Encyclopaedia Index

MULTI-BLOCK GRIDS AND FINE-GRID EMBEDDING

(by I.Poliakov and V.Semin)

Contents:

  1. Introduction to CCM/MBFGE.
  2. Mathematical Background of CCM/MBFGE method.
    2.1 Organization of the computational space.
    2.2 Description of the LINK scheme.
    2.3 MBFGE equations.
    2.4 Unstructured Linear equation solver.
    2.5 New Collocated (CCM-) method for CFD-problems.
    2.6 Alternative discretisation Schemes for convection.
  3. Activation and use of MBFGE/CCM method.
    3.1 MBFGE/CCM library.
    3.2 Grid generation.
    3.3 MBFGE/CCM activation.
    3.4 Activation of physical models and built-in source terms.
    3.5 Initial and boundary conditions.
    3.6 Use of alternative convection schemes.
    3.7 Activation and use of special options:
    3.7.1 Swirling flow simulation.
    3.7.2 Sliding grid simulation.
    3.7.3 Darcy flows.
    3.7.4 Shallow flow models.
    3.8 GROUND influence on CCM/MBFGE calculations.
    3.9 Use of IPSA with MBFGE/CCM.
    3.10 Use of self-adjustment of relaxation parameter.
    3.11 Automatic provision of friction at solid/liquid interface.
  4. CCM-switch in SATELLITE.
    4.1 New options of READCO and MBFLINK commands.
    4.2 How to convert existing Q1 or just created by SATELLITE menu to the Q1 for CCM method.
  5. Use of PHOTON to visualize results of the MBFGE calculations.
  6. References

1. Introduction to CCM/MBFGE.

CCM and MBFGE are alternative solvers of Navier-Stokes equations, which are available as optional in PHOENICS. The object library necessary to run CCM/MBFGE solvers resides in /phoenics/d_earth/ d_opt/d_mbfgem subdirectory.

CCM stands for Collocated Covariant Method. Essentially CCM is a segregated Navier-Stokes equation solver based on the collocated-grid arrangement, with covariant velocity projections as dependent variables.

The coupling of equations is achieved by a global iteration algorithm, which employs SIMPLE-like procedure [1] and Rhie-Chow-like interpolation [8]. As linear-equation solver CCM uses either LU-, or 2-step Jacobi pre-conditioned conjugate-residuals solver [3].

MBFGE stands for Multi-Blocking and Fine Grids Embedding. The MBFGE solver is based on the non-overlapping domain-decomposition method and provides the use of multi-block grids (MBFGE grids) in CFD simulations with PHOENICS.

The MBFGE grid for a problem is the grid which is built from a set of separate grids and linked together through common boundaries using link-scheme developed for MBFGE method. Separate grids in a MBFGE grid can be either the grids created for parts of complex computational domain, or fine grids embedded into the coarse ones, or any combination of these.

The MBFGE link-scheme has the following main features:

Computational grids for each of separate blocks are structured, however the global MBFGE grid is unstructured. To get a solution of linear equations on that unstructured grid, MBFGE method employs specially developed unstructured version of LU- or 2-step Jacobi preconditioned conjugate residuals solver.

As a result the speed of convergence is governed only by the overall number of cells and it is not affected by the number of linked domains or links set in the MBFGE grid.

The simulation of CFD problems with MBFGE method is based on the CCM algorithm, which was specially enhanced to treat interdomain links. The link treatment in CCM for MBFGE is developed in such a way, that it does not affect the global convergence.

Together with unstructured conjugate-residuals linear equation solver it provides for the convergence of a CFD problem being governed by exactly the same factors as for the one -domain CCM simulation (by the character of a problem, by overall number of computational cells, etc.), but not by the number of linked cells and the type of set links (one- to-one or one-to-many).

2. Mathematical Background of CCM/MBFGE method.

2.1 Organization of the computational space.

The CCM method is based on structured computational grids. It can employ all types of grids available in PHOENICS, namely:

It is supposed below that reader is well familiar with the organization of PHOENICS computational space.

The MBFGE method is based on the domain-decomposition approach to facilitate grid generation for application problems with a complex geometry. Essentially, user should subdivide the whole region into a set of rather simple domains, within each of which a separate computational grid is generated. The union of all domains (or grids) covers the whole region.

In the MBFGE method, subdomains are to be put together without overlapping; i.e. linked domains have only boundary surface (or surfaces) in common.

The domain-decomposition approach gives the following advantages and new features to PHOENICS simulations:

In the MBFGE method, all grids created for subdomains (local grids) are structured. Cells are topologically cartesian brick elements. All kinds of grids available in PHOENICS can be employed as local grids. Local grids are combined together to create global grid (or global computational space).

The MBFGE global computational space is it meanslogically unstructured, i.e. a neighbour in grid index space (say, cell J+1) may by far apart in physical space.

At present, the MBFGE method uses the (X,Y,Z) notation along with (I,J,K) notation to describe global computational space. This means, that all MBFGE grids are built as BFC grids in PHOENICS notation, irrespective of the actual topology of constituent grids.

The process of combining of local grids into global grid is here called 'stacking'. MBFGE stacking employs layers of dummy cells to separate local grids put into the global one (see picture). Later dummy cells are blocked DOMAIN 1 DOMAIN n to prevent their use in the computation. At present it is j+1 achieved by setting the volume j < porosity (VPOR) to zero. For user it means that VPOR should DOMAIN 2 DOMAIN m be stored if MBFGE method is used (i.e. Q1 should comprise  Dummy PIL command STORE(VPOR)). k k+1 < layers

For the problems without convection (for example, heat-conduction problems) MBFGE method enables to use arbitrary stacking of grids. However, for the CFD problems, MBFGE method permits only the stacking, which provides uniform directions of all local system of coordinates. It means that all local grids must have the same position of not only (X,Y,Z)-, but (I,J,K)-coordinates.

