nondimensional momentcoefregcient vs drift angle

HFF126

652

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

been presented The method is general for it allows a macruid deregnition by meansof arbitrary equations of state and it is applicable to macrows characterized by thewhole range of Reynolds Grashof and Mach numbers The implicitdiscretization presented is able to stand inregnite CFL numbers The keyfeature for obtaining unconditional stability is the proper construction of thesystem matrix of the linearized time advancing scheme The resulting linearsystem is solved at each linearization step with a redndashblack relaxation schemePushing the linear system solution until convergence makes the scheme anefregcient Newton method Moreover the redndashblack relaxation is intrinsicallyindependent of the arbitrary multi-block partition of the macrow domain andallows an easy multi-processor implementation with MPI technique Thesecond order TVD Roersquos scheme keeps its outstanding accuracy featuresprovided that the eigenvector decomposition is done in the preconditioned

Figure 8Rocket nozzle

temperature regeld

Efregcient parallelcomputations of

macruid macrows

653

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

world General preconditioned eigenvector matrices have been developed forarbitrary thermodynamics The numerical examples showed that thecomputational efregciency is independent of the class of macruid macrow problem

References

Augelli M Bignami GF Bruno C Calligarich E De Maria G Mulas M Musso CPellizzoni A Piperno W Piva R Procacci B Rosa-Clot M and Rubbia C (1999)ordfReport of the working group on a preliminary assessment of a new regssion fragmentheated propulsion concept and its applicability to manned missions to the planet Mars(Project 242)ordm ASI report March 15 Roma

Bijl H and Wesseling P (1996) ordfA method for the numerical solution of the almostincompressible Euler equationsordm TU Delft Report 96-37 Delft

Bijl H and Wesseling P (1998) ordfA unireged method for computing incompressible andcompressible macrows in boundary-regtted coordinatesordm J Comput Phys Vol 141 pp 153-73

Bijl H and Wesseling P (2000) ordfComputation of unsteady macrows at all speeds with a staggeredschemeordm ECCOMAS 2000 11ndash14 September Barcelona

Chorin AJ (1967) ordfA numerical methods for solving incompressible viscous macrow problemsordmJ Comput Phys Vol 2 pp 12-26

Choi D and Merkle CL (1985) ordfFully implicit solution of incompressible macrowsordm AIAA J Vol23 pp 1518-24

Choi D and Merkle CL (1993) ordfThe application of preconditioning to viscous macrowsordm J ComputPhys Vol 105 pp 207-23

Di Piazza I ` Prediction of free convection in liquid metals with internal heat generation andormagnetohydrodynamic interactionsrsquorsquo PhD thesis University of Palermo Italy

Di Piazza I and Mulas M (2001) ordfPreconditiong tests on a buoyancy-driven anular loop usingthe CFD code Karalisordm CRS4 TECH-REP 0126

DemirdzugraveicAcirc I Lilek Z and PericAcirc M (1993) ordfA collocated regnite volume method for predictingmacrows at all speedsordm Int J Numer Methods Fluids Vol 16 pp 1029-50

Guerra J and Gustafsson B (1986) ordfA numerical method for incompressible and compressiblemacrow problems with smooth solutionsordm J Comput Phys Vol 63 pp 377-96

Hirsch Ch (1990) Numerical Computation of Internal and External Flows Wiley New YorkVol 2

Lee D ``Local preconditioning of the Euler and NavierndashStokes equationsrsquorsquo PhD thesisUniversity of Michigan

Longo JF ``Effects of yaw model-scale ship macrowsrsquorsquo PhD thesis University of Iowa

Lee D and van Leer B (1993) ordfProgress in local preconditioning of the Euler and NavierndashStokesequationsordm AIAA Paper 93-3328

Le H Moin P and Kim J (1997) ordfDirect numerical simulation of turbulent macrow over abackward facing stepordm J Fluid Mech Vol 330 pp 349-74

van Leer B Lee W and Roe P (1991) ordfCharacteristic time-stepping or local preconditioning ofthe Euler equationsordm AIAA Paper 91ndash1552

Leonardi E Pili P and Varone A (2001) ordfDevelopment of Montecarlo codes for the simulationof the trasport of Fission Fragments and the computation of radiative and absorptioneffectsordm CRS4 TECH-REP 0185

Mulas M (2000) ordfProject 242 Fluid Dynamics regnal reportordm CRS4 TECH-REP 0094

HFF126

654

Merkle CL and Venkateswaran S (1998) ordfThe use of asymptotic expansions to enhancecomputational methodsordm AIAA Paper 98-2485

Mulas M and Talice M (1999) Numerical determination of hydrodynamic coefregcients formanoeuvring without free surface effects CRS4-TECH-REP 9919 1999

