Shallow Water - Derivation and Applications

Transcription

Shallow Water
Derivation and Applications
Christian Kühbacher
Technische Universität Dortmund
May 28, 2009
The 2d Shallow Water Equations
Derivation from basic conservation laws
Solutions to SWE problems
Overview
1
2
3
Overview
1
2
3
Basic properties
The shallow water equations (SWE) are
are a set of PDEs, that describe fluid-flow-problems
are derived from the physical conservation laws for the mass and
momentum
are valid for problems in which vertical dynamics can be neglected
compared to horizontal effects
are a 2d model, which is derived from a 3d model by depth averaging
Basic properties
momentum
Basic properties
momentum
Basic properties
momentum
Is “shallow water” a good name?
For two reasons the name shallow water is not exact
The fluid doesn’t have to be water
For example wheather forecasting was done with a modification of
the SWE
The fluid doesn’t necessarily have to be shallow
For example a tsunami wave on the ocean (approx. 5 km deep) can
be a SW wave
Is “shallow water” a good name?
For two reasons the name shallow water is not exact
The fluid doesn’t have to be water
For example wheather forecasting was done with a modification of
the SWE
The fluid doesn’t necessarily have to be shallow
For example a tsunami wave on the ocean (approx. 5 km deep) can
be a SW wave
Tsunami
Some examples
The SWE have been applied to
Tsunamis prediction
Atmospheric flows
Storm surges
Flows around structures (pier)
Planetary flows
Some examples
Tsunamis prediction
Atmospheric flows
Storm surges
Planetary flows
Some examples
Tsunamis prediction
Atmospheric flows
Storm surges
Planetary flows
Some examples
Tsunamis prediction
Atmospheric flows
Storm surges
Planetary flows
Some examples
Tsunamis prediction
Atmospheric flows
Storm surges
Planetary flows
Counterexamples
The SWE can not be applied, when
3d effects become essential
Waves become too short or too
high
Counterexamples
The SWE can not be applied, when
3d effects become essential
Waves become too short or too
high
Overview
1
2
3
Mass conservation
Conservation of mass
∂
̺ + ∇ · (̺v) = 0
∂t
̺ = Density
v = (u, v , w ) = Velocity
Mass conservation
∂
̺ + ∇ · (̺v) = 0
∂t
̺=Density
Mass conservation
∇·v = 0
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
Momentum conservation
Conservation of momentum
∂
(̺v) + ∇ · (̺v ⊗ v + pI − Π) = ̺g
∂t
̺ = Density
p = Pressure
Π = Viscosity
g = Body force vector (gravitation)
∂
(̺v) + ∇ · (̺v ⊗ v + pI − Π) = ̺g
∂t
̺ = Density
p = Pressure
Π=Viscosity
∂
(̺v ) + ∇ · (̺v ⊗ v + pI) = ̺g
∂t
̺ = Density
p = Pressure
∂
(̺v ) + ∇ · (̺v ⊗ v + pI) = ̺g
∂t
̺=Density
p = Pressure
∂
1
v + ∇ · (v ⊗ v) + ∇ · pI = g
∂t
̺
̺=Density
p = Pressure
∂
1
v + ∇ · (v ⊗ v) + ∇ · pI = g
∂t
̺
̺ = Density
p = Pressure
∂
1
v + ∇ · (v ⊗ v) + ∇ · pI = g
∂t
̺
̺ = Density
p = Pressure