It is also necessary to distinguish MBFGE stacking described above and that currently provided by SATELLITE. The SATELLITE stacking is only a variant of these permited by MBFGE and is as follows:

User is free to develop either add-on to SATELLITE, or his own grid generator which will provide more compact stacking than that available in SATELLITE ver. 2.

2.2 Description of the LINK scheme.

The MBFGE link scheme consists of the following elements:

By default, any (I,J,K)-cell in a structured grid is linked to the six immediate neighbours: (I-1,J,K), ..., (I,J,K+1). In other words it has six links, one on each of six faces. (It is not the case for cells at domain boundaries, but in that case missing links are substituted by boundary conditions.)

In the MBFGE grid, a cell might have on one (or more) of its faces not only the links mentioned above, but the link to a face of some (L,M,N) cell, or to faces of a group of cells. These cells may be positioned in any place of global computational space. Below only that kind of links as referred to as the MBFGE link or just link.

                   An  Bn 
                     Dom A          Dom B        
                                                 
                     J         Ae  Bw   J        
                     I             I    
                   As  Bs 

Two types of links are distinguished in the MBFGE link-scheme:

At present, only natural links can be used for simulation of CFD problems with MBFGE method. Non-natural links may only be used for scalar problems without convection, for example, heat-conduction, potential flows, etc. .

The MBFGE link-scheme is based on the non-overlapping approach. It means, that a cell at the linked boundary of a domain can be linked only either to one cell (link 'one-to-one'), or to the whole number of cells from the other domain (link 'one-to-many'). There is no limit on the number of cells connected to one cell; it could be 1-to-5, 1-to-17, etc.. However, the type of connections should be uniform across the linked surfaces of domains.

The linked surfaces in the MBFGE method are defined by special PIL command MBLINK (see p.2.2). Actually, this command generates pair of LINK patches (one for each linked domains) which spans define the size of common surfaces of linked domains. Link patches are processed by MBFGE solver to fill special arrays which provide information on the presence of links for a cell and, if some cell face has a link, the adresses of first and last cells linked to it. User can get these information using functions and subroutines described in p. 2.8.

As it was already mentioned, the type of cell-to-cell connection has to be uniform across the link patch. If it is desirable for a user to introduced mixed connections across the same common surface, it can be done, at present, only by setting several link patches.

2.3 MBFGE equations.

In general, MBFGE solver treats the conservation equations in exactly the same way as PHOENICS solver (see [6]), i.e. the finite- difference approximation of balance equations is transformed into finite-volume equations of the form

Ap Fp = An Fn + As Fs + Ae Fe + Aw Fw + Ah Fh + Al Fl + At Ft + S

where Fi's are values of a solved variable in neighboring cells; Ai's are coefficients and S is appropriate source term. The form of Ai's is exactly the same as in PHOENICS:

          Dens*Vel*Area       Exch_Co*Area/Dist     Dens*Vol/Dt
           ^Convection           ^Diffusion         ^Transience

Moreover, the values of coefficients for non-linked cell faces are calculated by the same subroutines as for PHOENICS solver. They stored in the same arrays and can be accessed by standard methods.

If a cell P (I,J,K) has a link at, say, SOUTH-face to some other cell (L,M,N), the coefficient As is recalculated by MBFGE solver taking into account the actual inter-cell connection. If the link is not one-to-one, but one to a number of cells, the term A*F takes the form of Ai*Fi, where the summation covers all cells linked to the face. The sum and Ai's values are calculated by MBFGE solver by balancing the fluxes passing through the linked faces.

Due to the fact, that position of linked cell or cells in the global computational space could be arbitrary, all variables in MBFGE method are solved for in the whole-field manner.

On orthogonal grids source term for scalar variables is the same as produced by PHOENICS solver after processing source patches. For collocated velocity projections it comprises additional source terms which are calculated by MBFGE solver. These are the pressure gradient and terms associated with the curvilinearity treatment.

On nonorthogonal grids MBFGE solver recalculates Ai's, as well as adds appropriate terms to the source term.

As for PHOENICS solver, the balance equation is cast into the correction form before the solution.

2.4 Unstructured Linear equation solver.

Finite-volume-equations, written for each cell in the global computational space, form a set of simultaneous algebraic equations. For one domain problems a matrix of that system has some regular structure (usually it has only a number of non-zero diagonals). This is not the case for a general multi-block problem.

A variety of iterative methods is available to solve systems of linear algebraic equations arising for one domain problems (SOR, Stone, MSIP, etc.). Most of them are based on the use of the existent regular structure of the system matrix, thus these methods can not be applied directly to solve the sytem of linear equations for a general MBFGE case. The problem could be tackled by the introduction of additional iterative processes to treat the wrongly positioned elements explicitly. However, this procedure may significantly reduce the global convergence speed.

The alternative way to solve the system of linear equations for a multi-domain problem consists in the use of a special solver, which is able to treat linear equations sytems with unstructured matrices immediately. The MBFGE method employs that kind of a solver. The unstructured linear equations solver is developed on the basis of preconditioned conjugate residuals algorithm [3].

By default, the LU-preconditioning is used. User can change it on the 2-step Jacobi preconditioning by setting LSG5=T. The LU- peconditioning usually provides faster convergence, than 2-step Jacobi, however it requires more additional memory. For this reason 2-step Jacobi preconditioning can be recommended when there is lack of computer memory available.

2.5 New Collocated (CCM-) method for CFD-problems.

For the CFD problems MBFGE solver employs CCM solver. CCM is a segregated Navier-Stokes equation solver based on the collocated grid arrangement and covariant velocity projections as dependent variables. The coupling of equations is achieved by a global iteration algorithm, which employs SIMPLE-like procedure [1] and Rhie-Chow-like interpolation [8].

