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UNIC-UNS USERS MANUAL

A General Purpose CFD Code

Using Unstructured Mesh

Compatible Grid Formats: Gridgen/Fieldview, V-Grid, Patran and PLOT3D

ENGINEERING SCIENCES INCORPORATED

Huntsville, AL 35802, USA

Copyright since 2000. All rights reserved.

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TABLE OF CONTENTS

pageAN INTEGRATED TOOL 2SOLUTION METHOD 3

Governing Equations 3Spatial Discretization 4Time Integration 6Pressure-Velocity-Density Coupling 6Higher Order Schemes 7Pressure Damping 9Boundary Conditions 9Linear Matrix Solver 10

Algebraic Multi-Grid Solver 10Mathematical Formulations For Radiative Transfer Equation (RTE) 13

MESH ADAPTATION REFINEMENT 17Identification of Cells for Refinement or Coarsening 17Grid Refinement 19Cell Coarsening 20Smoothing Strategy 20Parallelization 20Domain Decomposition 21Parallel Implementation 21Implementation of Parallel Computing Algorithm 22

VALIDATIONS 23Flow Solver Accuracy 23Solution Adaptation Mesh Refinement 26

HIGH-SPEED FLOW HEAT TRANSFER 32Hypersonic Heat Transfer Benchmark Test Case 32

CONJUGATE HEAT TRANSFER MODEL 35

POROSITY MODEL 40

UNIC-UNS INPUT GUIDE 45DOMAIN DECOMPOSITION & COMMUNICATION EXAMPLE 60REFERENCES 62

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AN INTEGRATED TOOL

The procedure of predicting the general fluid dynamics environment involves thegeneration of computational mesh and the solutions of the complex flowfield that mayinvolve radiative heat transfer effects. The integrated process is illustrated in Figure 1.

The governing equations, numerical methodologies, radiative transfer model, meshrefinement, solution-adaptation, and parallel strategy are described in the followingsections.

VGRIDUNIC-MESH

PATRAN

CAD

Pre Processor

RTE Solver Flow Solver Turbulence Equation

Species Equation

Energy Equation

Post Processor

Stop

Gray Medium

Diffusive Boundary

Gas Band Model

Fig. 1 Procedure for base-heating analysis

Grid Generation

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SOLUTION METHOD

Unstructured grid methods are characterized by their ease in handling completelyunstructured meshes and have become widely used in computational fluid dynamics.The relative increase of computer memory and CPU time with unstructured grid

methods is not trivial, but can be offset by using parallel techniques in which manyprocessors are put together to work on the same problem. Furthermore, whenunstructured grid method is facilitated with the multi-grid technique its convergence canbe greatly enhanced to reach a converged solution. Moreover, flexible mesh adaptationis another attraction for the unstructured grid method where high resolution can beachieved by mesh refinement in high gradient regions such as shock waves.

Traditionally, numerical methods developed for compressible flow simulationsuse an unsteady form of the Navier-Stokes or Euler equations [1-6]. These methods usedensity as one of the primary variables and pressure is determined via an equation ofstate. Their application in cases of incompressible or low Mach number flows is

questionable, since in low compressibility limit, the density changes are very small andthe pressure-density coupling becomes very weak. Some possible ways to circumventthis difficulty lie in the use of a fictitious equation of state or artificial compressibility [7,8]such that the Jocobian matrix is not ill-conditioned. On the other hand, methods forincompressible flows are mostly of the pressure correction type and use pressure as theprimary variable [9] for solving the continuity equation. These methods are wellestablished, with many variations being possible depending upon the choice of thedependent variables and their arrangement, computational grid, pressure correctionalgorithm, differencing schemes, etc. [9,10].

Numerical methods, which are applicable to flows for all speed regimes are of

special interest in the present research. An important step in this direction was reportedby Hirt et al. [11]. Their method suffered from oscillations due to pressure-velocitydecoupling. Recently, Karki and Patankar [12] and Chen [13] presented solutionmethods based on modified pressure-velocity coupling algorithms of the SIMPLE-typewhich include the compressibility effects and are applicable to flows at all speeds. Karkiand Patankar [12] use locally fixed base vectors. Their method suffers from sensitivity togrid smoothness, due to presence (directly or indirectly) of curvature terms in theequations. The sensitivity is manifested by increased total pressure loss wherever thegrid lines change their directions or surface areas.

In the present work, a cell-centered unstructured finite volume method was

developed to predict the all speed flows [13-18], in which the primary variables are theCartesian velocity components, pressure, total enthalpy, turbulence kinetic energy,turbulence dissipation and mass fractions of chemical species.

Governing Equations

The general form of mass conservation, Navier-Stokes equation, energyconservation and other transport equations can be written in Cartesian tensor form:

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S

xxU

xtjj

j

j

+

=

+

)()(

)( (1)

where is an effective diffusion coefficient, denotes the source term, is the fluiddensity and

S),,,,,,1( khwvu= stands for the variables for the mass, momentum, total

energy and turbulence equations, respectively. Detailed expressions for the k models and wall functions can be found in [19].

Spatial Discretization

The cell-centered scheme is employed here then the control volume surface canbe represented by the cell surfaces and the coding structure can be much simplified.The transport equations can also be written in integral form as

=+

dSdnFdt

r

r

(2)

where is the domain of interest, the surrounding surface, nr

the unit normal in

outward direction. The flux function Fr

consists of the inviscid and the viscous parts:

= VF

vr

(3)

The finite volume formulation of flux integral can be evaluated by the summationof the flux vectors over each face,

=

=

)(

,

ikj

jjiFdnF

r

r

(4)

where k(i) is a list of faces of cell i, Fi,j represents convection and diffusion fluxesthrough the interface between cell i and j,

j is the cell-face area.

The viscous flux for the face e between control volumes P and E as shown in

Figure 2 can be approximated as:

+

rr

rrn

rrn

E

PE

e

PE

PE

e rr

rr

r

rr

r

)( (5)

That is based on the consideration that

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)(PEePE

rr rr

(6)

where e is interpolated from the neighbor cells E and P.

The inviscid flux is evaluated through the values at the upwind cell and a linearreconstruction procedure to achieve second order accuracy

)(ueueue

rrrr

+= (7)

where the subscript u represents the upwind cell and e is a flux limiter used to prevent

from local extrema introduced by the data reconstruction. The flux limiter proposed by

Barth [20] is employed in this work. Defining ),min(),,max( minmax juju == , the

scalar associated with the gradient at cell u due to edge e ise

=

1

0if),1min(

0if),1min(

0

0

min

0

0

max

e

ue

u

e

ue

u

e (8)

where is computed without the limiting condition (i.e.0

e 1=

e )

P E

e

n

Figure 2. Unstructured control volume.

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Time Integration

A general implicit discretized time-marching scheme for the transport equationscan be written as:

S

tAA

t

n

p

NB

m

n

mm

n

pp

n

+

+=+ =

++)(

)(1

11

(9)

where NB means the neighbor cells of cell P. The high order differencing terms andcross diffusion terms are treated using known quantities and retained in the source termand updated explicitly.

The -form used for time-marching in this work can be written as:

SUAA

t

NB

m

mmpp

n

+=+

=1

)( (10)

)(1

n

p

NB

m

n

mm AAS

SU

+=

=

(11)

where is a time-marching control parameter which needs to specify. 5.0and1 == are for implicit first-order Euler time-marching and second-order time-centered time-marching schemes. The above derivation is good for non-reacting flows. For generalapplications, a dual-time sub-iteration method is now used in UNIC-UNS for time-accurate time-marching computations.

Pressure-Velocity-Density Coupling

In an extended SIMPLE [12-18] family pressure-correction algorithm, thepressure correction equation for all-speed flow is formulated using the perturbedequation of state, momentum and continuity equations. The simplified formulation canbe written as:

ppppDRT

p nnnn

u+=+==

= ++ 11 ;uuu;u;

rrrr

(12)

n)u()()u()u( rrr

=++

n

tt (13)

where Du is the pressure-velocity coupling coefficient. Substituting Eq. (12) into Eq.(13), the following all-speed pressure-correction equation is obtained,

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n)u()()()RT

u(

1 rr

=+

n

u

tpDp

t

p

RT (14)

For the cell-centered scheme, the flux integration is conducted along each faceand its contribution is sent to the two cells on either side of the interface. Once theintegration loop is performed along the face index, the discretization of the governingequations is completed. First, the momentum equation (9) is solved implicitly at thepredictor step. Once the solution of pressure-correction equation (14) is obtained, thevelocity, pressure and density fields are updated using Eq. (12). The entire correctorstep is repeated 2 or 3 times so that the mass conservation is enforced. The scalarequations such as turbulence transport equations, species equations etc. are thensolved sequentially. Then, the solution procedure marches to the next time level fortransient calculations or global iteration for steady-state calculations. Unlike forincompressible flow, the pressure-correction equation, which contains both convectiveand diffusive terms is essentially transport-like. All treatments for inviscid and theviscous fluxes described above are applied to the corresponding parts in Eq. (14)

Higher Order Schemes

The challenge in constructing an effective higher-order scheme is to determinean accurate estimate of flux at the cell faces. Barth and Jespersen [21] proposed amulti-dimensional linear reconstruction approach, which forms the basis for the presentscheme. In the cell reconstruction approach, higher-order accuracy is achieved byexpanding the cell-centered solution to each cell face with a Taylor series:

)(),,(),,(2

rOrqzyxqzyxqcccc

rr

++= (15)

where

[ ]Tdedktwvuq ,,,,,, (16)

This formulation requires that the solution gradient be known at the cell centers.Here a scheme proposed in [22] is employed to compute the gradients:

= dnqqd r

(17)

The general approach was to: 1) coalesce surrounding cell information to the vertices ornodes of the candidate cell, then 2) apply the midpoint-trapezoidal rule to evaluate thesurface integral of the gradient theorem

= dnqq r1

(18)

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over the faces of each cells. Here denotes the volume enclosed by the surface .

