Smoothed-Particle Hydrodynamics (SPH) Method for Multiphase Flow

Smoothed-Particle Hydrodynamics (SPH) is a mesh-free Lagrangian numerical method that overcomes the volume mesh related limitations of the finite volume methods while still being based on the Navier-Stokes equations. This method is particularly suited for applications with highly dynamic free surface flows and complex moving geometries.

The SPH modeling approach was initially developed by Gingold and Monaghan [240] and Lucy [263] to model astrophysical problems. Monaghan later adapted the approach to simulate free surface fluid flow [269]. In SPH, the fluid is discretized into dynamic elementary points, referred to as particles, without predefined connectivity between them. These particles move at flow velocity and their physical quantities are computed based on their positions. A fundamental concept in SPH is the kernel (smoothing) function, which considers the influence of neighboring particles on one another within a specified radius.

Unlike volume mesh-based methods that require explicit tracking of the interface between two phases, SPH directly simulates a free surface for interactions between two-phase fluids. Thereby, the particles represent the fluid with the higher density, typically a liquid, whereas the fluid with the lower density, typically a gas, is represented by the non-discretized space.

In Simcenter STAR-CCM+, the conservation equations of mass and momentum of the particles are computed within a Lagrangian framework.

Continuity Equation

The conservation of mass for an SPH fluid is given by the continuity equation:

Figure 1. EQUATION_DISPLAY

\frac{d ρ}{d t} + ρ \nabla \cdot v = 0

(1096)

where:

$ρ$ is the density of the fluid particle.
$v$ is the continuum velocity.
$\frac{d}{d t}$ is the Lagrangian time derivative.

Momentum Equation

The rate of change in momentum is balanced by the body forces that act on the continuum. The equation of conservation of momentum for SPH is given by:

Figure 2. EQUATION_DISPLAY

ρ \frac{d v}{d t} = - \nabla (p I) + \nabla T + f_{b}

where:

(1097)

$p$ is the pressure.
$I$ is the unity matrix.
$T$ is the viscous stress tensor. In SPH, a fluid is characterized by an explicit constitutive equation that connects the viscous stress tensor to the velocity field, using a constant viscosity. This linear relationship establishes a direct proportionality between the shear stress and the shear rate in the fluid. In the case of incompressible flow, it can be expressed as $T = 2 μ D$ , where $D = \frac{1}{2} (\nabla v + {(\nabla v)}^{T})$ .
$f_{b}$ is the resultant of body forces (gravity and centrifugal).

Equation of State

The SPH model implemented in Simcenter STAR-CCM+ is incompressible.

Figure 3. EQUATION_DISPLAY

ρ = ρ_{0}

(1098)

where $ρ_{0}$ is a constant density.

Gravity: The body forces due to gravity are added to the momentum equations as a source term:
Figure 4. EQUATION_DISPLAY

$f_{b} {= g ρ}_{0}$

(1099)

Volume Partitioning

When simulating SPH multiphase flow, the volume that is occupied by the liquid is subdivided into a set of dynamic elementary points. The gas phase within the solution domain is disregarded and no points are allocated within the gas volume. The liquid phase points move at the velocity of the flow:

Figure 5. EQUATION_DISPLAY

\frac{d r}{d t} = v

(1100)

where $r$ is the position vector. For volume partitioning, updating the volume of points requires a local equation. The rate of volume expansion for a control volume $V$ that moves along a velocity field $v$ can be expressed as:

Figure 6. EQUATION_DISPLAY

\frac{d V}{d t} = V \nabla \cdot v

(1101)

The volume of points remains constant in time, that is, there is no net volume change within the system over time for incompressible flow.