Institute of Fluid DynamicsInstitute of Fluid Dynamics

Patrick Jenny (IFD, ETH Zürich)

Manuel Torrilhon (SAM, ETH Zürich)

Stefan Heinz (Dept. of Mathematics, University of Wyoming)

Institute of Fluid Dynamics

Fokker Planck Equation for PDF Evolution

Institute of Fluid DynamicsPatrick Jenny

• Kinetic Description of Non-Equilibrium G as

• Consistency with Continuum Fluid Dynamics

• Collision Models

• Fokker Planck Model and Integration Scheme

• Knudsen Paradox

• Performance

• Conclusion

Outline

Institute of Fluid DynamicsPatrick Jenny

Kinetic Description of Non-Equilibrium Gas

Assume that the statistics of particles in a monatomic gas can be described by the mass density function

with the weights

In equilibrium, the velocity PDF is given by the Maxwell distribution:

Institute of Fluid DynamicsPatrick Jenny

Consistency with Fluid Dynam ics

with the weights

Institute of Fluid DynamicsPatrick Jenny

Consistency with Fluid Dynam ics

with the weights

Institute of Fluid DynamicsPatrick Jenny

Consistency with Fluid Dynam ics

with the weights

Institute of Fluid DynamicsPatrick Jenny

Consistency with Fluid Dynam ics

with the weights

Chapman-Enskog Expansion => Navier-Stokes

Institute of Fluid DynamicsPatrick Jenny

Collision Models

with the weights

Institute of Fluid DynamicsPatrick Jenny

with the weights

Collision Models

Institute of Fluid DynamicsPatrick Jenny

with the weights

Here we propose a Fokker-Planck operator:

motivation: leads to the possibility to use highly efficient numerical methods

Collision Models

Institute of Fluid DynamicsPatrick Jenny

Collision Models

Institute of Fluid DynamicsPatrick Jenny

Fokker-Planck Solution Algorithm

Institute of Fluid DynamicsPatrick Jenny

Fokker-Planck Solution Algorithm