This entails that cell-centered velocity projections Uc, Vc and Wc are solved-for variables. They are aligned with the grid lines. The grid-line direction in the center of a cell is defined by line connecting the centers of appropriate opposite faces. In CCM/MBFGE notation, the first-phase velocities are UC1, VC1 and WC1, thus Q1 should comprise SOLVE(UC1,VC1,WC1) command.

The face-centered velocity projections have exactly the same meaning, as in the staggered PHOENICS solver; they are convection-flux velocities. However, in CCM they are not solved for, but calculated using Rhie-Chow like interpolation procedure. The face-centered velocity projections are stored in U1, V1 and W1 respectively.

Cartesian velocity components UCRT, VCRT and WCRT calculated for curvilinear grids are attributed to the cell-centers.

2.6 Alternative discretisation Schemes for convection.

The general transport equations solved by PHOENICS includes transient, convection, diffusion and source terms. To derive finite volume equations, certain numerical schemes are used for discretisation of these terms (see [6] or appropriate entries in POLIS). By default, the numerical schemes used in MBFGE/CCM for the approximation of the diffusion and convection contributions are exactly the same as in standard staggered PHOENICS solver.

Thus the diffusion-convection interaction is accounted for by the "hybrid-interpolation" scheme (see DIFCUT entry to PHENC, or [6]), which resolves into the classical upwind differencing scheme (UDS- scheme) for convection dominant flows.

The UDS scheme is very robust; but in the CFD simulation of the convection-dominated physical problems it usually suffers from the severe numerical diffusion, which can prevent from receiving the accurate solution, especially for the problems with rather strong gradients of variables. The collocated MBFGE/CCM solver includes a set of optional alternative convection schemes to alleviate the influence of the numerical diffusion by introducing the treatment of the convection terms with a high accuracy. The schemes available at present are briefly described below.

The straightforward approach to increase the approximation order of the UDS-scheme (for example, to second order in SOU-scheme [10], or third order in QUICK-scheme [11]) resulted in algorithms which tend to give rise to random oscillations of the solution in the regions of strong gradients. This fact is in the agreement with the conclusion of the Godunov's (1959) theorem [12] concerning the inevitable production of oscillations by linear convective scheme of accuracy higher than first order. The MBFGE/CCM solver includes as an option QUICK-scheme.

In recent years, a set of the non-linear algorithms has been proposed by various authors to suppress the parasitic oscillations. All these methods are based on the analysis of the local solution behavior and can be described as a sequel to the Godunov's first- order Lagrangean scheme [12].

The first robust monotonic second-order convective scheme was the monotonic piecewise linear scheme (MINMOD-scheme) of Van Leer [13], where the slab averaging of the solved variable is used instead of mesh point values as in Gogunov's scheme and monotonicity is enforced by suitably adjusting the second-order terms. Better results in resolving the sharp gradient regions were achieved by the monotonic second-order upwind algorithm (SUPER-BB-scheme) of Roe [14], which has been originally developed for the prediction of the inviscid compressible flows. Both schemes belong to the class of TVD-schemes (TVD stands for Total Variation Diminishing) and are included into the CCM/MBFGE convection schemes' set.

Among the recently proposed procedures to express and develop TVD- schemes it is worth mentioning the notion of NVD (Normalized Variable Diagrams) of Leonard [15]. The NVD-representation can significantly simplify the test on whether some convective scheme is TVD or not. Using the NVD-technique Leonard developed the new SHARP-scheme (Simple High-Accuracy Resolution Program), which is a monotonic version of his QUICK-scheme. Similar to SHARP is the simpler SMART-scheme of Gaskell and Lau [16], which is based on piecewise characteristics. The MBFGE/CCM solver includes as an option SMART-scheme.

The alternative convection schemes are implemented in the CCM/MBFGE solver using the deferred correction method. It means that all the additional terms, arising in the general transport equation after it is discretized using some higher order convection scheme instead of default UDS-scheme, are calculated on the solution from the previous iteration and added to the source term.

All the convection schemes available in the CCM/MBFGE solver (i.e. QUICK, MINMOD, SUPER-BB and SMART schemes) can be applied to any solved variable (including collocated velocities), provided that the convective term is active for it. Details of their activation can be found in p. 2.6.

3. Activation and use of MBFGE/CCM method.

3.1 MBFGE/CCM library.

To help and simplify the familiarisation with and learning of the CCM/MBFGE method, SATELLITE library includes CCM/MBFGE library. This library can be accessed either through submenu

'Library' of SATELLITE menu, or by typing SEELIB(F) on the SATELLITE command mode prompt. The library file resides in D_SATELL/D_OPT/D_MBFGE subdirectory of main PHOENICS directory.

Cases included into CCM/MBFGE library can be loaded either through submenu 'Library' of SATELLITE menu, or by entering LOAD(Fnnn), where 'nnn' is the case number, on the SATELLITE command mode prompt.

The CCM/MBFGE library consists of two parts: CCM library and MBFGE library. First part includes cases set on one domain grids to run with CCM solver. Most of the cases included in it are set to be simulated either with CCM solver, or PHOENICS staggered solver (to chose the latter, user should set PIL variable LCCM defined in Q1 to FALSE). The MBFGE-library includes cases set on multiblock grids (including grids with fine grid/grids embedding). These cases can be run with MBFGE solver. It is recommended to the user to study them not only as examples of MBFGE solver activation, but as the examples of the MBFGE grid generation.

