Volume of fluid methods for
compressible multiphase flow
Keh-Ming Shyue
Department of Mathematics National Taiwan University
Taiwan
Objective
Discuss a class of volume of fluid (vs. level set, MAC,
particles) methods for interface problems with application to compressible multiphase flow
1. Adaptive moving grid approach (this lecture) Cartesian grid embedded volume tracking
Useful also for turbulent flow (Glimm & Abarzhi’s talk)
Moving mapped grid interface capturing
Simple method if mesh handling is robust
2. Eulerian interface sharpening approach (next lecture) Artificial interface compression method
Anti-diffusion method
Incompressible 2-phase flow
One model for free-surface (or 2-phase) flow can be described by incompressible Navier-Stokes equations:
∇ · ~u = 0 (Continuity)
∂t (ρ~u) + ∇ · (ρ~u ⊗ ~u) + ∇p = ∇ · τ + ρ~g + ~fσ (Momentum)
∂tα + ~u · ∇α = 0 (Volume fraction transport ) Material quantities in 2-phase coexistent region are often
computed by α-weighted average (α ∈ [0, 1]) as
ρ = αρ1 + (1 − α) ρ2, ǫ = αǫ1 + (1 − α) ǫ2, where
τ = ǫ ∇~u + ∇~uT
, f~σ = −σκ∇α with κ = ∇ ·
∇α
|∇α|
Incompressible 2-phase flow
Accurate advection scheme for volume fraction α is of fundamental importance. Typical methods employed are
Simple line interface calculation (SLIC) Donor acceptor scheme
Flux corrected transport method Youngs (PLIC) method
Level set method
CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes)
THINC (Tangent of Hyperbola for INterface Capturing) or its variant
Compressible flow in porous medium
One model for gas flow in variable porous medium can be described by compressible Euler equations:
∂t (αρ) + ∇ · (αρ~u) = 0
∂t (αρ~u) + ∇ · (αρ~u ⊗ ~u) + ∇ (αp) = p∇α
∂t (αρE) + ∇ · (αρE~u + αp~u) = 0
Embeded discontinuous porosity α ∈ [0, 1]
Well-balanced & entropy solution due to non-conservative term
Baer-Nunziato 2-phase model
Known 2-phase mixture model for deflagration-to-detonation transition (DDT) in reactive granular materials
(α1ρ1)t + ∇ · (α1ρ1~u1) = 0
(α1ρ1~u1)t + ∇ · (α1ρ1~u1 ⊗ ~u1) + ∇(α1p1) = p0∇α1 + λ(~u2 − ~u1) (α1ρ1E1)t + ∇ · (α1ρ1E1~u1 + α1p1~u1) = p0 (α2)t + λ~u0 · (~u2 − ~u1) (α2ρ2)t + ∇ · (α2ρ2~u2) = 0
(α2ρ2~u2)t + ∇ · (α2ρ2~u2 ⊗ ~u2) + ∇(α2p2) = p0∇α2 − λ(~u2 − ~u1)
(α2ρ2E2) t + ∇ · (α2ρ2E2~u2 + α2p2~u2) = −p0 (α2)t − λ~u0 · (~u2 − ~u1) (α2)t + ~u0 · ∇α2 = µ (p2 − p1)
αk = Vk/V : volume fraction (α1 + α2 = 1), ρk: density,
~uk: velocity, pk: pressure, Ek = ek + ~u2k/2: specific total energy, ek: specific internal energy, k = 1, 2
Baer-Nunziato 2-phase model
Quantities p0 & ~u0 denote interfacial pressure & velocity Baer & Nunziato (1986)
p0 = p2, ~u0 = ~u1 Saurel & Abgrall (1999)
p0 = P2
k=1 αkpk, ~u0 = P2
k=1 αkρk~uk
P2
k=1 αkρk
Quantities λ & µ (> 0) are velocity & pressure relaxation parameters that give rates to equilibrium for velocitiy &
pressure, respectively
Reduced 2-phase model
For some problems, it is simple to use reduced 2-phase model of Kapila, Menikoff, Bdzil, Son, & Stewart (2001)
∂t (α1ρ1) + ∇ · (α1ρ1~u) = 0