It is possible to further simplify the method for triangle (2D) or tetrahedron (3D)cells such that Eq. (18) need not be evaluated explicitly. The simplification stems from

the useful geometrical invariant features of triangle and tetrahedron. These features areillustrated for an arbitrary tetrahedral cell in Figure 3. Note that a line extending from acell-vertex through the cell-centroid will always intersect the centroid of the opposingface. Furthermore, the distance from the cell-vertex to the cell centroid is always three-fourths of that from the vertex to the opposing face (For a triangle, the comparable

ration of distance is two-thirds). By using these invariants along with the fact that r isthe distance between them, Eq. (16) can be evaluated as:

rr

qqqqr

r

qrq

nnnn

c

++

=4

)(3

14321

r

(19)

Thus Eq. (15) can be approximated for tetrahedral cells by the simple formula:

[ ]43213,2,1

)(3

14

1nnnncf

qqqqqq +++= (20)

where as illustrated in Figure 3, the subscripts n1, n2 and n3 denote the nodescomprising face f1,2,3of cell c and n4corresponds to the opposite node. This modifiedscheme is analytically equivalent to that in [22]. and results in a factor of two reductionsin computational time of the flow solver.

The nodal quantities qn are determined in the manner described in [22]. Accordingly,estimates of the solution are determined at each node by a weighted average of thesurrounding cell-centered solution quantities. It is assumed in the nodal averagingprocedure that the known values of the solution are concentrated at the cell centers,and that the contribution to a node from the surrounding cells is inversely proportional tothe distance from each cell centroid to the node:

=

==

N

i i

N

i i

ic

n

rr

qq

11

, 1 (21)

where

( ) ( ) ( )[ ]21

2,

2,

2, nicnicnici zzyyxxr ++= (22)

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n2

- Cell-centroid, c

- Face-centroid, f

n3

n1

n4

Figure 3. Reconstruction stencil for tetrahedral cell-centered scheme.

Until recently, the sole approach for the reconstruction was a pseudo-Laplacianaveraging scheme presented in [23]. This scheme offers the advantage of second-order

accuracy in reconstructing data from surrounding cells to a node. However, there is aneed to artificially clip the weighting factors between 0 and 2 [24] to avert a violation ofthe positivity principle, which is necessary for solution stability. This artificial clippingprocess does, unfortunately, compromise the formal second-order accuracy of thescheme to some extent. Recent experience of applying the pseudo-Lapacian scheme toNavier-Stokes computations has surfaced some anomalous behavior, which needsfurther investigation. Meanwhile, for the present work, we are temporarily reverting tothe inverse-distance averaging of Eq. (21), which represents only acceptable accuracy,but will never violate the principle of positivity.

Pressure Damping

Following the concept of Rhie and Chow [10] developed for structured gridmethod to avoid the even-odd decoupling of velocity and pressure fields, a pressuredamping term can also introduced for unstructured grid method when evaluating theinterface mass flux. This form is written as:

=PE

PE

e

PE

PEe

ee

rr

rrp

rr

ppDuUU

rr

rr

rr (23)

where eee DupU and, are interpolated from the neighboring cells E and P, respectively.

The last term on the right hand side is a higher order pressure damping term projectedin the direction PE.

Boundary Conditions

Several different types of boundaries may be encountered in flow calculations,such as inflow, outflow, impermeable wall and symmetry. In the case of viscousincompressible flows, the following boundary conditions usually apply:

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The velocities and temperature are prescribed at the inlet; Zero normal gradient for the parallel velocity component and for all scalar quantities,

and zero normal velocity component are specified at symmetry planes or axes;

No-slip condition and prescribed temperature or heat flux are specified at the walls;

Zero (or constant non-zero) gradient of all variables is specified at the outlet.

In the case of compressible flows, some new boundary conditions may apply:

1. Prescribed total conditions (pressure, temperature) and flow direction at inflow;2. Prescribed static pressure at outflow;3. Supersonic outflow.

Some of these boundary conditions are straightforward to implement the detailedimplementation is described in [13].

In order to facilitate the variations of inlet operating conditions in rocket engineapplications, an inlet data mapping tool is developed. This inlet data-mapping toolaccepts multiple patches of surface data. Each individual surface data patch is astructured grid that contains flowfield data such as velocity, pressure, temperature,turbulence quantities and species concentrations, etc. At start up, the UNIC code readsinput data file and restart flowfield data, then it will check the existence of the inletboundary condition file, bcdata.dat. If inlet data is obtained from the inlet boundarycondition file, data mapping procedure is then performed to incorporate the new inletconditions.

Linear Matrix Solver

The discretized finite-volume equations can be represented by a set of linearalgebra equations, which are non-symmetric matrix system with arbitrary sparsitypatterns. Due to the diagonal dominant for the matrixes of the transport equations, theycan converge even through the classical iterative methods. However, the coefficientmatrix for the pressure-correction equation may be ill conditioned and the classicaliterative methods may break down or converge slowly. Because satisfaction of thecontinuity equation is of crucial importance to guarantee the overall convergence, mostof the computing time in fluid flow calculation is spent on solving the pressure-correctionequation by which the continuity-satisfying flow field is enforced. Therefore thepreconditioned Bi-CGSTAB [25] and GMRES [26] matrix solvers are used to efficiently

solve, respectively, transports equation and pressure-correction equation.

Algebraic Mult i-Grid Solver

An algebraic multi-grid method (AMG) is also developed for the solution of thealgebra equation resulting from the pressure-correction equation. The AMG method isoften used to low speed flows with long domains, which are subject to long-wave errors.

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The algorithm of the AMG method is outlined below. The final linear equation has theform of

(24)a ai i inb

nb i nb of i

b=

where nb means the neighbor cell of i. Assume the solution on coarse mesh cell I which contains the fine mesh cell isi I i/ . The correction on fine mesh from coarse

mesh is

(25) i i I~

/= + i

r

B

a

ri

In order to enforce the residual sum in I be zero,

(26)ri

~

i in I = 0

The equation on coarse mesh comes out as

(27)a ai I i inb

NB nb i / /i in I nb of ii in I i in I

=

Rewrite the equation on coarse mesh in the form of

(28)A AI I INB

NB I =

NB of I

Then the come directly from Equation (27) asA A BI INB

I, and

(29)A a aI i inb=

i in I i in Inb in I

(30)AINB

inb=

i in Inb in NB

BI=

i in I (31)

The algebraic multi-grid (AMG) method contains restriction and prolongationprocesses. In the restriction process, the equation coefficients and source terms oncoarse mesh are generated based on those on fine mesh from Equations (29-31). Whilein the prolongation process, variable on fine mesh is modified by that on coarse mesh

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according to Equation (25). It can be seen that there is no actual coarse grid mesh inalgebraic multi-grid method, but only the coarse grid equations followed completely fromthe fine grid equations. This solver strongly based on physical conservative conceptrather than pure mathematical property. It requires the governing equation on fine gridto be in conservation form.

The coarsening process is merging the adjacent cells to form larger finite volumewith the limitation on minimum and maximum number of cells to group together. Themerging cell picked is the one that has strong connection with the cells in the group.The neighbor cell is considered as strongly connected with cell ifn i

(32)a c ain

i> max

where presents the largest neighboring coefficient among , c is a constant

and the value of 1/3 works well. The minimum and maximum fine cell number in one

coarse volume is set to 5 and 9 for 2-D, and 9 and 13 for 3-D, respectively.

aimax ai

nb

There are two types of fixed cycle multi-grid sequences, V cycle and W cycle. W

cycle is more efficient because each grid level has the chance to pass its residual downto the coarse grid level twice and receive the corrections twice. The paths of V cycleand W cycle are shown in Figure 4, where the restriction process goes down at d,reach the bottom at b and then the prolongation process goes up at u. In thisresearch, W cycle is used and the coarse grid level is set to no more than six. Thebottom level is reached when the mesh cell number is less than 100 or the level isnumber 6 whichever comes first. The Incomplete Lower Upper (ILU) factorizationscheme is used to solve the linear equations. It takes two sweeps in the down

processes and 3 sweeps in the up processes. At the bottom, GMRES method is usedto ensure the accurate solution at the coarsest grid level. Several cycles may beneeded to reduce the residual by two orders of magnitude in each time step. For all thecalculation in this study, the cycle number is less than 10 and mostly around 5.

d

b

d

d

u

u

u

d

b

d

d u

b b b

u

u

u

u

u d u

Figure 4. AMG solver V and W cycle diagram.

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Mathematical Formulations For Radiative Transfer Equation (RTE)

Consider the RTE in a Cartesian coordinate system as shown in Fig. 5a. The

balance of energy passing in a specified direction through a small differential volumein an absorbing-emitting and scattering medium can be written as:

')'()',(4

)(),()(),()(4'

, dIIII b +++= =

rrrr (33)

where the subscript represents the wave-number; I( , )r is the spectral radiativeintensity, which is a function of position and direction; is the blackbody radiative

intensity at the temperature of the medium;

Ib, ( ) r

and are the spectral absorption and

scattering coefficients, respectively; and ( ' ) is the scattering phase functionfrom the incoming ' direction to the outgoing direction . The term on the left handside represents the gradient of the intensity in the direction . The three terms on theright hand side represent the changes in intensity due to absorption and out-scattering,emission and in-scattering, respectively.

If the wall bounding the medium emits and reflects diffusely, then the radiativeboundary condition for Eq. (1) is given by

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angles with N and N representing numbers of control angle in polar angle andazimuthal angle directions, respectively. These MB discrete solid angles arenonoverlapping and their sum is 4. Unlike the selection of a quadrature scheme in thediscrete ordinates method (DOM), there is no specific restriction in selecting controlangles in the FVM. However, the control angles are usually chosen in a manner that

best capture the physics of a given problem. This is analogous to the selection ofcontrol volumes.

Figure 5. (a) Coordinate system for radiative transfer equation, (b) a representative

Control-volume, and (c) a representative control angle.

