particle velocity time series

The goal is to obtain a separate velocity time series for the particle (bubble/dropplet), based on the velocity time series of the surrounding fluid and by taking into account the most relevant forces.

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.