Multi-Block Gridding Technique for FLOW-3D Flow Science, Inc. July 2004

FSI-02-TN59-R2 Multi-Block Gridding Technique for FLOW-3D Flow Science, Inc. July 2004 1. Introduction A major new extension of the capabilities of FLOW-3D -- the multi-block grid model -- has been incorporated into the code starting with Version 8.0. Multi-block gridding in FLOW-3D will enable more efficient use of the software s resources when modeling complex flow phenomena. Each block spans a certain region of the whole flow domain and contains the standard structured rectangular mesh. Two types of mesh blocks can be used: the nested blocks and the linked blocks. Data transfer between any two mesh blocks is facilitated by special boundary (or ghost) cells. Solution quantities are interpolated from the real cells of a donor block into the boundary cells of the acceptor block. The interpolation technique varies depending on the variable at hand. Conserved quantities, like concentrations and thermal energy, are interpolated using the piecewise constant method. A special variant of this method is used for fluid fraction interpolation, where a reconstruction of the interface is performed to locate the interface within the donor cells before the interpolation. Pressure and velocities are calculated using linear interpolation to preserve the gradients. A mixture of the Neumann- and Dirichlet-type boundary conditions is used for the solution of the Poisson equation for pressure. A weighing factor defines the contribution of each type of the boundary condition to the final solution, ensuring continuity of both pressure and velocities across the inter-block boundaries, convergence and local conservation of mass. 2. Defining mesh blocks The order of the mesh namelists in the input file defines the mesh block numbers that distinguish one block from another. The multi-block model is implemented in such a manner as to minimize the dependence of the numerical solution on mesh block ordering. There is no limit on the number of mesh blocks that can be used in a simulation other imposed by the hardware itself. Any two mesh blocks cannot partially overlap each other. They either do not overlap at all, i.e., they are adjacent or completely separated blocks, or one of them completely overlaps the other. Two adjacent mesh blocks are called linked. A linked block communicates with adjacent blocks via its boundaries. If a mesh block is completely overlapping another block, the former is called a nested block, while the latter is called

the containing block. There may be more that one containing mesh block for each nested one. A nested block can also be linked to another nested block. Multi-level nesting is allowed, where a nested block can be nested in another, larger, nested mesh block, which may be nested in an even larger block, etc. A nested block receives data from the smallest containing block at its boundaries, and passes the data back to that containing mesh block in the part of the real domain of the nested block adjacent to the boundaries of the latter. A mesh cell belongs to the real domain, if the numerical solution is obtained in this cell by solving the discretized flow equations. During an inter-block data transfer, a mesh block from which data is being passed is called the donor block and the other block is called the acceptor. A given mesh block can be an acceptor or donor at different times during data transfer. There are no specific requirements or limitations on how the meshes in adjacent or overlapping blocks must match at the boundaries. However, the interpolation procedure employed in the inter-block data transfer algorithm introduces additional truncation errors into the solution. These errors are common to any spatial interpolation algorithm. Besides, data is passed between mesh blocks at discrete stages of the solution procedure, (see section 4 below for details), adding to the truncation errors in temporal discretization of the solution. These errors are usually small, but may become significant if there are large spatial and temporal variations in the flow near the inter-block boundaries. Following the guidelines below will help minimize those errors maintain good accuracy and stability of the multi-block algorithm: 1. Avoid having large difference in resolution between any two adjacent or nested blocks. A ratio of 2 to 1 in any coordinate direction should be considered as the maximum acceptable to avoid significant accuracy loss in inter-block data transfer. 2. Avoid placing inter-block boundaries in flow areas where large gradients of solution quantities are expected, including areas with significant variations in geometry. 3. When using nested blocks, user-defined mesh planes (lines, in two dimensions) in the containing blocks should be positioned at the boundaries of the nested blocks. This should be easy to achieve since all mesh blocks are rectangular. The pre-processor carries out the following diagnostics and initialization: 1. Makes sure the mesh blocks are not partially overlapping. If they are then the simulation will stop with an error message. 2. Automatically sets inter-block boundary conditions (of type 9). They are similar to the fixed-pressure-type boundaries in that they contain two layers of boundary cells. 3. The communications pattern, or hierarchy, is set up to define how the linked blocks are passing the data to each other, and how the nested block data is passed to the linked blocks. Each nested block passes its data only to the smallest containing mesh block. 2

