Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
Conduction–convection problems
1. General differential equation
1.1. Momentum equation: ϕ = u
1.2. Control–volume discretization equation for simple situation
2. Four basic rules for discretization equation
2.1. Consistency at control–volume faces
2.2. Positive coefficients
2.3. Negative–slope linearization of source term
2.4. Sum of the neighbor coefficients
3. Heat conduction
3.1. Conduction–type problems
d
k dT
3.2. Steady 1–D conduction dx
dx + S = 0
3.3. Grid spacing
3.4. Interface conductivity
3.5. Limiting cases
Conduction–convection problems
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
4. Convection and diffusion
4.1. Steady 1–D convection and diffusion
4.2. Upwind scheme (upstream–difference, donor–all method)
5. Exact solution
6. Exponential scheme
7. Hybrid scheme
8. Power–law scheme
9. Generalized formulation
9.1. Example ϕE = 1; ϕW = 0
9.2. Discretization equation of general diffusion equation in 3D
10. Flow field calculation
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
11. Additional features
11.1. Pressure–gradient terms
11.2. Continuity equation
12. Solution: staggered grid
13. Momentum equation
13.1. How to get guessed pressure field p∗ to get to “correct” pressure field?
13.2. Pressure–correction equation
13.3. Special discussions
13.4. SIMPLER algorithm
14. Examples
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
1. General differential equation
∂
∂t
(% φ) + div (% ~u φ) = div (Γ grad φ) +S
{z
}
|
{z
}
|
Convection term
φ
%
Γ
S
~u
=
=
=
=
=
Dif f usion term
dependent variable
density
diffusion coefficient
source term
velocity field
Remarks: 1. % is related to variables like mass fraction and temperature (Equation of state)
2. Additional constraint for flow field → mass conservation of continuity equation:
∂%
+ div (% ~u) = Scon
∂t
3. If diffusion terms are not governed by gradient of relevant variable
φ → contribution in S
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
1. General differential equation
1.1. Momentum equation: φ = u
∂
∂t
∂p
(% u) + div (% ~u u) = div (µ grad u) − ∂x
+ B x + Vx
u
p
Bx
Vx
=
=
=
=
x–direction velocity
pressure
x–direction body force per unit volume
additional viscous terms not included in div (µ grad u)
All relevant differential equations for heat and mass transfer, fluid flow, turbulence and related phenomena can be brought into standard form
→ general φ equation
→ general numerical procedure ?
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
1. General differential equation
• Calculational domain is divided into nonoverlapping control volumes (one
control volume surrounding each grid point)
• Differential equation is integrated over each control volume
• Piecewise profiles expressing variation of φ between grid points are used for
integrals
⇒ discretization equation for φ: express conservation principle for φ for finite
control volume
→ guarantees always conservation (exact balance)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
1. General differential equation
1.2. Control–volume discretization equation for simple situation
d
k dT
steady one–dimensional heat conduction: dx
dx + S = 0
(δx)w
Grid:
(δx)e
w
W
(x–direction)
e
P
E
∆x
dashed lines = faces of control volume
volume of control volume (assume ∆y = ∆z = 1): ∆x × 1 × 1
k dT
dx e
→ integral over control volume:
−
k dT
dx w
+
Re
S dx = 0
w
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
1. General differential equation
T
Profiles:
Assumption necessary!
e.g.: stepwise profile
piecewise-linear
profile
W w
stepwise profile: problem for
dT
dx
P
at control–volume faces
e
E
x
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
1. General differential equation
Discretization equation: piecewise–linear profile used
⇒
ke ·(TE −TP )
(δx)e
P −TW )
− kw (T(δx)
+ S̄ ∆x = 0
w
↓
average value
of S over
control volume
⇔ a P TP = aE · T E + aW · T W + b
in general:
with: aE =
ke
(δx)e
• a P TP =
P
aw =
kw
(δx)w
aP = aE + aW
b = S̄ ∆x
anbTnb + b
all
neighbors
• other interpolation functions for profiles give same general form
• different profiles possible for different quantities (S̄, ke) or different terms (T ; dT
dx ; . . .)
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
2. Four basic rules for discretization equation
2.1. Consistency at control–volume faces
Flux across a control volume face must be the same for both directions (integral
balance)
T
quadratic profile can produce inconsistency at interface
interface
flux inconsistency: assume that fluxes at faces of
given control volume are all governed by center–
point conductivity kP
slope from
left
W
P
slope from
right
E
EE x
P −TE
→ heat flux at e: kP T(δx)
(point P)
P −TE
kE T(δx)
(point E)
e
e
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
2. Four basic rules for discretization equation
Source term: usually linearized as
S̄ = Se + SP TP
↓
↓
constant
part
linearization
coef f icient with respect
to TP (not S evaluted at P !)
⇒ a P T P = a E · T E + a W T W + bi
with aE =
ke
(δx)e
aW =
kw
(δx)w
aP = aE + aW − SP ∆x
b = Se ∆x
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
2. Four basic rules for discretization equation
2.2. Positive coefficients
φ at a grid point is influenced by values at neighboring grid points through
convection and conduction
→ increase of φ at one point should increase φ at neighboring points
→ if TE ↑ =⇒ TP ↑ ←→ aE and aP must have same sign
in general: neighbor coefficients anb and center–point coefficient aP must have
same sign (we choose positive sign)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
2. Four basic rules for discretization equation
2.3. Negative–slope linearization of source term

