Forces and Fluid Coupling

Overview

Inertial particles are modeled as Lagrangian objects that move according to Newton’s second law. Within this framework, the time-evolution of the particle acceleration vector is linked to the sum of the external forces acting on the particle via the particle mass. The particle mass is calculated automatically from the instantaneous particle density and diameter.

The external forces considered within the calculation are user-selected and defined. Particle momentum and forces can be either one-way or two-way coupled to the fluid.

Particle trajectories are governed by Newton’s second law, such that:

\[m_i \frac{d \mathbf{v}_i}{dt}=\mathbf{F}_i^p+\mathbf{F}_i^f+\mathbf{F}_i^c+\mathbf{F}_i^g+\mathbf{F}_i^u\]

where \(m_i\) is the mass of particle \(i\); \(\mathbf{v}_i\) is the particle velocity vector; \(\mathbf{F}_i^p\) is net particle-particle interaction force; \(\mathbf{F}_i^f\) is the fluid-particle interaction force; \(\mathbf{F}_i^c\) is a custom user-defined force; \(\mathbf{F}_i^g\) is the net gravity force on the particle, which includes the effects of buoyancy; and \(\mathbf{F}_i^u\) is the user-defined force.

The fluid-particle interaction force on particle \(i\) is given by:

\[\mathbf{F}_i^f=\mathbf{F}_p+\mathbf{F}_v+\mathbf{F}_s+\mathbf{F}_d\]

where \(\mathbf{F}_p\) is the pressure gradient force, \(\mathbf{F}_v\) is the virtual mass force, and \(\mathbf{F}_s\) is the Saffman lift force, and \(\mathbf{F}_d\) is the drag force.

Gravity/Buoyancy Force

This force represents the net effect of gravity and buoyancy. The gravitational force is calculated from the particle mass and system gravity. The buoyancy force is equal to the weight of the continuous phase fluid displaced by the particle, as calculated from the fluid density, particle volume, and system gravity. These forces cannot be deactivated independently.

On

The gravity and buoyancy force are considered.

Off

The force is not considered.

Virtual Mass

Virtual mass refers to the virtual increase in the effective mass of a particle moving through fluid due to the inertia of the surrounding fluid. It accounts for the the motion of the particle, affecting its overall dynamic response in fluid environments. This force is turned on by default and is particularly important when modeling bubbles.

The virtual mass force is given as:

\[\mathbf{F}_m = \left(2.1 - \frac{0.132}{0.12 + A_c^2}\right) V_p \rho_f \left[\frac{\left(\dot{\mathbf{u}} - \dot{\mathbf{v}}\right)}{2}\right]\]

with

\[A_c = \frac{\left(\mathbf{u} - \mathbf{v}\right)^2}{d_p \frac{d}{dt}\left(\mathbf{u} - \mathbf{v}\right)}\]

where \(\rho_f\) is the density of the fluid, \(d_p\) is the bubble diameter, and \(\mathbf{u}\) is the fluid velocity.

On

The virtual mass is considered. This is the default setting.

Off

The virtual mass is not considered.

Lift (Saffman)

The Saffman lift force arises when a particle experiences a cross-flow induced by the rotation of the particle. This flow results in a lateral force perpendicular to the fluid flow. This force is turned off by default and is typically negligible for unresolved particles.

The Saffman lift force is given as:

\[\mathbf{F}_s = 1.61 d_p^2 \rho_f \sqrt{\frac{\nu_f}{|\mathbf{\omega}|}} \left[ \left(\mathbf{u} - \mathbf{v}\right) \times \mathbf{\omega_c} \right]\]

where \(\nu_f\) is the kinematic viscosity of the fluid and \(\mathbf{\omega}\) is the fluid vorticity.

On

The Saffman lift force is considered.

Off

The Saffman lift force is not considered. This is the default setting.

Drag Force Model

The drag force is the resistance encountered by particles as they move through the fluid medium. The functional form of this drag force can depend on the particle density. This is an important force that should be included in all simulations.

