A conservative FE-discretisation of the Navier-Stokes equation
description
Transcript of A conservative FE-discretisation of the Navier-Stokes equation
![Page 1: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/1.jpg)
A conservative FE-discretisation of the Navier-Stokes equation
JASS 2005, St. Petersburg
Thomas Satzger
![Page 2: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/2.jpg)
2
Overview
• Navier-Stokes-Equation– Interpretation– Laws of conservation
• Basic Ideas of FD, FE, FV
• Conservative FE-discretisation of Navier-Stokes-Equation
![Page 3: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/3.jpg)
3
Navier-Stokes-Equation
• The Navier-Stokes-Equation is mostly used for the numerical simulation of fluids.
• Some examples are- Flow in pipes- Flow in rivers- Aerodynamics- Hydrodynamics
![Page 4: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/4.jpg)
4
Navier-Stokes-Equation
The Navier-Stokes-Equation writes:
gupuuut
1
Equation of momentum
0 u
Continuity equation
with u
Velocity field
Pressure fieldp
Density
Dynamic viscosity
![Page 5: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/5.jpg)
5
Navier-Stokes-Equation
The interpretation of these terms are:
gupuuut
1
Derivative of velocity field
Convection
Pressure gradient
Diffusion
Outer Forces
![Page 6: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/6.jpg)
6
Navier-Stokes-Equation
The corresponding for the components is:
1122
2
121
2
11
221
111
1gu
xu
xp
xu
xuu
xuu
t
2222
2
221
2
22
222
112
1gu
xu
xp
xu
xuu
xuu
t
022
11
ux
ux
for the momentum equation, and
for the continuity equation.
![Page 7: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/7.jpg)
7
Navier-Stokes-Equation
With the Einstein summation jiji
m
jjiji xayxay
1
and the abbreviation we get:tt
i
i x
iijjiijjii gupuuu
1
)2,1( i
0 jju
for the momentum equation, and
for the continuity equation.
![Page 8: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/8.jpg)
8
Navier-Stokes-Equation
Now take a short look to the dimensions:
iijjiijjii gupuuu
1
2s
m
2
1
s
m
ss
m
2
1
s
m
ms
m
s
m
22
1
3 s
m
msm
kgm
m
kg
22
1
3 s
m
ms
m
m
kgsm
kg
![Page 9: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/9.jpg)
9
Navier-Stokes-Equation - Interpretation
We see that the momentum equations handles with accelerations. If we rewrite the equation, we get:
iijjiijjii gupuuu
1
This means:
Total acceleration is the sum of the partial accelerations.
![Page 10: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/10.jpg)
10
Navier-Stokes-Equation - Interpretation
Interpretation of the Convection
1x
2x
fluid particle
iiu pi
1ijj u
igijj uu
Transport of kinetic energy by moving the fluid particle
![Page 11: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/11.jpg)
11
Navier-Stokes-Equation - Interpretation
Interpretation of the pressure Gradient
1x
2x
fluid particle
iiu ijj uu ijj u
igpi
1
Acceleration of the fluid particle by pressure forces
![Page 12: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/12.jpg)
12
Navier-Stokes-Equation - Interpretation
Interpretation of the Diffusion
iiu ijj uu pi
1ig
ijj u
1x
2x
fluid particle
Distributing of kinetic Energy by friction
![Page 13: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/13.jpg)
13
Navier-Stokes-Equation - Interpretation
Interpretation of the continuity equation 0 jju
• Conservation of mass in arbitrary domain
h
h
11u 12
u
21u
22u
this means:
influx = out fluxhuhu
11 21 huhu 22 21
01212 2211
h
uu
h
uu
for 0h we get
02211 uu
![Page 14: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/14.jpg)
14
Navier-Stokes-Equation - Laws of conservation
Conservation of kinetic energy:
We must know that the kinetic energy doesn't increase, this means:
Proof:
duuduE iikin 212
21
duu
duuuuduuduudt
dE
dt
d
iti
iititiiitiikin 21
21
21
0kinEdt
d
![Page 15: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/15.jpg)
15
Navier-Stokes-Equation - Laws of conservation
With the momentum equation it holds
Using the relations (proof with the continuity equation)
and
0Re
1 puuuu iijjijjii
puuuuuu
puuuuduuEdt
d
iiijjiijji
iijjijjiitikin
Re
1
Re
1
ijjiji uuuu
ijjiijijijji
ijijijii
product
ruleijii
Gauß
ii
uuuuuuuuu
uuuuuuuuuuuu
2
0
![Page 16: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/16.jpg)
16
Navier-Stokes-Equation - Laws of conservation
Additionally it holds
Therefore we get
Due to Greens identity we have
00
Re
1
puuuuuuE
dt
diiijjiijjikin
jj
continuity
equationjjjj
product
rulejj
Gauß
upupuppuup0
0
0 ijij
Green
ijji uuuu
![Page 17: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/17.jpg)
17
Navier-Stokes-Equation - Laws of conservation
This means in total
We have also seen that the continuity equation is very important for energy conservation.
