A conservative FE-discretisation of the Navier-Stokes equation

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A conservative FE-discretisation of the
Navier-Stokes equation

• JASS 2005, St. Petersburg
• Thomas Satzger

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Overview

• Navier-Stokes-Equation
• Interpretation
• Laws of conservation
• Basic Ideas of FD, FE, FV
• Conservative FE-discretisation of
  Navier-Stokes-Equation

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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

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Navier-Stokes-Equation
The Navier-Stokes-Equation writes
Equation of momentum
Continuity equation
with
Velocity field
Pressure field
Density
Dynamic viscosity
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Navier-Stokes-Equation
The interpretation of these terms are
Outer Forces
Diffusion
Pressure gradient
Convection
Derivative of velocity field
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Navier-Stokes-Equation
The corresponding for the components is
for the momentum equation, and
for the continuity equation.
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Navier-Stokes-Equation
With the Einstein summation
and the abbreviation
we get
for the momentum equation, and
for the continuity equation.
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Navier-Stokes-Equation
Now take a short look to the dimensions
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Navier-Stokes-Equation - Interpretation
We see that the momentum equations handles with
accelerations. If we rewrite the equation, we
get
This means
Total acceleration is the sum of the partial
accelerations.
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Navier-Stokes-Equation - Interpretation
Interpretation of the Convection
fluid particle
Transport of kinetic energy by moving the fluid
particle
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Navier-Stokes-Equation - Interpretation
Interpretation of the pressure Gradient
Acceleration of the fluid particle by pressure
forces
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Navier-Stokes-Equation - Interpretation
Interpretation of the Diffusion
Distributing of kinetic Energy by friction
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Navier-Stokes-Equation - Interpretation
Interpretation of the continuity equation

• Conservation of mass in arbitrary domain

this means influx out flux
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Navier-Stokes-Equation - Laws of conservation
Conservation of kinetic energy We must know
that the kinetic energy doesn't increase, this
means Proof
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Navier-Stokes-Equation - Laws of conservation
With the momentum equation it holds Using
the relations (proof with the continuity
equation) and
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Navier-Stokes-Equation - Laws of conservation
Additionally it holds Therefore we get Due
to Greens identity we have
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Navier-Stokes-Equation - Laws of conservation
This means in total We have also seen that
the continuity equation is very important for
energy conservation.
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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.

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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

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Basic Ideas of FD, FE, FV
Comparison of FD, FE and FV
Finite Difference
Finite Volume
Finite Element
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Basic Ideas of FD, FE, FV
Advantages and Disadvantages Finite
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 Conserv
ation laws aren't always complied by the
discretisation. This can lead to problems in
stability of the solution.
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Conservative FE-Elements
We use a partially staggered grid for our
discretisation.
We write
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Conservative FE-Elements
The FE-approximation is an element of an
finite-dimensional function space with the
basis The approximation has the
representation
whereby
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Conservative FE-Elements
If we use a Nodal basis, this means we can
rewrite the approximation
and
and
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Conservative FE-Elements
Every approximation should have the following
properties continuous conservative 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.
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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.
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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.
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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.
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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)
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Conservative FE-Elements
The bilinear interpolation isn't conservative
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Conservative FE-Elements
The bilinear interpolation isn't conservative
It is easy to show that
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Conservative FE-Elements
The bilinear interpolation isn't conservative
Basis on the box
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Conservative FE-Elements
These basis function for the bilinear
interpolation are called Pagoden. The
picture shows the function on the whole support.
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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?
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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?
Divide the box in four triangles.
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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?
Divide the box in four triangles.
Make on every triangle an linear interpolation.
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the
following relations
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Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the
following relations
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Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the
following relations
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Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the
following relations
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Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the
following relations
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Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the
following relations
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Conservative FE-Elements
What's the right velocity in the middle?
We must have at every point in the box the
following relations
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
Till now we have
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Conservative FE-Elements
Till now we have With the discrete
continuity equation we get
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Conservative FE-Elements
Till now we have With the discrete
continuity equation we get Therefore we
choose
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Conservative FE-Elements
What's the right velocity in the middle?
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Now we calculate the basis.
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Conservative FE-Elements
Linear interpolation provides the basis.
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Conservative FE-Elements
View on conservative elements in 3D
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Conservative FE-Elements
View on conservative elements in 3D
Partially staggered grid in 3D
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Conservative FE-Elements
We also search for a conservative interpolation
of the velocities.
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Conservative FE-Elements
We also search for a conservative interpolation
of the velocities.
Divide every box into 24 tetrahedrons, on which
you make a linear interpolation