The drag force is given by:

\[\mathbf{F}_d = \frac{\pi}{8}C_D d_p^2 \rho_f \left|\mathbf{u} - \mathbf{v}\right| \left(\mathbf{u} - \mathbf{v}\right),\]

where \(C_D\) is the drag coefficient. Three options are available for defining \(C_D\):

Disabled

No drag force on the particles.

Free Particle

The analytic representation Brown and Lawler describing the drag force on a single bubble in fluid is applied.

\[C_D = \frac{24}{Re_p}\left(1 + 0.15Re_p^{0.681}\right) + \frac{0.407}{1 + \frac{8710}{Re_p}}\]

Packed Bed

The analytic representation of Rong, Dong and, Yu for describing the drag force on a bubble in a packed bed is applied:

\[C_D = \left(0.63 + \frac{4.8}{\sqrt{Re_p}}\right) \epsilon_f^{\left(2 - \beta\right)}\]

with

\[\beta=2.65 \left( \epsilon_f+1 \right)-(5.3-3.5 \epsilon_f) \epsilon_f^2 e^{\frac{-\left(1.5-\log Re_p^2\right)}{2}},\]

where \(\epsilon_f\) is the fluid volume fraction.

In the Free Particle and Packed Bed models, \(Re_p\) is the particle Reynolds number calculated using the particle diameter, local fluid viscosity, and the velocity of the particle relative to the fluid.

Nonspherical

The analytic representation from Haider and Levenspeil describing the drag force on a single, nonspherical particle in fluid is applied. The form of the equation is similar to the Free Particle representation, but the coefficients are dependent on the surface area ratio, \(\phi\).

\[C_D = \frac{24}{Re_p}\left(1 + A \cdot Re_p^{B}\right) + \frac{C}{1 + \frac{D}{Re_p}}\]

where

\[\phi = \frac{\text{Actual Particle Surface Area}}{\text{Equivalent Volume Sphere Surface Area}}\]

\[A = exp(2.3288 - 6.4581\phi + 2.4486\phi^2)\]

\[B = 0.0964 + 0.5565\phi\]

\[C = exp(4.905 - 13.8944\phi + 18.4222\phi^2 - 10.2599\phi^3)\]

\[D = exp(1.4681 +12.2584\phi - 20.7322\phi^2+ 15.8855\phi^3)\]

Nonspherical Area Ratio

This value is \(\phi\) from the equation above, and it represents the ratio of the actual particle surface area to the surface area of a sphere with the same diameter. This value must be in the range [0:1], since a sphere has the minimum surface area for a given volume.

Two-way Coupling

Two-way coupling describes the mutual interaction between fluid flow and particle motion. In a two-way coupled system, the effects of the particle forces on the fluid are considered explicitly. In a one-way coupled system, the effects of fluid forces on the particles are considered but the effects of particle forces on the fluid are not.

There are two approaches for describing two-way coupling. The Density approach is a simpler two-way coupling algorithm that only considers the effects of buoyancy/gravity when calculating the local fluid body force. That is, the body force on the fluid voxel is only a function of the particle volume fraction within the voxel. The Third Law approach presents a more rigorous application of the conservation of momentum. Small time steps may be required to keep the simulation stable if large body forces are realized in the fluid.

Two-way coupling is important for systems with high particle number densities which contain, on average, more than one particle per fluid voxel. One-way coupling is sufficient in systems with lower particle number densities.

Disabled

Particles do not inform fluid dynamics. This implies the particles are one-way coupled to the fluid.

Density

A local body force, \(a_b\), is applied to each fluid voxel in accordance with the local solid-phase volume fraction:

\[a_b=g\alpha \left[ \frac{ \rho_g}{\rho_f}-1\right ],\]

where \(g\) is gravity, \(\alpha\) is the solid-phase volume fraction of the fluid voxel hosting the particle, \(\rho_g\) is the density of the solid phase, and \(\rho_f\) is the density of the liquid phase.

Third Law

The sum of all forces applied to the solid particles in each voxel is applied (in opposite direction) to the fluid.