The CCM/MBFGE library includes the 'macro' case F150, which contains PIL commands to activate solution for collocated covariant velocity projections. This case can be loaded in Q1, after setting NX, NY and/or NZ greater than 1 as appropriate, by the following PIL commands:

NOWIPE= T
LOAD(F150)

3.2 Grid generation.

Both CCM and staggered PHOENICS solvers are based on structured computational grids. They can employ grids of three distinct kinds, namely:

The details of the grid specification, as well as grid generation using SATELLITE can be found under appropriate entries of PHOENICS manuals (TR100, TR200 and TR219) and encyclopedia. In addition to SATELLITE, user may also employ any other available to him grid generator to build BFC grids in XYZ-format.

MBFGE solver is based on the computational grids which are built as a union of structured grids created for each of the blocks using the standard grid-generation technique. Grids are connected through common surfaces.
At present, overlapping is not permitted. Cells' faces at the interface can be connected either one-to-one, or one- -to-many ('many' must be whole number of cells).

NOTE !, that at the surfaces defined as a LINK (details see below) all cells has to be connected in the same way, i.e. either one-to -one, or one-to-two, or one-to-three, etc.. If it is desirable to have regions with different connections along the same common surface, user should introduce several LINKs instead of one.

Note the difference between PHOENICS specification of cartesian or cylindrical polar grids and that of curvilinear grids, which is important for MBFGE method. The first two kinds of grids are defined using fraction arrays; there is no special arrays to store coordinate of cells' vertices (X,Y,Z arrays). On the contrary, BFC grids are defined using X, Y, Z -arrays, which provides the way to store grid information for all domains of multiblock grid. Thus:

At present, the generation of the MBFGE grid can be schematically represented as three successive steps:

Let us consider these steps in details, implying that SATELLITE will be used for the second and third steps. It is possible for the user to create and use alternative grid generator and/or GROUND coding for SATELLITE, which will incorporate all steps. In the latter case, it is desirable to obtain additional information, which is available from CHAM development team.

The SATELLITE treats grids for the subdomains independently while stacking them into one MBFGE grid, thus all subgrids have to be present in the working directory as separate XYZ-files. XYZ-files should be named as

NAME1, NAME2, ..., NAMEn

where NAME (string of up to four letters) is common name for all subgrids of a multiblock grid; n is the total number of blocks. The numbering order of subgrids is not important.

All NAMEi files has to be created at the first step or copied from a library of previously created grids. If SATELLITE grid generator is used to build subgrids the steps are as follows:

Note, that all subgrids can be created in the single Q1. It can be achieved by dumping each created subgrid to disk by DUMPC(NAMEi) PIL-command. All MBFGE examples in the CCM/MBFGE library had been created using that method.

In case of an alternative grid generator user should follow its instructions to create subgrids and write them to disk in PHOENICS XYZ format.

Once all subgrids had been created, it is possible to proceed to the second step. As a result of this step, SATELLITE will stack separate subgrids together and create single MBFGE computational grid. It is achieved by the following PIL-commands:

BFC= T; NUMBLK= n; READCO(NAME+)

READCO() command will read in NUMBLK XYZ-files (with names NAME1, ..., NAMEn) and put them them into single grid file using rules described in pp. 2.1 and 2.2.

By default subgrids are stacked by SATELLITE either along Z-axis, or along X-axis for cases set in X-Y plane. It is possible to change the stacking direction by specifying the new one in the argument of READCO:

READCO(NAME+X), or READCO(NAME+Y), or READCO(NAME+Z)

Note, that for 2D cases the alternative direction must be present.

The third step consists in the definition of necessary links, which is achieved by the MBLINK commands. The formats of MBLINK command are as follows:

MBLINK(m,SOUTH,n,NORTH) or MBLINK(j,IN,k)

First command defines LINK between domain 'm' at its SOUTH-side and domain 'n' at its NORTH-side; while the second defines domain 'j' as embedded into domain 'k'.

After processing READCO(NAME+) and MBLINK() commands, SATELLITE creates PATCHes with MBD... and MBL... names accordingly (see pp. 2.1 and 2.2). User can check positions of defined PATCHes by investigating contents of RESULT file after the run of EAREXE (to initialize Q1 contents print-out into RESULT file user should put ECHO=T in it). It can be recommended to check positions of LINK PATCHes introduced by SATELLITE before proceeding to the setting and simulation of the problem itself.

To summarize the MBFGE grid generation consider possible scheme of Q1 which incorporates three steps described above:

All examples of Q1 in MBFGE library are built according to that scheme.

3.3 MBFGE/CCM activation.

The PIL commands used for the CCM/MBFGE activation may be divided into four groups. First group comprises settings, which has to be used to activate CCM/MBFGE solver and are the same (with minor variations) for all MBFGE or CCM cases.

The second group includes setttings which are used to define grid geometry type.The third group consists of switches to activate special techniques available in CCM/MBFGE (high order convection schemes, the NX=1 simulation of swirling flows, etc.). The first and second groups are described in this paragraph. The third type settings are detailed in pp. 3.6 and 3.7.

All other PIL-commands which are usually used in Q1 and are not specific to CCM/MBFGE (for example, initialization of variables, activation of various physical models, etc.) may be attributed to the fourth group. Their description can be found under appropriate entries of PHENC. Some notes on their use for MBFGE can be found in pp. 3.4 and 3.5.

PIL commands to activate CCM/MBFGE solver are as follows:

3.4 Activation of physical models and built-in source terms.

Most of physical models, built-in or GREX-coded sources for scalar variables and models to introduce thermo-physical properties of media available for PHOENICS solver are also available for CCM/ MBFGE solver. User should refer to the approprite encyclopedia entry to find the description of PIL commands necessary to activate the model under question. The list of available models includes:

At present CHEMKIN, GENTRA, stresses in solids model and radiation model based on the view-factors calculation are not available for use with CCM/MBFGE.

There are certain differences in the activation of the following physical models:

More details of the coriolis and centrifugal, as well as buoyancy forces activation can be found in MBFGRN.FOR file, which resides in /phoenics/d_earth/d_opt/d_mbfge subdirectory.

