dispersed phase turbulence

The goal is to obtain the correct coupling between the continuous and dispersed phase velocity time series. So far, it appeared that the dispersed phase time series is decoupled too strongly from the continuous phase time series.

Based on Eq. 5 in Balachandar and Eaton (2010) the forces acting on the particle \bold{F_p} (neglecting Basset history and gravitational force) are:

\bold{F_p} = 3 \pi \mu_f d(\bold{u}-\bold{v}) \phi(Re) + m_f \frac{D\bold{u}}{dt} + m_f C_M\left(\frac{D\bold{u}}{dt} - \frac{d\bold{v}}{dt}\right)

Where \bold{u} is the fluid velocity, \bold{v} the particle velocity, m_f and m_p are the mass of the fluid and the particle, C_M the added mass coefficient (assumption: C_M = 0.5 (cf. p. 102 Rushe, 2002), \mu_f is the dynamic viscosity of the fluid, d the particle diameter. \phi(Re) is term originating from the drag coefficient C_D:

\bold{C_D} = \frac{24}{Re}\phi(Re) = \frac{24}{Re}\left(1 + 0.15 Re ^{0.687}\right)

The particle velocity \bold{v} is then obtained by solving the momentum equation for the particle:

m_p \frac{D\bold{v}}{dt} = \bold{F_p} 

Since Eq. 5 in Balachandar and Eaton (2010) is only applicable for a spherical particle in a uniform cross flow, the spatial derivates can be omitted, and setting and pressure gradient force to 0 (du/dt=0) results in the following equation:

m_p \frac{d\bold{v}}{dt} = 3 \pi \mu_f d(\bold{u}-\bold{v}) \phi(Re) - m_f C_M \frac{d\bold{v}}{dt}

Further simplified and setting m_p = V_P \rho_f and m_f = V_p \rho_f with V_P being the particle volume, the equation simplifies to:

V_p\left(\rho_p + \rho_f C_M\right)\frac{d\bold{v}}{dt} = 3 \pi \mu_f d(\bold{u}-\bold{v}) \phi(Re)

First-order Euler scheme:

V_p\left(m_p + m_f C_M\right)\frac{\bold{v}^{t+1}-\bold{v}^{t}}{\Delta t} = 3 \pi \mu_f d(\bold{u}^{t}-\bold{v}^{t}) \phi(Re)

Solving for \bold{v}^{t+1}:

\bold{v}^{t+1} = \bold{v}^{t} + \frac{\Delta t}{V_p\left(m_p + m_f C_M\right)} \left(3 \pi \mu_f d(\bold{u}^{t}-\bold{v}^{t}) \phi(Re)\right)

Literature:
Balachandar, S., & Eaton, J. K. (2010). Turbulent dispersed multiphase flow. Annual review of fluid mechanics, 42, 111-133. https://doi.org/10.1146/annurev.fluid.010908.165243.