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The video above shows a simulation orange and blue smoke using the MAC method.
About the MAC method
When using the Marker and Cell method, the fluid is modeled as a velocity field, which is “a vector field that defines the motion of a fluid at a set of points in space” (Cline, Cardon). In order to simulate a moving fluid with the MAC method, the velocity field is changed and evolved over time by moving marker particles through a space dictated by this velocity field. In addition to marker particles being used, a level set may be used to manipulate the velocity field.
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It is impossible to store the velocity information of every possible point in a given space, the compromise of having a rigid grid with discrete points is made instead. The velocities between these points is then calculated via interpolation, creating the simulation of a fluid. Navier-Stokes equations are the basis of the rules which manipulate the velocity field in a MAC simulation . Basically, at each point of the grid there is an addition and subtraction using the values of a small neighborhood around said point.
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The Navier-Stokes Equations
The Navier-Stokes equations are “a set of two differential equations that describe the velocity field of a fluid over time” (Cline, Cardon). The first equation, ∇· u = 0, is the mass conservation equation which states that the amount of fluid moving into any volume is equal to the amount moving out of the volume. In the simulation, a pressure term is used to solve the mass conservation equation.
The second Navier-Stokes equation, du/dt = -(∇ · u) u - 1/p + v∇2u + F, governs how the vector field of the fluid changes over time. It accounts for the motion of the fluid through a given space while accounting for any internal and external forces that act on the fluid.
The left side of the equation, du/dt is the derivative of the velocity of the fluid with respect to time.
The right side of the equation is made up of the following:
the convection term -(∇ · u) u which concerns the conservation of the momentum of the fluid,
the pressure term - 1/p which captures the forces generated by the pressure differences within the fluid,
the viscosity term v∇2u which handles the friction between the particles of the fluid that can slow down or speed up overall movement,
and the external force term F which factors in forces like gravity and contact with other objects that have an effect on the motion of the entire fluid.
Since the formula used is a derivative of the entire motion of the vector field for the fluid, it can only describe one dimension of the three dimensions in space at a time. These calculations are made essentially three times for each particle, and then the components are summed together to get the final result of the velocity of a particle in a specific cell on the grid.
The Cells in the Grid
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The MAC grid method uses cells like the one shown in Figure 1, with each cell having a width h. Each cell has a pressure p and also three components for its velocity: ux, uy, and uz. While the pressure is defined at the center of the cell, the three components of the velocity are placed at the centers of three of the cell faces. The ux component is on the x-minimum face, the uy component is on the y-minimum face, and the uz component is on the z-minimum face. Harlow and Welch found that centering the velocity’s components on the centers of the minimum faces for each dimension produced more stable simulations as opposed to storing the components all in the center.