Multiplying Eq. (33) by a representative control volume V (Fig. 5b) and a controlangle m(Fig. 5c), carrying out the integration, and transforming the left-hand side ofthe equation from the volume integral to a surface integral by the divergence theorem,Eq. (33) then becomes

m

P

mmmM

m

m

b

m

m

i

m

i

bswi

m

i

m

i

m

i

tnei

m

i

VIII

dnAIdnAImm

+++=

=

==

]

4

)([

)()(

''

1'

'

,,,,

(35)

where the subscripts e, w, etc. indicate the values on the eastern, western, etc. surfacesof the volume; The subscript P represents the value at the central node of the control

volume and A represents the control volume surface area; mm'

is the averaged

scattering phase function from the control angle mto the control angle m. DividingEq. (35) by m , we have

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VIIIDAIDAI Pmmm

M

m

m

b

mm

ii

bswi

m

i

m

ii

tnei

m

i +++= ===

]4

)([ ''

1'

'

,,,,

(36)

where

m

i

m

m

m

i

m

i dnnDm == )(

1

(37)

In Eq. (37), Dim

is the product of a surface unit normal vector and the averaged intensity

direction m.

To close the above equation, relations are needed between the intensities on thecontrol volume surfaces and the nodal intensities. One appropriate closure relation forcomplicated geometries is based on the step scheme, which sets the downstreamsurface intensities equal to the upstream nodal intensities. Use of the step schemeavoids the negative intensities, overshoots, and undershoots which may occur in other

radiation schemes such as diamond scheme, positive scheme, etc. Furthermore it hasmuch less connection with the neighboring nodes, and thus it is particularly suitable forparallel computation. The effect of the step scheme on communication costs is notsignificant. The final discretized equation for the FVM can be written as

mm

B

m

B

m

T

m

T

m

S

m

S

m

N

m

N

m

W

m

W

m

E

m

E

m

P

m

P bIaIaIaIaIaIaIa ++++++= (38)

where the intensities with the subscripts E, W, ... denote the eastern, western, etc.nodal intensities, and

)()0,min()0,max(,,,,

++= == VDADAam

i

bswi

i

m

i

tnei

i

m

P (39a)

TNEItneiDAam

ii

m

I ,,and,,)0,min( === (39b)

BSWIbswiDAam

ii

m

I ,,and,,)0,max( === (39c)

P

mmm

M

m

m

b

m IIVb ]4

[ ''

1'

' += =

(39d)

The preceding discretization is carried out along only one control angle at anode. The same procedure should be applied to all of the MB control angles at all of the

MA nodes. This forms MAMB systems of non-symmetric algebraic equations. Asolution of these equations only represents radiative contribution at a single wave-number. The radiative divergence is the quantity used in the energy equation and itshould consist of radiative contributions from all wave-numbers. The radiativedivergence is expressed in terms of the radiative intensities as

dIIddIdIqMB

m

mm

bbr ]4[][10

,

4

0

,

0

4

0

=

=+= (40)

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A typical radiatively participating gas consists of many lines whose absorptioncoefficients vary rapidly with wave number. Thus, it becomes a very difficult and time-consuming to evaluate the radiative properties over the actual band contour and includethem into the RTE. To avoid this difficulty, the spectrum can be divided into MC bandsand the radiative properties are assumed constant over each band. The integrated

quantity in Eq. (40) is found as the summation over all bands of the individualcontribution for each band, that is

j

MB

m

mm

jbjj

MCj

j

r IIq = =

=

=

]4[11

(41)

The above approach to the spectral problem essentially corresponds to thespectral discretization and it represents a good compromise between accuracy andcomputational time. The number of MC can change from one (gray gas model) to severalhundred (narrow band model). Obviously, the use of higher MC number provides moreaccurate results.

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MESH ADAPTATION REFINEMENT

Unstructured meshes are particularly flexible for local grid refinement andcoarsening. The fundamental idea behind adaptation is to modify the grids to betterresolve the features in the flow fields and hence to achieve accurate numerical

solutions. In this study, the modification of the grids is carried out by a hanging nodeapproach. In this approach [18], a new node in the grid refinement process is producedby subdividing an edge. This new node is a hanging node if it is not a vertex of all thecells sharing that edge. The present face based flux evaluation method provides anideal environment for implementing a hanging node adaptation scheme - solver simplyvisits each of new child faces instead of original parent face. In addition, hanging nodegrid adaptation provides the ability to efficiently operate on grids with a variety of cellshapes, including hybrid grids. The basic structure of an adaptive solution procedureconsists of: (a) solving the governing equations on the current grid; (b) identifying cellsfor refinement or coarsening; (c) subdividing the cells identified for refinement; (d)coalescing the cells identified for coarsening; and (e) refining additional cells to maintain

a smooth grid density variation required for solver to guarantee the stability andaccuracy.

Identif ication of Cells for Refinement or Coarsening

The grid adaptation requires a formulation to detect and locate features ofinterest of the flow field, which is generally called as adaptation function. An errorindicator or sensor is used in the adaptation criterion to identify regions of highersolution gradients, and it is the key to the success of grid adaptation. Such indicatorcan be specified based on the physics of the solution fields. It is very important for theerror indicator to detect a variety of flow features yet sensitive enough to detect weak

features. Therefore, the formulation of the error indicator must be flexible to identify theflow features based on density, pressure, velocity magnitude, vorticity, turbulencekinetic energy or their combinations.

By assuming as a suitable flow property, the adaptation parameter at cell i isevaluated by [18]

21 ,max

jjjjii

E=

= (42)

where j represents the cells adjacent to cell i. The standard deviation of error indicator is

then calculated as [18]:

=

=N

i

iNE E

1

21 (43)

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The summation is only conducted for cells, which satisfy maxEEi > , thereforethe undisturbed fields are not included to determine the standard deviation, where is aspecified small value and is equal to 0.05 in the present study.

To better resolve the multiple flow field features, a combined adaptation function

can be obtained by summation each normalized error indicator ( kkiE ) multiplied by a

weighting factor ( k ) as follows [18]:

= kkiki EA (44)

Then, the minimum and maximum thresholds for adaptation function can be assigned,and cells are subsequently marked for refinement or coarsening based on the value ofadaptation function whether the cells fall into the specified region or not.

The effectiveness of adaptive schemes, as it has been pointed out in the openliterature, depends very much on the selection of flowfield functions, from whichgradients are linked to the strength for mesh refinement. Naturally, this selection againdepends on what types of flow problems are under investigation. For low-speed (orincompressible) flows, what interest the researcher or the engineering designer mostwould be the resolutions of the boundary layers and/or the shear layers. Conversely, forhigh-speed flows, we would like to also include density, pressure and temperature in theformulations of the function to be used to guide the mesh adaptation. Presently, sevenfunctions are provided as options selectable from the input data file, unic.inp, of theUNIC-UNS code. These functions can be selected by setting the adaptive-functionparameter, IADPF, as given in the following:

IADPF Flow Functions

1 Combination of Density, Pressure and Velocity Magnitude

2 Density

3 Pressure

4 Temperature

5 Combination of Density and Mach Number

6 Turbulence Kinetic Energy

7 Vorticity Magnitude

These functions have been tested and their detailed formulation may needfurther adjustment for different types of flow problems to reveal their effectiveness. Forrocket engine performance and base heating problems, the calculated thrust and heatfluxes are used to guide the sensitivities of the functions.

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Grid Refinement

The cells are refined by either isotropically or non-isotropiclly subdividing eachcell identified for refinement. The sub-division of the supported cell shapes is describedbelow (see Figure 6):

A triangle is split into 4 triangles; A quadrilateral is split into two or four quadrilateral according to anisotropic

refinement functionality;

A tetrahedron is split into eight tetrahedra. The subdivision consists of trimming eachcorner of the tetrahedron, and then subdividing the enclosed octahedron byintroducing the shortest diagonal;

A hexahedron is split into 2, 4 or 8 hexahedra in terms of anisotropic refinementfunctionality;

A prism is split into 8 prisms; A pyramid is split into 6 pyramids and 4 tetrahedra.

The basic steps of the subdivision process include orienting the geometricentities (nodes, edges and faces) of cell marked for subdivision in a consistent way. Tomaintain accuracy, neighboring cells are not allowed to differ by more than one level ofrefinement. This prevents the adaptation from producing excessive cell volume changesand ensures the positions of the parent and child cell centroids are similar, thus keepingthe accuracy of the flux evaluations.

TriangleRectangle

Tetrahedron Pyramid

1 2

Figure 6. Cell sub-dividing strategy in the grid refinement method.

3

1 2

34

1

2

3

4

1

2

3

4

5

Prism

1

2

3

4

5

6

Hexahedron

1

2

3

4

5

6

7

8

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Cell Coarsening

The mesh is coarsened by reintroducing inactive parent cells. This process isequivalent to coalescing the child cells of the previously subdivided parent cell. An

inactive parent cell is recovered if all its children are marked for coarsening. The basicprocedure is to visit each inactive cell, locate its children using the face siblinginformation, remove the appropriate nodes, edges, faces and cells, reintroduce theparent cell with its associated entities into the active data structure, and assign the cellvariables to the parent using the volume weighted averaging of childrens cell variables.The original grid is ultimately reclaimed. This is a particularly attractive feature fortransient computations, especially transient solutions with limiting cycles that themaximum cell number is bounded.

Smoothing Strategy

To assure a smooth variation of cell volume, additional cells are refined based onthe number and/or relative position of neighboring cells that have been subdivided. Atriangle cell is refined if it has more than one refined neighboring cells. A quadrilateralcell is refined if it has two refined opposing cells or more than two refined neighboringcells. A tetrahedral cell is refined if it has more than two refined neighboring cells. Ahexahedral cell is refined if it has two refined opposing cells, or more than three refinedneighboring cells. And finally a pyramid or prism cell is refined if it has more than tworefined neighboring cells. The smoothing process is effective that a cell is dividedmandatorily due to the refining state of its neighboring cells, even though its value ofadaptation function is less than the value specified. When a cell is coarsened, the samerule is followed to ensure that no excessive cell volume variations occur.

Parallelization

Compared with a structured grid approach, the unstructured grid algorithm ismore memory and CPU intensive because links between nodes, faces, cells, needs tobe established explicitly, and many efficient solution methods developed for structuredgrids such as approximate factorization, line relaxation, SIS, etc. cannot be used forunstructured methods.