Merkle CL Sullivan JY Buelow PEO and Venkateswaran S (1998) ordfComputation of macrowswith arbitrary equation of stateordm AIAA J Vol 36 No 4 pp 515-21

Peyret R and Viviand H (1985) ordfPseudo unsteday methods for inviscid or viscous macrowcompuatationsordf in Casci C (Eds) Recent Advances in the Aerospace Sciences PlenumNew York pp 41-71

Patnaik G Guirguis RH Boris JP and Oran ES (1987) ordfA barely implicit correction for macrux-corrected transportordm J Comput Phys Vol 71 No 1 pp 1-20

Roe PL (1981) ordfApproximate riemann solvers parameters vectors and difference schemesordmJ Comput Phys Vol 43 pp 357-72

Spalart PR and Allmaras SR (1992) ordfA one-equation turbulence model for aerodynamicsmacrowsordm AIAA Paper 92-0439

Turkel E (1984) ordfAcceleration to a steady state for the Euler equationsordm ICASE Rep pp 84-32

Turkel E (1987) ordfPreconditioned methods for solving the incompressible and low speedcompressible equationsordm J Comput Phys Vol 72 pp 277-98

Venkateswaran S Weiss S and Merkle CL (1992) ordfPropulsion related macrowregelds using thepreconditioned NavierndashStokes equationsordm AIAA Paper 92-3437

Weiss JM and Smith WA (1995) ordfPreconditioning applied to variable and constant densitymacrowordm AIAA J Vol 33 No 11 pp 2050-7

Weiss JM Maruszewski JP and Smith WA (1997) ordfImplicit solution of the NavierndashStokesequations on unstructured meshesordm AIAA Paper 97-2103

Wesseling P van der Heul DR and Vuik C (2000) ordfUnireged methods for computingcompressible and incompressible macrowsordm ECCOMAS 2000 11ndash14 September Barcelona

Appendix Eigenvectors and eigenvaluesThe Jacobian matrices of primitive variables with respect to conservative variables and viceversa are given by

M =shy Q

shy Qv=

rp 0 0 0 rT

urp r 0 0 urT

vrp 0 r 0 vrT

wrp 0 0 r wrT

rp 2 (1 2 rhp) ru rv rw HrT + rhT

(28)

Efregcient parallelcomputations of

macruid macrows

655

M 2 1 =shy Qv

shy Q=

rhT + rT (H 2 V 2)

d

rT

du

rT

dv

rT

dw 2

rT

d

2u

r1

r0 0 0

2v

r0

1

r0 0

2w

r0 0

1

r0

2 rp(H 2 V 2) + 1 2 rhp

d2

rp

du 2

rp

dv 2

rp

dw

rp

d

(29)

where d is given by d = rrphT + rT (1 2 rhp) The modireged versions Mm and M2 1m are obtained

from non modireged ones (equations (28) and (29)) substituting rp with rmp so that d becomes d m

accordingly

The system eigenvalues elements of Lp of equation (13) are l123 = V n unx + vny + wnz

and

l45 = Vnd + d m

2 d mV 2

n

d 2 d m

2 d m

2

+rhT

d m

Non-preconditioned eigenvalues are recovered when d = d m noticing that the speed of sound isgiven by c = rhT curren d The matrix Lv

p of left eigenvectors is given by

Lvp =

2rT

d mnx 0

A

Dm nz 2A

Dm ny 2A

Tnx

2rT

d mny 2

A

Dm nz 0A

Dm nx 2A

Tny

2rT

d mnz

A

Dm ny 2A

Dm nx 0 2A

Tnz

12 + B 1

2

A

Dm nx12

A

Dm ny12

A

Dm nz 0

12 2 B 2 1

2

A

Dm nx 2 12

A

Dm ny 2 12

A

Dm nz 0

(30)

where

Dm = V 2n

d 2 d m

2 d m

2

+rhT

d m A =

r 2 hT

d m B =

1

4

V n

Dm

d 2 d m

d m(31)

In case of no preconditioning

d m = d Dm = D = c A = r c 2 and B = 0 (32)

It has to be noticed that when deriving the eigenvector matrices the regrst three elements of theregfth column of Lv

p take the following form

HFF126

656

rrT hT

d m (1 2 r hp)(33)

which would generate a 00 term when using incompressible macruids with b = x = 0 Howeverfor any pure substance the above term is equivalent to r 2hT curren d mT

In the matrix Lvp the regrst three rows represent linear combinations of the entropy and the two

shear waves all of them propagating with characteristic speed given by l123 = V n The lasttwo rows represent the two acoustic waves with caracteristic speeds given by l4 and l5 Whenthe ideal gas law is considered (with no preconditioning) the well known left eigenvector matrixLv in primitive variables is recovered