∂t (α2ρ2) + ∇ · (α2ρ2~u) = 0
∂t (ρ~u) + ∇ · (ρ~u ⊗ ~u) + ∇p = 0
∂t (ρE) + ∇ · (ρE~u + p~u) = 0
∂tα1 + ~u · ∇α1 = ρ2c22 − ρ1c21 P2
k=1 ρkc2k/αk ∇ · ~u
Model is zero-order approximation of Baer-Nunziato model with stiff mechanical relaxation
Positivity & sharpness of α is important for accurate solution
Cartesian embedded boundary method
Towards volume tracking for sharp interface resolution, we consider conservation laws
∂tq + ∇ · ~f (q) = 0
with embedded boundary (i.e., interface with rigid BCs)
q ∈ lRm: vector of m conserved quantities
f = (f~ 1, f2, . . . , fd) ∈ lRm×d: flux matrix; fj ∈ lRm
Γ: embedded boundary
Γ →
Finite-volume discretization
Finite volume formulation of solution QnS gives approximate value of cell average q over cell S at time tn
QnS ≈ 1 M(S)
Z
S
q(x, tn) dV = 1
κS (∆x)d Z
S
q(x, tn) dV M(S): measure (area in 2D
or volume in 3D) of cell S κS = M(S)/ (∆x)d: volume fraction
hkS = M(∂Sk)/∆x(d−1): area fraction
Assume uniform discretiza- tion ∆x = ∆x , j = 1, . . . , d
A B
C D
Finite-volume discretization
Divergence theorem over each control volume leads to finite volume approximation for
∇ · ~f (q)
S ≈ 1
M(S) Z
S
∇ · ~f (q) dV = 1 M(S)
Z
∂S
f (q) · ~n dA~
= 1
κS∆x
X
k∈∂S
hkSf (q)~ k · ~nk = 1 κS∆x
X
k∈∂S
hkSF˘k
≡ DS · ~f (q)
Away from boundaries,
method reduces to standard conservative finite difference discretization
τ∆x =
∇ · ~f
S − DS · ~f = O(∆xp−1/κ ) if F˘k = O(∆p)
A B
C D
Finite-volume discretization
For conservation laws, On each control volume S, a fully discretized finite volume method can be written as
Qn+1S = QnS − ∆tDS · ~f (q) = QnS − ∆t κS∆x
X
k∈∂S
hkSF˘k
which satisfies discrete conservation identity X
∪S
κSQn+1 = X
∪S
κSQn − ∆t
∆x
X
∪S
X
k∈∂S
hkSF˘k
Method is subject to stringent CFL stability constraint
∆t = O
(κmin)1/d∆x
|λ|max
≪ 1 if κmin ≪ 1 (Small-cell problem)
Small-cell problem
Methods proposed for small-cell problem include Non-conservation approach: Glimm & coworker
Flux-redistribution approach: Chern, Colella, & Modiano Cell-merging approach: Quirk, Powell, Causon
H-box approach: Berger, Helzel, LeVeque (see below)
Q¯
R( ¯Q) Q¯R
Q¯L
Flux calculation: Wave propagation
On 2D uniform grid, high-resolution in fluctutation form Qn+1ij = Qnij − ∆t
∆x1
A+1 ∆Qi−1
2,j + A−1 ∆Qi+1
2,j
− ∆t
∆x2
A+2 ∆Qi,j−1
2 + A−2 ∆Qi,j+1
2
− ∆t
∆x1
F˜i+1 1
2,j − ˜Fi−1 1 2,j
− ∆t
∆x2
F˜i,j+2 1 2
− ˜Fi,j−2 1 2
A+1 ∆Q
A−1 ∆Q
A−2 A+1 ∆Q
A+2 A−1 ∆Q
A+2 A+1 ∆Q
A−2 A−1 ∆Q
Flux calculation: Wave propagation
Fluctuation terms in wave propagation form are obtained from solving Riemann problems at each cell edge, yielding speeds λj,m & waves Wj,k, j = 1, 2, k = 1, . . . , mw,
A±1 ∆Q =
mw
X
k=1
λ1,k±
W1,k, A±2 ∆Q =
mw
X
k=1
λ2,k±
W2,k
A±2 A±1 ∆Q =
mw
X
p=1
λ1,p±
A±2 W1,p =
mw
X
p=1
λ1,p± Xmw
q=1
λ2,q±
W2,q
F˜1 := F˜1 + 1 2
mw
X
k=1
λ1,k
1 − ∆t
∆x1
λ1,k
Wf1,k
To aviod oscillations near discontinuities, a wave limiter is applied leading to limited waves Wf
Wave propagation with boundary
First order upwind with normal-wave update