As a result, numerical simulation of three-dimensional flow fields remains veryexpensive even with todays high-speed computers. As it is becoming more and moredifficult to increase the speed and storage of conventional supercomputers, a parallelarchitecture wherein many processors are put together to work on the same problemseems to be the only alternative. In theory, the power of parallel computing is unlimited.It is reasonable to claim that parallel computing can provide the ultimate throughput forlarge-scale scientific and engineering applications. It has been demonstrated thatperformance that rivals or even surpasses supercomputers can be achieved on parallelcomputers.

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Domain Decomposition

To implement a parallel CFD computation strategy, the computational domainneeds to be partitioned into many sub-domains. Each sub-domain then occupies oneprocessor of a parallel computer. Many partitioning algorithms have been developed to

partition an unstructured grid. These algorithms include Recursive Coordinate Bisection(RCB), Recursive Spectral Bisection (RSB), and Recursive Graph Bisection (RGB)methods. A publicly available package developed at University of Minnesota, METIS[28], can partition high quality unstructured meshes efficiently. Grids with 1 millionvertices can be partitioned in 256 parts in less than 20 second on a Pentium Propersonal computer. In this work, both of RCB and RGB are employed and implementedin the solver.

In performing parallel computation, domain decomposition method is currentlyused in the present code. By default, METIS recursive graph bisection (RGB) method isemployed for efficient decomposition. In some cases (e.g. internal flow problems with

long domain), however, the recursive coordinate bisection (RCB) method seems toprovide better overall convergence for the flowfield solutions. To make these twomethods readily accessible in the present CFD code, an input control parameter,IPART, is introduced to control the selection of domain decomposition methods. TheRCB method is also activated as a built-in option for domain decomposition so that itcan be used when the METIS library is not available on a computer system. Theselections are shown below.

IPART Decomposition Method

1 Built-in Recursive Coordinate Bisection (RCB)

2 METIS Recursive Graph Bisection (RGB)

When the IPART = 2 option is selected, the METIS module must be executed togenerate the RGB database before running the flow solver in parallel mode.

Parallel Implementation

In a parallel computation, the governing equations are solved in all sub-domains,which are assigned to different computers processor [29]. Exchange of data betweenprocessors is necessary to enforce the boundary conditions at the divided interfaces.The communication overhead must be kept well below the computational time.Currently, many communication software packages, such as PVM and MPI, have been

developed for distributed computing.

The Parallel Virtual Machine (PVM) software system [30] is developed at theUniversity of Tennessee and Oak Ridge National Laboratory (ORNL). It is a standardmassage-passing interface and enables distributed computing across a wide variety ofcomputer types, including massively parallel processors (MPPs). It is built around theconcept of a virtual machine, which is a dynamic collection of (homogenous or

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heterogeneous) computational resource managed as a large single parallel computer.PVM is implemented for data communication among processors in this work.

MPI stands for Message Passing Interface [31]. The goal of MPI, simply stated,is to develop a widely used standard for writing message-passing programs. As such,

the interface attempts to establish a practical, portable, efficient, and flexible standardfor message passing. The main advantages of establishing a message-passingstandard are portability and ease-of-use. In a distributed memory communicationenvironment in which the higher level routines and/or abstractions are built upon lowerlevel message passing routines the benefits of standardization are clear. Furthermore,the definition of a message passing standard provides vendors with a clearly definedbase set of routines that they can implement efficiently or in some cases providehardware support for, thereby enhancing scalability. The MPI implementation of thepresent CFD program is described in Appendix A.

Implementation of Parallel Computing Algorithm

Generally, there are two schemes for parallelization, explicit and implicit. Explicitschemes are relatively easy to parallelized, since all operations are performed on datafrom preceding time steps. It is only necessary to exchange the data at the interfaceregions between neighboring sub-domains after each step is completed. The sequenceof operations and results are identical on one and many processors. The most difficultpart of the problem is usually the solution of the elliptic Poisson-like equation for thepressure or pressure-correction equation. Implicit methods are more difficult toparallelize. While calculation of the coefficient matrix and source vectors uses only olddata and can be efficiently performed in parallel, solution of the linear equation systemsneeds special attention to parallelize.

With the explicit block coupling in parallel computing, the solution may notconverge in some cases. Thus in this solver the implicit block coupling is implementedin the AMG method, which greatly enhance the stability and capability of the solver tohandle complex problems.

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VALIDATIONS

To validate the accuracy of the methodologies used in the present flow solverand demonstrate its capabilities, several representative test cases for invisicid andviscous flow are presented below.

Flow Solver Accuracy

Flows over an arc bump are studied. Different types of flow cases (subsonic,transonic and supersonic) in a channel with a circular arc bump were chosen asvalidation examples for inviscid flow calculations. The width of the channel is the sameas the length of the bump and the total length of the channel is three times the axiallength of the bump.

First, a subsonic flow case was considered. At the inlet, it is assumed that theflow has uniform properties and the upstream far-field variable values (except for

pressure) are specified. At the outlet, all variables are extrapolated except for pressure,which is prescribed. At the upper and lower wall, the flow tangency and zero mass fluxthrough the boundary are prescribed. Fig. 7 shows the pressure contours for a givenMach number at the inlet, Min= 0.5. This result compares favorably with those found inother publication [12].

Next, a transonic flow case was considered. The mesh and the treatment ofboundary conditions are identical to those described for subsonic flow. For the giveninlet Mach number Min= 0.675, the pressure contours are shown in Fig. 8. The normalshock location and strength predicted agree well with the results in Ref. 12.

Finally, a supersonic flow was analyzed. The inlet Mach number is uniform andequals to 1.65. The flow is also supersonic at the outlet. Thus, all variables areprescribed at inlet and are extrapolated at the outlet. Fig. 9 shows the shock patternpredicted of this case. The presentation solution also compares well with the resultsgiven in Ref. 12.

Figure 7. Pressure contours with subsonic flow.

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Figure 8. Pressure contours with transonic flow.

Figure 9. Pressure contours with supersonic flow.

Next, low speed flows past a circular cylinder is considered. The flow feature ofthis case depends on the Reynolds numbers. For Reynolds numbers below 50, steadywake flow has been observed experimentally. When the Reynolds number increasedbeyond 100, unsteady wake flow with cyclic vortex shedding patterns emerges. In thepresent study, cases with Reynolds numbers of 40 and 300 were considered. Figure 10shows the streamline plot of the predicted wake flow at Reynolds number 40. Thepredicted wake length (defined as the length of the recirculation zone behind the

cylinder) is 2.22 times the cylinder diameter, which compares well with the measureddata of 2.21. For the second case with Reynolds number 300, the wake pattern isinitially symmetric for some period of time. The bifurcation started later, due to theaccumulated numerical noise, with growing amplitude in oscillations, and finally reachesa periodic vortex shedding pattern as shown in Figure 11. The predicted Strouhalnumber (St) is 0.205, which is in good agreement with the measured value of 0.2. Thepredicted mean drag coefficient is 1.385, which compares well with other predictions.

Figure 10. Laminar flow (Re = 40) past a circular cylinder.

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(a) Pressure Contours

(b) Vorticity Contours

(c) Velocity Vectors

Figure 11. Unsteady wake flow behind a circular cylinder at Reynolds number 300.

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Solution Adaptation Mesh Refinement

To demonstrate the capabilities of the adaptation with the solver, several caseswith different flow conditions and geometries were investigated and their results arepresented below.

The first test case is an incompressible laminar flow past three circular cylinders.The flow Reynolds number 40 (based on the free stream velocity and the diameter ofthe cylinder) is used and the flow is believed to be stable at this flow condition. Theinitial hybrid unstructured mesh is shown in Fig. 12. Quadrilateral cells are generated atthe vicinity of the cylinders to better resolve the flow boundary layers and achieve highgrid quality.

The initial solution was obtained from the initial mesh when it was converged.Then the level-2 adaptive mesh was generated based on the current solution and isshown in Fig. 13. It can be observed that the grids closed to cylinders and at the wake

region are enriched. The level-3 mesh adaptation is achieved based on the solution ofthe level-2 adaptation mesh, and is illustrated in Fig. 14. The final solutions of velocitycontours and vectors are presented in Figs. 15 and 16. The flow fields are symmetricabout the centerline due to the symmetric geometry and stable flow conditions. Theresults obtained indicate the ability to achieve higher resolution flow fields throughautomatic mesh adaptation.

An inviscid transonic (M = 0.799) flow around a NACA 0012 profile at an angle ofattack (2.27 degree) is considered in the second test case. Fig. 17 illustrates a series ofadaptive meshes and the corresponding solutions from the beginning to the end of anadaptation loop. In all the following cases, the color of meshes represents the contoursof pressure. Comparing the pressure contours between the initial grid (Fig. 17(a)) andthe final adapted grid (Fig. 17(b)), one can see a dramatic improvement in the resolutionof the shock wave on the upper surface of the airfoil.

The third test case is a supersonic flow over a bump as described in previousvalidation cases. Here, we would like to look at the effect of mesh refinement onsolution improvements. Fig. 18 illustrates the effects of adaptive meshes from the initialto the end of an adaptation loop. The numerical solution on the initial mesh as shown inFig. 18(a) is very diffusive and less accurate at capturing the complicated interactions ofshock-to-shock, and shock-to-boundary. However, adapting the grid dramaticallyimproves the accuracy of the numerical simulation as shown in Fig. 18(b). It can also beobserved that two oblique shocks are formed at both corners of the bump. The leadingedge shock reflects from the top wall and intersects with the shock leaving the trailingedge. It is interesting to note that the shock position changes only slightly with gridrefinement. Both solutions satisfy the conservation of the equations. Refined grid givesbetter shock resolutions.

