Virtual Mass

The inertia of surrounding fluid influences the acceleration of a particle that is submerged in the flow. Inviscid flow theory allows this effect to be represented as a “virtual mass” or “added mass” equal to a constant multiplied by the mass of fluid that the particle displaces.

Including this “virtual mass force” can make accelerating flows more realistic [496]. For example, the virtual mass term can influence the trajectory of bubbles in a steady but swirling flow. Because it narrows the range of response timescales across different phases, the virtual mass term can also help non-accelerating flows to converge, by making them less sensitive to momentum or pressure relaxation factors.

A two-phase formulation for the virtual mass force is from Auton et al. [430]. This formulation can also be used in the multiphase context where there is one continuous phase and several dilute, dispersed phases. For non-dilute systems, it can still give a useful first approximation, especially if the virtual mass coefficient includes a correction for the concentration of particles.

The virtual mass force acting on phase $i$ due to acceleration relative to phase $j$ is:

Figure 1. EQUATION_DISPLAY

{F_{i j}}^{V M} = C_{VM} ρ_{c} α_{d} (a_{j} - a_{i})

(1996)

where:

$c$ is the continuous phase in the phase interaction $i j$ .

$d$ is the dispersed phase in the phase interaction $i j$ .

$C_{VM}$ is the virtual mass coefficient for interaction $i j$ .

$a_{i}, a_{j}$ are the acceleration of phase $i$ and $j$ , respectively.

The acceleration term for virtual mass is based on the rate of change of velocity of constant mass particles, which in the stationary coordinate frame becomes the material derivative:

Figure 2. EQUATION_DISPLAY

a_{i} = {(\frac{D v}{D t})}_{i}

(1997)

However, for applications where the particle mass can increase or decrease due to interphase mass transfer, $a_{i}$ must be generalized. This is accomplished by starting with the momentum balance for the phase that is contained within a constant control volume. The momentum fluxes across phase-to-phase interfaces can be identified as similar terms to the momentum fluxes across intersections between the phase and the outer surfaces of the control volume. This leads to the following definition of the total inertial term for phase $i$ in reaction to all other applied forces:

Figure 3. EQUATION_DISPLAY

ρ_{i} α_{i} a_{i} = \frac{\partial}{\partial t} ρ_{i} α_{i} v_{i} + \nabla \cdot (ρ_{i} α_{i} v_{i} v_{i}) - \sum_{i}^{} m_{ij} v_{ji}

(1998)

where:

$m_{ij}$ is the interphase mass transfer rate per volume from phase $i$ to phase $j$ .

$v_{ij}$ is the interphase value of velocity.

The following closure satisfies conservation of momentum in the sense that the momentum flux leaving one phase balances the momentum flux entering the other phase:

Figure 4. EQUATION_DISPLAY

v_{ij} = {\begin{matrix} \begin{matrix} v_{j} m_{ij} > 0 \end{matrix} \\ \begin{matrix} v_{i} m_{ij} < 0 \end{matrix} \end{matrix}

(1999)

The full acceleration term, including momentum fluxes due to interphase mass transfer, then becomes:

Figure 5. EQUATION_DISPLAY

a_{i} = {(\frac{D v}{D t})}_{i} + \frac{1}{ρ_{i} α_{i}} \sum_{l}^{} m_{ij} ({v_{i} - v}_{ij})

(2000)

Spherical Particle Virtual Mass Coefficient

From inviscid flow theory [496], the virtual mass coefficient for a spherical particle accelerating in an unbounded three-dimensional fluid is:

Figure 6. EQUATION_DISPLAY

C_{VM, sphere} = 0.5

(2001)

This function is limited for stable solutions across a wide range of volume fraction by specifying a minimum free stream fraction $γ_{m i n}$ with a default value of zero.

