Lagrange vs. Euler Fluid Equations...1 1 M Müller – Fluid Equations ETH Zurich Fluid Equations...
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1 1 M Müller – Fluid Equations ETH Zurich Fluid Equations Matthias Müller Seminar – Wintersemester 02/03 Movie 2 M Müller – Fluid Equations ETH Zurich Outline Euler vs. Lagrange Euler Equations Two Quantities - Two Equations Conservation of Mass Density Flow Rate Conservation of Momentum Newton’s Second Law of Motion Material Derivative Internal and External Forces Implementation Simplified Model for Compressible Fluids Finite Differences and Euler Integration 3 M Müller – Fluid Equations ETH Zurich Lagrange vs. Euler • displacement field u(x) • follows particle • we know where original particle is at any time u(x) • velocity field v(x) • is fixed in space • we don’t know where original particle is v(x) Eulerian (velocity) approach Lagrangian (displacement) approach 4 M Müller – Fluid Equations ETH Zurich Lagrange vs. Euler typically • computed on a mesh • Finite Element Method • elastic objects u(x) typically • computed on a (regular) grid • Finite Differences Method • fluids v(x) Eulerian (velocity) approach Lagrangian (displacement) approach
Transcript of Lagrange vs. Euler Fluid Equations...1 1 M Müller – Fluid Equations ETH Zurich Fluid Equations...
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1M Müller – Fluid Equations
ETH Zurich
Fluid Equations
Matthias MüllerSeminar – Wintersemester 02/03
Movie
2M Müller – Fluid Equations
ETH ZurichOutline
Euler vs. LagrangeEuler Equations
Two Quantities - Two EquationsConservation of Mass
Density Flow RateConservation of Momentum
Newton’s Second Law of MotionMaterial DerivativeInternal and External Forces
ImplementationSimplified Model for Compressible FluidsFinite Differences and Euler Integration
3M Müller – Fluid Equations
ETH ZurichLagrange vs. Euler
• displacement field u(x)• follows particle• we know where original particle
is at any time
u(x)
• velocity field v(x)• is fixed in space• we don’t know where original
particle is
v(x)
Eulerian (velocity) approachLagrangian (displacement) approach
4M Müller – Fluid Equations
ETH ZurichLagrange vs. Euler
typically• computed on a mesh• Finite Element Method• elastic objects
u(x)
typically• computed on a (regular) grid• Finite Differences Method• fluids
v(x)
Eulerian (velocity) approachLagrangian (displacement) approach
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5M Müller – Fluid Equations
ETH ZurichTwo Continuous Quantities
Scalar field ρ(x,t) [kg/m3]
ρ(x,t)
Vector field v(x,t) [m/s] v(x,t)
For incompressible fluids: ρ(x,t) ≡ 1
=
),,,(
),,,(
),,,(
),,,(
tzyxw
tzyxv
tzyxu
tzyxv
6M Müller – Fluid Equations
ETH ZurichTwo Equations
ρ(x,t)
v(x,t)0)( =⋅∇+∂∂
vρρt
Conservation of mass
local increase flow out
vgv 2∇++−∇= µρρ pDt
D
Conservation of momentum
“m⋅a / volume”
“force / volume”
At every point x:
7M Müller – Fluid Equations
ETH ZurichDensity Flow Rate
dxdy
dz
8M Müller – Fluid Equations
ETH ZurichDensity Flow Rate
w(x,y,z+dz)⋅dx⋅dy⋅ρ(x,y,z+dz) mass flow rate out
-w(x,y,z)⋅dx⋅dy⋅ρ(x,y,z) mass flow rate out
w(w(x,y,zx,y,z))
dxdy
dz