The fourth test case is an inviscid, subsonic flow past a three-element airfoil thathas been undergone extensive testing in the Low Turbulence Pressure Tunnel located

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at NASA Langley Research Center [32]. Adaptive simulations were performed for a

free-stream Mach number of M = 0.2 and an angle of attack (16.2 degree). Fig. 19demonstrates a series of meshes from the initial to the end of an adaptation loop. Theinitial coarse grid contains 10,403 node and 20,294 triangles. After three levels of gridadaptation the numbers of final grid nodes and cells increase to 17, 611 and 32,957,

respectively. Comparing to the pressure contours of the level-2 refined grid (Fig. 19(a)),the solution from the level-3 adapted mesh is greatly improved as shown in Fig. 19(b).

The last case considered is a transient turbulent flow past a half cylinder disc.For this case, the solution adaptive mesh refinement procedure is called for every timestep. Thus, the refined mesh follows the development of the flowfield. Figure 20 showsa series of adaptive mesh development due the unsteady feature of the wake flow.When the flow reaches its cyclic solutions, the maximum number of cells varies within abounded value. Based on this test, for this kind of flow problems, a maximum cellnumber limit to 3 times the original grid size is a good number to be specified in theinput data file.

Figure 12. Mesh of level-1 (initial) for flow past multiple cylinders.

Figure 13. Mesh of level-2 adaptation.

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Figure 14. Mesh of level-3 adaptation.

Figure 15. Velocity contours. Figure 16. Velocity vectors.

(a) Original Grid (b) Level-3 Refined Grid

p: 3.60 4.59 5.57 6.56 7.55 8.54 9.52 10.51 11.50 12.48

Figure 17. Transonic flow around NACA 0012 with adaptation.

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(a) Original Grid

(b) Level-3 Refinement

p: 5.50 6.67 7.83 9.00 10.16 11.33 12.49 13.66 14.82 15.99

Figure 18. Supersonic flow over a bump with adaptation.

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(a-1) (a-2)

(b-1) (b-2)

p: 5.73 6.06 6.39 6.72 7.06 7.39 7.72 8.05 8.38 8.71

Figure 19. Subsonic flow around a three-element airfoil with mesh adaptation.

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(a) (b)

(c) (d)

(e) (f)

Figure 20. Test of transient adaptive mesh refinement for a wake flow behind a halfcylinder.

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HIGH-SPEED FLOW HEAT TRANSFER

Hypersonic Heat Transfer Benchmark Test Case

This test case investigates the interaction between an impinging shock wavegenerated by a wedge and the bow shock around a circular cylinder. This is a type IVshock-chock interaction in Edneys classification. The geometry is obtained from thedrawing in the test case specification of the Houston High Speed Flow Database(Chapter 11). Umesh is used for mesh generation. The grid size used in computation is5,248. The following figure shows the mesh system.

Figure 21. Mesh system used in the T11-97 test case CFD computation.

The free stream conditions are:

M = 9.2T = 797 K

= 2.19 x 10-3kg/m3

YN2 = 0.74YN = 0.00YO2 = 0.05YO = 0.17YNO = 0.04Twall = 300 K

Finite-rate chemistry model with 9 species and 20 reactions is used in thecomputation. The calculated Mach number contour, impinging jet pattern and datacomparisons for the wall pressure and heat flux data are shown in the following figures.Good correlation between the data and the current model is revealed in Figures 24 &25.

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Figure 22. Mach number contours predicted for the T11-97 test case.

Impinging jet due to shock-shock

interaction, which causes high

heating at blunt-body surface.

Figure 23. Velocity Vectors Near the Blunt Body Leading Edge

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Figure 24. Blunt body nose surface pressure data comparisons.

Figure 25. Blunt body nose wall heat flux comparisons.

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CONJUGATE HEAT TRANSFER MODEL

Conjugate Heat Transfer Modeling

The present conjugate heat transfer model solves the heat conduction equationfor the solid blocks separately from the fluid equations. The solid heat conductionequation can be written as:

sv

ii

QQx

T

xt

CT+=

(45)

where Qvand Qs represent source terms from volumetric and boundary contributions,

respectively. and Cdenote the thermal conductivity and capacity of the solid material,respectively. The temperature value at the fluid-solid interface is obtained by enforcing

the heat flux continuity condition. Then a source term for the cell next the boundary isassigned and with the cell link coefficient to the boundary eliminated. This way thesolution of the temperature field in the solid domain satisfies the flux conservationcondition. The boundary heat flux is,

b

wdx

dTq

= where qwis the heat flux from the fluid side. (46)

Numerically, it is expressed as:

n

bc

w y

TT

q

=

or

nwcb

yqTT = (47)

with ynrepresents the normal distance from the boundary to the center of the solid cellnext to the boundary. The source term for the cell is expressed as:

n

cbs

y

TTQ

= (48)

In the original formulation, to avoid numerical oscillations, an under-relaxationparameter is assign when evaluating the boundary temperature from Eq. (47). This stepaffects the source term accuracy using Eq. (48). Also, the source term calculated fromEq. (48) can be very large when a case is freshly started, depending of the initialcondition specified. This may also contribute to numerical instability in the solutionprocedure. An implicit treatment of the boundary condition, which will be described next,is required for stability and accuracy.

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Implicit Treatment of the Solid/Gas Interface Temperature

The gas/solid interface temperature, which is stored at interior boundary points,is calculated using heat flux continuity condition. For solution stability and accuracy, the

gas/solid interface boundary temperature is updated using the transient heat conductionequation. The heat conduction equation on the fluid-solid interface can be written as:

( Tkydt

dTC

= ) (49)

It is discretized as:

=

HL

y

TTk

yt

TTC 0100

1'

2

1 (50)

where the superscript represents the next time-level solution. Subscripts, 0 and 1represent cells at the interface and one point into the solid, respectively. HL is the wallheat flux from the solid wall to the fluid, which has negative value of the wall functionpoint heat flux. The factor on the left hand side of the above equation is because onlyhalf of the solid cell is involved in the control volume. Therefore, the followingexpression is obtained for updating the interface temperature.

( ) HLBATTAT += 100 1' (51)

where

yC

tB

yC

KtA

=

=

2

22

(52)

(53)

In principle, this scheme is good for transient cases. For steady-state solutions,acceleration factor should be applied to force better convergence of heat conduction inthe solid. Based on numerical experiment, A and B calculated above is multiplied by afactor of 10 to have good results. However, this factor for steady-state acceleration iscase dependent and can be reduced, especially when there is large heat source

present as in the nuclear thermal propulsion case or when mesh skewness is high.

A Conjugate Heat Transfer Model Example Case

An example test case of heat exchanger is used to test the conjugate heattransfer model without heating power. The domain is 50 cm long with the heatexchanger located between 6 cm and 30 cm stations. The heat exchanger consists ofone circular tube along the axis and 64 rectangular channels distributed inside the

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region. A quarter of the problem is modeled using the symmetry property. The entiredomain is initialized at 35 ATM pressure and 1500 degree gas temperature and 100m/sec gas velocity at the inlet boundary. The heat exchanger temperature is initializedas 300 K. The outer wall temperature is fixed at 300 K. The input section pertinent to theconjugate heat transfer model is listed below for reference.

$CONJ UGATE HEAT TRANSFER ==( Not e: Del et e # t o act i vat e) =====#NBLOCK

1# I BLOCK NTHERcht - gr d. f mt

1 3# THER_TMP THER_CON THER_DEN THER_CP

300 1. 000E+02 1. 000E+03 1. 000E+031000 1. 200E+02 1. 000E+03 1. 000E+035000 2. 000E+02 1. 000E+03 1. 000E+03

## I DBC TM_FAR HT_COEF

1 0. 0 0. 02 0. 0 0. 03 0. 0 0. 04 0. 0 0. 05 0. 0 0. 06 0. 0 0. 07 0. 0 0. 0

#$POWER DI STRI BUTI ON# NPOWER

1# NSBLK i ax_h t her t m yt h_cnt zt h_cnt

1 1 0. 0E06 0. 0 0. 0

# I SBLK( I , I =1, NSBLK)1#

In setting up this case, a mesh for the entire domain is generated and the grid file(unic.xyz), connectivity file (unic.con) and input file (unic.inp) are prepared the sameway. Then, a sub-grid is generated to represent the conjugate heat transfer domain(cht-grd.fmt listed in unic.inp). Notice that because of the conjugate heat transfer model,internal wall boundary is created based on the sub-grid cht-grd.fmt, one additional wallboundary is added to the boundary list in unic.inp. Material properties are prescribed asdefined in the UNIC-UNS INPUT GUIDE section. A converged solution is obtained in5000 time steps with 1.0E-04 sec time step size. The predicted whole-domain

temperature, heat exchanger temperature, pressure and Mach number are shown inFigures 26, 27, 28 and 29, respectively.

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Figure 26. Temperature contours (K) on the symmetry planes of a heat exchanger flow.

Figure 27. Close-up view of the heat exchanger temperature contours.

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Figure 28. Pressure contours (ATM) on the symmetry planes of a heat exchanger flow.

Figure 29. Mach number contours on the symmetry planes of a heat changer flow.

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POROSITY MODEL

A Porosi ty Model for the Nuclear Thermal Propulsion Appl ication

For the two-temperature porosity model, separate thermal conductivitycoefficients for the flow and the solid parts shall be used. Since the flow channels areisolated, zero thermal conductivity should be assigned. For the solid part, the heatconduction is modified by a factor of (1-porosity). Lastly, the heat transfer between theflow and solid is modeled by using the empirical correlation of heat transfer coefficientfor circular pipes as a function of the pipe flow Reynolds number, multiplied by anempirical constant. This empirical constant is then tuned based on the solutioncomparisons of the porosity model and the conjugate heat transfer model. The porositymodel is described below.

