The Surface Marker and Micro Cell Method

Introduction

The surface marker and micro cell (SMMC) method can be used to simulate two-dimensional, incompressible, free-surface fluid flow problems. The SMMC method is capable of simulating transient free surface fluid flow problems that include the impact of a free surface with a solid obstacle or the convergence of two free surfaces.

This website is divided into two main sections. The first is devoted to a brief discussion of the development of the SMMC method and contains links to the full texts of five papers associated with the development of the method. The second section contains links to the complete FORTRAN computer code for the implementation of the SMMC method and to the twenty-nine associated subroutines.

An interested reader is welcome to use the code for the SMMC method to simulate free surface fluid flow problems of interest or to use it as a point of departure for future research. The authors of the SMMC method were motivated in part to make the SMMC computer code available because the computer code for the Simplified Marker and Cell (SMAC) method, which was made available by the Los Alamos National Laboratory, represented an important point of departure for their own research in free surface fluid flow.

Development of the SMMC Method

The SMMC method was developed during the 1990s at SMU in Dallas, Texas. In order to gain an in-depth understanding of the method and its significant features, one should study five publications. Links to the complete texts of these five papers are found in this section.

The Surface Marker Method (1991) includes fundamental modifications of the original marker and cell (MAC) method which result in significant reductions in the computer time and mass storage required for the solution of transient, free surface fluid flow problems. The MAC method, which was developed in the late 1960s, and its successor, the simplified marker and cell method (SMAC), which was presented in 1970, both require markers initially evenly distributed throughout the fluid. During each computational step in the MAC and SMAC methods, all markers must be moved, and all cells containing markers must be reflagged. Consequently, for the MAC and SMAC methods, approximately three-fourths of the total computing time is required to solve the pressure potential Poisson equation and to move markers and reflag cells. In the surface marker method, only one string of markers that stretches along the free surface is required and only the cells adjacent to the free surface must be reflagged. As a result, the time required to move markers and reflag cells is significantly less for the surface marker method. A second significant increase in efficiency was realized with the surface marker method by replacing the successive overrelaxation method with the preconditioned conjugate gradient method for the solution of the pressure potential Poisson equation.

Accurate velocity boundary conditions are critical to the successful simulation of free surface fluid flow. In Velocity Boundary Conditions for the Simulation of Free Surface Fluid Flow (1995), it is shown that previous approaches for the assignment of free surface velocity boundary conditions in marker and cell methods artificially introduce asymmetry and can even cause the simulation to break up. New procedures that improve the accuracy of the treatment of the free surface velocity boundary conditions and which do not artificially introduce asymmetry are presented. It also is shown that use of the new velocity boundary conditions enables the simulation of problems with realistic values of viscosity and gravity, problems for which the breakup of the solution occurs if previous approaches are employed.

Deficiencies associated with the simulation of impacts of fluid free surfaces with solid boundaries were identified and addressed in Simulation of Impacts of Fluid Free Surfaces with Solid Boundaries (1994). New procedures are introduced that affect the movement of markers in cells adjacent to a solid boundary, the flags of the cells that comprise a solid boundary, and the pressure boundary conditions for a cell in which impact occurs. Combined with fundamental changes in the sequence of steps in the computational cycle, these new procedures allow the intentional treatment of impact. As a result, improved estimates are obtained of the pressure associated with the cells adjacent to a boundary along which impact occurs. Consequently, more appropriate adjustments are made of the tentative internal velocities associated with such cells. In addition, a special procedure is presented for the adjustment of the tentative internal velocity between two surface cells. Finally, a new cell type termed a corner cell is defined and a procedure for its treatment is presented. Numerical examples illustrate the previous deficiencies associated with the treatment of impact as well as the effectiveness of the new procedures. Validation of the new procedures is achieved by comparison of simulation and experimental results for flow over a containment dike.

In The Introduction of Micro Cells to Treat Pressure in Free Surface Fluid Flow Problems (1995), an entirely new method of calculating the pressure field for the simulation of free surface fluid flow is presented. A critical feature of the mew method is the introduction of a finer mesh of cells in addition to the regular mesh of finite volume cells. The smaller (micro) cells are used only near the free surface, while the regular (macro) cells are used throughout the computational domain. The advancement of the free surface is accomplished by use of surface markers, while the discrete representation of the free surface for the purpose of the application of pressure boundary conditions is accomplished by use of micro cells. In order to exploit the advantages offered by micro cells, a new general equation governing the pressure field is derived. Micro cells enable the identification and treatment of multiple points on the free surface in a single surface cell and of points on the free surface in a macro cell has no empty macro cell neighbors. Both of these situations are likely to occur repeatedly in a free surface fluid flow simulation, but neither situation has been explicitly taken into account in previous methods. The new capabilities enabled by use of surface markers and micro cells are essential for the successful simulation of free surface fluid flow problems that involve multivalued free surfaces and breaking fluid fronts.