0kinEdt
d
![Page 18: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/18.jpg)
18
Basic Ideas of FD, FE, FV
We can solve the Navier-Stokes-Equations only numerically.Therefore we must discretise our domain. This means, we regard our Problem only at finite many points.There are several methods to do it:
•Finite Difference (FD)One replace the differential operator with the difference operator, this mean you approximate by
or an similar expression.
f
h
xfhxfxf
![Page 19: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/19.jpg)
19
Basic Ideas of FD, FE, FV
• Finite Volume (FV)- You divide the domain in disjoint subdomains- Rewrite the PDE by Gauß theorem- Couple the subdomains by the flux over the
boundary• Finite Elements (FE)
- You divide the domain in disjoint subdomains- Rewrite the PDE in an equivalent variational
problem- The solution of the PDE is the solution of the
variational problem
![Page 20: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/20.jpg)
20
Basic Ideas of FD, FE, FV
Comparison of FD, FE and FV
Finite Difference
Finite Element
Finite Volume
![Page 21: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/21.jpg)
21
Basic Ideas of FD, FE, FV
Advantages and DisadvantagesFinite Difference:
+ easy to programme- no local mesh refinement- only for simple geometries
Finite Volume:+ local mesh refinement+ also suitable for difficult geometries
Finite Element:+ local mesh refinement+ good for all geometries
BUT:Conservation laws aren't always complied by the discretisation. This can lead to problems in stability of the solution.
![Page 22: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/22.jpg)
22
Conservative FE-Elements
We use a partially staggered grid for our discretisation.
h
h
u
v
u
v
u
v
u
v
p
We write: N for the number of grid pointsiu
ivfor the horizontal velocity in the i-th grid point
for the vertical velocity in the i-th grid point
![Page 23: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/23.jpg)
23
Conservative FE-Elements
The FE-approximation is an element of an finite-dimensional function space with the basis
The approximation has the representation
Nff 21,...,
2: IRfi whereby
N
iNiiiih fvfuu
1
![Page 24: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/24.jpg)
24
Conservative FE-Elements
If we use a Nodal basis, this means
we can rewrite the approximation
0
1point grid th-iif
0
0points gridother allNif
1
0point grid th-iNif
0
1points gridother allif
and
and
N
i vNi
uNii
vi
uii
h
h
f
fv
f
fu
v
u
1 ,
,
,
,
![Page 25: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/25.jpg)
25
Conservative FE-Elements
Every approximation should have the following properties:continuousconservative
In the continuous case the continuity equation was very important for the conservation of mass and energy.
If the approximation complies the continuity pointwise in the whole area, e.g. , then the approximation preserves energy.
hu
0 hu
![Page 26: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/26.jpg)
26
Conservative FE-Elements
Now we search for a conservative interpolation for the velocities in a box.
We also assume that the velocities complies the discrete continuity equation.
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
![Page 27: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/27.jpg)
27
Conservative FE-Elements
Now we search for a conservative interpolation for the velocities in a box.
We also assume that the velocities complies the discrete continuity equation.
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
hvv
huu
224331
![Page 28: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/28.jpg)
28
Conservative FE-Elements
Now we search for a conservative interpolation for the velocities in a box.
We also assume that the velocities complies the discrete continuity equation.
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
hvv
huu
224331 h
vvh
uu
22
2142
![Page 29: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/29.jpg)
29
Conservative FE-Elements
Now we search for a conservative interpolation for the velocities in a box.