4. The hierarchy information is written into the prpout file. Search for the word information to find it. 5. A new array is created and written into the flsgrf file, called cell type. The array is an integer variable that defines whether a mesh cell is a normal real cell, in which solution values are obtained by solving the conservation equations (value = mesh_block_number_the_cell_belongs_to), a standard mesh boundary cell (value = 0), or a cell on to which data will be interpolated from another block (value = - block_number_from_which_data_will_be_interpolated a negative value!). Cells with the negative cell type value are called passive, while the normal real cells are called active, and the boundary cells remain boundary, or ghost, or fictitious cells. 6. Creates composite 2d mesh and solid component plots, which are written to the prpplt file. Figure 1 shows an example of five mesh blocks, two linked and three nested, color coded by mesh block numbers. 7. Computes and prints to the prpout file open areas between every pair of communicating mesh blocks. In general, open areas at the two sides of an interblock boundary differ due to differences in mesh resolution of the two mesh blocks connected at that boundary. As a result, the global conservation of mass is not maintained, even if there is good local conservation. It is important to keep the differences between the corresponding open areas small, e.g., below 1 %, to maintain reasonable conservation of fluid mass. The area printout in the preprocessor includes the differences between respective open areas as percent of the average. An cumulative difference of all open areas is also reported. The logic used to define the hierarchy (item 3 above) is fairly intuitive: each nested block passes its solution data to the next in size containing block. If the later block is also nested it will in turn pass its data to the next in size containing block, etc. The data in the active cells of each nested block represent the real solution. This logic implies that the smallest blocks most likely have the finest grid and, therefore, the most accurate solution of all overlapping blocks in this area. This solution is then used to calculate the variables in the passive cells of the underlying containing blocks (see section 4 for more details). Note that the solution from a nested donor block is overlaid only to those cells in the containing acceptor block that are located completely within the donor block. All other real cells in the acceptor block are treated as active cells. Each nested block passes data to one and only one containing block, and each nested block receives data at its boundaries from the same containing block. This communications hierarchy can not be controlled by the user. 3

1 3 2 4 5 Figure 1. Sample multi-block mesh setup (color denotes block number). Blocks 1 (dark blue) and 2 (light blue) are linked. Block 3 (green) is nested in the containing block 2, while block 4 (yellow) is nested within both blocks 2 and 3. Finally, block 5 (red) is nested in block 2 and linked to block 3. 3. Inter-block boundary conditions A new type of boundary condition has been added to manage data transfer between mesh blocks. The boundary is called the inter-block boundary and is designated with number 9. These boundaries contain two layers of fictitious cells, providing up to second-order accuracy in the calculation of advection terms at the boundary. Type 9 boundaries are always used between any two linked blocks and for all nested block boundaries and are automatically set by the pre-processor. Any standard type of FLOW-3D boundary condition can be used at a mesh block boundary not adjacent to another mesh block. Such mesh boundary is called an external boundary. Additional checks are made in the pre-processor to make sure the type and location of the nested block boundaries are consistent with those of the containing block(s). If a nested block boundary location is the same as the location of a mesh boundary of a containing block (like the left boundary of blocks 2 and 3 in Fig. 1), then the former assumes the same type as the latter, i.e., the larger block takes precedence. 4

4. Solution procedure Physical quantities are calculated in each block by numerically solving the flow equations. Solution is passed from one block to another via inter-block boundaries using an interpolation technique that ensures convergence in the whole computational domain. The data exchange between mesh blocks is done differently for pressures and velocities than other solution variables. Linear interpolation is used for pressures and velocities, with the aim of maintaining good accuracy in representing pressure gradients and fluid flow at inter-block boundaries. Pressures and velocities are exchanged between blocks at each pressure iteration pass though a block. Fluid fraction, thermal energy, scalars and turbulent parameters are passed from one block to another using the piecewise constant overlay procedure similar to the one used when overlaying solution data from one mesh to another during restarts. Special interface tracking algorithm is used for fluid fraction interpolation in the presence of sharp fluid interfaces. The main property of these procedures is that the overlaid quantity is conserved. The pressure/velocity data transfer algorithm employs linear spatial interpolation. For any boundary point that receives interpolated data (interpolation point), an eight-point interpolation stencil in a donor block, and linear interpolation coefficients are calculated. Figure 2 shows a 3-D interpolation stencil with its base (or reference) point as (i, j, k) and interpolation point P (i p, j p, k p ). The interpolation formula is: Φ 8 p = c j j = 1 Φ j where Φ j is the quantity Φ at one point of a interpolation stencil, Φ p is the interpolated value at point P, and c j are the interpolation coefficients, where c 1 = (1 ξ )(1 η)(1 ζ ) c 2 = ξ (1 η)(1 ζ ) c = ξη(1 ) c c c c 3 ζ = (1 ξ ) η(1 ) = (1 ξ )(1 η) ζ 4 ζ 5 6 = ξ (1 η) ζ 7 = ξηζ c 8 = (1 ξ )ηζ 5