In general, if some physical model (available as part of PHOENICS installation or coded by user for PHOENICS solver) is implemented as an algorithm user should consult CHAM development team to check for its applicability to CCM/MBFGE.

There are the following rules for the models based on additional source terms to PHOENICS equations, which can be used to check on whether or not some model in question can be applied without modifications to CCM or MBFGE:

3.5 Initial and boundary conditions.

In CCM/MBFGE solver initial and boundary conditions, as well as additional sources, are introduced in exactly the same manner, as these for PHOENICS solver. Thus, user should specify appropriate PATCHes and set COVAL() statements for solved variable. However, there are certain differences described below:

For MBFGE grids user may use MPATCH() command instead of standard PATCH(). It has the following format:

MPATCH(NB,NAME,TYPE,IXF,IXL,IYF,IYL,IZF,IZL,ITF,ITL)

Here NB is the number of the domain at which the patch NAME is set. Other arguments have exactly the same meaning as for PATCH(). The difference is in the fact that IXF,IXL,...,IZL should be specified as (IX,IY,IZ) local for NB-domain. SATELLITE will process MPATCH and transform IXF,IXL,...,IZL into global (IX,IY,IZ) according to the stacking used by it. NOTE, in the RESULT file all MPATCHes are printed out as standard PATCHes with global (IX,IY,IZ).

3.6 Use of alternative convection schemes.

At present, the following high-order numerical schemes are included in CCM/MBFGE solver as optional (instead of default UDS-scheme) for the approximation of convection terms:

Details of the mathematical formulation of these schemes can be found either in the appropriate references, or in the entry SCHeMe of PHOENICS encyclopedia.

Current implementation has the following main features:

To activate the alternative convection treatment in the MBFGE/CCM solver user should set LSG7=T in Q1. Than it is necessary to provide the information on which scheme must be applied to what solved variable. It should be done by introducing into Q1 the following lines:


SCHMBEGIN
VARNAM VAR1 SCHEME MINMOD
VARNAM VAR2 SCHEME SUPERB
VARNAM VAR3 SCHEME SMART
VARNAM VAR4 SCHEME QUICK
SCHMEND

Here SCHMBEGIN and SCHMEND are keyword-brackets; VARNAM and SCHEME fixed keywords; VAR1, ..., VAR4 are the names of solved variables, and MINMOD, SUPERB, SMART, or QUICK are keywords to designate the appropriate numerical algorithm, which will be applied to the variable, provided that convection term for it is active.

It is necessary to note the following:

User is advised to study various examples of schemes' activations and use, which can be found in MBFGE/CCM library.

3.7 Activation and use of special options:

3.7.1 Swirling flow simulation.

The CCM/MBFGE module provides special option for the simulation of laminar or turbulent swirling flows taking place in axisymmetric geometries.

The problem should be defined and set as 2D problem, using Z, as axis of symmetry, Y, as radial direction and X, as circumferential direction (NX=1). Along with VC1 and WC1 velocity projections, which are normally solved for in the cases set in Y-Z plane, user should activate solution for UC1. Set no-slip or other suitable boundary conditions for UC1.

To activate treatment of UC1 as swirling velocity component user should specify in Q1 file LSG6= T. Other CCM/MBFGE variables (for example, LSG4 to activate the traetment of nonorthogonality, etc.) should be set according to the problem under consideration.

Examples of cases set to simulate swirling flows using described option can be found in CCM/MBFGE library: cases 105 and 110 for CCM; cases 208 and 215 for MBFGE.

NOTE !, that swirling flow problems can be simulated as 3D problem (NX>1), without using LSG6 variable. However, if the flow is uniform in circumferential direction, the speed of convergence is very low. For these reason 3D simulation of swirling flows can be recommended only if there are flow variations in circumferential direction.

3.7.2 Sliding grid simulation.

The MBFGE method provides the special type of a link to simulate the flows which take place in domains one part of which is moving in respect to the other. That type of flows can be found, for example, in stirring reactors, turbines, etc..

The fact that one domain is moving in respect to the other is accounted for by the SLIDING link and special patch, which covers the moving domain. Sliding link is introduced by changing the standard part of the link patch, i.e. MBL, to MBS. For example, the following link between domain M and N is sliding:

MPATCH(m,MBSm.n,NORTH,1,NXm,NYm,NYm,1,NZm,1,LSTEP)
MPATCH(n,MBSn.m,SOUTH,1,NXn, 1, 1,1,NZn,1,LSTEP)

At present, only one sliding link per multi-block grid is permitted by MBFGE solver. Moreover, the sliding link has to be introduced between the NORTH face of one domain and SOUTH face of the other domain. It means, for example, that the geometry of the multi-block grid to represent the rotation of one domain in respect to the other should be build using X as circumferential coordinate, Y as readial direction and Z as axis of rotation (see MBFGE library cases 218, 219 and 220).

There two patches with fixed names 'SLIDRT' and 'SLIDMV' to mark the moving domain (see cases 218, 219, 220 and 221):

MPATCH(n,SLIDRTn,CELL,1,NXn,1,NYn,1,NZn,1,LSTEP)

MPATCH(n,SLIDMVn,CELL,1,NXn,1,NYn,1,NZn,1,LSTEP)

Both patches should be set to cover the whole extent of the moving domain. Patch 'SLIDRT' marks domain as rotating domain. Values of angular velocity and position of the axis of rotation are set through PIL variables ANGVEL, ROTAXA, ROTAYA, ROTAZA, ROTAXB, ROTAYB and ROTAZB (see ROTA entry to PHENC and library cases). The angular velocity vector is directed along Z axis; angular velocity is positive when rotation of cartesian X-axis to Y-axis is counter clock-wise, as seen from the end of the vector.

