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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
JASS 2005, St. Petersburg
Thomas Satzger
Navier-Stokes-Equation
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
for the momentum equation, and
for the continuity equation.
Navier-Stokes-Equation
and the abbreviation
we get:
for the momentum equation, and
for the continuity equation.
Navier-Stokes-Equation
Navier-Stokes-Equation - Interpretation
This means:
Total acceleration is the sum of the partial accelerations.
Navier-Stokes-Equation - Interpretation
fluidparticle
Transport of kinetic energy by moving the fluid particle
Navier-Stokes-Equation - Interpretation
fluid particle
Acceleration of the fluid particle by pressure forces
Navier-Stokes-Equation - Interpretation
fluid particle
Distributing of kinetic Energy by friction
Navier-Stokes-Equation - Interpretation
for
we get
this means:
influx = out flux
Navier-Stokes-Equation - Laws of conservation
Navier-Stokes-Equation - Laws of conservation
Navier-Stokes-Equation - Laws of conservation
Navier-Stokes-Equation - Laws of conservation
Basic Ideas of FD, FE, FV
Basic Ideas of FD, FE, FV
Basic Ideas of FD, FE, FV
Finite Difference
Finite Volume
Finite Element
Basic Ideas of FD, FE, FV
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 write:
Conservative FE-Elements
whereby
Conservative FE-Elements
and
and
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
(1)
Conservative FE-Elements
Conservative FE-Elements
It is easy to show that
Conservative FE-Elements
Basis on the box
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Divide the box in four triangles.
Conservative FE-Elements
Divide the box in four triangles.
Make on every triangle an linear interpolation.
Conservative FE-Elements
Conservative FE-Elements
We must have at every point in the box the following relations:
Conservative FE-Elements
We must have at every point in the box the following relations:
Conservative FE-Elements
We must have at every point in the box the following relations:
Conservative FE-Elements
We must have at every point in the box the following relations:
Conservative FE-Elements
We must have at every point in the box the following relations:
Conservative FE-Elements
We must have at every point in the box the following relations:
Conservative FE-Elements
We must have at every point in the box the following relations:
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
Conservative FE-Elements
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