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

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A conservative fe discretisation of the navier stokes equation

A conservative FE-discretisation of the Navier-Stokes equation

JASS 2005, St. Petersburg

Thomas Satzger


Overview

Overview

  • Navier-Stokes-Equation

    • Interpretation

    • Laws of conservation

  • Basic Ideas of FD, FE, FV

  • Conservative FE-discretisation of Navier-Stokes-Equation


A conservative fe discretisation of the 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


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation

The Navier-Stokes-Equation writes:

Equation of momentum

Continuity equation

with

Velocityfield

Pressurefield

Density

Dynamic viscosity


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation

The interpretation of these terms are:

Outer Forces

Diffusion

Pressure gradient

Convection

Derivative of velocity field


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation

  • The corresponding for the components is:

for the momentum equation, and

for the continuity equation.


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation

  • With the Einstein summation

and the abbreviation

we get:

for the momentum equation, and

for the continuity equation.


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation

  • Now take a short look to the dimensions:


A conservative fe discretisation of the navier stokes equation

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.


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Interpretation

  • Interpretation of the Convection

fluidparticle

Transport of kinetic energy by moving the fluid particle


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Interpretation

fluid particle

  • Interpretation of the pressure Gradient

Acceleration of the fluid particle by pressure forces


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Interpretation

fluid particle

  • Interpretation of the Diffusion

Distributing of kinetic Energy by friction


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Interpretation

for

we get

  • Interpretation of the continuity equation

  • Conservation of mass in arbitrary domain

this means:

influx = out flux


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Laws of conservation

  • Conservation of kinetic energy:

  • We must know that the kinetic energy doesn't increase, this means:

  • Proof:


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Laws of conservation

  • With the momentum equation

  • it holds

  • Using the relations(proof with the continuity equation)

  • and


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Laws of conservation

  • Additionally it holds

  • Therefore we get

  • Due to Greens identity we have


A conservative fe discretisation of the navier stokes equation

Navier-Stokes-Equation - Laws of conservation

  • This means in total

  • We have also seen that the continuity equation is very important for energy conservation.


A conservative fe discretisation of the navier stokes equation

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.


A conservative fe discretisation of the navier stokes equation

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


A conservative fe discretisation of the navier stokes equation

Basic Ideas of FD, FE, FV

  • Comparison of FD, FE and FV

Finite Difference

Finite Volume

Finite Element


A conservative fe discretisation of the navier stokes equation

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.


A conservative fe discretisation of the navier stokes equation

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:


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

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

  • The approximation has the representation

whereby


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • If we use a Nodal basis, this means

  • we can rewrite the approximation

and

and


A conservative fe discretisation of the navier stokes equation

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.


A conservative fe discretisation of the navier stokes 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.


A conservative fe discretisation of the navier stokes 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.


A conservative fe discretisation of the navier stokes 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.


A conservative fe discretisation of the navier stokes 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)


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • The bilinear interpolation isn't conservative


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • The bilinear interpolation isn't conservative

It is easy to show that


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • The bilinear interpolation isn't conservative

Basis on the box


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • These basis function for the bilinear interpolation are called

  • Pagoden.

  • The picture shows the function on the whole support.


A conservative fe discretisation of the navier stokes equation

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?


A conservative fe discretisation of the navier stokes equation

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.


A conservative fe discretisation of the navier stokes equation

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.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?

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


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?

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


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?

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


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?

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


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?

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


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?

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


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?

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


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Till now we have:


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Till now we have:

  • With the discrete continuity equation

  • we get


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Till now we have:

  • With the discrete continuity equation

  • we get

  • Therefore we choose


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • What's the right velocity in the middle?


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

  • Now we calculate the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

Linear interpolation provides

the basis.


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

View on conservative elements in 3D


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

View on conservative elements in 3D

Partially staggered grid in 3D


A conservative fe discretisation of the navier stokes equation

Conservative FE-Elements

We also search for a conservative interpolation of the velocities.


A conservative fe discretisation of the navier stokes equation

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


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