We also assume that the velocities complies the discrete continuity equation:
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
hvv
huu
224331 h
vvh
uu
22
2142
043314321 vvvvuuuu (1)
![Page 30: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/30.jpg)
30
Conservative FE-Elements
The bilinear interpolation isn't conservative
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
1 2 3 4
1 2 3 4
, 1 1 1 1
, 1 1 1 1
h
h
x y x y x y x yu x y u u u u
h h h h h h h h
x y x y x y x yv x y v v v v
h h h h h h h h
![Page 31: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/31.jpg)
31
Conservative FE-Elements
The bilinear interpolation isn't conservative
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
1 2 3 4
1 2 3 4
, 1 1 1 1
, 1 1 1 1
h
h
x y x y x y x yu x y u u u u
h h h h h h h h
x y x y x y x yv x y v v v v
h h h h h h h h
It is easy to show that
0general in
hh vy
ux
![Page 32: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/32.jpg)
32
Conservative FE-Elements
The bilinear interpolation isn't conservative
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
, , ,,1, 3, 4,2,
,2 4,,1 4,
2 3 4
2
3
1
1 3
, 1 1 1 1
, 1 1 1
f x y f x y f x yf x yu u uu
f x yvxv
h
y
h
f f
x y x y x y x yu x y u u u u
h h h h h h h h
x y x y x yv x y v v v
h h h h h h
4
, ,4, 4 4,
1
x y f x yv v
x yvh h
Basis on the box
![Page 33: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/33.jpg)
33
Conservative FE-Elements
These basis function for the bilinear interpolation are calledPagoden.
The picture shows the function on the whole support.
, i iu v
h
h
![Page 34: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/34.jpg)
34
Conservative FE-Elements
Now we are searching a interpolation of the velocities which complies the continuity equation on the box.
How can we construct such an interpolation?
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
![Page 35: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/35.jpg)
35
Conservative FE-Elements
Now we are searching a interpolation of the velocities which complies the continuity equation on the box.
How can we construct such an interpolation?
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
Divide the box in four triangles.
![Page 36: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/36.jpg)
36
Conservative FE-Elements
Now we are searching a interpolation of the velocities which complies the continuity equation on the box.
How can we construct such an interpolation?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
Divide the box in four triangles.Make on every triangle an linear interpolation.
![Page 37: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/37.jpg)
37
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
![Page 38: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/38.jpg)
38
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
We must have at every point in the box the following relations:
, , 0 , ,
, ,
u x y v x y u x y v x yx y x y
v x y u x yy x
![Page 39: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/39.jpg)
39
Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the following relations:
, , 0 , ,
, ,
u x y v x y u x y v x yx y x y
v x y u x yy x
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
![Page 40: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/40.jpg)
40
Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the following relations:
, , 0 , ,
, ,
u x y v x y u x y v x yx y x y
v x y u x yy x
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
![Page 41: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/41.jpg)
41
Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the following relations:
, , 0 , ,
, ,
u x y v x y u x y v x yx y x y
v x y u x yy x
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
![Page 42: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/42.jpg)
42
Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the following relations:
, , 0 , ,
, ,
u x y v x y u x y v x yx y x y
v x y u x yy x
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
![Page 43: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/43.jpg)
43
Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the following relations:
, , 0 , ,
, ,
u x y v x y u x y v x yx y x y
v x y u x yy x
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
![Page 44: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/44.jpg)
44
Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the following relations:
, , 0 , ,
, ,
u x y v x y u x y v x yx y x y
v x y u x yy x
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
![Page 45: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/45.jpg)
45
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
5 33u u
![Page 46: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/46.jpg)
46
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 35 33 2
u uhu u
h
![Page 47: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/47.jpg)
47
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 3 1 35 33 2 2
u u v vh hu u
h h
![Page 48: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/48.jpg)
48
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 3 1 35 3 3 1 3 1 3 1 3 1 33
1 1
2 2 2 2
u u v vh hu u u u u v v u u v v