0
1
∂
v + ∇ · (v ⊗ v) + ∇ · pI =  0 
∂t
̺
−g
̺ = Density
p = Pressure


0
∂
1
v + ∇ · (v ⊗ v) + ∇ · pI =  0 
∂t
̺
−g
̺ = Density
p = Pressure


0
1
∂
v + ∇ · (v ⊗ v) = − ∇ · pI +  0 
∂t
̺
−g
̺ = Density
p = Pressure
Simplified form
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
∂
∂
∂
1 ∂
∂
w +u w +v
w +w w =−
p−g
∂t
∂x
∂y
∂z
̺ ∂z
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
∂
∂
∂
1 ∂
∂
w +u w +v
w +w w =−
p−g
∂t
∂x
∂y
∂z
̺ ∂z
Solveability
We have obtained four PDEs for the four unknowns
Given appropriate initial and boundary conditions, these equations
can be solved
Boundary conditions have to be applied to the surface
BUT
The position of the surface is not
known a priori!
Solveability
can be solved
BUT
known a priori!
Solveability
can be solved
BUT
known a priori!
Variables
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
∂
∂
∂
1 ∂
∂
w +u w +v
w +w w =−
p−g
∂t
∂x
∂y
∂z
̺ ∂z
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
1 ∂
d
w =−
p−g
dt
̺ ∂z
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
d
1 ∂
p−g
0= w =−
dt
̺ ∂z
Main simplification
Primary assumption in SW-Theory
Horizontal scales (wave length l) are much larger, than vertical scales
(undistubed water heigth h)
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
|
{z
} | {z }
≈W
h
≈ Ul
⇓
h
W ≈U ,
l
where
h
≪1
l
Main simplification
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
|
{z
} | {z }
≈W
h
≈ Ul
⇓
h
W ≈U ,
l
where
h
≪1
l
Main simplification
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
|
{z
} | {z }
≈W
h
≈ Ul
⇓
h
W ≈U ,
l
where
h
≪1
l
Boundary layer theory
Neglegt vertical accelerations
Boundary layer assumption
d
w =0
dt
So you can think of the SWE as a boundary layer.
This explains the name “shallow”.
Boundary layer theory
Neglegt vertical accelerations
Boundary layer assumption
d
w =0
dt
So you can think of the SWE as a boundary layer.
This explains the name “shallow”.
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
d
1 ∂
p−g
0= w =−
dt
̺ ∂z
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
1 ∂
p−g
0= −
̺ ∂z
Hydrostatic pressure
∂
p = −̺g
∂z
Integrate this equation
Hydrostatic pressure distribution
p=g
Z
s
̺ dz + pa
z
Hydrostatic pressure
∂
p = −̺g
∂z
Integrate this equation
Hydrostatic pressure distribution
p=g
Z
s
̺ dz + pa
z
Pressure gradient
Assuming constant density along the z-axis, we obtain
p = ̺g (s − z) + pa
Pressure gradient
∂
∂
p = ̺g s
∂x
∂x
∂
∂
p = ̺g s
∂y
∂y
Pressure gradient
Assuming constant density along the z-axis, we obtain
p = ̺g (s − z) + pa
Pressure gradient
∂
∂
p = ̺g s
∂x
∂x
∂
∂
p = ̺g s
∂y
∂y
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
1 ∂
u+u u+v
u+w u =−
p
∂t
∂x
∂y
∂z
̺ ∂x
∂
∂
∂
1 ∂
∂
v +u v +v
v +w v =−
p
∂t
∂x
∂y
∂z
̺ ∂y
1 ∂
0= −
p−g
̺ ∂z
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
∂
u+u u+v
u + w u = −g s
∂t
∂x
∂y
∂z
∂x
∂
∂
∂
∂
∂
v +u v +v
v + w v = −g s
∂t
∂x
∂y
∂z
∂y
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
∂
u+u u+v
u + w u = −g s
∂t
∂x
∂y
∂z
∂x
∂
∂
∂
∂
∂
v +u v +v
v + w v = −g s
∂t
∂x
∂y
∂z
∂y
Horizontal mean velocity
The pressure gradient
∂
∂
p = ̺g s
∂x
∂x
∂
∂
p = ̺g s
∂y
∂y
is independent of the variable z
is responsible for horizontal accelerations
The horizontal velocities u, v
are independent of the variable z
can be interpreted as vertical mean velocities
∂
∂
∂z u = ∂z v = 0
∂
∂
p = ̺g s
∂x
∂x
∂
∂
p = ̺g s
∂y
∂y
∂
∂
∂z u = ∂z v = 0
∂
∂
p = ̺g s
∂x
∂x
∂
∂
p = ̺g s
∂y
∂y
∂
∂
∂z u = ∂z v = 0
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
∂
u+u u+v
u + w u = −g s
∂t
∂x
∂y
∂z
∂x
∂
∂
∂
∂
∂
v +u v +v
v + w v = −g s
∂t
∂x
∂y
∂z
∂y
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
u+u u+v
u = −g s
∂t
∂x
∂y
∂x
∂
∂
∂
∂
v +u v +v
v = −g s
∂t
∂x
∂y
∂y
Simplified form
Mass continuity
∂
∂
∂
u+
v+
w =0
∂x
∂y
∂z
∂
∂
∂
∂
u+u u+v
u = −g s
∂t
∂x
∂y
∂x
∂
∂
∂
∂
v +u v +v
v = −g s
∂t
∂x
∂y
∂y