Figure 7. EQUATION_DISPLAY

C_{VM} = {[{(C_{VM, sphere})}^{- 2} + C_{VM, \max} {(α_{c}, γ_{m i n})}^{- 2}]}^{- 1 / 2}

(2002)

Zuber Virtual Mass Coefficient

Lamb [496] provided an inviscid flow solution for a case of an inner sphere accelerating across the center of an outer stationary sphere. Zuber [578] used this solution to estimate the virtual mass coefficient for an infinite array of particles. The cube of the sphere diameter ratio represents the volume fraction.

The result in terms of the volume fraction of a single dispersed phase is:

Figure 8. EQUATION_DISPLAY

C_{VM, Zuber} = 0.5 \frac{1 + 2 α_{d}}{1 - α_{d}}

(2003)

This function is adapted for multiphase computation by replacing $α_{d}$ by the sum of dispersed phase volume fractions:

Figure 9. EQUATION_DISPLAY

\sum α_{d} = 1 - α_{c}

(2004)

and then applying a limiter for robustness as $α_{c} \to 0$ :

Figure 10. EQUATION_DISPLAY

C_{VM, Zuber} = 0.5 \frac{3 - {2 α}_{c}}{α_{c}}

(2005)

Figure 11. EQUATION_DISPLAY

C_{VM} = {[C_{VM}^{- 2} + C_{VM, \max} {(α_{c}, γ_{m i n})}^{- 2}]}^{- (1 / 2)}

(2006)

The default value of the notional “free stream” fraction is $γ_{m i n} = - 10$ . This allows the Zuber virtual mass coefficient to represent the effect of total particle concentration on the virtual mass force in the useful range $0 < \sum α_{d} < 0.66$ before limiting is applied.

In a simulation, the selected virtual mass coefficient can be plotted as a field.

The default option is the spherical particle method that is based on a standard coefficient of 0.5 for spherical particles. For stability, the coefficient is reduced at high dispersed phase concentrations.

The Zuber option is suitable for improved accuracy in modeling the effect of increasing particle concentration, or multiple dispersed phases, in accelerating flows.

The virtual mass coefficients that are produced by these two methods are compared in the figure below, each using their own default value for free stream fraction $γ_{m i n}$ as a robustness limit.

Virtual Mass Terms in Rotating Frame

The Virtual Mass Force term that is defined above is modified as follows when used in multiple reference frame (MRF) applications. This modification is based on an absolute velocity formulation, which solves directly for velocities in the inertial frame, $I$ , using momentum equations that are derived in the rotating frame, $R$ .

Figure 12. EQUATION_DISPLAY

{F_{i j}}^{V M} = C_{VM} ρ_{c} α_{d} [{a_{j}}^{I R} - {a_{i}}^{I R}] + C_{VM} ρ_{c} α_{d} ω \times [{v_{j}}^{I} - {v_{i}}^{I}]

(2007)

where $ω$ is the angular velocity of the rotating frame, and the phase-k acceleration in the rotating frame is defined in terms of the absolute velocity by:

Figure 13. EQUATION_DISPLAY

{a_{k}}^{I R} = {[\frac{D}{D t} {v_{k}}^{I}]}_{k}^{R} + \frac{1}{ρ_{k} α_{k}} \sum_{l}^{} m_{k l} ({v_{k}}^{I} - {v_{k l}}^{I})

(2008)

Figure 14. EQUATION_DISPLAY

{[\frac{D}{D t} ϕ]}_{k}^{R} = \frac{\partial ϕ}{\partial t} + {v_{k}}^{R} \cdot \nabla ϕ

(2009)

Virtual Mass Stress

The terms above describe the virtual mass effects of mean flow acceleration such as occur in developing flows or due to swirl. However, virtual mass effects are also present in the local instantaneous response of particles to turbulent fluctuations. The resulting contribution to the Reynolds-averaged phase momentum equations can be described by a further virtual mass stress term.