ξ = ( x η = ( y ζ = ( z p p p x ) /( x i k j i+ 1 y ) /( y z ) /( z j + 1 k + 1 x ) i y ) z ) k j The interpolation coefficients are calculated for all interpolation points and saved, including the interpolation point indices and their block numbers, and all corresponding stencil base points and their block numbers. The solver can recover this information at any time when physical quantity interpolation is needed. An eight-point stencil may not provide correct interpolation near obstacles and at free surface because part of its points may be located inside the obstacle or in a void. Special treatments have been developed for such cases based on the assumption of hydrostatic pressure distribution. When there are no moving obstacles, all interpolation information is calculated in the preprocessor and written to a data file, which is then read by the solver and does not change during the calculations 1. Pressure and velocity interpolation is made for all the stencils in that block and transferred to boundaries of other blocks before further calculations are conducted in these blocks. For moving object problems, interpolation information is not generated in the preprocessor but generated in the solver at each time step. This is because for moving obstacle problem, the interpolation information may change with time. (i+1,j+1,k+1) STENCIL P(i p,j p,k p ) (i,j+1,k) z y (i+1,j+1,k) (i,j,k) x (i+1,j,k) Figure 2. Interpolation point P and interpolation stencil. The interpolation stencil base point is (i, j, k). 1 When the option LPR > 3, a diagnostic report will be printed, listing all interpolation information. This special output is primarily used by developers and support engineers. 6

With this interpolation method, many tests have been conducted, including large industrial problems. Satisfactory results have been obtained. The interpolation is found efficient and accurate for many flow problems. An iterative solution method is used to solve for pressure and velocity in incompressible flow. After each iterative pass through a mesh block, these variables must be updated at inter-block boundaries of all adjacent blocks. This ensures proper coupling of the solution between all blocks. When solving the continuity equation for pressure, two types of boundary conditions (BC) can be used at an inter-block boundary: Neumann type - when the pressure gradient is defined, - or Dirichlet type when the pressure itself is defined. In FLOW-3D, the Neumann type condition translates into defining the normal velocity at the boundary. Each type of BC has its cons and pros. The Neumann type gives a better conservation of mass, but, generally, has slower convergence and can result in discontinuous pressure field. The present method applies both methods at the same time at all inter-block boundaries. A weighing factor, 0.0 α 1. 0, is used to define the portion of each type of BC. At iteration k+1 the normal velocity U n and pressure P in a boundary cell are computed as U P = U + α ( U U k + 1 k k + 1 k n n n _ int n = P + (1 α )( P k + 1 k k + 1 k int ) P ) where U n_int and P int are the interpolated normal velocity and pressure, respectively. When α=0.0, then a pure Dirichlet type BC is used and when α=1.0, then the Neumann type BC is employed. The default value of α is 0.25. The overall solution procedure produces a unique numerical solution in the physical domain covered by the various mesh blocks. In other words, when mesh blocks overlap, covering the same physical space, the numerical equations are solved in only one of the blocks. All other overlapping blocks obtain value of the primary quantities via the solution interpolation or keep the initial values, depending on the location of a cell. Mesh cells containing such overlays are the passive cells described in the previous section. The enforcement of a unique solution greatly simplifies the solution process, its postanalysis and, of course, it saves CPU time. In agreement with the hierarchy described in section 2, the real solution is found in the smallest mesh block covering a region. The short and long prints produced by the solver include fluid volume, energy and other quantities, integrated over all blocks, excluding the passive cells. Volume error also excludes the passive cells. When LPR > 1 in XPUT, the short prints will also include a separate line for each mesh block. 7

5. Post-processing Spatial 2D and 3D plots have been extended to handle multiple mesh blocks on the same plot, including particle and obstacle plotting. Text output, probes and 1-D plots can only be used for one mesh block at a time. The GUI allows users to select the mesh block(s) for post-processing. All spatial limits definitions in the input file (e.g., plotting limits, history point coordinates, particle source definitions) must be done in terms of coordinates rather than cell indices when the multi-block model is employed. 6. Examples We will consider three examples using 2-D geometry for greater clarity. The input files and the results for the example problems are included with this report. A. Inkjet droplet formation In this test we compared a two-block solution to a single-block one using roughly the same resolution. The two mesh blocks are positioned one above the other in a linked configuration (Fig. 3). Also, we ran a test with two linked blocks where the lower mesh block has a coarser grid than the upper one. The inlet boundary condition (BC) at the bottom of the lower mesh block is fixed-pressure (type 5), with a time-dependent pressure to generate the pulse needed to produce a droplet. The outlet BC at the top is the continuative boundary (type 3). The mesh boundaries connecting the two blocks are of type 9. All other boundaries are of the symmetry type. The purpose of this test is to see how well fluid fraction, pressure and velocity data is transferred through the inter-block boundaries. A secondary objective is to see how the solution behaves at such boundaries when one of them has much coarser mesh than the other. Figure 4 shows the formation of the droplets in all three meshes shown in Fig. 3, at three different times. It can be seen that the predicted shape and displacement of the droplet are close in the fine-mesh single- and double-block calculations. The average speed and volume of the droplet for cases A and B differ by less than 0.25%. The largest differences are observed when the droplet hits the outlet boundary. The two-block model (case B) has 18% fewer cells than the single block model, while it ran only 7% faster. This can perhaps be explained by the fact that the number of active cells (i.e., cells containing fluid) in the two cases is roughly the same. For the coarse mesh calculation, the droplet separation is delayed by about 0.1 sec. The tail of the droplet is not defined as well in the coarse grid as it is in the examples with finer grids. However, little distortion of the shape of the droplet can be observed as the 8

