Fluid Dynamics

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Fluid Dynamics

• David Marshburn
• Comp 259
• April 17, 2002

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Fluid properties

• Imagine a volume of fluid
• position, velocity, acceleration
• viscosity µ
• density ?

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Fluid velocity

• Velocity is the physical property simulated for
  fluids
• Why? Were usually interested in what the
  fluids carrying. Advection.
• Velocity is denoted by u in fluid dynamics
  literature (even in graphics).

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Taxonomy of Fluids

• Compressible vs. incompressible
• constancy of density
• Rotational vs. irrotational
• whether small volumes have angular velocity
• Viscous vs. inviscid
• whether shear forces are present
• Newtonian vs. non-Newtonian
• model for viscous force
• We will derive a model for incompressible,
  Newtonian, irrotational, viscous fluids.

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Fluid dynamics

• In the beginning, there was Newton F ma
• So what forces are there on a fluid?

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Forces on a Fluid

• Imagine a small volume of fluid (so we get
  forces per unit volume)
• external or body forces (e.g., gravity)
• relative pressure
• viscous friction from other bits of fluid
  sliding by
• inertia (not really a force, but needs some
  special treatment)

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Getting rid of volume

• We want Newtons 2nd in terms of forces per unit
  volume, so F/V m/V a
• but, m/V is just the density ?, so f/? a

Well talk about forces per unit volume hereafter.
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Body forces

• Gravity
• Rigid objects
• Other forces external to the fluid
• Denote the conglomeration of these forces by g, a
  force per unit mass.

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Pressure

• Pressure (denoted p, a force per unit volume) in
  one tiny bit of fluid is relative to the pressure
  in neighboring tiny bits.

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Viscosity

• Friction from other bits of fluid sliding by.

From Chorin Marsden. B and B are two blobs of
fluid
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Viscosity

• For instance, we want the difference in
  z-velocity as we look in the x direction.
• This generalizes in all dimensions to the
  Laplacian.

Note that this is the Laplacian for a
vector-valued field, not a scalar-valued field.
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Acceleration

• Our little bit of fluid is moving along at some
  velocity u.
• Two components of acceleration
• temporal change in velocity
• motion of the bit of fluid

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Acceleration

• Temporal change in velocity

• Movement of the bit of fluid (inertia)

Where the ui are the velocities in the x, y and z
directions
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Navier-Stokes equation 1

• Putting this all together

Inertia
Viscosity
Acceleration
Pressure
External forces
? is µ/? and is called the kinematic viscosity.

• This is conservation of energy.

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Navier-Stokes equation 2

• Were talking about incompressible fluids..
• So, the velocity into our little bit of fluid
  must be the same as the velocity out

• This is conservation of mass.
• That the divergence is 0 states incompressibility.

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No-slip condition

• At the rigid, stationary boundaries of a fluid,
  velocity is zero. (experimentally and
  mathematically)
• At non-stationary boundaries, the fluid velocity
  must be the same as that of the boundary.

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Questions?

• Any questions about how we got to the
  Navier-Stokes equations?

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Solving these

• So, we have some differential equations
• We have four equations and four unknowns
• Whats the problem?
• Second order
• Non-linear

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Foster/Metaxas 1996

• Realistic Animation of Liquids
• A finite differencing approximation with
  correction.
• Regular, rectilinear discretization

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Foster/Metaxas 1996

• Finite differencing approximation (1 dimension
  shown)

• The point is that is the energy-conservation
  equation with all the differentials replaced by
  finite differences.

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Foster/Metaxas 1996

• Conservation of mass isnt assured.
• Correction Relax pressure and velocity until
  all cells satisfy both Navier-Stokes equations
  (to within some tolerance).

means unconserved mass
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Foster/Metaxas 1996

• Each cell looks at its neighbors
• So, stuff shouldnt move more than one cell in a
  time step.
• Two possibilities
• The largest velocity anywhere in the system
  determines an adaptive time step
• For some fixed time step, the simulation
  eventually blows up.

This causes instability.
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Stam 1999

• Stable Fluids
• Important features
• Semi-Lagrangian advection.
• Implicit solvers
• Projection

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Stam 1999

• Semi-Lagrangian advection (called the method of
  characteristics).
• Resolves the non-linearity
• To find the velocity as some point, trace the
  velocity field backwards in time from that point
  along the path p.

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Stam 1999

• Method of Characteristics
• A characteristic is a curve through a vector
  field on which a constant field element
  propagates.
• Given the equation
• Turn the PDE into some ODEs by taking
  uu(x(s),t(s)) and using the chain rule to find
  du/ds0
• Integrate with your favorite scheme.

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Stam 1999

• Implicit solver for

• In implicit form, this is

• Write this down as a finite difference equation
  and solve with the POIS3D linear solver from
  FISHPACK.

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Stam 1999

• Projection to ensure that the mass conservation
  condition is met.
• The Helmholtz-Hodge Decompostion (a result from
  vector algebra) any vector field1 can be
  uniquely decomposed as

• w and u are vector fields, u is divergence-free,
  and q is a scalar field.
• Solve for q and subtract if from the result.

1There are some well-behaved constraints on the
field.
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Stam 1999

• These methods are chained together to solve the
  Navier-Stokes equations.
• Stability stable for any time step
• In the advection step, the largest velocity
  generated is bounded by the maximum velocity in
  the earlier field.

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References

• Chorin, Alexandre J. and Jerrold E. Marsden, A
  Mathematical Introduction to Fluid Mechanics.
  3rd ed. Springer 1993.
• Acheson, D.J. Elementary Fluid Dynamics. Oxford
  University Press 1990.
• Foster, Nick, and Dimitri Metaxas. Realistic
  Animation of Liquids. Graphics Models and Image
  Processing. 58(5)471-483, 1996.
• Stam, Jos. Stable Fluids. SIGGRAPH 1999.