Transcript Taxonomy of Fluids

Fluid Dynamics
David Marshburn
Comp 259
April 17, 2002
1
Fluid properties
• Imagine a volume of fluid…
• position, velocity, acceleration
• viscosity μ
• density ρ
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Fluid velocity
• Velocity is the physical property simulated
for fluids…
• Why? We’re usually interested in what the
fluid’s 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 Newton’s 2nd in terms of forces per
unit volume, so…
F/V = m/V a
• but, m/V is just the density ρ, so…
f/ρ = a
We’ll 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.
f external   g
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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.
f pressure  p
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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 zvelocity as we look in the x direction.
• This generalizes in all dimensions to the
Laplacian.
f viscosity    u
2
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
u
t
• Movement of the bit of fluid (inertia)
u
u
u
ux
 uy
 uz
 u   u
x
y
z
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:
u
1
2
 u   u   p   u g
t

Inertia
Acceleration
Viscosity
Pressure
External forces
ν is μ/ρ and is called the kinematic viscosity.
• This is conservation of energy.
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Navier-Stokes equation #2
• We’re talking about incompressible fluids..
• So, the velocity into our little bit of fluid
must be the same as the velocity out…
u  0
• 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
• What’s the problem?
– Second order
– Non-linear
u
1
2
 u   u   P   u  f
t

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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 isn’t assured.
• Correction: Relax pressure and velocity
until all cells satisfy both Navier-Stokes
equations (to within some tolerance).
u  0
means unconserved mass
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Foster/Metaxas 1996
• Each cell looks at its neighbors…
• So, stuff shouldn’t 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 u  u
• 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.
u
 u   u  0
• Given the equation: t
• Turn the PDE into some ODEs by taking
u=u(x(s),t(s)) and using the chain rule to
find du/ds=0
• Integrate with your favorite scheme.
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Stam 1999
• Implicit solver for:
u
2
  u
t
• In implicit form, this is:
I t 2 ui1  ui
• 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  u  q
• w and u are vector fields, u is divergencefree, 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, Joe, “Stable Fluids.” SIGGRAPH 1999.
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