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