Solve Riemann problem in direction normal at cell edge,
& use resulting waves to update nearby cell averages Qn+1S := Qn+1S − M (Zp ∩ S)
M(S) Wp
Z↓p
A B
C
D Z↓p
A B
C D
Wave propagation with boundary
First order upwind with transverse-wave update
Use transverse portion of equation, solve Riemann
problem in transverse direction, & use resulting waves to update neighboring cell averages as usual
Stability of method is typically improved, while conservation of method is maintained
Z↓pq
A B
C
D Z↓pq
A B
C D
Wave propagation with boundary
High resolution with piecewise linear wave update
wave before propagation after propagation
a) b)
c) d)
αprp/2
αprp/2
λp∆ t
λp∆ t
Moving boundary tracking
With a prescribed moving-object velocity ~ub at current time, tracked boundaries are advanced in a Lagrangian manner over time step ∆t
Final moving boundary locations are set by where new segments intersect grid lines; interpolation is required with care to initialize cut-cell values
DC BA new object↓
↑
old object
Embedded boundary condition
Rigid embedded boundary (stationary or moving), reflection principle is used to assign states for fictitious subcells, i.e.,
z := zE (z = ρ, p, α)
~u := ~uE − 2(~uE · ~n)~n + 2(~ub · ~n), where ~ub is velocity of object
If motion of object is governed by Newton’s second law, velocity of object ~un+1b at time step tn+1 is computed by
~un+1b = ~unb + ∆t
MbF~bn,
where Mb is mass of object & F~bn is force on object
Moving piston problem
Numerical results agree well with exact solution Piston moves with speed ~ub = (0, 300)m/s
0 0.5 1
0.5 1 1.5 2
x (m) ρ (kg/m3 )
0 0.5 1
0 0.05 0.1 0.15 0.2 0.25 0.3
x (m)
u (km/s)
0 0.5 1
0.5 1 1.5 2 2.5
x (m)
p (Bar)
piston
Cylinder lift off
Cylinder motion is governed by Newton’s law
Pressure contours are shown with a 1000 × 200 grid
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.05 0.1 0.15 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.05 0.1 0.15 0.2
0 0.05 0.1 0.15 0.2
t = 0
t = 0.1641s
t = 0.30085s
Cylinder lift off
A convergence study of center of cylinder & relative mass loss at final stopping time t = 0.30085s
Mesh size Center of cylinder Relative mass loss 250 × 50 (0.618181, 0.134456) −0.257528
500 × 100 (0.620266, 0.136807) −0.131474 1000 × 200 (0.623075, 0.138929) −0.066984
Results are comparable with numerical appeared in literature
Falling rigid object in water tank
−2 0 2
−3
−2
−1 0 1 2 3
−2 0 2
−3
−2
−1 0 1 2 3
−2 0 2
−3
−2
−1 0 1 2
Density Pressure 3 Volume fraction air
water t = 1ms
−2 0 2
−3
−2
−1 0 1 2 3
−2 0 2
−3
−2
−1 0 1 2 3
−2 0 2
−3
−2
−1 0 1 2 3
air
water t = 3ms
VOF interface tracking
Given volume fractions, VOF tracking consists of:
1. Volume fraction update
Take a time step on current grid using wave propagation method to update volume fractions α governed by
∂tα + ~u · ∇α = 0
at next time step; ~u underlying velocity field
2. Interface reconstruction (by PLIC or MOF, see Z. Jia’s talk at 17:40-18:00, 9/15)
Standard PLIC method does:
Compute interface normal, using method such as Gradient or least squares of Youngs or Puckett Determine interface location by iterative bisection
Interface reconstruction: Example
Cell-averaged volume fractions (left) & reconstructed interface (right); a set of disconnected line segments
0 0
0 0
0 0
0 0
0 0
0
0 0
0 0
0 0
0 0
0
1 0.29
0.68 0.09
0.51 0.51
0.09 0.68
0.29 interface↓
Interface reconstruction: Example
With α in previous slide & ~u = (1, 1) updated volume fraction over ∆t = 0.06 shown here ( κ1 = 5.7 × 10−3 &
κ2 = 1.3 × 10−3)
New reconstructed interface location
0
0 0 0 0
0 0 0 0
0 0
0 0
0 0
0 0
0
1 0.01
0.38 0.11
0.25 0.72
0.06 0.85
0.74 κ1 κ2
↑
↓
old interface
new interface
Solution update
Solution is double-valued on cells subdivided by VOF tracked interface; Again use finite volume formulation of solution QnS for approximate value of solution q
QnS ≈ 1 M(S)
Z
S
q(x, tn) dV
C E
D F
G H
Wave propagation method
Normal- & transverse-edge wave propagation
↓
↓
Zp Zp
↓ ↓
Zpq Zpq
Stability issues
Choose time step ∆t based on uniform mesh size ∆x as ν = |λ|max∆t
∆x ≤ 1
Use wave propagation method described above which expands range of influence of solution to maintain
stability of method even in presence of small Cartesian cut cells
Use smoothing operator (such as, h-box approach of Berger et al. ) locally for cell averages in irregular cells for accuracy
Shock in air & R22 bubble
Shock in air & R22 bubble
Shock in air & R22 bubble
Shock in air & R22 bubble
Quantitative assessment of prominent flow velocities:
Velocity (m/s) Vs VR VT Vui Vuf Vdi Vdf Haas & Sturtevant 415 240 540 73 90 78 78 Quirk & Karni 420 254 560 74 90 116 82 Our result (tracking) 411 243 538 64 87 82 60 Our result (capturing) 411 244 534 65 86 98 76
Vs (VR, VT) Incident (refracted, transmitted) shock speed t ∈ [0, 250]µs (t ∈ [0, 202]µs, t ∈ [202, 250]µs ) Vui (Vuf) Initial (final) upstream bubble wall speed t ∈ [0, 400]µs (t ∈ [400, 1000]µs)
Vdi (Vdf) Initial (final) downstream bubble wall speed t ∈ [200, 400]µs (t ∈ [400, 1000]µs)
Underwater explosion
Flying Aluminum plate
Track vacuum-Al boundary Impact velocity 400m/s
Pressure contours are shown
Al target Al flyer
vacuum
time = 0.5µs time = 1µs time = 2µs time = 4µs
Moving mapped grid method
Towards sharp interface resolution, one may also consider moving mapped grid in that hyperbolic conservation laws
∂q
∂t +
Xd j=1
∂fj(q)
∂xj = 0
in transformed coordinates (x, t) 7→ (ξ, τ ) becomes conservation laws with patially varying fluxes:
∂
∂τ (Jq) +
Xd j=1
∂
∂ξj Jq ∂ξj
∂t +
Xd k=1
Jfk(q) ∂ξj
∂xk
!
= 0
To close system, conditions on grid motion ∂tξ & so grid metrics ∂ξj/∂xk with J = det (∂ξ/∂x)−1 should be set
Computing geometric quantities
To set ∂tξ, ∂ξj/∂xk, & J, one may use Hui’s unified coordinate approach
∂xi
∂τ = hui & ∂
∂τ
∂xi
∂ξj
+ ∂
∂ξj
−∂xi
∂τ
= 0
for i, j = 1, 2, . . . , d & compute J = det (∂ξ/∂x)−1 directly.
Quantity h may be kept as a constant or varied with time by geometric measure
Variant ALE approach, see next talk by B. Chen or B.