The instantaneous virtual mass force (seen by the dispersed phase) reads:

Figure 15. EQUATION_DISPLAY

F_{v m s} = ρ_{c} C_{VM} α_{d} (\frac{\partial v_{c}}{\partial t} + v_{c} \cdot \nabla v_{c} - \frac{\partial v_{d}}{\partial t} - v_{d} \cdot \nabla v_{d})

(2010)

where subscripts $c$ and $d$ stand for continuous and dispersed phases, respectively.

Reynolds averaging of Eqn. (2010) yields several terms. While acceleration of the mean flow is accounted by the Virtual Mass Force, the fluctuation covariances, given by:

Figure 16. EQUATION_DISPLAY

〈 α_{d} v_{d}^{"} \cdot \nabla v_{d}^{"} 〉, 〈 α_{c} v_{c}^{"} \cdot \nabla v_{c}^{"} 〉

(2011)

yield, after some algebra:

Figure 17. EQUATION_DISPLAY

〈 F_{v m s} 〉 = ρ_{c} C_{VM} {\bar{α}}_{d} (\frac{1}{{\bar{α}}_{c}} \nabla\cdot {\bar{α}}_{c} σ_{c} - \frac{1}{{\bar{α}}_{d}} \nabla\cdot {\bar{α}}_{d} σ_{d})

(2012)

where $σ_{c}$ and $σ_{d}$ are the Reynolds Stresses of the corresponding phases, and $〈 F_{v m s} 〉$ is the Virtual Mass Stress term.

Virtual Mass Coefficient Limit

In the pursuit of treating the virtual mass force as an objective interphase force, Cook and Harlow [442] formulated an alternative virtual mass term by collapsing a three-field (dispersed, entrained continuous and bulk continuous phases) model into two-fields.

The Cook and Harlow formulation becomes singular when the virtual mass associated with the dispersed phase exceeds the total mass of the continuous phase. This is not necessarily a physical constraint and it does not apply to other virtual mass formulations, such as those based on Auton et al. [430]. Nevertheless the Cook and Harlow concept of an entrained fluid fraction and a free stream fluid fraction can be used as a starting point when considering the maximum particle concentrations for which the virtual mass force is a relevant correction to the multiphase momentum equations. It is assumed that there is one continuous phase and that the same model for virtual mass coefficient $C_{VM}$ is used for all dispersed phases. The fraction of “free stream” fluid, that is the fraction that is not entrained on the dispersed phases as “added mass” is:

Figure 18. EQUATION_DISPLAY

γ = \frac{α_{c} - C^{V M} \sum α_{d}}{α_{c}}

(2013)

Then an upper bound on the virtual mass coefficient is provided by a plausible assumption that the fraction of free fluid $γ$ is not less than some minimum $γ_{m i n}$ :

Figure 19. EQUATION_DISPLAY

C_{VM, \max} (α_{c}, γ_{m i n}) = \frac{α_{c}}{1 - α_{c}} (1 - γ_{m i n})

(2014)

Choosing $γ_{m i n} = 0$ gives a useful operating range of continuous phase volume fraction $1 \leq α_{c} \leq 0.5$ for which a typical virtual mass coefficient of $C_{VM} = 0.5$ can be applied before the constraint starts to limit the coefficient. Beyond this range, the constraint makes the virtual mass force go to zero as the continuous phase volume fraction goes to zero, which helps maintain stability.

The virtual mass coefficient is based on the kinetic energy content in the fluid field, and it is not always helpful to consider it only in terms of entrained mass. Strong virtual mass effects, exceeding the Cook and Harlow limit $γ_{m i n} = 0$ , can indeed be realistic in close-packed particles [578] or in slug flows [479]. The physical interpretation of such high virtual mass coefficients is that a relative acceleration difference of the particle can cause a fluid backflow through confined gaps between particles, or between particles and walls, at local peak velocities many times higher than the mean relative slip velocity between phases. Under these conditions, relax the constraint by using negative values of $γ_{m i n}$ to see how far accuracy can be improved without the solution becoming unstable.