To assess the suitability of the Density approach, consider a single fluid voxel containing \(N\) identical particles with particle diameter \(d_b\), particle density \(\rho_b\), particle area \(A_b\), and particle volume \(V_b\). Take the continuous phase fluid density to be \(\rho_f\) and the fluid voxel volume to be \(V_l\).

If the primary forces on the particles are gravity/buoyancy \(F_b\) and drag \(F_d\), the body force \(F_T\) on the fluid in the voxel is calculated from:

\[F_T=F_b+F_d=\sum_i^N [g V_b \left(ρ_b-ρ_f\right)+\frac{1}{2} ρ_f A_b C_d V_s^2 ] ,\]

where \(C_d\) is the drag coefficient on the particles and \(V_s\) is the particle slip velocity, and the sum is over all particles in the voxel. Since particles are assumed to be identical, this expression can be re-cast into:

\[F_T=g N_b V_b (ρ_b-ρ_f )[1+\lambda]\]

where \(N_b\) is the number of particles in the voxel and

\[\lambda=\frac{ρ_f A_b C_d V_s^2}{2g V_b \left(ρ_b-ρ_f\right) }.\]

For spherical particles in turbulent liquids with order-of-magnitude larger densities, this expression can be simplified to:

\[\lambda \approx \frac{V_s^2}{gd}.\]

For fluids, we are interested primary in the volumetric body force, such that

\[f_T=\frac{F_T}{V_b} =g \epsilon \left(ρ_b-ρ_f\right)\left[1+\lambda\right]\]

where \(\epsilon\) is the solid-phase volume fraction within the voxel that varies between 0 and 1.

The \(\lambda\) term represents the ratio between drag force and buoyancy force. When \(\lambda\) is much smaller than 1, the body force on the voxel due to buoyancy is much greater than the body force on the fluid voxel due to fluid drag. As, such the volume fraction approach to two-way coupling is sufficient. This condition is typically realized for millimeter sized particles in agitated tanks. For larger particles moving through quiescent tanks, however, the complete form of Newton’s third law should be considered.

Fluid-Particle Force UDF

N | This UDF defines custom vectors for fluid and external forces acting on particles. Six output variables must be specified within the UDF: fx_fluid, fy_fluid, fz_fluid, fx_external, fy_external, and fz_external.

The fx_fluid, fy_fluid, and fz_fluid variables represent particle forces due to the fluid (e.g., custom drag, lift, or drag forces). These forces are included in the two-way force coupling modeled through Newton’s Third Law. The fx_external, fy_external, and fz_external variables represent forces acting on the particles from external sources (e.g., electrophoresis or magnetophoretic forces). If two-way coupling is enabled, these external forces are coupled back to the fluid motion through particle drag. The x, y, and z components of each force must be explicitly defined.

This is a particle-based local UDF, calculated on a particle-by-particle basis using the local particle/fluid properties.

Download Sample File: Fluid-Particle Force - DEM

Download Sample File: Fluid-Particle Force - Inertial Particles

Viscosity Coupling

When this option is enabled, the effective local fluid viscosity \(v_e\) is calculated from the local particle volume fraction via a modified form of Euler’s equation:

\[\nu_e=\nu_0\left(1+\frac{0.5[\mu]\epsilon_p}{\frac{\left(1-\epsilon_p\right)}{\epsilon_{cp}}}\right)^2\]

where \(v_0\) is the fluid viscosity, \(\epsilon_p\) is the local partical volume fraction, \(\epsilon_{cp}\) is the the maximum (critical) particle packing fraction, and \([\mu]\) is the so-called intrinsic viscosity. The fluid viscosity and local particle packing fraction are calculated at runtime on a voxel-by-voxel basis. The maximum particle packing fraction and intrinsic viscosity are set to 0.64 and 2.5, respectively. More sophisticated relationships between viscosity and solids concentration can be defined via a viscosity UDF. If a custom UDF that links fluid viscosity to particle solid fraction is defined, the automatic option presented here should be disabled.

Off

Do not automatically consider the effects of particle volume fraction on local fluid viscosity.

On

Automatically consider the effects of particle volume fraction on local fluid viscosity using the Euler equation presented above.

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