Tian’s talk at 11:50-12:10, 9/15
∂J
∂τ +
Xd ∂
∂ξj
J ∂ξj
∂t
= 0
Reduced 2-phase model (revisit)
Physical balance laws
∂
∂τ
α1ρ1J α2ρ2J ρuiJ
ρEJ
+
Xd j=1
∂
∂ξj J
α1ρ1Uj α2ρ2Uj ρuiUj + p∂x∂ξj
i
ρEUj + pUj − p∂ξ∂tj
= 0
∂α1
∂τ +
Xd j=1
Uj ∂α1
∂ξj = ρ2c22 − ρ1c21 P2
k=1 ρkc2k/αk
Xd j=1
∂Uj
∂ξj Uj = ∂tξj + Pd
i=1 ui∂xiξj: ξj’s contravariant velocity
Mixture equation of state: p = p(α2, α1ρ1, α2ρ2, ρe) with isobaric closure: p1 = p2 = p
Numerical approximation
To ensure free-stream (constant states) preservation, which is one of basic properties that a moving mesh method
should have, a consistent discretization of
∂J
∂τ +
Xd j=1
∂
∂ξj
J ∂ξj
∂t
= 0
is required which may be difficult for 2D quadrilateral or 3D hexahedral mesh
In this regards, non-conservative form is often employed
∂q
∂τ + 1 J
Xd j=1
∂
∂ξj
Xd
k=1
Jfk(q) ∂ξj
∂xk + Jq∂ξj
∂t
!
= 0
PLIC-VOF adaptive moving mesh
We remark that Menciger & ˘Zun (JCP 2011) developed a PLIC-VOF adaptive moving grid method in that monitor function D employed in elliptic grid generator, say in form proposed by Ceniceros & Hou (JCP 2001),
∇ξ · (D∇ξxj) = 0, j = 1, . . . , d
for x(ξ), make use of volume fraction α explicitly, i.e.,
∂D
∂τ = ε∇2D with D(x, 0) = 1 + βM (∇α) & D|∂Ω = 0 where M (z) = |z|R
dV / R
|z|dV
This can be used readily in moving-mesh method developed by T. Tang & many others
Woodward-Colella’s problem
0 0.5 1
0 2 4 6 8
ρ
0 0.5 1
−10 0 10 20
0 0.5 1
0 100 200 300 400
u p
t = 0.016 t = 0.016
t = 0.016
0 0.5 1
0 5 10 15 20
ρ
Fine grid h0=0 h0=0.99
0 0.5 1
−5 0 5 10 15
0 0.5 1
0 200 400 600
u p
t = 0.032 t = 0.032
t = 0.032
0 2 4 6 8
ρ
−5 0 5 10 15
0 200 400 600
u p
t = 0.038 t = 0.038
t = 0.038
Oscillating water column
Initially, in closed shock tube, water column moves at u = 1 from left to right, yielding air compression at right
& air expansion at left
Subsequently, pressure difference built up across water column resulting deceleration of column of water to
right, makes a stop, & then acceleration to left; a reverse pressure difference built up across water column redirecting flow from left to right again
Eventually, water column starts to oscillate
air
air water
u →
Oscillating water column
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time
oscilw grid
0 1 2 3 4 5 6 7 8 9 10
0.8 1 1.2 1.4 1.6 1.8
pL
0.8 1 1.2 1.4 1.6 1.8
pR
mmesh mesh0
mmesh mesh0
Water-vapor cavitation
Initially, in closed shock tube, flow is homogeneous (contains α = 10−6 gas in bulk liquid) at standard
