Intro to Computational Fluid Dynamics. Brandon Lloyd COMP 259 April 16, 2003. Image courtesy of Prof. A. Davidhazy at RIT. Used without permission. . Overview. Understanding the Navier-Stokes equations Derivation (following [Griebel 1998]) Intuition Solving the Navier-Stokes equations
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April 16, 2003
Image courtesy of Prof. A. Davidhazy at RIT. Used without permission.
div - divergence
2 - Laplacian
… - Hand waving / Lengthy math compression
Discretizing the viscosity term spreads velocity among immediate neighbors. Unstable when time step too small, grid spacing too large, or viscosity is high.
Solution: Instead of using an explicit time step use an implicit one.
This leads to a large but sparse linear system.
The tentative velocity field is not necessarily divergence free and thus does not satisfy the continuity equation.
Three methods for satisfying the continuity equation:
Since we have not yet added the pressure term, we can use pressure to ensure that the velocities are divergence free.
u>0 increased pressure and subsequent outflux
u<0 decreased pressure and subsequent influx
Another approach involves solving for a pressure correction term over the whole field such that the velocities will be divergence free and then update the velocities at the end.
Discretize in time
Satisfy continuity eq.
We end up with the Poisson equation for pressure.
This is another sparse linear system. These types of equations can be solved using iterative methods.
Use pressures to update final velocities.
The Helmholtz-Hodge Decomposition Theorem states that any vector field can be decomposed as:
where u is divergence free and q is a scalar field defined implicitly as:
We can define an operator P that projects a vector field onto its divergence free part:
Applying P to both sides of the momentum equation yields a single equation only in terms of u:
Thus for the last step :
Look familiar? The scalar field q is actually related to pressure!
All three methods are equivalent!
The staggered grid provides velocities immediately at cell boundaries, is convenient for finite differences, and avoids oscillations.
Consider problem of a 2D fluid at rest with no external forces. The continuous solution is:
On a discretized non-staggered grid you can have:
The movement of the free surface is not explicit in the Navier-Stokes equations.
Three methods for tracking the free surface:
Due to [Harlow and Welch 1965].
Track massless marker particles to determine where the free surface is located.
Markers are transported according to the velocity field.
Cells with markers are fluid cells. Fluid cells bordering empty cells are surface cells.
There are boundary conditions that must be satisfied at the surface.
Extended by [Chen et al. 1997] to track particles only near the surface.
Image from [Griebel 1998].
Proposed by [Foster and Fedkiw 2001]
Front tracking uses a combination of a level set and particles to track the surface.
The particles are used to define an implicit function. An isocontour of this function represents the liquid surface.
The isocontour yields a smoother surface than particles alone.
Using the level set method, the isocontour can be evolved directly over time by using the fluid velocities.
Particles and level set evolution have complementary strengths and weaknesses
Combine the two techniques by giving particles more weight in areas of high curvature. Particles escaping the level set are rendered directly as splashing droplets.
Presented by [Enright et al 2002].
Implicit surface loses detail on coarse grids.
Particles keep the surface from crossing them but can’t keep it from drifting away.
Add particles to both side of the implicit surface.
Escaped particles indicate the location of errors in the implicit surface so it can be rebuilt.
Extrapolated velocities at the surface give more realistic motion.
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