h h
![Page 49: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/49.jpg)
49
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 3 1 35 3 3 1 3 1 3 1 3 1 33
1 1
2 2 2 2
u u v vh hu u u u u v v u u v v
h h
5 42u u
![Page 50: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/50.jpg)
50
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 3 1 35 3 3 1 3 1 3 1 3 1 33
1 1
2 2 2 2
u u v vh hu u u u u v v u u v v
h h
2 45 42 2
h u uu u
h
![Page 51: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/51.jpg)
51
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 3 1 35 3 3 1 3 1 3 1 3 1 33
1 1
2 2 2 2
u u v vh hu u u u u v v u u v v
h h
2 4 2 45 42 2 2
h u u h v vu u
h h
![Page 52: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/52.jpg)
52
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 3 1 35 3 3 1 3 1 3 1 3 1 33
1 1
2 2 2 2
u u v vh hu u u u u v v u u v v
h h
2 4 2 45 4 4 2 4 2 4 2 4 2 42
1 1
2 2 2 2
h u u h v vu u u u u v v u u v v
h h
![Page 53: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/53.jpg)
53
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
1 3 1 35 3 3 1 3 1 3 1 3 1 33
1 1
2 2 2 2
u u v vh hu u u u u v v u u v v
h h
2 4 2 45 4 4 2 4 2 4 2 4 2 42
1 1
2 2 2 2
h u u h v vu u u u u v v u u v v
h h
free are and
41 55 uu
![Page 54: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/54.jpg)
54
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
4 3 4 35 3 3 4 3 4 3 3 4 3 44
1 1
2 2 2 2
v v u uh hv v v v v u u u u v v
h h
![Page 55: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/55.jpg)
55
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
51u
52u53
u
54u
51v
52v53
v
54v
1
ux
1
ux
4
ux
4
ux
3
vy
3
vy
2
vy
2
vy
4 3 4 35 3 3 4 3 4 3 3 4 3 44
1 1
2 2 2 2
v v u uh hv v v v v u u u u v v
h h
2 1 2 15 1 1 2 1 2 1 1 2 1 21
1 1
2 2 2 2
h v v h u uv v v v v u u u u v v
h h
free are and
23 55 uv
![Page 56: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/56.jpg)
56
Conservative FE-Elements
Till now we have:
5 1 3 1 33
1
2u u u v v 5 2 4 2 42
1
2u u u v v
5 3 4 3 44
1
2v u u v v 5 1 2 1 21
1
2v u u v v free are and
23 55 uv
free are and 41 55 uu
![Page 57: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/57.jpg)
57
Conservative FE-Elements
Till now we have:
With the discrete continuity equation
we get
5 1 3 1 33
1
2u u u v v 5 2 4 2 42
1
2u u u v v
5 3 4 3 44
1
2v u u v v 5 1 2 1 21
1
2v u u v v free are and
23 55 uv
free are and 41 55 uu
1 2 3 4 1 2 3 4 0 (1)u u u u v v v v
5 53 2u u 5 54 1
v v
![Page 58: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/58.jpg)
58
Conservative FE-Elements
Till now we have:
With the discrete continuity equation
we get
Therefore we choose
5 1 3 1 33
1
2u u u v v 5 2 4 2 42
1
2u u u v v
5 3 4 3 44
1
2v u u v v 5 1 2 1 21
1
2v u u v v free are and
23 55 uv
free are and 41 55 uu
1 2 3 4 1 2 3 4 0 (1)u u u u v v v v
5 53 2u u 5 54 1
v v
5 5 5 5 5 1 2 3 4 1 2 3 41 2 3 4
5 5 5 5 5 1 2 3 4 1 2 3 41 2 3 4
1: : : : :
41
: : : : :4
u u u u u u u u u v v v v
v v v v v u u u u v v v v
![Page 59: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/59.jpg)
59
Conservative FE-Elements
What's the right velocity in the middle?
1u 2u
3u 4u
h
h
1v 2v
3v 4v
h
h
12
34
12
34
5u 5v
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
![Page 60: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/60.jpg)
60
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
Ni u i N uh
i ii v i N vh i
f fuu v
f fv
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
![Page 61: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/61.jpg)
61
Conservative FE-Elements
Now we calculate the basis.
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
, ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
i i
![Page 62: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/62.jpg)
62
Conservative FE-Elements
Now we calculate the basis., ,
, ,1i
Ni u i N uh
ii i N vih v
f fu
u
vv
f f
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
h
h
h
h
,i uf ,i vf
0 0 0
0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
![Page 63: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/63.jpg)
63
Conservative FE-Elements
Now we calculate the basis.
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
, ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
1 1, u v 2 2, u v
3 3, u v 44, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
![Page 64: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/64.jpg)
64
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
1 1, u v 2 2, u v
3 3, u v 44, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
45 1 2 3 1 2 3 4
5 1 2 3 4 1 2 3 4
:
1:
4
1
4u u u u v v v v
v
u
u u u u v v v v
![Page 65: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/65.jpg)
65
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
1 1, u v 2 2, u v
3 3, u v 44, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
45 1 2 3 1 2 3 4
5 1 2 3 4 1 2 3 4
:
1:
4
1
4u u u u v v v v
v
u
u u u u v v v v
1
4
![Page 66: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/66.jpg)
66
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
1 1, u v 2 2, u v
33, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
![Page 67: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/67.jpg)
67