Lv =

(g 2 1)nx 0 rcnz 2 rcny 2rc 2

Tnx

(g 2 1)ny 2 rcnz 0 rcnx 2rc 2

Tny

(g 2 1)nz rcny 2 rcnx 0 2rc 2

Tnz

12

12rcnx

12rcny

12rcnz 0

12 2 1

2rcnx 2 12rcny 2 1

2rcnz 0

(34)

The matrix Mm Rvp is given by equation (36) without preconditioning and when the ideal gas law

is applied it reduces to R the Euler right eigenvector matrix in conservative variables

R =1

c 2

nx ny nz 1 1

unx uny 2 cnz unz + cny u + cnx u 2 cnx

vnx + cnz vny vnz 2 cnx v + cny v 2 cny

wnx 2 cny wny + cnx wnz w + cnz w 2 cnz

V 2

2nx + c(vnz 2 wny)

V 2

2ny + c(wnx 2 unz)

V 2

2nz + c(uny 2 vnx) H + cV n H 2 cV n

(35)

MmRvp

=d m

rhT

2TrT

rnx 2

TrT

rny 2

TrT

rnz 1 1

2TrT

runx 2

TrT

runy 2 Dmsz 2

TrT

runz + Dmsy u + (F + Dm )nx u + (F 2 Dm )nx

2TrT

rvnx + Dmsz 2

TrT

rvny 2

TrT

rvnz 2 Dmsx v + (F + Dm )ny v + (F 2 Dm )ny

2TrT

rwnx 2 Dmsy 2

TrT

rwny + Dmsx 2

TrT

rwnz w + (F + Dm )nz w + (F 2 Dm )nz

2 THrT

r+ hT nx + Dm (vnz 2 wny ) 2 T

HrT

r+ hT ny + Dm (wnx 2 unz) 2 T

HrT

r+ hT nz + Dm (uny 2 vnx ) H + (F + Dm)V n H + (F 2 Dm )V n

(36)

where F = 2 12 V n(d 2 d m curren d m)

Efregcient parallelcomputations of

macruid macrows

657

rrT hT

d m (1 2 r hp)(33)

which would generate a 00 term when using incompressible macruids with b = x = 0 Howeverfor any pure substance the above term is equivalent to r 2hT curren d mT

In the matrix Lvp the regrst three rows represent linear combinations of the entropy and the two

shear waves all of them propagating with characteristic speed given by l123 = V n The lasttwo rows represent the two acoustic waves with caracteristic speeds given by l4 and l5 Whenthe ideal gas law is considered (with no preconditioning) the well known left eigenvector matrixLv in primitive variables is recovered

Lv =

(g 2 1)nx 0 rcnz 2 rcny 2rc 2

Tnx

(g 2 1)ny 2 rcnz 0 rcnx 2rc 2

Tny

(g 2 1)nz rcny 2 rcnx 0 2rc 2

Tnz

12

12rcnx

12rcny

12rcnz 0

12 2 1

2rcnx 2 12rcny 2 1

2rcnz 0

(34)

The matrix Mm Rvp is given by equation (36) without preconditioning and when the ideal gas law

is applied it reduces to R the Euler right eigenvector matrix in conservative variables

R =1

c 2

nx ny nz 1 1

unx uny 2 cnz unz + cny u + cnx u 2 cnx

vnx + cnz vny vnz 2 cnx v + cny v 2 cny

wnx 2 cny wny + cnx wnz w + cnz w 2 cnz

V 2

2nx + c(vnz 2 wny)

V 2

2ny + c(wnx 2 unz)

V 2

2nz + c(uny 2 vnx) H + cV n H 2 cV n

(35)

MmRvp

=d m

rhT

2TrT

rnx 2

TrT

rny 2

TrT

rnz 1 1

2TrT

runx 2

TrT

runy 2 Dmsz 2

TrT

runz + Dmsy u + (F + Dm )nx u + (F 2 Dm )nx

2TrT

rvnx + Dmsz 2

TrT

rvny 2

TrT

rvnz 2 Dmsx v + (F + Dm )ny v + (F 2 Dm )ny

2TrT

rwnx 2 Dmsy 2

TrT

rwny + Dmsx 2

TrT

rwnz w + (F + Dm )nz w + (F 2 Dm )nz

2 THrT

r+ hT nx + Dm (vnz 2 wny ) 2 T

HrT

r+ hT ny + Dm (wnx 2 unz) 2 T

HrT

r+ hT nz + Dm (uny 2 vnx ) H + (F + Dm)V n H + (F 2 Dm )V n

(36)

where F = 2 12 V n(d 2 d m curren d m)

Efregcient parallelcomputations of

macruid macrows

657