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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

- Navier-Stokes-Equation
- Interpretation
- Laws of conservation

- Basic Ideas of FD, FE, FV
- Conservative FE-discretisation of Navier-Stokes-Equation

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

Navier-Stokes-Equation

The Navier-Stokes-Equation writes:

Equation of momentum

Continuity equation

with

Velocityfield

Pressurefield

Density

Dynamic viscosity

Navier-Stokes-Equation

The interpretation of these terms are:

Outer Forces

Diffusion

Pressure gradient

Convection

Derivative of velocity field

Navier-Stokes-Equation

- The corresponding for the components is:

for the momentum equation, and

for the continuity equation.

Navier-Stokes-Equation

- With the Einstein summation

and the abbreviation

we get:

for the momentum equation, and

for the continuity equation.

Navier-Stokes-Equation

- Now take a short look to the dimensions:

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.

Navier-Stokes-Equation - Interpretation

- Interpretation of the Convection

fluidparticle

Transport of kinetic energy by moving the fluid particle

Navier-Stokes-Equation - Interpretation

fluid particle

- Interpretation of the pressure Gradient

Acceleration of the fluid particle by pressure forces

Navier-Stokes-Equation - Interpretation

fluid particle

- Interpretation of the Diffusion

Distributing of kinetic Energy by friction

Navier-Stokes-Equation - Interpretation

for

we get

- Interpretation of the continuity equation

- Conservation of mass in arbitrary domain

this means:

influx = out flux

Navier-Stokes-Equation - Laws of conservation

- Conservation of kinetic energy:
- We must know that the kinetic energy doesn't increase, this means:
- Proof:

Navier-Stokes-Equation - Laws of conservation

- With the momentum equation
- it holds
- Using the relations(proof with the continuity equation)
- and

Navier-Stokes-Equation - Laws of conservation

- Additionally it holds
- Therefore we get
- Due to Greens identity we have

Navier-Stokes-Equation - Laws of conservation

- This means in total
- We have also seen that the continuity equation is very important for energy conservation.

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.

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

Basic Ideas of FD, FE, FV

- Comparison of FD, FE and FV

Finite Difference

Finite Volume

Finite Element

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:
- Conservation laws aren't always complied by the discretisation. This can lead to problems in stability of the solution.

Conservative FE-Elements

for the number of grid points

for the horizontal velocity in the i-th grid point

for the vertical velocity in the i-th grid point

- We use a partially staggered grid for our discretisation.

We write:

Conservative FE-Elements

- The FE-approximation is an element of an finite-dimensional function space with the basis
- The approximation has the representation

whereby

Conservative FE-Elements

- If we use a Nodal basis, this means
- we can rewrite the approximation

and

and

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.

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.

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.

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.

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)

Conservative FE-Elements

- The bilinear interpolation isn't conservative

Conservative FE-Elements

- The bilinear interpolation isn't conservative

It is easy to show that

Conservative FE-Elements

- The bilinear interpolation isn't conservative

Basis on the box

Conservative FE-Elements

- These basis function for the bilinear interpolation are called
- Pagoden.
- The picture shows the function on the whole support.

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?

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.

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.

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

We must have at every point in the box the following relations:

Conservative FE-Elements

- What's the right velocity in the middle?

We must have at every point in the box the following relations:

Conservative FE-Elements

- What's the right velocity in the middle?

We must have at every point in the box the following relations:

Conservative FE-Elements

- What's the right velocity in the middle?

We must have at every point in the box the following relations:

Conservative FE-Elements

- What's the right velocity in the middle?

We must have at every point in the box the following relations:

Conservative FE-Elements

- What's the right velocity in the middle?

We must have at every point in the box the following relations:

Conservative FE-Elements

- What's the right velocity in the middle?

We must have at every point in the box the following relations:

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- Till now we have:

Conservative FE-Elements

- Till now we have:
- With the discrete continuity equation
- we get

Conservative FE-Elements

- Till now we have:
- With the discrete continuity equation
- we get
- Therefore we choose

Conservative FE-Elements

- What's the right velocity in the middle?

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

- Now we calculate the basis.

Conservative FE-Elements

Linear interpolation provides

the basis.

Conservative FE-Elements

View on conservative elements in 3D

Conservative FE-Elements

View on conservative elements in 3D

Partially staggered grid in 3D

Conservative FE-Elements

We also search for a conservative interpolation of the velocities.

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