Patch 'SLIDMV' marks domain as domain which moves along the surface of other domain with transitional velocity U. Its value is passed to MBFGE solver through PIL variable RSG2. Positive U value means that the domain is moving in the positive direction of X-axis.

In transitional cases and rotational cases set for only a sector of axisymmetric geometry user should introduce XCYCLE condition. It is achieved by using READCO(Y+) command and choosing the same number of cells across the sliding link (see library cases).

3.7.3 Darcy flows.

Current release of CCM/MBFGE method includes special treatment to provide for Darcy flows simulation. There is no special switch to turn it on. The treatment is automatically activated once user set patches on UC1, VC1 or WC1 to introduce flow resistance.

Case 108 in CCM/MBFGE library gives an example of the Darcy flow simulation

3.7.4 Shallow flow models.

The CCM/MBFGE module provides two special models for the simulation of shallow flows. The 3D flow can be defined as shallow, if its geometrical size in one dimension (or depth) is much smaller than the sizes in two other dimensions. The water flows in rivers or sea bays are examples of shallow flows. In general, that kind of flows can be simulated using full 3D Navier-Stokes equations by CCM/MBFGE solver. However, the fact that the flow depth is much smaller than the characteristic width and length, can be used to simplify the Navier-Stokes equations and provide more robust algorithm.

NOTE !, that at present, the coordinate direction adopted as 'shallow' in CCM/ MBFGE module is that along Z-axis. It means that only the cases set using Z to define flow depth can be treated by CCM/MBFGE shallow flow option.

There are two logical parameters to activate shallow water models available in the CCM/MBFGE module. The parameters and the models are described below:

NOTE !, that 2D shallow flow model is implemented for CCM problems only (i.e. for one domain simulations) and can not be used for multiblock grids. The 3D shallow flow model (LSG10=T) can be used as for CCM, as for MBFGE cases.

Examples of the use of the 2D and 3D shallow flow models for a 3D flow simulation can be found in the CCM/MBFGE library.

3.8 GROUND influence on CCM/MBFGE calculations.

The CCM/MBFGE module can be influenced by user in practically the same manner as all others PHOENICS modules. Thus the presentation of this paragraph is based on the presumption that the reader is well familiar with main rules of FORTRAN programming for PHOENICS. If it is not the case, user should start reading from appropriate paragraphs of PHOENICS lectures, manuals or encyclopedia entries.

Two kind of FORTRAN module are accessible to the user: the MAIN program of EARTH, and GROUND/GREX subroutines. There are only two parameters in the MAIN program, which are specific for dimensioning the arrays used in MBFGE module:

By default, NDM = 10 and NDMB = 6. The user may increase/decrease these values and rebuild EARTH in the standard way.

All the ways to influence EARTH through FORTRAN coding introduced into GROUND/GREX subroutines are also available to the user of CCM/ MBFGE method, i.e. user may modify solved equations by adding new source terms or modifying coefficients; introduce necessary thermo- dynamic properties of media; set complex boundary conditions; and etc. The general rules of the use of GROUND groups; of retrieving/ placing information from/to F-array; and of the work with PHOENICS arrays' indices are exactly the same as in standard EARTH. However there are several remarks worth mentioning:

There is a set of special subroutines to provide the user with the tools for the treatment of links in GROUND coding. It includes the following subroutines and functions:

LDMN(IX,IY,IZ) - returns TRUE if (IX,IY,IZ)-cell belongs to one of the domains. It can be also used for CCM, than it returns TRUE if 1<=IX=<NX, 1<="IY=<NY," and 1<=IZ=<NZ.
LLINK0(IJK) - returns TRUE if IJK-cell has at least one link. IJK is a whole-field index of (IX,IY,IZ)-cell, i.e. IJK = IY + (IX-1)*NY + (IZ-1)*NX*NY.
LLINK(IDIR,IJK) - returns TRUE if IJK-cell has a link in the IDIR direction. IDIR has the same meaning as for PHOENICS solver, namely IDIR=1 corresponds to NORTH-direction; 2 - SOUTH; 3 - EAST; 4 - WEST; 5 - HIGH; and 6 - LOW.
LINKD(IDIR,IJK,NF,NL) - returns the direction of the linked face for the cells (from NF to NL) linked to the IDIR-face of a current IJK-cell. NOTE !, call to LINKD() must be preceded by the call to GETFLI(), moreover the signs of NF, NL are important.
SSCLNK(IDIR,IJK,ISC0) - returns an average value of the scalar variable for the cells linked to the IDIR-face of a current IJK-cell. Here ISC0 is absolute zero-location index of the variable; for slab- wise arrays it is zero-location index of the first slab.

Sign of IJK defines type of the variable: if IJK>0 then ISC0-variable is slab-wise, else it is whole-field. There are two types of averaging: if ISC0>0 then SSCLNK() returns volume average; else it returns an arithmetic average.

GETFLI(IDIR,IJK,NF,NL) - returns first NF and last NL indices of cells linked to the IDIR-face of the current IJK-cell (NF, NL, IJK are whole-field indices). NOTE !, that GETFLI() can be called only if LLINK(IDIR,IJK) is true. NOTE !, that NF, NL could be negative for 'unnatural' links.
GETIJK(IJK,NX,NY,IX,IY,IZ) - returns IX, IY and IZ for a whole- field index IJK. NOTE !, IJK must be positive.

There is also set of auxiliarily subroutines, which can be used in GROUND coding for CCM/MBFGE module. These subroutines provide check for the presence of blocked cells in the computational region and they can be subdivided into two groups: the subroutines to carry out check for a current cell, and that to check neighboring cells.