even if aW0 > 0 



(aP can get negative)
aP ≥ 0
via SP term




→ S̄ = Se + SP TP
with SP ≤ 0
SP > 0: physical situation can get unstable
without
heat removal
⇒
TxP ↑



SP ↑
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
2. Four basic rules for discretization equation
2.4. Sum of the neighbor coefficients
!
aP =
P
anb
If differential equation remains satisfied after adding a constant to φ
T gets determined by boundary condition
comment: discretization equation without source linearization obeys this rule,
but not the equation with source linearization (correct, because if
source term depends on φ, both φ and φ + e are no longer solutions
of the differential equation)
If S = 0 and Tnb = constant =⇒ TP must get the source value
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
3. Heat conduction
for general algorithm: omit convection term (included later)
3.1. Conduction–type problems
heat conduction, potential flow, mass diffusion, flow through porous media,
electromagnetic field theory, diffusion models of thermal radiation, . . .
d
3.2. Steady 1–D conduction dx
k dT
dx + S = 0
⇒ a P TP = aE TE + aW TW + b
ke
aE = (δx)
e
kW
(δx)w
aW =
aP = aE + aW − SP ∆x
b = Se ∆x
• piecewise–linear profile for T (x) dT
dx
• S = Se + SP TP −→ stepwise profile
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
3. Heat conduction
3.3. Grid spacing
nonuniform grids gives better computing power efficiency: fine grid, where T (x)
variation is steep
comment: nonuniform grids do not lead to less accuracy if grid spacing is ”correctly” adapted to T (x)
no general recipe for best mesh spacing available
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
3. Heat conduction
3.4. Interface conductivity
(δx) e+
(δx) e -
P
nonuniform conductivities:
e.g. nonhomogeneity of material, T –
dependence of k
E
e
(δx) e
simple–mined ansatz (linear variation of k between P and E):
(δx)
ke = fe kP + (1 − fe) kE
fe = (δx)e+
e
midway position of e: → fe = 0.5
ke = arithmetic mean of kP and kE
⇒ leads to rather incorrect implications
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
3. Heat conduction
much better ansatz:
main objective is good representation of heat flux qe at interface: qe =
→ desired expression for ke is the one with correct q
⇒ assume control volume around P has material with kP
E
kE
→ composite slab between P and E (without S): qe =
−1
(δx)
1−fe
fe
⇒ ke = k + k
fe = (δx)e+
P
E
ke ·(TP −TE )
(δx)e
TP −Te
(δx)e− /kP +(δx)e+ /kE
e
⇒ midway interface e: fe = 0.5
ke−1 = 0.5 kP−1 + kE−1
(harmonic mean)
↔ arithmetic mean of simple ansatz is wrong
⇒ aE =
h
(δx)e−
kP
+
i
(δx)e+ −1
kE
→ aE represents conductance of material between P and E (similar procedure
for a )
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
3. Heat conduction
3.5. Limiting cases
1. kE → 0: ke → 0
heat flux at the face of an insulator should get zero
(arithmetic–mean would give nonzero )
2. kP kE : ke → kfEe
interface conductivity independent of kP (material around P offers negligible
resistance) (arithmetic–mean would keep kP –effect on ke )
P −TE )
ke is not kE , but kfEe : qe = ke(T
(δx) +
e
kP k E :
• TP will prevail up to interface e
• temperature drop will take place over (δx)e+
→ factor fe compensates use of nominal distance (δx)e in original
equation
⇒ this formulation can handle abrupt changes in conductivity without need of fine grid
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
4. Convection and diffusion
only term missing for general equation: convection term → created by fluid
flow → obtain a solution in presence of given flow field (calculation of flow field
see later)
remark: flow field must satisfy continuity equation
∂%
+ ∂x∂ j (%uj ) = 0 (without sources)
∂t
∂φ
∂
∂
∂
→ general diffusion equation: ∂t (%φ) + ∂xj (%uj φ) = ∂xj Γ ∂xj + S
∂φ
∂φ
∂φ
∂
can be rewritten: % ∂t + %uj ∂xj = ∂xj Γ ∂xj + S (*)
⇒ for given %, uj , Γ, S: any φ and φ+constant satisfy (*)
⇒ basic rule 4 (sum of coefficients) apply
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
4. Convection and diffusion
4.1. Steady 1–D convection and diffusion
dφ
d
d
u = velocity in x–direction
dx (%uφ) = dx Γ dx
continuity equation:
d
dx
(%u) = 0 ←→ %u = constant