For The Continuity, Navier-Stokes and Energy (Total Enthalpy) Equations, can

be written in a Cartesian tensor form with porosity factor, , included:

( )

( ) ( )[ ]

( ) ( )

QVft

Q

x

VV

x

VV

xVpH

xpHU

xH

t

Lx

PVVgUU

xU

t

Uxt

k

kj

k

j

k

j

r

jr

j

j

i

jiijji

j

i

i

i

=

+

=

++

+

=

+

3

2

2

11

,

0

2

(54)

(55)

(56)

For the solid heat conduction,

( )

QT

xt

TCss

j

sss =

)1()1(

(57)

where L and Q represent the drag loss and heat transfer source due to the porousmaterial. For regular porous media, Erguns equation for the drag loss can be used.That is,

( ) id uU

dL

+

=3

Re

115075.1

1

(58)

For the present fuel elements heat exchanger configuration, drag loss in circular pipescan be used. That is,

if uUcL = 2

1 (59)

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where25.0

3Re

160791.0

=d

Acf

For the heat exchange source term,

( fTTCUp

cQ gsp

r

f =3/22

1 ) (60)

where subscripts s and g denote the solid and gas domain, respectively. The remainingparameter on the right-hand-side of Eq. (60), f, is an empirical parameter that isidentified by comparing solutions of the porosity model and the detailed conjugate heattransfer model.

Boundary Condit ions between the Gas and Porous Regions

Since the porosity model is solved the same way as the flow equations, theboundary conditions between the gas and porous regions remain the same withoutspecial treatment. Thermal conductivity at the boundary of the porous regions iscalculated the same way as the interior points so that it is consistent throughout theporous regions.

Boundary Conditions between the Porous and CHT-Solid Regions

The temperature boundary condition between the porous and CHT-solid regionsis formulated using the heat flux continuity method with thermodynamics properties ofboth regions. This option is added to the subroutine handling the gas/solid interfacetemperature updates (i.e. subroutine CONTACT in u1.f).

A Porosi ty Model Example Case

An example test case of nozzle flow is used to test the porosity model withheating power applied. The chamber is 30 cm long with the porous region located from

6 cm to 30 cm location. The porous region has the property of 0.4 porosity and 2 mmpore size simulating array of circular tubes running in the axial direction. There is aouter wall layer of the porous region that is modeled with the conjugate heat transfermodel. The chamber is initialized at 35 ATM pressure, 300 degree temperature and 100m/sec inlet gas velocity. A heating source with total power of 10 megawatts isdistributed within the porous region with prescribed profiles in the axial and radialdirections. The input section pertinent to the porosity model is listed below for reference.

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#============================================================$CONJ UGATE HEAT TRANSFER ==( Not e: Del et e # t o act i vat e) =====#NBLOCK

1# I BLOCK NTHERcht - gr d. f mt

1 3

# THER_TMP THER_CON THER_DEN THER_CP300 1. 000E+02 1. 000E+03 1. 000E+031000 1. 500E+02 1. 000E+03 1. 000E+035000 2. 500E+02 1. 000E+03 1. 000E+03

## I DBC TM_FAR HT_COEF

1 0. 0 0. 02 0. 0 0. 03 0. 0 0. 04 0. 0 0. 05 0. 0 0. 06 0. 0 0. 07 0. 0 0. 0

##$POWER DI STRI BUTI ON

# NPOWER1# NSBLK i ax_h t hert m yth_cnt zt h_cnt

1 1 1. 0E05 0. 0 0. 0# I SBLK( I , I =1, NSBLK)

1##============================================================$POROSI TY MODEL I NPUT ==( Not e: Del et e # t o act i vat e) ========#NNPORO

1# I DPORO DPORO POROSI TY I PDI R I POROH POROTM ynt p_cnt znt p_cnt nporopor o- gr d. f mt

1 2000. 0 0. 40000 1 3 10. 0E06 0. 0 0. 0 3# PORO_TMP PORO_CON PORO_DEN PORO_CP

300 1. 000E+02 1. 000E+03 1. 200E+031000 1. 200E+02 1. 000E+03 1. 200E+035000 2. 000E+02 1. 000E+03 1. 200E+03

#$POWER CURVES# 2 f i l es namespwd_curve1. datpwd_curve2. dat#

In setting up this case, a mesh for the entire domain is generated and the grid file(unic.xyz), connectivity file (unic.con) and input file (unic.inp) are prepared the sameway. Then, a sub-grid is generated to represent the conjugate heat transfer domain(cht-grd.fmt listed in unic.inp) and another sub-grid is also generated to represent theporous region (poro-grd.fmt listed in unic.inp). Notice that because of the conjugate heattransfer model, as in the previous case, one more wall boundary is added to theboundary list in unic.inp. Material properties are prescribed as defined in the UNIC-UNSINPUT GUIDE section. A converged solution is obtained in 8000 time steps with 1.0E-05 sec time step size. The predicted pressure, Mach number, gas temperature andpores-solid temperature are shown in Figures 30, 31, 32 and 33, respectively. The gastemperature shows only slight delay in temperature rise downstream than the pores-

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solid temperature. For this case, the original gas-solid heat transfer coefficient for theporosity model is used without modification (i.e. f = 1.0 in Eqn. 60) that causes high heattransfer between the gas and the solid.

Figure 30. Pressure contours (ATM) on the symmetry planes of a nozzle flow with aporous heating region in the chamber.

Figure 31. Mach number contours on the symmetry planes of a nozzle flow with aporous heating region in the chamber.

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Figure 32. Gas temperature contours (K) on the symmetry planes of a nozzle flow with

a porous heating region in the chamber.

Figure 33. Pores-solid temperature contours (K) on the symmetry planes of a nozzleflow with a porous heating region in the chamber.

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UNIC-UNS INPUT GUIDE

List of the README file:

===============================================================

INSTRUCTION FOR UNIC UNSTRUCTURED-GRID CFD CODE

---- Unified Finite Volume Method for Fluid Flow Calculations ------

Revised 10/08/2002

===============================================================If you have any question, please contact:===============================================================-----------------------------------------------------------------------------------------------------------

UNIC Department Voice: (256) 883-6233 (O)

Engineering Sciences, Inc. (256) 883-6445 (O)1900 Golf Rd, Suite D Fax: (256) 883-6267Huntsville, AL 35802 E-Mail: unic@esi-al.com

----------------------------------------------------------------------------------------------------------

Introduction

The unstructured grid flow solver has been developed for all-speed flow calculationusing pressure-based method at Engineering Sciences Inc. (ESI). The unstructuredgrid method has the advantages of automated grid generation in very complex domain

and flexible mesh adaptation in high gradient region over the structured grid method.The general grid topology allows the use of even the traditional structured grids andoptimized grids in viscous boundary layers. A high-order upwind scheme with fluxlimiter has been incorporated for convection terms. The convergence of the linearalgebraic equations is accelerated through a preconditioned conjugate gradient matrixsolver. Numerical calculations have been conducted for compressible/incompressible,steady/transient, laminar/turbulent, spray combustion and chemical reacting flows.

A parallel implementation using MPICH-1.2.1 is also developed.

Model Data Files

unic.inp General input data.curve.bc Used for 2-D unstructured grids to specify boundary condition.unic.sgb Used for 2/3-D structured grids to specify boundary condition.fastg.unf unformatted, single block, unstructured FAST grid file.fastg.fmt formatted, single block, unstructured FAST grid file.

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unicg.unf unformatted, multi block, unstructured UNIC-UNS grid file for generalgrid topology.

unicg.fmt formatted, multi block, unstructured UNIC-UNS grid file for general gridtopology.

plot3dg.unf unformatted, multi block, structured PLOT3D grid file.

plot3dg.fmt formatted, multi block, structured PLOT3D grid file.patrang.fmt formatted, PATRAN unstructured grid file. Boundary index is assignedthrough pressure load.

fieldvw.fmt formatted, Gridgen unstructured grid file saved in Fieldview format.

Important Pre-/Post-Processing and User Files

xcont Connects multi-block grids into a single block mesh (PLOT3D grids needunic.sgbboundary index file to proceed with this tool)

xprep Generates model data base files: unic.con, unic.xyz, and unic.per(if periodic bc involved).xconv Converts unic.flofor plotting.xunic Mainprogram for UNIC-UNS.

Test Cases

# cyln Laminar 2-D incompressible flow past a cylinder.# mfoil Inviscid 2-D compressible flow past a multi-element air foil.# naca12 Turbulent 2-D compressible flow past NACA0012 air foil.

Uses hybrid grids.# moon Laminar 2-D flow past a half cylinder.

Uses 2-block structured grids.# cascade 2-D cascade case with periodic bc.# port3 3-D portflow case.

Database and Solut ion Data Files

unic.con unformatted, direct access grid connection data file Generated by xprep.unic.xyz unformatted, direct access grid xyz location data file Generated by

xprep.unic.per formatteddata file for periodic bc. Generated by xprep.unic.flo Output unformatted, direct access flow solution file.unic.ini Input unformatted, direct access flow solution file.

* Copy unic.flo tounic.ini for restart runs.

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fieldview.unf unformattedFIELDVIEW unstructured data file.fieldview.fmt formattedFIELDVIEW unstructured data file.plot3dq1.unf unformatted, multi-block,unstructured PLOT3D solution file

(rho, rho*u, rho*v, rho*w, rho*e0).plot3dq2.unf unformatted, multi-block,unstructured PLOT3D solution file

(pressure,temperature,turbulent k,eddy viscosity,Ma/vof).plot3dq3.unf unformatted, multi-block,unstructured PLOT3D solution file(mass fractions defined by user).

*Note: Solution files are generated by xconv from unic.flo. They are the same forstructured and unstructured grids. Model data base files include: unic.con and unic.xyz,and unic.per if periodic bc is involved

To generate model data base files

Method 1: (only for structured multi-block mesh from Umesh or other grid generators)

Enter the command "uinit" in an appropriate directory. Follow the instructions forrunning uinit to finish input data and flowfield specifications. Then, select option #4towrite out UNIC-UNS data files (unic.sgb, unicg.unf, unic.con, unic.xyz, unic.001 andunic_ini). unic.001 is to be copied to unic.inp and unic_ini is to be copied to unic.inibefore start running xunic.