atmospheric condition & exists interface separating flow with opposite motion (u = 100 m/s)
Result in pressure drop & formation of cavitation zone in middle; shocks form also from both ends
Eventually, shock-cavitation collision occurs
−u u
← Inteface
Water-vapor cavitation
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1x 10−3
time
cavi grid
Underwater explosions
t=0.2ms
t=0.4ms
t=0.8ms
t=1.2ms
Moving mesh Eulerian mesh air
water
Underwater explosions
Moving mesh (coarsen by factor 5)
−1 0 1
−1
−0.5 0 0.5
−1 0 1
−1
−0.5 0 0.5
time = 0.2ms time = 0.4ms
−1
−0.5 0 0.5
−1
−0.5 0 0.5
time = 0.8ms time = 1.2ms
Accuracy test in 2D
Consider 2D compressible Euler equations with ideal gas law as governing equations
Take smooth vortex flow with initial condition
ρ =
1 − 25(γ − 1)
8γπ2 exp (1 − r2)
1/(γ−1)
, p = ργ,
u1 = 1 − 5
2π exp ((1 − r2)/2) (x2 − 5) , u2 = 1 + 5
2π exp ((1 − r2)/2) (x1 − 5) ,
& periodic boundary conditions as an example, where r = p
(x1 − 5)2 + (x2 − 5)2
Accuracy test in 2D
Grids used for this smooth vortex flow test
0 5 10
0 2 4 6 8 10
0 5 10
0 2 4 6 8 10
0 5 10
0 2 4 6 8
Grid 1 Grid 2 10 Grid 3
kEzk1,∞ = kzcomput − zexactk1,∞ denotes discrete 1- or maximum-norm error for state variable z
Results shown below are at time t = 10 on N × N mesh
Accuracy results in 2D: Grid 1
N E1(ρ) Order E1(u1) Order E1(u2) Order E1(p) Order
40 0.6673 2.3443 1.7121 0.8143
80 0.1792 1.90 0.6194 1.92 0.4378 1.97 0.2128 1.94 160 0.0451 1.99 0.1537 2.01 0.1104 1.99 0.0536 1.99 320 0.0113 2.00 0.0384 2.00 0.0276 2.00 0.0134 2.00
N E∞(ρ) Order E∞(u1) Order E∞(u2) Order E∞(p) Order
40 0.1373 0.3929 0.1810 0.1742
80 0.0377 1.87 0.1014 1.95 0.0502 1.85 0.0482 1.85 160 0.0093 2.02 0.0248 2.03 0.0123 2.03 0.0119 2.02 320 0.0022 2.07 0.0062 2.00 0.0030 2.04 0.0029 2.04
Accuracy results in 2D: Grid 2
N E1(ρ) Order E1(u1) Order E1(u2) Order E1(p) Order
40 0.9298 2.6248 2.1119 1.2104
80 0.2643 1.81 0.7258 1.85 0.5296 2.00 0.3277 1.89 160 0.0674 1.97 0.1833 1.99 0.1309 2.02 0.0845 1.96 320 0.0169 2.00 0.0458 2.00 0.0327 2.00 0.0212 1.99
N E∞(ρ) Order E∞(u1) Order E∞(u2) Order E∞(p) Order
40 0.1676 0.4112 0.2259 0.2111
80 0.0471 1.83 0.1242 1.73 0.0645 1.79 0.0586 1.85 160 0.0126 1.91 0.0333 1.90 0.0162 2.02 0.0149 1.97 320 0.0033 1.93 0.0085 1.97 0.0040 2.00 0.0038 1.98
Accuracy results in 2D: Grid 3
N E1(ρ) Order E1(u1) Order E1(u2) Order E1(p) Order
40 4.8272 4.7734 5.3367 5.4717