Depth averaging
Z
b
s
∂
∂
∂
u+
v+
w dz = 0
∂x
∂y
∂z
Mass continuity
∂
∂
∂
h+
(hu) +
(hv ) = 0
∂t
∂x
∂y
Depth averaging
Z
b
s
∂
∂
∂
u+
v+
w dz = 0
∂x
∂y
∂z
Mass continuity
∂
∂
∂
h+
(hu) +
(hv ) = 0
∂t
∂x
∂y
Simplified form
Mass continuity in conservative form
∂
∂
∂
h+
(hu) +
(hv ) = 0
∂t
∂x
∂y
∂
∂
∂
∂
u+u u+v
u = −g s
∂t
∂x
∂y
∂x
∂
∂
∂
∂
v +u v +v
v = −g s
∂t
∂x
∂y
∂y
Simplified form
Mass continuity in conservative form
∂
∂
∂
h+
(hu) +
(hv ) = 0
∂t
∂x
∂y
Momentum equations in conservative form
∂
(hu) +
∂t
∂
(hv ) +
∂t
∂
1
∂
∂
(hu 2 + gh2 ) +
(huv ) = −gh b
∂x
2
∂y
∂x
∂
∂
1
∂
(huv ) +
(hv 2 + gh2 ) = −gh b
∂x
∂y
2
∂y
Final form of the equations I
SWE I
∂
∂
∂
h+
(hu) +
(hv ) = 0
∂t
∂x
∂y
∂
1
∂
∂
∂
(hu) +
(hu 2 + gh2 ) +
(huv ) = −gh b
∂t
∂x
2
∂y
∂x
∂
∂
1
∂
∂
(hv ) +
(huv ) +
(hv 2 + gh2 ) = −gh b
∂t
∂x
∂y
2
∂y
h
u, v
b
:
:
:
Water heigth
Depth-averaged velocity in x- and y-direction
Function describing the bottom profile
Final form of the equations I
SWE I
∂
∂
∂
h+
(hu) +
(hv ) = 0
∂t
∂x
∂y
∂
1
∂
∂
∂
(hu) +
(hu 2 + gh2 ) +
(huv ) = −gh b
∂t
∂x
2
∂y
∂x
∂
∂
1
∂
∂
(hv ) +
(huv ) +
(hv 2 + gh2 ) = −gh b
∂t
∂x
∂y
2
∂y
h
u, v
b
:
:
:
Water heigth
Depth-averaged velocity in x- and y-direction
Function describing the bottom profile
Final form of the equations II
SWE II
∂
∂
∂
Q+
F (Q) +
G(Q) = Sb (Q)
∂t
∂x
∂y
Q
F (Q)
G(Q)
Sb
=
=
=
=
(h, hu, hv )⊤
(hu, hu 2 + 21 gh2 , huv )⊤
(hv , huv , hv 2 + 21 gh2 )⊤
(0, −ghbx , −ghby )⊤
Vector of conservered variables
Flux in x-direction
Flux in y-direction
Source term for bottom profile
Final form of the equations II
SWE II
∂
∂
∂
Q+
F (Q) +
G(Q) = Sb (Q)
∂t
∂x
∂y
Q
F (Q)
G(Q)
Sb
=
=
=
=
(h, hu, hv )⊤
(hu, hu 2 + 12 gh2 , huv )⊤
(hv , huv , hv 2 + 21 gh2 )⊤
(0, −ghbx , −ghby )⊤
Vector of conservered variables
Flux in x-direction
Flux in y-direction
Source term for bottom profile
Extensions
Coriolis forces
Wind shear stress
Multilayer models
Additional transport equations (heat, dissolved substances, sediment
particles)
The Boussinesq Equations (to model shorter and steeper waves) can
be considered as an extension to the SWE
Extensions
Coriolis forces
Wind shear stress
Multilayer models
particles)
Extensions
Coriolis forces
Wind shear stress
Multilayer models
particles)
Extensions
Coriolis forces
Wind shear stress
Multilayer models
particles)
Extensions
Coriolis forces
Wind shear stress
Multilayer models
particles)
Final remarks
The SWE are not completely 2d
The resulting 2d velocity field is not divergence-free
There are 3d models, but as the detph-averaging is a very important
part of the SWE, they are usually just called boundary layer models
Final remarks
Final remarks
Overview
1
2
3
Melitta test case
Our first test case will show us, that
the SWE can produce reasonable
solutions
Its solution is well known from the
famous tv ad of melitta :-)
Melitta
Melitta
Melitta
Melitta
Analytical solutions
1d dambreak problem
∂
∂t
h
hu
∂
+
∂x
h0 :=
hu
hu 2 + 12 gh2
hl
hr
= 0.
x <0
x >0
u0 := 0
1d dambreak
Generalisation to 2d
There are 2d versions of the dambreak problem.
First we will consider a water revervoir break, then we will observe the
solution of an asymmetric dambreak problem.
Generalisation to 2d
There are 2d versions of the dambreak problem.
First we will consider a water revervoir break, then we will observe the
solution of an asymmetric dambreak problem.
2d circular dambreak
2d circular dambreak
2d asymmetric dambreak
2d asymmetric dambreak
Flow around a pillar
Floodwave flowing around a pillar
Computational grid
Flow around pillar
Flow around pillar
Flow around pillar
Flow around pillar
Flow around pillar
Flow around pillar
Nonlinear hyperbolic charakter
In our last test case we observe the nonlinear hyperbolic character of the
SWE.
The end

Shallow Water - Derivation and Applications

Transcription

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