integrating over control volume: (%uφ)e − (%uφ)w =
convection term: (piecewise–linear profile)
→ φe =
1
2
(φE + φP )
φw =
1
2
1
2
(φP + φW )
⇐⇒ assumed midway interfaces
⇒ 12 (%u)e (φE + φP ) − 21 (%u)w (φP + φW ) =
|
dφ
Γ
dx
dφ
− Γ
dx
e {z
w}
already discussed
(piecewise − linear prof ile)
Γe (φE −φP )
(δx)e
P −φW )
− Γw (φ(δx)
w
Γe and Γw obtained as discussed for interface conductivity: harmonic mean for
midway location
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
4. Convection and diffusion
new symbols: F = %u
Γ
D = δx
strength of convection
diffusion conductance (≥ 0)
F is positive for u > 0
⇒
and negative for u < 0
aP φP = a E φE + a W φW
aw = Dw + F2w
aE = De − F2e
aP = De + F2e + Dw − F2w = aE + aW + (Fe − Fw )
Remarks:
• Fe = Fw (continuity) ⇒ aP = aE + aW
• discretization equation (piecewise–linear profile for φ) is central–
difference scheme
• problems, if coefficient gets negative (|F | > 2D)
e.g.: φE = 100, φW = 200, → φP = 250
if this happens: unphysical solutions
F
)
←→ central–difference limited to low Reynolds numbers (low D
• zero diffusion (Γ = 0) ⇒ aP = 0 → solution unsuitable for iterative
methods
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
4. Convection and diffusion
4.2. Upwind scheme (upstream–difference, donor–all method)
weak point in preliminary discussion is that convected property φe is average of
φE and φP
diffusion term: unchanged
convection term: value of φ at interface is equal to value of φ at grid point on
upwind side
→ φe = φP if Fe > 0
φe = φE if Fe < 0
←→ Feφe = φP Fe, 0 − φE −Fe, 0
( A, B : greater of A and B)
⇒ a P φP = a E φE + a W φW
aE = De + −Fe, 0
a W = DW + FW , 0
aP = aE + aW + (Fe − FW ) = De + Fe, 0 + Dw + −Fw , 0
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
4. Convection and diffusion
• this forms guarantees always positive coefficients
• main physics idea: ”tank–and–tube” model
control volume =
b
stirred tanks connected
in series by short tubes
flow through tubes ←→ convection
conduction through tank walls ←→ diffusion
⇒ fluid flowing in each connecting tube has temperature that prevails in tank
on upstream side
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
4. Convection and diffusion
for discretization equation approximate φ ∼ x profile necessary
←→ • linear profile (preliminary derivation): only valid for small |P |
dφ
dx
• |P | large →
nearly 0 at x =
L
2
⇒ upwind scheme overestimates diffusion in this case
⇒ exact solution possible by using complete expression (→ exponential scheme)
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
5. Exact solution
d
dx
(%uφ) =
d
dx
Γ dφ
dx
% · u = constant (continuity equation)
Γ = constant (boundary conditions: x = 0, φ = φ0; x = L, φ = φL)
exact solution:
φ
φL
φ−φ0
φL −φ0
=
exp(P x/L)−1
exp(P )−1
P = Péclet number = ratio of strength of convection and diffusion
P = % uΓ L
-P>>1
P = 0 → pure diffusion (conduction)
φ ∼ x variation linear
P > 0 → φ more influenced by upstream value φ0
P < 0 → reversed picture
P=0
P=-1
P=1
φ0
P>>1
L
x
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
6. Exponential scheme
consider total flux: J = %uφ − Γ dφ
dx
d
d
⇒ dx
Γ dφ
←→ dJ
(%uφ) = dx
= 0
dx
dx
integration over control volume: Je − Jw = 0
use exact solution as profile between P an E (φP and φE replacing φ0 and
φL; (δx)e replacing L)
(%u)e (δx)e
φP −φE
= DFee
⇒ Je = Fe φP + exp(Fe)−1 ; Pe =
Γe
JE is independent of location of interface (clear, because we started from
dJ
dx = 0)
similar for w
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
6. Exponential scheme
⇒ a P φP = a E φE + a W φW
aE =
Fe
exp(Fe /De )−1 ;
aW =
Fw exp(Fw /Dw )
exp(Fw /Dw )−1
aP = aE + aW + (Fe − Fw )
exact scheme for steady 1–D, but:
• exponentials are expensive to compute
• scheme not exact for 2 or 3D, nonzero
sources, . . .
⇒ needed: easy–to–compute scheme with qualitative behavior of exponential
scheme
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
7. Hybrid scheme
aE
De
=
aE
De
Pe
exp(Pe )−1
P
aE
=1- e
De
2
( Pe = 0)
exact
aE
= - Pe
De
( Pe → - ∞ )
-2
-4
0
2
Pe
4
aE
= 0 ( Pe→ ∞)
De
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
7. Hybrid scheme
→ approximation to exact curve:

aE
=
−P
Pe < −2 : D

e

e
aE = De −Pe, 1 − P2e , 0
aE
Pe
−2 ≤ Pe ≤ 2 : De = 1 − 2

←→ aE = −Fe, De − F2e , 0

aE
Pe > 2 : D e = 0
hybrid scheme:
• identical to central–difference scheme for −2 ≤ Pe ≤ 2
• outside this: upwind scheme (diffusion = 0)
←→ aP φP = aE φE + aW φW
aE = −Fe, De − F2e , 0
FW
2 ,0
a W = FW , D W +
aP = aE + aW + (Fe − FW )
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
8. Power–law scheme
departure of hybrid scheme from exact curve is rather large at Pe = ±2
→ power–law scheme (extremely good approximation)
Pe < −10: aCEe = −Pe
−10 ≤ Pe < 0:
aE
De
= (1 + 0.1Pe)5 − Pc
0 ≤ Pe ≤ 10:
aE
De
= (1 + 0.1Pe)5
↔ a E = De
aE
= 0
Pe > 10: D
e
5
0.1|Fe|
0, 1 − De
+ 0, −Fe
recommended scheme: accurate and fast
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
9. Generalized formulation
⇒ a P φP = a E φE + a W φW
aE = DeA(|Pe|) + −Fe, 0
aW = D + W A(|PW |) + FW , 0
aP = aE + aW + (Fe − Fw )
different schemes: different functions