80 1.5740 1.62 1.5633 1.61 1.5660 1.77 1.5634 1.81 160 0.4536 1.79 0.4559 1.78 0.4537 1.79 0.4560 1.78 320 0.1215 1.90 0.1221 1.90 0.1222 1.89 0.1221 1.90
N E∞(ρ) Order E∞(u1) Order E∞(u2) Order E∞(p) Order
40 0.4481 0.4475 0.4765 0.4817
80 0.1170 1.94 0.1181 1.92 0.1196 1.99 0.1191 2.02 160 0.0434 1.43 0.0431 1.45 0.0442 1.43 0.0440 1.44 320 0.0117 1.89 0.0119 1.86 0.0119 1.89 0.0118 1.89
Future perspectives
Extension of Cartesian embedded boundary method in 3D will be based on either FronTier or Chombo
framework developed by Stony Brook, or ANAG, LBL Free-stream perserving PLIC-VOF moving mesh
method is under development
Low Mach solver for cavitation problems is ongoing
Thank you
Basic grid-metric relations
Assume existence of inverse transformation
t = τ, xj = xj(~ξ, t) for j = 1, 2, . . . , d,
To find basic grid-metric relations between different coordinates, employ elementary differential rule
∂(τ, ~ξ)
∂(t, ~x) = ∂(t, ~x)
∂(τ, ~ξ)
−1
yielding in d = 3 case, for example, as
1 0 0 0
∂tξ1 ∂x1ξ1 ∂x2ξ1 ∂x3ξ1
∂tξ2 ∂x1ξ2 ∂x2ξ2 ∂x3ξ2
= 1 J
J 0 0 0
J01 J11 J21 J31 J02 J12 J22 J32
Basic grid-metric relations
Here
J =
∂(x1, x2, x3)
∂(ξ1, ξ2, ξ3)
= det
∂(x1, x2, x3)
∂(ξ1, ξ2, ξ3)
, J11 =
∂(x2, x3)
∂(ξ2, ξ3)
, J21 =
∂(x1, x3)
∂(ξ3, ξ2)
, J31 =
∂(x1, x2)
∂(ξ2, ξ3) ,
J12 =
∂(x2, x3)
∂(ξ3, ξ1)
, J22 =
∂(x1, x3)
∂(ξ1, ξ3)
, J32 =
∂(x1, x2)
∂(ξ3, ξ1) ,
J13 =
∂(x2, x3)
∂(ξ1, ξ2)
, J23 =
∂(x1, x3)
∂(ξ2, ξ1)
, J33 =
∂(x1, x2)
∂(ξ1, ξ2) ,
J0j = −
Nd
X
i=1
Jij∂τxi, j = 1, 2, 3,
and so grid-metric relations between different coordinates
∇ξj = (∂tξj, ∇~xξj) = (∂tξj, ∂x1ξj, ∂x2ξj, ∂x3ξj) = 1
J (J0j, J1j, J2j, J3j)
Basic grid-metric relations
Note in two dimensions d = 2, we have
∂ξ1
∂t , ∂ξ1
∂x1 , ∂ξ1
∂x2
= 1 J
−∂x1
∂τ
∂x2
∂ξ2 + ∂x2
∂τ
∂x1
∂ξ2 , ∂x2
∂ξ2 , −∂x1
∂ξ2
∂ξ2
∂t , ∂ξ2
∂x1 , ∂ξ2
∂x2
= 1 J
∂x1
∂τ
∂x2
∂ξ1 − ∂x2
∂τ
∂x1
∂ξ1 , −∂x2
∂ξ1 , ∂x1
∂ξ1
J = ∂x1
∂ξ1
∂x2
∂ξ2 − ∂x1
∂ξ2
∂x2
∂ξ1
Thus to have G = 0 fulfilled, grid-metrics should obey
∂J
∂τ + ∂
∂ξ1
J ∂ξ1
∂t
+ ∂
∂ξ2
J ∂ξ2
∂t
= 0
∂
∂ξ1
J ∂ξ1
∂x1
+ ∂
∂ξ2
J ∂ξ2
∂x1
= ∂
∂ξ1
∂x2
∂ξ2
+ ∂
∂ξ2
−∂x2
∂ξ1
= 0
∂
J ∂ξ1
+ ∂
J ∂ξ2
= ∂
−∂x1
+ ∂
∂x1
= 0
Basic grid-metric relations
Here we have
G = q
∂J
∂τ +
Xd j=1
∂
∂ξj
J ∂ξj
∂t
+
Xd j=1
fj
" d X
k=1
∂
∂ξk
J ∂ξk
∂xj
#
Note with the use of basic grid-metric relations (see next), it is known that
∂J
∂τ +
Xd j=1
∂
∂ξj
J ∂ξj
∂t
= 0 (geometrical conservation law) Xd
k=1
∂
∂ξk
J ∂ξk
∂xj
= 0 ∀ j = 1, 2, . . . , d (compatibility condition)
and hence G = 0