The principle of Random Galilean Transformation (RGT) Invariance is applied to the random motion of particles in a turbulent gas to construct a kinetic equation for the transport of the particle phase space probability <W(\mathbf{ v},\mathbf{ x},t)> where \mathbf{ v} and \mathbf{ x} are the velocity and position of a particle at time t. The essential problem is to find closed expressions for the phase space dispersion current \left\langle \boldsymbol{ f}W\right\rangle , where \boldsymbol{ f} is the fluctuating aerodynamic force at \mathbf{ v} and \mathbf{ x} at time t. The simplest form consistent with RGT invariance, the correct equation of state ancl form for the inter-phase momentum transfer tern is shown to be \left\langle \boldsymbol{ f}W\right\rangle =-\left(\boldsymbol{ \mu}\cdot\frac{ \partial}{ \partial\mathbf{ v}}+\boldsymbol{ \lambda}\cdot\frac{ \partial}{ \partial\mathbf{ x}}\right)\left\langle W\right\rangle in which \text{ \textbf{ \ensuremath{ \boldsymbol{ \,\mu}}}\ensuremath{ =<\boldsymbol{ f}(t)\mathbf{ v}(t)>}} and \boldsymbol{ \lambda}=<\boldsymbol{ f}(t)\boldsymbol{ \mathbf{ x}}(t)>.This approach to modeling gas-solid flows is currently being used to investigate the behavior of radioactive aerosols inside gas-cooled nuclear reactors.
Categories: physics.flu-dyn