Blob Shape Criteria

When bodies of liquid break up, ligaments and droplets form where ligaments can subsequently break up into droplets. To evaluate the shape of large VOF resolved droplets and ligaments, also known as blobs, certain criteria can be applied. A blob can span multiple cells.

The Resolved VOF to Lagrangian Transition model is used to resolve the large-scale (primary) breakup with VOF, whereas the Lagrangian model handles the small-scale (secondary) breakup. The blob shape criteria help to ensure that only VOF droplets, not ligaments, transition to Lagrangian Multiphase.

Blob Sphericity

Blob sphericity of blob

i

is defined as:

Figure 1. EQUATION_DISPLAY

ϕ_{i} = \frac{A_{sphere, i}}{A_{blob, i}}

(2637)

where $A_{sphere, i} = π^{\frac{1}{3}} {(6 V_{blob, i})}^{\frac{2}{3}}$ is the surface area of the perfect sphere having the same volume $V_{blob, i}$ as blob $i$ and $A_{blob, i}$ is the actual surface area of blob $i$ , computed as:

Figure 2. EQUATION_DISPLAY

A_{blob, i} = \sum_{j \in b l o b, i} | {\nabla α}_{j}^{'} | V_{j}

(2638)

where:

$\sum_{j \in b l o b, i}$ is the summation over the cells j which are in blob i.
$V_{j}$ is the volume of the cell j.
$α_{j}^{'} = {\begin{matrix} 1 & i f & c e l l j i s p a r t o f t h e b l o b \\ 0 & i f & c e l l j i s n o t p a r t o f t h e b l o b \end{matrix}$

If a blob attaches to a boundary, the above evaluation is no longer valid because it does not take into account the effects of the boundary. As a result the blob sphericity is set to 0 for those blobs.

Theoretically, blob sphericity should range from [0-1]. However, if a blob is poorly resolved, the actual blob surface area computed from the gradient of the blob phase volume fraction becomes underestimated and the computed blob sphericity could exceed 1. In these situations, Simcenter STAR-CCM+ will impose a blob sphericity value of 1 for such blobs.

Blob Inertia Tensor

The tensor of inertia of a blob is defined as:

Figure 3. EQUATION_DISPLAY

I = \sum_{k} m_{k} ({| r_{k} |}^{2} 1 - r_{k} \otimes r_{k})

(2639)

where $m_{k}$ is the phase mass in the blob cell $k$ , $r_{k}$ is the vector pointing from the blob centroid to the centroid of the blob cell $k$ and $1$ is the unity tensor.

There are three eigenvalues and eigenvectors for this tensor. Each eigenvalue represents the momentum of inertia in the direction of the corresponding eigenvector. For a perfect sphere, the three eigenvalues are identical due to the nature of isotropy of a perfect sphere. If the shape is anisotropic, the three eigenvalues have different values.

Simcenter STAR-CCM+ uses the ratio of minimum to maximum eigenvalue of the blob inertia tensor to evaluate how isotropic the blob is:

Figure 4. EQUATION_DISPLAY

γ = \frac{λ_{2}}{λ_{0}}

(2640)

where $λ_{0}$ is the maximum eigenvalue and $λ_{2}$ is the minimum eigenvalue. If the blob shape is isotropic, including a perfect sphere, $γ = 1$ and $γ < 1$ otherwise.

If a blob attaches to a boundary, the above evaluation is no longer valid because it does not take into account the effects of the boundary. As a result, the blob inertia tensor eigenvalue ratio is set to 0 for such blobs.