My First Fluid Project

Ryan Schmidt

OutlineMAC MethodHow far did I get?What went wrong?Future Work

The MAC MethodMarker-and-Cell – Harlow&Welch 1965

Standard technique for simulating incompressible fluids w/Navier-Stokes fluid equations

LANL Technical Report (access restricted!!!)

Navier-Stokes Fluid Dynamics

Velocity field u, Pressure field pViscosity v, density d (constants)External force f

Navier-Stokes Equation:

Mass Conservation Condition:

Navier-Stokes EquationDerived from momentum conservation condition4 Components:

Advection/ConvectionDiffusion (damping)

PressureExternal force (gravity, etc)

System of Nonlinear partial differential equations

Incompressibility Condition

We want incompressible fluids*Velocity field u has zero divergence

Mass conservation over any subregionFlow in == flow outIncompressible fluid

Comes from continuum assumption

*gasses assumed to be locally incompressible

Spatial DiscretizationStaggered grid for uCentered grid for p

(Cells)

Equation DiscretizationCentral differences for spatial derivativesForward difference for time derivativeu component:

Mathematical TrickeryAdvection form different in literature:

These two are equivalent if the fluid is incompressible. Proof:

MarkersCell resolution very coarse (20-150)Want higher resolution surfaceAlso need to track which cells contain fluid

Solution: ‘Marker’ particlesMassless particles that flow freely in u fieldDo not contribute to computationVery fast to process

MAC AlgorithmInitialize u,p grids (easier said than done)

Forward-difference u to get new velocities

Enforce zero-divergence condition

Rinse and repeat

Enforcing Zero Divergence2 possibilities:

Iterative procedureProjection method of Stam99

Iterative Procedure – Pressure Iteration

Individually set each cell divergence to 0Calculate pressure change and modify velocities

Repeat over entire grid until maximum cell divergence < predefined tolerance

Pressure IterationFor each cell calculate change in pressure

Now update cell:

Bad Formatting?Does this:

Mean this?:

Inverse dependence onBut set to If << , Di,j will be small?

If not, system explodes!

How far did I get?

Well…

It’s not pretty…

Symmetry?Tried to reproduce experiments in literature

Correct Physical Constants! d=1, v=0.01, g=981 for breaking dam

Inflow supposed to be symmetric…

What went wrong?

Initial Conditions ?!?System becomes unstable as soon as there is any large amount of divergence

How do we specify initial conditions that will give us motion w/o immediately causing unstable divergence?

(I don’t know…)Inflow is simple case, but it still doesn’t work…

Boundary Conditions Many, many cases

Too many to have special cases of finite difference equation

Solution: construct velocities & pressures in boundary cells so that standard finite difference equation comes out right

I may have them wrong…Not sure when to apply them

Unclear how order of application affects velocties…

Wall BoundariesNormal velocity is 0

Prevents flow into boundary cellAlso have to set internal pressure

No-slipzero tangential velocity

Free-slip free tangential velocity

Wall Boundary ProblemAssumption is made that there is only one adjacent fluid cell

What if there is morethan one?

Cannot do both…

Free-Surface BoundariesHave to make sure that divergence in surface cells is 0

Lots of casesI think this is where my problem is28 cases and counting…

Asymmetry?

Outer Tangential Velocities

Interpolation in surface cells reaches out into empty cells

Finite difference equations may as well

Need to have same velocity set there

Future WorkGo back and check boundary conditions

Harass Nick Foster

Finish report and put it on the web, hope that someone reads it and has some insight

Thanks!

Questions?