Current cell is treated as blocked if the value of VPOR<=0 or value of PRPS>=SOLPRP. The cell neighboring to the current cell is treated as blocked if it has either VPOR<=0, or PRPS>=SOLPRP, or if the value of face-porosiry array in the appropriate direction is zero. All subroutines are pared; the other subroutine in a pair does not check for PRPS value if the variable under consideration is a temperature (TEM1 or TEM2). The latter subroutine is for the conjugate heat transfer problems.

The set of auxiliary logical functions is as follows (note, that functions with DD in the name treat temperature separately):

LSLDC(IJKS), or LSLDD(INDVAR,IJKS) - returns true if the current (IX,IY,IZ)-cell is blocked (IJKS is slab-wise index of the cell, i.e. IJKS = IY + (IX-1)*NY + (IZ-1)*NFM). INDVAR is block-location index.
LSLDCP(IJKS,NDFPOR,IPLUS), or LSLDDP(INDVAR,IJK,NDFPOR,IPLUS) - returns true if the cell neighboring to the current cell in 'plus' direction is blocked.
LSLDCM(IJKS,NDFPOR,IPLUS), or LSLDDM(INDVAR,IJKS,NDFPOR,IPLUS) - returns true if the cell neighboring to the current cell in 'minus' direction is blocked.

Here NDFPOR is block-location index of the face porosity array for a treated direction; i.e. it may be either NPOR, or EPOR, or HPOR. Integers NPOR, EPOR and HPOR can be retrieved from /IDATA/ common block. IPLUS is the appropriate shift, i.e. IPLUS=1 for NORTH-SOUTH direction; IPLUS=NY for EAST-WEST; and IPLU=NFM for HIGH-LOW. For example, call to LSLDCP(IJKS,EPOR,NY) will check whether the cell in EAST-direction is blocked or not.

3.9 Use of IPSA with MBFGE/CCM.

The CCM/MBFGE module now enables the use IPSA algorithm, including the shadow phase option, to solve the system of algebraic equations for two-phase flows. IPSA stands for Inter-Phase Slip Algorithm. Detailed description of IPSA; as well as the way to activate it in Q1 can be found in [17, 18].

The IPSA realization in CCM/MBFGE solver differs from that in default PHOENICS solver by the follows:

Present version of CCM/MBFGE realization of IPSA (dated 1.12.95) has the following restrictions:

All advanced two-phase models will be available in the next release of CCM/MBFGE solver.

There is no special CCM library to examplify the use of CCM solver for two-phase flows. All cases from the PHOENICS main two-phase library can be easily run with CCM solver using CCM switch (see chapter 3) to introduce collocated velocity components.

3.10 Use of self-adjustment of relaxation parameter.

The CCM/MBFGE module provides the optional self-adjustment of relaxation parameters for solved variables. The procedure makes use of the false time step relaxation method and is based on the estimation of the characteristic change times for all solved variables at each iteration step. The minimum time is then chosen as the false time step for all variables; this is applied at the next iteration. Note, that the algorithm assumes that pressure is always solved using linear under-relaxation.

To activate this procedure the user should include into Q1 prepared for the CCM/MBFGE solver the line LSG11=T. The self-adjustment will affect all variables which are solved using a false time step relaxation, i.e. variables for which PIL command RELAX has the format RELAX(phi,FALSDT,Dt).

It should be noted, that by default SATELLITE activates false time step relaxation for all variables except pressure (P1) and volume fractions (R1,R2 and RS), for which it introduces the linear under-relaxation. Thus the user should include necessary commands RELAX(volume fraction,FALSDT,Dt) into Q1 in order to make use of the self-adjustment algorithm for them.

3.11 Automatic provision of friction at solid/liquid interface.

For laminar flows CCM/MBFGE module automatically introduces the friction at the surface of solids defined using PRPS array (see PROPS entry to PHENC). The friction is introduced at the surface of all materials with indeces grater or equal to SOLPRP, except the totally non-participating material VACPRP.

For turbulent flows it is possible to activate the EGWF option of PHOENICS (Earth Generated Wall Functions, see EGWF entry to PHOENC); which also works for CCM/MBFGE solver. At present only two types of wall functions are availabe: WALLCO=GRND1 and WALLCO= GRND2.

NOTE, that laminar or turbulent friction is not introduced if a WALL-type patch is found at the phase interface. It is assumed that appropriate friction is introduced by this patch.

4. CCM-switch in SATELLITE.

CCM-switch is a unifying name for a group of facilities available in SATELLITE (version 2.1.4 and higher) to simplify the use of CCM/MBFGE solver. This can help the user in two major aspects:

CCM-switch facilities are optional, thus the user can create MBFGE grid from separate subgrids and set problem in Q1 by himself using commands and settings described above (Chapter 2). However, CCM- switch provides the capability to use the existing SATELLITE menu system to create Q1; and then convert it into Q1 for the CCM/MBFGE solver.

4.1 New options of READCO and MBFLINK commands.

As it is mentioned in chapter 3.2, the generation of a MBFGE grid can be schematically represented as three successive steps:

Starting from SATELLITE version 2.1.4 and higher the last two steps may be carried out by the single PIL-command READCO, instead of a group of READCO and appropriate MBLINK commands. This option is activated by adding 'L' as the last character of the READCO argument:

BFC= T; NUMBLK= n; READCO(NAME+L)

The call to READCO(NAME+L) causes it to carry out the two stage work. The result of the first stage is analogous to that, which may be achieved by READCO(NAME+) in the standard format; e.g. it will read in 'n' XYZ-files (NAME1, ... , NAMEn) and stack them into single computational space (see pp. 3.2).

At the second stage READCO scans all domains to find their relative positions in XYZ-space and define possible LINKs between them. The LINKs could be as of the multi-blocking type, as well as of the fine grid-embedding type. READCO introduces LINKs on the basis of the presence of a physical contact between subdomains. For a fine grid embedding it means that one subdomain lies inside the other. While in the multi-blocking case, external surfaces of subdomain/s or their parts are treated as linked if the distance between them is less than 1% of the minimum size of the adjacent cells.