1 − 0.5|P |





1


0, 1 − 0.5|P |
A(|P |) =



0, (1 − 0.1|P |)5




|P |/ exp(|P |) − 1
A(|P |)
central difference
upwind
hybrid
power law
exponential (exact)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
9. Generalized formulation
WENDELSTEIN 7-X
9.1. Example: φE = 1; φW = 0
A(|P|)
1.0
• all schemes except central difference give
physically realistic solution
upwind
0.5
• central–difference scheme produces some
values outside 0−1 range of boundary values (reasonable solutions for P < 2)
exponential
power law
hybrid
grid Péclet number P < 2 → grid refinement (uneconomic)
|P|
central
difference
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
9. Generalized formulation
9.2. Discretization equation of general diffusion equation in 3D
aP φP = a E φE + a W φW + a N φN + a S φS + a T φT + a B φB + b
aE
aW
aN
aS
aT
=
=
=
=
=
DeA(|Pe|) + −Fe, 0
Dw A(|Pw |) + Fw , 0
DnA(|Pn|) + −Fn, 0
DsA(|Ps|) + Fs, 0
DtA(|Pt|) + Ft, 0
Fe
Fw
Fn
Fs
Ft
=
=
=
=
=
(%
(%
(%
(%
(%
u)e
u)w
v)n
v)s
w)t
∆y ∆z
∆y ∆z
∆z ∆x
∆z ∆x
∆x ∆y
=
=
=
=
=
Γe ∆y ∆z
(δx)e
Γw ∆y ∆z
(δx)w
Γn ∆z ∆x
(δy)n
Γs ∆z ∆x
(δy)s
Γt ∆x ∆y
(δz)t
Db =
Γb ∆x ∆y
(δz)b
De
Dw
Dn
Ds
Dt
%0P ∆x ∆y ∆z
∆t
a0P =
b = Se ∆x ∆y ∆z + a0P φ0P
Fb = (% w)b ∆x ∆y
aP = aE + aW + aN + aS + aT + aB + a0P − SP ∆x ∆y ∆z
Pe =
Fe
D ;
power–law: A(|P |) =
0, (1 − 0.1|P |)5
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
10. Flow field calculation
How to calculate the flow field?
problem:
• not non–linearity (convection coefficient %u =
b k being function of
T)
• unknown pressure field: pressure gradient forms part of source
term for momentum equation → no obvious equation for obtaining
pressure (for given pressure field: no problem)
problem: correct pressure field satisfies continuity equation and
momentum equation (problem for iterative methods; direct solution of complete set of equations does not have
this)
⇒ how to convert indirect information in continuity equation into
direct algorithm for calculation of pressure
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
11. Additional features
11.1. Pressure–gradient terms
new feature:
−dp
dx
integrated over control volume
resulting contribution: pw − pe (net pressure force)
• assume piecewise–linear profile for pressure
• midway faces c and w:
pw − p e =
pw +pP
2
E
− pP +p
=
2
pW −pE
2
→ momentum equation contains pressure difference between 2 alternate grid
points, not between adjacent ones
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
11. Additional features
⇒ pressure is taken from coarser grid
⇒ zig–zag fields are felt like uniform pressure field
1D:
100 500
x
x
2D:
100
5
100
5
100
100
x
500
x
100 500
x
x
300 100 300 100 300
27 5 27 5 27
300 100 300 100 300
27 5 27 5 27
300 100 300 100 300
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
11. Additional features
11.2. Continuity equation
similar problem as in 9.2.16.1.
1–D:
du
dx
=0
(simplest situation)
→ (integration over control volume): ue − uw = 0
piecewise–linear profile for u
midway faces →
uP +uE
2
P
− uW +u
= uE − uW = 0
2
→ zig–zag fields satisfy above equation
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
12. Solution: staggered grid
variables do not have to be calculated on same grid
→ staggered grid for velocities (on all faces of control volumes)
→ staggered grid removes zig–zag problems, because velocities of adjacent
velocity components enter
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
13.1. How to get guessed pressure field p∗ to get to ”correct” pressure field?
correct pressure: p = p∗ + p0 (p0 = pressure correction)
←→ u = u∗ + u0; v + v ∗ + v 0; w = w ∗ + w0
X
0
⇒ a e ue =
anbu0nb + (p0P − p0E ) Ae
|
{z
}
now drop
this term
(discussion later)
→ aeu0e = (p0P − p0E ) Ae ←→ ux0e = de (p0P − p0E ) with de =
y
velocity–correction formula:
=
b ue = u∗e + de (p0P − p0E )
same for: vn = vn∗ + dn (p0P − p0N )
wt = wt∗ + dt (p0P − p0T )
Ae
ae
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
13.2. Pressure–correction equation
turn continuity equation into equation for pressure correction
assume (for derivation, discussion later): density % does not directly depend on
pressure
continuity equation:
discretized equation:
∂%
∂t
+ ∂(%u)
+ ∂(%v)
+ ∂(%w)
= 0
∂x
∂y
∂z
+ (%u)c − (%u)w ∆y ∆z
+ (%v)n − (%v)s ∆z ∆x + (%w)t − (%w)b ∆x ∆y = 0
(%p−%◦P )
∆x ∆y ∆z
∆t
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
