The importance of random galilean transformation invariance in modelling dispersed particle flows

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(\boldsymbolμ\cdot\frac{\partial}{\partial$\mathbf{v}$}+\boldsymbolλ\cdot\frac{\partial}{\partial$\mathbf{x}$}\right)\left\langle W\right\rangle in which \text{\textbf{\ensuremath{\boldsymbol{\,μ}}}\ensuremath{=<\boldsymbol{f}(t)\mathbf{v}(t)>}} and \boldsymbolλ=<\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.