The Surface Marker and Micro Cell Method ( 1997) combines the new procedures presented in the four papers referenced above and presents a complete new method for the simulation of two-dimensional, incompressible, free surface fluid flow problems. The SMMC method is capable of simulating transient free surface fluid flow problems that include multivalued free surfaces, impact of free surfaces with solid obstacles, and converging fluid fronts such as those that occur during wave breaking. The first method to address the challenges associated with the simulation of free surface fluid flow was the MAC method, which was published in 1965 and which introduced the concept of massless markers that move with the fluid. The SMMC method, like the MAC method, employs massless markers to track the movement of the fluid. The SMMC method is therefore a marker and cell method, but it otherwise differs from the MAC method in essentially every significant detail. Surface markers, rather than markers distributed throughout the fluid, are used in the SMMC method. New procedures are used to insure that only physically meaningful velocity information is used to move the surface markers and advect the free surface. Physically motivated new procedures to carefully approximate momentum fluxes, as opposed to straightforward use of finite difference approximations of terms in the Navier-Stokes equations, are used for the calculation of the tentative internal velocity field. To avoid the physically unrealistic use in previous methods of the continuity equation to calculate tentative surface velocities, as well as the resulting inconsistency between the treatments of pressure and continuity in surface and full cells, a new approach also is used for the determination of tentative surface velocities. In connection with converging fluid fronts, new procedures that incorporate multivalued velocities insure the use of physically relevant information in the calculation of the velocity and pressure fields. In order to assign appropriate values at different stages of the computational cycle to velocities between converging fluid fronts, the physical purpose for which the velocity is needed is taken into account. For the same velocity, one value is appropriate for a given purpose, such as marker movement, but inappropriate for a different purpose, such as momentum transfer or the development of a physically appropriate pressure field. It is recognized that for a particular discrete velocity that is associated with the convergence of two fluid fronts, as many as five different values are appropriate for different purposes during a single computational cycle. To demonstrate both the capabilities and the validity of the SMMC method, simulation and experimental results are compared for water sloshing in a tank. The comparison reveals good agreement between the shapes and locations of the simulated and experimental free surfaces even after repeated occurrences of impact of the water with the tank walls and of the convergence of fluid fronts as the water sloshes back and forth in the tank. A grid refinement study also demonstrates the convergence of the new method.

 

SMMC Computer Code

The FORTRAN code for the implementation of the SMMC method is SMMC_PLUS.f, which calls twenty-nine associated subroutines. The name and purpose of each subroutine is listed below in the order of appearance in SMMC_PLUS.f:

            Subroutine                                Purpose

            domain.f                                   Define the solution domain and initialize variables.

            obst.f                                        If applicable, define fixed obstacles.

            outlet.f                                      If applicable, define outlets.

            inlet.f                                        If applicable, define inlets.

            fluid_rect.f                                If applicable, define rectangular fluid filled regions.

            fluid_drop.f                               If applicable, define a circular fluid filled region.

            miscel.f                                     Define various system and output parameters.

            dtilde.f                                      Compute the tilde accelerations.

            move_marker.f                         Move markers.

            tilde_velocity.f                          Compute the tilde velocities.

            pre_flag.f                                  Assign check flags to appropriate macro and micro cells.

            flag_micro.f                              Assign micro cell flags.

            droplet.f                                   Track the movement of droplets (fluid region smaller than one macro cell.)

            flag_macro.f                             Assign macro cell flags.

            new_velocity.f                          Assign new fluid cell velocities.

            separation_wall.f                       Treat separation of fluid from a fixed obstacle.

            pcg_setup.f                               Prepare for use of the preconditioned conjugant gradient method.

            pcg_solver.f                              Compute the pressure potential field.

            final_velocity.f                           Compute the final velocities.

            surface_velocity.f                      Assign the surface velocities.

            converging_surfaces.f           Treat converging fluid fronts.

            output_control.f                        Control program output.

            print_marker_positions.f           Output marker position information.

            volume_check.f                        Check the fluid volume.

            print_velocity_info.f                  Output velocity information.

            print_pressure_info.f                 Output pressure information.

            print_macro_cell_info.f             Output macro cell information.

            print_micro_cell_info.f              Output micro cell information.

            symmetry_check.f                     If appropriate, check for solution symmetry.

The macro compile can be used to create the executable file smmc.x.