fluid passes from the coarse mesh block to the fine one. The average cell size ratio for the two grids is about 1.8. In the present test, the small errors in the VOF function overlay affect the dynamics of the fluid since surface tension forces depend on the shape of the free surface. Given the complexity and speed of the flow, it can be concluded that errors introduced by the solution overlay procedure at the inter-block boundaries are quite small. B. Weir flow In this example we use four mesh blocks, one nested block and three linked ones to track the dynamics of a stream of water coming down under gravity (Fig. 5). The fluid enters at the left-most edge of the domain from a fixed pressure head and cascades down the two steps produced by the irregular domain size. The right-most domain boundary is of an outflow type. The nested block is used to produce better resolution at the point of stream separation. The bottom boundary of each linked block is a wall type boundary. The calculated flow is shown in Fig. 6 at four different times. One thing that stands out on these contour and vector plots is a seeming discontinuity of the pressure and free surface values across the inter-block boundaries. However, this is due to a deficiency in data interpolation by the post-processor at mesh boundaries during plotting and is something we hope to improve before the final release of the multi-block model. Otherwise, the results are quite reasonable. 9

Z Z Z A B C R R R Figure 3. Mesh definitions for the inkjet problem: (A) single block uniform mesh; (B) two linked blocks with fine mesh, and (C) two linked blocks, one with coarse and the other with fine grid. The cell sizes in the fine mesh blocks in all three cases are approximately same. The left boundary of the nested mesh block is of type 9. The data is passed to it from the containing block. The boundary (acceptor) cells of the nested block overlap the boundary (donor) cells on the left of the containing block. C. Moving obstacle case In this simulation two mesh blocks were used, with the second block having three times higher grid resolution in the region behind the missile near the symmetry axis than the main block does (Fig. 7). The second (nested block) has bottom, top and right mesh boundaries as inter-block boundaries (i.e., type 9). All other boundaries in this block are symmetry boundaries. The average cell size in the nested block is three times smaller than that in the containing block. 10

T=2.1 sec T=3.5 sec T=4.3 sec A B C Figure 4. Calculated droplet formation for meshes A, B and C shown in Fig. 2. Colors denote fluid fraction contours. 11

z Figure 5. Four-block setup with three linked blocks and a nested one. x The adiabatic bubble model is used in this calculation. The bottom boundary of the nested block divides the initial void region below the missile in two sections. This is designed to test the new algorithm for tracking void regions across multiple mesh blocks. In the present case, the lower part of the void region is tracked in the larger and coarser containing mesh block, and the upper part is tracked in the finer mesh of the nested block. A separate calculation was also performed with a single mesh block identical to the containing grid of the two-block case. The results for the two cases are compared in Fig. 8. The flow pattern 0.15 seconds after the start of the simulation is shown in Fig 8. The solution is shown for both the two-block and one-block cases for comparison. It can be seen that the fluid rushes in behind the missile and divides the initial void region into several smaller regions. Each void has its own volume and pressure calculated at each time step. The single-block calculation predicts a faster filling of the cavity by the fluid. This may be attributed to the coarseness of the grid. 12

t=0.3 sec 3626 2935 t=0.5 sec 20372 16896 2245 13421 1554 9945 863 6469 173 2994-517 -481 t=0.6 sec 69986 58127 t=0.8 sec 17668 9539 46268 1409 34410-6719 22551-14848 10693-22977 -1165-31106 Figure 6. Weir-type flow simulation results for the four mesh blocks shown in Fig.5. Color denotes pressure. 13

0.90 missile 0.54 0.18 z -0.18-0.54-0.90 0.0 0.37 r Figure 7. Two-block setup for the inkjet problem showing block outlines, and the obstacle blue denotes block 1, and red is block 2. Block 2 is a nested block overlapping the main block 1. The cylindrical axis is on the left. 0.90 ( = 1.69E+01) 0.90 ( = 1.03E+01) -0.90-0.90-0.37 0.0 0.37-0.37 0.0 0.37 Figure 8. Predicted free surface locations in the two-block (left) and single-block calculations at t=0.15 sec. 14