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
1 1, u v 2 2, u v
33, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
5 1 2 4 1 2 3 4
5 1 2 3 4 1 2
3
3 4
:
1
1
:
4
4
u u u u v v v v
v u u u u
u
v v v v
![Page 68: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/68.jpg)
68
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
1 1, u v 2 2, u v
33, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
5 1 2 4 1 2 3 4
5 1 2 3 4 1 2
3
3 4
:
1
1
:
4
4
u u u u v v v v
v u u u u
u
v v v v
1
4
![Page 69: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/69.jpg)
69
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
5 1 3 4 1 2 3 4
5 1 2 3 4 1 4
2
2 3
:
1
1
:
4
4
u u u u v v v v
v u u
u
u u v v v v
1 1, u v 22, u v
3 3, u v 4 4, u v
h
h
5 5, u v0
0
h
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
![Page 70: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/70.jpg)
70
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
hi
Ni u i N u
ii v i N vh i
f fuu v
f fv
5 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
1
1
:
4
4
u u u u v v v v
v u u u v
u
u v v v
11, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
0
h
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
1
4
![Page 71: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/71.jpg)
71
Conservative FE-Elements
Now we calculate the basis., ,
, ,1i
Ni u i N uh
ii i N vih v
f fu
u
vv
f f
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
1
4
![Page 72: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/72.jpg)
72
Conservative FE-Elements
Now we calculate the basis., ,
, ,1i
Ni u i N uh
ii i N vih v
f fu
u
vv
f f
5 1 2 3 4 1 2 3 4
5 1 2 3 1 2 3 44
1:
41
4:
u u u u u v v v v
v u u u v v v vu
1 1, u v 2 2, u v
3 3, u v 44, u v
h
h
5 5, u v0
0
h
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
1
4
1
4
![Page 73: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/73.jpg)
73
Conservative FE-Elements
Now we calculate the basis., ,
, ,1i
Ni u i N uh
ii i N vih v
f fu
u
vv
f f
5 1 2 3 4 1 2 3 4
5 1 2 4 13 2 3 4
1:
41
4:
u u u u u v v v v
v u u u v v v vu
1 1, u v 2 2, u v
33, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
1
4
1
4
1
4
![Page 74: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/74.jpg)
74
Conservative FE-Elements
Now we calculate the basis., ,
, ,1i
Ni u i N uh
ii i N vih v
f fu
u
vv
f f
1 1, u v 22, u v
3 3, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
1
4
1
4
1
4
1
4
5 1 2 3 4 1 2 3 4
5 1 3 4 1 2 32 4
1:
41
4:
u u u u u v v v v
v u u u v v vu v
![Page 75: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/75.jpg)
75
Conservative FE-Elements
Now we calculate the basis., ,
, ,1i
Ni u i N uh
ii i N vih v
f fu
u
vv
f f
5 1 2 3 4 1 2 3 4
5 2 3 4 1 31 2 4
1:
41
4:
u u u u u v v v v
v u u u v v vu v
11, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i uf ,i vf
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
![Page 76: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/76.jpg)
76
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
Ni u i N uh
ii v i N vh i
i
f fuu
f fvv
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i N uf ,i N vf
0 0 0
0 0
0 0 0
0
0 0 0
0 0
0 0 0
1
i i
![Page 77: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/77.jpg)
77
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
Ni u i N uh
ii v i N vh i
i
f fuu
f fvv
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
,i N uf ,i N vf
0 0 0
0 0
0 0 0
0
0 0 0
0 0
0 0 0
1
i i
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
![Page 78: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/78.jpg)
78
Conservative FE-Elements
Now we calculate the basis., ,
, ,1
Ni u i N uh
ii v i N vh i
i
f fuu
f fvv
5 1 2 3 4 1 2 3 4
5 1 2 3 4 1 2 3 4
1:
41
:4
u u u u u v v v v
v u u u u v v v v
1 1, u v 2 2, u v
3 3, u v 4 4, u v
h
h
5 5, u v
h
h
h
h
, ,i N u i vf f , ,i N v i uf f
0 0 0
0 0
0 0 0
0
0 0 0
0 0
0 0 0
1
i i
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
![Page 79: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/79.jpg)
79
Conservative FE-Elements
h
h
h
h
, ,i u i N vf f , ,i v i N uf f
0 0 0
0 0
0 0 0
1
0 0 0
0 0
0 0 0
0
i i
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
Linear interpolation providesthe basis.
![Page 80: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/80.jpg)
80
Conservative FE-Elements
View on conservative elements in 3D
![Page 81: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/81.jpg)
81
Conservative FE-Elements
View on conservative elements in 3D
Partially staggered grid in 3D
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
p
hh
h
![Page 82: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/82.jpg)
82
Conservative FE-Elements
We also search for a conservative interpolation of the velocities.
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
p
hh
h
![Page 83: A conservative FE-discretisation of the Navier-Stokes equation](https://reader035.fdocuments.in/reader035/viewer/2022062305/56815af2550346895dc8ae71/html5/thumbnails/83.jpg)
83
Conservative FE-Elements
We also search for a conservative interpolation of the velocities.
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
, , u v w
hh
h
Divide every box into 24 tetrahedrons, on which you make a linear interpolation