Method 2:

Type the command "xprep" in an appropriate directory. It will use file curve.bc for 2-Dunstructured grids or unic.sgb for structured grids.

xprepconvert the following files:= 3: FAST UNFORMATTED fastg.unf= 4: FAST FORMATTED fastg.fmt= 5: UNIC-UNS UNFORMATTED unicg.unf= 6: UNIC-UNS FORMATTED unicg.fmt= 7: PLOT3D UNFORMATTED plot3dg.unf= 8: PLOT3D FORMATTED plot3dg.fmt=10: PATRAN FORMATTED patrang.fmt=12: FIELDVIEW FORMATTED fieldvw.fmt (Gridgen Output)

to= 1: UNIC-UNS MODEL DATA BASE FILES, UNFORMATTED, DIRECT

ACCESS unic.con, unic.xyz, unic.per(for periodic bc)

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To run UNIC-UNS

The following is for versions using the PVM method. For MPI version, runningscript is simply: mpirun np # xunic > out1 &. (where # stands for number processors)

1. Modify your .cshrc file by inserting the following lines for all parallel computers:(Skip steps 1, 2 & 4 for PC Windows systems)

#set f or PVMset env PVM_ROOT / home8/ pvm3i f ( $?PVM_ROOT) t hen

set env PVM_ARCH `$PVM_ROOT/ l i b/ pvmget ar ch`set pat h=( $pat h $PVM_ROOT/ l i b)

endi f

2. Prepare a .rhostsfile to include the hosts to be used.cpc1.esi.inccpc2.esi.inccpc3.esi.inccpc4.esi.inccpc5.esi.inccpc6.esi.inc

3. Create executable xunicfile in your source directory:

4. Prepare a hostfile contains the host names. Type pvmd hostfile & to launch thePVM. (This step is not need for sequential runs.)

5. In the working directory, type 'xunic' for sequential runs and 'xunic -p' for parallelruns. (Note: Parallel model may not work properly on PC Windows environmentdue to PVM basic functions that does not allow assignment of specific workingdirectories)

To restart UNIC-UNS

Simply copy unic.flo tounic.ini, then resubmit the job for succeeding runs. These twofiles are direct access files. The same files are used for serial and parallel runs.

To convert unic.flo for plotting

Type the command "xconv" in appropriate directory. xconvreads input from conv.inp.Or, without converting, you can just use the tecplot.datoutput file for TECPLOT.

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Data structure for FAST grids

C. . . . . NODE: NUMBER OF NODES; NBFACE: NUMBER OF BOUNDARY FACESC. . . . . NELE: NUMBER OF ELEMENTS OR CELLSC. . . . . X, Y AND Z( ) : NODE LOCATI ONSC. . . . . I BFACE( 3: 5) : BOUNDARY FACE NODE I DENTI FI CATI ONS

C. . . . . I BFACE( 2) : BOUNDARY FACE SURFACE GROUPC. . . . . I DNODE( ) : CELL NODE I DENTI FI CATI ONSOPEN( 8, FI LE=' f ast g. unf ' , FORM=' UNFORMATTED' )READ( 8) NODE, NBFACE, NELEKNODE=4 ! 3D TETRAHEDRON, 4 NODES PER CELLI F( NELE. EQ. 0) THEN

KNODE=3 ! 2D TRANGULAR, 3 NODES PER CELLNELE=NBFACENBFACE=0

ENDI FREAD( 8) ( X( I ) , I =1, NODE) , ( Y( I ) , I =1, NODE) , ( Z( I ) , I =1, NODE) ,

& ( ( I BFACE( J , I F), J =3, 5) , I F=1, NBFACE) ,& ( I BFACE( 2, I F) , I F=1, NBFACE) ,& ( I DNODE( I ) , I =1, NELE*KNODE)

C

Data structure for UNIC-UNS grids

OPEN( 8, FI LE=' uni cg. unf ' , FORM=' UNFORMATTED' )READ( 8) NZON, I DI M ! NUMBER OF ZONE AND DI MENSI ON ( 2D OR 3D)READ( 8) NELE, NODE, NBFACE, NI NODE ! NI NODE: MAXI MUM I NDEX FOR I DNODE( )READ( 8) ( X( I ) , Y( I ) , Z( I ) , I =1, NODE) ! NODE LOCATI ONREAD( 8) ( I NODE( I ) , I =1, NELE+1) ! I NODE( I : I +1) : CELL NODE NUMBERREAD( 8) ( I DNODE( I ) , I =1, NI NODE) ! CELL NODE I DREAD( 8) ( ( I BFACE( J , I ) , J =1, 6) , I =1, NBFACE) ! BOUNDARY FACES

READ( 8) ( I PI D( I ) , I =1, NELE) ! MATERI AL I NDEX ( 0: Fl ui d; >0: Sol i d)C

Sample Input Data File#===============================================================================# A sampl e of i nput data f i l e#===============================================================================#TI TLE: FLOW PAST NACA0012 AI R FOI L ( M=0. 799 ALFA=2. 26 DGREE)#===============================================================================#I DI M: DI MENSI ON OF PROBLEM ( =2: 2D; =3: 3D)#I GRI D: GRI D DATA FORMAT (UNI C- UNS ONLY READ I TS MODEL DATA# BASE FI LES)

# =1: UNI C- UNS DATA BASE FI LES, UNFORMATTED, DI RECT ACCESS# ( Def aul t )# uni c. con, uni c. xyz, uni c. per ( f or per i odi c bc)#I AX: =1: PLAI N; =2: AXI SYMMETRI C#I CYC: TYPE OF CYCLI C BC ( NOT I N USE)#LCONG: GRI D CONNECTI ON ( NOT I N USE)#I DCASE: CASE I DENTI FI CATI ON#LDBUG: =0: NO OUTPUT OF DEBUG I NFORMATI ON; =1, 2: YES#- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -# IDIM IGRID IAX ICYC LCONG IDCASE LDBUG

2 1 1 0 0 2 0

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#===============================================================================#NSTEP: NUMBER OF TI ME STEP OR GLOBAL I TERATI ON#I TPNT: SOLUTI ON OUTPUT FREQUENCY# >0: UPDATA uni c. f l o EACH I TPNT STEPS ( al so TECPLOT out put f i l e)#

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#===============================================================================#XREF: REFRENCE LENGTH ( METERS)#VI SC: GAS PHASE REFERENCE VI SCOSI TY ( KG/ ( M*S)#AMC: REFRENCE MACH NUMBER ( TO I NDI CATE COMPRESSI BLE OR I NCOMPRESSI BLE)#- - - - - - - - - - - - - - - - - - - - - - - - -# XREF VISC AMC

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1.0000 1.84E-5 0.7990

#===============================================================================#SPECI FY EQUATI ONS TO BE SOLVED AND CG SOLVER ( =0: DEACTI VE, >=1: ACTI VE)#U, V, W: U, V, AND W VELOCI TY EQUATI ON ( 0 OR 1)#P: NUMBER OF PRESSURE CORRECTI ONS ( =0: NOCORRECTI ON)#TM: ENERGY EQUATI ON ( 0 OR 1)#DK: TURBULENT KI NETI C ENERGY EQUATI ON# =1: STANDARD K- E MODEL# =2: EXTENDED K- E MODEL# =3: TWO- LAYER MODEL ( NOT YET AVAI LABLE)# =4: LOW- RENOLDS MODEL ( NOT YET AVAI LABLE)# =5: ALGEBRA STRESS MODEL ( NOT YET AVAI LABLE)# =6: RENOLDS STRESS MODEL ( NOT YET AVAI LABLE)#DE: TURBULENT KI NETI C ENERGY DI SSI PATI ON EQUATI ON ( 0 OR 1)

#FM: MASS- FRACTI ON EQUATI ON ( 0 OR 1)#VF: VOF EQUATI ON ( 0 OR 1)#10: RESERVED FOR RADI ATI VE TRANSFER MODEL#11: RESERVED FOR VI BRATI ONAL ENERGY EQUATI ON#12: RESERVER FOR ELETRONE ENERGY EQUATI ON#I ERCG: CG MATRI X SOLVER CONTROL# = 1: ALL EQUATI ONS SOLVED BY Bi CGSTAB SOLVER# = 2: PRESSURE EQUATI ON SOLVED BY GMRES SOLVER, OTHERS BY Bi CGSTAB# = 3: PRESSURE EQUATI ON SOLVED BY MULTI - GRI D SOLVER, OTHERS BY Bi CGSTAB# =- 3: ALL EQUATI ONS SOLVED BY MULTI - GRI D SOLVER#- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -# SOLVE: U V W P TM DK DE FM VF 10 11 12 IERCG

- - - - - - - - - - - - - - - - - - - - -

1 1 0 1 1 1 1 0 0 0 0 0 2

#===============================================================================

##BOUNDARY CONDI TI ON ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++#NDBC: NUMBER OF BOUNDARI ES TO BE DEFI NED#I DBC: BOUNDARY I NDEX#I BTY: TYPE OF BOYNDARY CONDI TI ONS# =< 1: I NLET OR FREESTREAM; 2: OUTLET;# 3: SYMMETRY OR I NVI SCI D WALL; 4: WALL#- - - - - I NLET B. C. SPECI FI CATI ON# = 0: I NLET FI X EVERYTHI NG ( SUPERSONI C I NLET) ;# = 1: FOR SUBSONI C I NLET FI X TOTAL P AND T, EXTROPLATE P# FOR I NCOMPRESSI BLE FLO, EXTROPLATE P# =- 1: I NLET FI X MASS FLOW RATE, EXTROPLATE P# =- 2: I NLET FI X EVERYTHI NG EXCEPT P AND DENSI TY# =- 3: FOR FREESTREAM FAR- FI ELD FI X TOTAL P AND T, SOLVE FOR U, V, W#PRAT( or PEXI T) : SPECI FI ES FREESTREAM/ OUTLET PRESSURE CONDI TI ON OPTI ON ( i n ATM)

# = 0. 0: FOR OUTLET MASS CONSERVATI ON# =- 1. 0: FOR SUPERSONI C OUTLET BC# > 0. 0: FI X OUTLET STATI C OR FREESTREAM TOTAL PRESSURES ( i n ATM)#TRAT: FREESTREAM TOTAL TEMPERATURE CONDI TI ON ( i n K)#XPFI X: X- COORDI NATE FOR PRESSURE ANCHORI NG LOCATI ON ( i n gr i d uni t )# ( Act i ve when I BTY = 2 and PRAT > 0. 0)# OR = 9999. 0 FOR FI XED UNI FORM PRESSURE AT OUTLET#YPFI X: Y- COORDI NATE FOR PRESSURE ANCHORI NG LOCATI ON ( i n gr i d uni t )#ZPFI X: Z- COORDI NATE FOR PRESSURE ANCHORI NG LOCATI ON ( i n gr i d uni t )#TMEX: ENERGY EQUATI ON BOUNDARY CONDI TI ONS# =- 1. 0 FOR FI XED TEMPERATURE CONDI TI ON