use velocity–corrections formulas:
→ discretization equation for p0:
aP · p0P = aE · p0E + aW · p0W + aN · p0N + aS · p0S + aT · p0T + aB · p0B + b
aE = %e de ∆y ∆z, . . .
aP = aE + aW + aN + aS + aT + aB
(%◦ −% ) ∆x ∆y ∆z ∗
+ (%u )w − (%u∗)e ∆y ∆z
b = P P ∆t
+ (%v ∗)s − (%v ∗)n ∆z ∆x + (%w ∗)b − (%w ∗)t ∆x ∆y
term b in pressure–correction equation: essentially the negative of the left–hand
size of discretized continuity equation evaluated in terms of ”starred” velocities
b = 0 → starred velocities satisfy correction
correction and continuity equation
→ correct velocities
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
⇒ SIMPLE algorithm (semi–implicit method for pressure–linked equation)
1. guess pressure field p∗
2. solve momentum equation for u∗, v ∗, w ∗
3. solve p0 equation
4. calculate p from p0 and p∗
5. calculate u, v, w from u∗, v ∗, w ∗
6. solve discretization equation for other φ’s
7. treat correct pressure p as now guessed
pressure p∗
8. goto 2.; repeat until convergence
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
Pressure–correction
continuity equation:
∂%u∗
∂x
∗
∂%
∂t
∂%v
+ ∂%u
∂x + ∂y = 0
∂%
∂t
=0
0
u = u ∗ + u0
v = v∗ + v0
0
∂%u
∂%v
+ ∂%v
∂y + ∂x + ∂y = 0
0
0
∂% ∂p
∂%u∗ ∂%v ∗ ∂% ∂p
∂x
+
+ ∂x + ∂y∂y = 0
| ∂x {z ∂y }
b
0
∂% ∂p
+ ∂x∂y
+
0
∂% ∂p
∂y
∂y
= 0 ←→ 2nd order equation for p0
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
13.3. Special discussions
1.
• keeping this term would involve pressure correction at all grid points
• semi–implicit method: omission of this term
• omission of this term: for converged solution no error (b = 0, p 0 = 0)
• b as indicator for convergence (b → 0)
however: rate of convergence depends on p0 equation
underrelaxation necessary for p: p = p∗ + αP · p0
(αP = 0.5 − 0.8)
2. correct velocity field → first give wrong pressure field
convergence later through iterations
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
improved convergence → SIMPLER (SIMPLE revised)
P
motivation: omission of
anbu0nb leads to exaggerated pressure correction →
underrelaxation essential
neighbor–point velocity corrections removed → pressure correction has entire
burden for velocity–correction
pressure–correction equation is good for velocity correction, but rather bad for
pressure correction
⇒ basic idea: employ pressure–correction equation only for velocity–correction
and use other means for improved pressure field
pressure equation (momentum equation): ue =
define pseudo–velocity u
be: u
be =
P
anb unb +b
ae
P
anb unb +b
ae
+ de (pP − pE )
(b
ue contains no pressure)
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
similarly: vn = vbn + dn (pP − pN )
wt = w
bt + dt (pP − pT )
u
b, vb, w
b appears in place of u∗, v ∗, w ∗
p appears in place of p0
→ equation for pressure:
a P pP = a E pE + a W pW + a N pN + a S pS + a T pT + a B pB + b
aE , . . . given as before
(%◦P −%P ) ∆x ∆y ∆z +
(%b
u
)
−
(%b
u
)
∆y ∆z
b =
w
∆t
e
+ (%b
v)s − (%b
v )n ∆z ∆x + (%w)
b b − (%w)
b t ∆x ∆y
• equation for b is only difference between pressure equation and pressure–
correction equation
• no approximation introduced for derivation of pressure equation
⇒ correct velocity field would result in correct pressure
Conduction–convection problems
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
13. Momentum equations
13.4. SIMPLER algorithm
revised algorithm solves pressure equation to obtain pressure field and solves
pressure–correction equation only to correct velocities
1. start with guessed velocity field
2. calculate coefficient for momentum equation and calculate u
b, vb, w
b
3. calculate coefficient for pressure equation and solve it for pressure field
4. take this pressure field as p∗; solve momentum equation for u∗, v ∗, w ∗
5. calculate mass source b and solve p0–equation
6. correct velocity field, but do not correct pressure
7. solve equations for other φ’s
8. return to 2.; repeat until convergence
remark: SIMPLER does not use guessed pressure fields, but extracts pressure
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
14. Examples: momentum
Conduction–convection problems
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
14. Examples: source
R. Schneider
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
14. Examples: circle
Conduction–convection problems
Max-Planck-Institut für Plasmaphysik, EURATOM Association
WENDELSTEIN 7-X
14. Examples: wall
R. Schneider