As a result of the call to READCO(NAME+L), SATELLITE will create a set of PATCHes with MBD... and MBL.. names necessary to define MBFGE grid (see pp. 2.1 and 2.2). This means that there is no need in later calls to MBLINK in Q1.

The present limitations of READCO(NAME+L) command are as follows:

NOTE, that the maximum number of scanned subdomains can be easily increased. However, the time necessary to scan and detect LINKs for a larger number of subdomains becomes significant. Due to this it should be recommended to employ the standard way of LINKs setting (see p.3.2), i.e. use a group of READCO(NAME+) and appropriate MBLINK() commands. If, meanwhile, the user wants to increase the number of scanned subdomains; he/she should contact CHAM for the modified SATELLITE.

The MBLINK() command included in the SATELLITE version 2.1.4 and higher comprises significantly improved algorithm of the search for linked surfaces between subdomains. As a result, MBLINK is capable of correct setting of all natural LINKs. The maximum number of LINKs between each two of subdomains is also 30.

4.2 How to convert existing Q1 or just created by SATELLITE menu to the Q1 for CCM method.

Paragraph 3.3 describes the PIL commands and variables necessary to activate CCM/MBFGE solver, once the MBFGE grid is created. The user may still introduce them into Q1 to set problem for CCM/MBFGE method. However, with the SATELLITE version 2.1.4 and higher he/she may also use the alternative way:

This will cause SATELLITE to convert the set case into the case for CCM/MBFGE solver; i.e. SATELLITE will activate solution for the appropriate collocated velocity components, including that for second phase; convert initial and boundary conditions set for U1, V1,... into these for UC1, VC1,...; activates logicals LSG3, LSG4 and LSG6 (see chapter 2), if necessary.

SATELLITE does not copy introduced changes into the processed Q1. All the additional information is passed to EARTH through the EARDAT file. To check the set case, the user should use ECHO= T in Q1 and study the RESULT file produced after the EARTH run.

At present, the CCM-switch option of SATELLITE has the following restrictions:

The user can try out CCM-switch option by loading cases from the PHOENICS core library or suitable optional library and converting them into these for CCM/MBFGE solver with CCM=T command.

5. Use of PHOTON to visualize results of the MBFGE calculations.

Several new commands had been introduced into PHOTON to facilitate the visualization of the MBFGE simulations' results. These commands are analogous to these for one domain; and differ only in the name and first argument. They are as follows:

MGRID N ... (same as GRID );
MVEC N ... (same as VECTOR );
MCON N ... (same as CONTOUR).

Here N is the number of block (domain) for which grid, vector field or variable contour will be drawn; while the other arguments have exactly the same meaning, as for standard commands, and should be used according to PHOTON manual.

Note, that PHOTON retrieves information on the positions of blocks in the global computational space from the PATGEO file. For this reason, this file should be stored by user along with PHI and XYZ files, if it is necessary to keep results of a run for a future presentation.

6. References.

  1. S.V. Patankar and D.B. Spalding, A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows, Int. J. Heat Mass Transfer, vol. 15, p. 1782, 1972.
  2. L.A. Hageman and D.M. Young, Applied Iterative Methods, Academic Press, New York, 1981.
  3. I.N. Poliakov and V.A. Semin, Development and Evaluation of New Linear Equation Solvers for PHOENICS, The PHOENICS J. of Computational Fluid Dynamics and its Applications, vol. 7, no. 1, p. 34-57, 1994.
  4. M.R. Hestenes and E.L. Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, J. Res. Natl. Bur. Stand. Sect. B 49, pp. 409-436, 1952.
  5. J.M. Ortega, Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, New York and London, 1988.
  6. D.B. Spalding, A Guide to the PHOENICS Input Language, CHAM/ TR100.
  7. F.H. Harlow and J.E. Welch, Numerical calculation of time- dependent viscous incompressible flow of fluid with free surface, Physics of Fluids, 8, 2182, 1965.
  8. C.M. Rhie and W.L. Chow, Numerical Study of the Turbulent Flow Past Airfoil with trailing Edge Separation, AIAA J., vol. 21, pp. 1525- 1532, 1983.
  9. I.N. Poliakov and V.A. Semin, An Introduction into the Method for Implementing Multi-Block Grids and/or Grids with Refinements in PHOENICS, CHAM/TR401, The PHOENICS Journal. of Computational Fluid Dynamics and its Applications, vol. 7, no. 2, p. 34-57, 1994.
  10. R.F. Warming and R.M. Beam, Upwind second-order difference schemes and applications in aerodynamic flows, AIAA J.,14, 1241- 1249 (1976).
  11. B.P. Leonard, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Comput. methods appl. mech. eng., 19, pp.59-98 (1979).
  12. S.K. Godunov, A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations, Mat. Sbornik 47, 3, pp.271-306, (1959). Translated as JPRS 7225 by US Dept. of Commerce, November 1960.
  13. B. Van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J.Comput. Phys., vol. 32, pp.101-136, (1979).
  14. P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, pp.357-372, (1981).
  15. B.P. Leonard, Simple high-accuracy resolution program for convective modeling of discontinuities, Int. J. Numer. Meth. Fluids, 8, pp.1291-1318, (1988).
  16. P.H. Gaskell and A.K. Lau, Curvature compensated convective transport: SMART, a new boundedness preserving transport algorithm, Int. J. Numer. Meth. Fluids, 8, pp.617-641, (1988).
  17. PHOENICS Instruction Course Notes, CHAM TR/300.
  18. D.B. Spalding, Numerical Computation of Multiphase Fluid Flow and Heat Transfer, in Recent Advances in Numerical Methods in Fluids, pp. 139-167, Editors C. Taylor and K. Morgan, (1981).

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