w(w(x,y,z+dzx,y,z+dz))
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9M Müller – Fluid Equations
ETH ZurichDensity Flow Rate
= [w(x,y,z+dz)⋅dx⋅dy⋅ρ(x,y,z+dz)- w(x,y,z)⋅dx⋅dy⋅ρ(x,y,z)] /(dx⋅dy⋅dz)
= [w(x,y,z+dz)⋅ρ(x,y,z+dz) -w(x,y,z)⋅ρ(x,y,z)] /dz
= ∂/∂z (w⋅ρ)
w(x,y,z+dz)⋅dx⋅dy⋅ρ(x,y,z+dz) mass flow rate out
-w(x,y,z)⋅dx⋅dy⋅ρ(x,y,z) mass flow rate outw(w(x,y,zx,y,z))
dxdy
dz
w(w(x,y,z+dzx,y,z+dz))
net density flow rate out
10M Müller – Fluid Equations
ETH ZurichTotal Density Flow Rate
= ∇⋅ (ρv)= div (ρv)
w(w(x,y,zx,y,z))
dxdy
dz
w(w(x,y,z+dzx,y,z+dz)) total net density flow rate out
= ∂/∂x (u⋅ρ) + ∂/∂y (v⋅ρ) + ∂/∂z (w⋅ρ)
density increase rate inside= ∂ρ/∂t
0)( =⋅∇+∂∂
vρρt
mass conservation:
0=⋅∇ v
incompressible:
11M Müller – Fluid Equations
ETH ZurichMomentum Conservation
Newton’s second law per unit volume (Navier-Stokes, simplified!):
“m⋅a per volume” = “force per volume”
Material derivative of velocity (following the fluid):
vgv 2∇++−∇= µρρ pDt
D
t
v
tw
zv
yu
x
t
t
tt
z
zt
y
yt
x
xttztytx
tDt
D
∂∂+∇⋅=
∂∂+⋅
∂∂+⋅
∂∂+⋅
∂∂=
∂∂⋅
∂∂+
∂∂⋅
∂∂+
∂∂⋅
∂∂+
∂∂⋅
∂∂=
∂∂=
vv
vvvv
vvvvv
v)),(),(),((
12M Müller – Fluid Equations
ETH ZurichExternal Forces
Newton’s second law per unit volume:“m⋅a per volume” = “force per volume”
External forces:
vgv 2∇++−∇= µρρ pDt
D
• gravity force = mass ⋅ acceleration per volume• other user applied forces (force per volume!)
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13M Müller – Fluid Equations
ETH ZurichPressure
Newton’s second law per unit volume:“m⋅a per volume” = “force per volume”
vgv 2∇++−∇= µρρ pDt
D
p
p
p
p
f
f
f
z
y
x
zzyzyxzx
zyzyyxyx
zxzyxyxx
z
y
x
∇=
=
++++++
=
=
,
,
,
,,,
,,,
,,,
int
στττστττσ
f
=
=p
p
p
zzyzx
yzyyx
xzxyx
00
00
00
pressure
στττστττσ
σ
14M Müller – Fluid Equations
ETH ZurichPressure
Newton’s second law per unit volume:“m⋅a per volume” = “force per volume”
vgv 2∇++−∇= µρρ pDt
D
What is p?• for isotermal fluids:• pV = const• p = k/V = kρ
15M Müller – Fluid Equations
ETH ZurichViscosity
Newton’s second law per unit volume:“m⋅a per volume” = “force per volume”
vgv 2∇++−∇= µρρ pDt
D
++++++
=∇
zzyyxx
zzyyxx
zzyyxx
www
vvv
uuu
,,,
,,,
,,,
2 µµ v
Simplified viscosity force for incompressible fluids:
Scalar µ is the fluid’s viscosity
16M Müller – Fluid Equations
ETH ZurichPutting it all together
vgvvv 2∇++∇−=
∇⋅+
∂∂ µρρρ kt
0)( =⋅∇+∂∂
vρρt
Rearrange to get rates of change:
vgvvv 2∇++∇−∇⋅−=
∂∂
ρµρ
ρk
t
)( vρρ ⋅−∇=∂∂t
So far we have:
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17M Müller – Fluid Equations
ETH ZurichFinite Difference Integration
On a uniform grid using finite differences we get:
vgvvv 2∇++∇−∇⋅−=
∂∂
ρµρ
ρk
t
)( vρρ ⋅−∇=∂∂t
h
vw
h
vvh
uu
t
kjikjikjikjikjikjikjikji
kjikjikjikjikji
1,,1,,,,,,,1,,1,,,,,
,,1,,1,,,,,,
−−−−
−−
−−
−−
−−=
∂∂
ρρρρ
ρρρ
Similar for change of velocity(see “A Fluid-Based Soft-Object Model”, IEEE Computer Graphics and Applications July 2002)
18M Müller – Fluid Equations
ETH ZurichTime Integration
Time step using Euler Integration:
tt
tt
kjikjikji
kjikjikji
∂∂
∆+=
∂∂
∆+=
,,,,,,
,,,,,,
vvv
ρρρ
Done! ☺
How to get a surface?• let particles flow with velocity field• particles define potential• use levelset method