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# = 0. 0 FOR ADI ABATI C CONDI TI ON ( Zer o Gr adi ent ) - - DOES NOT APPLY TO I NLET#I BTYRD: = 0: NON- REFLECTI VE SURFACE FOR RADI ATI ON# > 0: REFLECTI VE SURFACE FOR RADI ATI ON#EMI TB: BOUNDARY EMI SSI VI TY FOR RADI ATI ON ( 0. 0 TO 1. 0)#- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -#NDBC

3#IDBC IBTY PRAT TRAT XPFIX YPFIX ZPFIX TMEX IBTYRD EMITB

1 -1 0.0 300.0 0.0 0.0 0.0 -1.00 0 1.0

2 2 1.0 300.0 9999.0 0.0 0.0 -1.00 0 1.0

3 4 0.0 300.0 0.0 0.0 0.0 -1.00 1 1.0

#New Sample Input for the Conjugate Heat Transfer Model

#NBLOCK: number of sol i d- bl ock groups#Bl ock- f i l e- names ( bl k01_t h. f mt and bl k02_t h. f mt )#I BLOCK: sol i d bl ock i ndex#NTHER: number of dat a poi nt s i n t he t hermodynami cs data t abl e#THER_TMP: t emper at ure ( K) i n t he ther modynami cs dat a t abl e#THER_CON: t her mal conduct i vi t y f or t he t her modynami cs data tabl e#THER_DEN: mater i al densi t y f or t he t her modynami cs data tabl e#THER_CP: mater i al speci f i c heat f or t he t her modynami cs data t abl e#I DBC: boundar y i ndi ces

#TM_FAR: f ar - f i el d t emperat ur e ( K)#HT_COEF: f ar - f i el d heat t r ansf er coef f i ci ent#NPOWER: number of f uel assembl i es f or power di st r i but i on speci f i cat i ons#NSBLK: number of sol i d i nput l i nes i nvol ved i n t hi s assembl y#i ax_h = 1, t hen ( yt h_cnt, zt h_cnt) = ( Y, Z)#i ax_h = 2, t hen ( yt h_cnt, zt h_cnt) = ( Z, X)#i ax_h = 3, t hen ( yt h_cnt, zt h_cnt) = ( X, Y)#t her t m: t ot al power i nput ( i n wat t s) t o t he cur r ent sol i d bl ock#yth_cnt , zt h_cnt : coor di nat es as def i ned by t he i ax_h speci f i ed (i n gr i d uni t )#I SBLK( I , I =1, NSBLK) : l i st of sol i d pr oper t y i nput l i nes i n the conj ugat e heat#t r ansf er model sect i on.

$CONJ UGATE HEAT TRANSFER ==( Not e: Add # t o deact i vat e) ==#NBLOCK

2

# I BLOCK NTHERbl k01_t h. f mt1 3

# THER_TMP THER_CON THER_DEN THER_CP

Thermodynamics Table #1 300 1. 000E+02 1. 000E+03 1. 000E+03

1000 1. 100E+02 1. 000E+03 1. 000E+033000 1. 200E+02 1. 000E+03 1. 000E+03

# I BLOCK NTHERbl k02_t h. f mt

2 2# THER_TMP THER_CON THER_DEN THER_CP

Thermodynamics Table #2 300 1. 500E+02 1. 000E+03 1. 000E+03

4000 2. 000E+02 1. 000E+03 1. 000E+03# I DBC TM_FAR HT_COEF

1 0. 0 0. 0

2 0. 0 0. 03 0. 0 0. 0#$POWER DI STRI BUTI ON# NPOWER

1# NSBLK i ax_h t hert m yth_cnt zt h_cnt

NPOWER sets of input1 1 1. 0E05 0. 0 0. 0

# I SBLK( I , I =1, NSBLK)1

#

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#New Input for the Porosity Model

#Cur r ent l y, t he i nput f or mat f or t he por osi t y model i s kept si mi l ar t o i t s or i gi nal#f or mat t o avoi d conf l i ct wi t h t he power generat i on i nput met hod f or t he conj ugate#heat t r ansf er model . Ot her more general i nput f ormat can be desi gned l ater.

#Definitions:

#$POROSI TY t he por osi t y model t i t l e l i ne

#NNPORO number of por ous r egi ons#ntp- por o. f mt mul t i - zone PLOT3D mesh f i l e f or t he por ous r egi on#I DPORO por ous r egi on i ndex#DPORO por ous si ze ( i n mi cr on)#POROSI TY por osi t y ( voi d) of t he por ous r egi on#I PDI R por e di r ecti on. 0: f or Er gon equat i on; 1: f or pi pes i n x-di r ecti on#I POROH 1: f i xed t emper ature; 2: const ant energy i nput ; 3: NTP power i nput#POROTM f i xed temperature( I POROH=1) ; or to tal energy i nput ( I POROH=2 or 3)#ynt p_cnt, znt p_cnt power i nput axi s l ocat i on#nporo number of dat a poi nt i n t he t her modynami cs dat a t abl e#PORO_TMP t emper at ur e i n deg- K#PORO_CON t her mal conduct i vi t y#PORO_DEN mat er i al densi t y#PORO_CP t hermal capaci t y

$POROSITY MODEL INPUT ==(Note: Delete # to activate)========

#NNPORO

1

# IDPORO DPORO POROSITY IPDIR IPOROH POROTM yntp_cnt zntp_cnt nporo

ntp-poro.fmt

1 10000.0 0.50000 1 3 1.0e+05 0.0 0.0 3

# PORO_TMP PORO_CON PORO_DEN PORO_CP

300 1.000E+02 1.000E+03 1.200E+03

1000 1.500E+02 1.000E+03 1.200E+03

3000 1.800E+02 1.000E+03 1.200E+03

#

#Al so, f or sol vi ng t he t hermal conduct i on ef f ect of t he porosi t y model ( t wo-#t emperat ur e model ) , a new var i abl e, t m_por o(i ) , i s i nt r oduced t o save t he#t emper at ur e f i el d i n t he por ous mat er i al . Thi s var i abl e i s saved i n uni c_l oad. out#di r ect access f i l e. I n t he Tecpl ot out put , t hi s var i abl e i s cur r ent l y saved at t he

#TMv l ocat i on. To r est ar t , uni c_l oad. out shoul d be copi ed t o uni c_l oad. i ni t o r ead i n#t m_por o( i ) . Not e that , uni c_l oad. out and uni c_l oad. i ni r epl ace t he or i gi nal l oad-#bal anci ng dat a f i l e, uni c. l oad.

#===============================================================================##I NI TI AL FI ELD & BOUNDARY VALUE ++++++++++++++++++++++++++++++++++++++++++++++++#NZON: NUMBER OF ZONES#NDBC: NUMBER OF BOUNDARI ES TO BE DEFI NED#I SETBC: =0: BC VALUE USED ONLY AT I NI TI AL CONDI TI ON# =1: UPDATE BC VALUE AT RESTART#I ZON: ZONE I NDEX#UI N, VI N, WI N: I NI TI AL VELOCI TY COMPONENTS (M/ SEC)#PI N: I NI TI AL PRESSURE ( ATM)#TMI N: I NI TI AL TEMPERATURE ( K)

#DKI N: >0: I NI TI AL TURBULENCE KI NETI C ENERGY ( M/ SEC) **2; 0: TKE DI SSI PATI ON RATES; 0: BOUNDARY TURBULENCE KI NETI C ENERGY ( M/ SEC) **2; 0: TKE DI SSI PATI ON RATES;

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#NZON NDBC ISETBC

1 3 0

#IZON UIN VIN WIN PIN TMIN DKIN DEIN VOFIN

1 271.45 10.71 0.00 1.00 300.00 1.0000 -0.10 0.00

#IDBC UBC VBC WBC PBC TMBC DKBC DEBC VOFBC

1 271.45 10.71 0.00 1.00 300.00 1.0000 -0.10 0.00

2 0.00 0.00 0.00 1.00 300.00 1.0000 -0.10 0.003 0.00 0.00 0.00 1.00 300.00 1.0000 -0.10 0.00

## FOR SPECIES MASS FRACTION (IF NGAS>8, ADD MORE LINES)

#IZON FM01 FM02 FM03 FM04 FM05 FM06 FM07 FM08

1 0.0399 0.2234 0.00 0.00 0.7367 0.00 0.00 0.00

#IDBC FM01 FM02 FM03 FM04 FM05 FM06 FM07 FM08

1 0.0399 0.2234 0.00 0.00 0.7367 0.00 0.00 0.00

2 0.0399 0.2234 0.00 0.00 0.7367 0.00 0.00 0.00

3 0.0399 0.2234 0.00 0.00 0.7367 0.00 0.00 0.00

#

#SPECIES PROPERTIES ============================================================NGAS: NUMBER OF CHEMI CAL SPECI ESI CEC: 1: FOR CEC DATA; 2: FOR CEA DATA( I F $NGAS TI TLE I S READ I NSTEAD OF $NGAS_I CEC, I CEC = 2 I S ASSUMED)( THEN, FOLLOWED BY#SPECIES PROPERTIES

============================================================

SPECI ES DATA LI ST)

$NGAS_I CEC ! ( CEA f ormat i n t hi s case; I CEC = 1 f or CEC f ormat )6 2

H2O CODATA 1989. J RNBS 1987 v92 p35. TRC t uv- 25 10/ 88.2 g 8/ 89 H 2. 00O 1. 00 0. 00 0. 00 0. 00 0 18. 01528 - 241826. 000

200. 000 1000. 0007 - 2. 0 - 1. 0