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Overview Fluids for • Some terms… • Incompressible Navier-Stokes Equations Computer Graphics • Boundary conditions Bedřich Beneš, Ph.D. • Lagrange vs. Euler Purdue University • Eulerian approaches Department of Computer Graphics Technology • Lagrangian approaches • Shallow water • Conclusions © Bedrich Benes
Some terms Some terms • Advect: • Lagrangian: evolve some quantity forward in time methods that move fluid mass using a velocity field. For example (for example by advecting particles) particles, mass, etc. • Eulerian: • Convect: fluid quantities are defined on a grid transfer of heat by circulation of that is fixed movement of fluid. (the quantities can vary over time)
© Bedrich Benes © Bedrich Benes
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Equations Equations Fluids are governed by the incompressible • , , velocity of the fluid [ / ] Navier-Stokes Equations • (rho) fluid density / ] water ~1000 ∙ p υ ∙ (1) air ~1.3 ∙ 0 (2) • pressure [Pa] force per unit area that the fluid exerts © Bedrich Benes © Bedrich Benes
Equations Equations The momentum equation ( ) • 0, 9.81,0 accel. due gravity [ / ] How the fluid accelerates due to the forces
(assuming: acting on it gravity and other points to you, is up, is right) external forces drag 1 ∙ p υ ∙ • (upsilon) kinematics viscosity how difficult it is to stir viscosity
convection pressure acceleration (internal
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Equations Equations The momentum equation ( ) • Balance of momentum. gravity Internal + external forces = change in momentum.
• Conservation of energy. Kinetic + internal energy = const.
acceleration convection pressure drag
© Bedrich Benes © Bedrich Benes
Equations Equations Conservation of mass Conservation of mass Advecting mass through the velocity field 0 cannot change total mass.