High-Resolution Interface Capturing (HRIC)

An important quality of a system of immiscible phases (for example, air and water) is that the fluids always remain separated by a sharp interface. The High-Resolution Interface Capturing (HRIC) scheme is designed to mimic the convective transport of immiscible fluid components, resulting in a scheme that is suited for tracking sharp interfaces.

A simple higher-order scheme (for example, central differencing scheme (CDS) or second-order upwind scheme (SOU)) would fail in approximating large spatial variations of phase volume fractions, which are best represented by the Heaviside unit step function. The front sharpening aspect of the downwind scheme is just what is needed when the interface is perpendicular to the flow. However, when the interface is parallel to the flow direction, the downwind scheme tends to wrinkle it. The correction to those situations is to introduce additional blending with upwind (default HRIC) or QUICK (modified HRIC) contributions.

The HRIC and the modified HRIC schemes are based on the normalized variable diagram (NVD). The corrected normalized face value $ξ_{f}$ for both schemes is calculated as:

Figure 1. EQUATION_DISPLAY

ξ_{f} = F (θ) ξ_{f}^{C o m p r e s s i v e} + (1 - F (θ)) ξ_{f}^{B l e n d}

(2590)

where $ξ_{f}^{C o m p r e s s i v e}$ is the normalized compressive face value, $ξ_{f}^{B l e n d}$ is the normalised face value of the blending partner. $F (θ)$ is the blending function, which depends on the angle $θ$ between the normal to the interface $n_{i}$ and the cell-face surface vector $a_{f}$ . The following diagram shows the interface between two fluids and the notation used.

The following table of the NVD representations provides a brief summary of the two HRIC schemes available in Simcenter STAR-CCM+ :


Property	HRIC	Modified HRIC
$ξ_{f}^{C o m p r e s s i v e}$	Figure 2. EQUATION_DISPLAY ${\begin{matrix} ξ_{C} & if & ξ_{C} < 0 o r ξ_{C} > 1 \\ (\frac{2}{0.3} - \frac{1}{0.09} ξ_{C}) ξ_{C} & if & {0 \leq ξ}_{C} < 0.3 \\ 1 & if & {0.3 \leq ξ}_{C} \leq 1 \end{matrix}$ (2591)	Figure 3. EQUATION_DISPLAY ${\begin{matrix} ξ_{C} & if & ξ_{C} < 0 o r ξ_{C} > 1 \\ \frac{1}{ξ_{D}} ξ_{C} & if & {0 \leq ξ}_{C} < ξ_{D} \\ 1 & if & {ξ_{D} \leq ξ}_{C} < 1 \end{matrix}$ (2592) $ξ_{D} = 0.5$ is the default
$ξ_{f}^{B l e n d}$	First-order upwind	QUICK (formally third-order), bounded from above by $ξ_{f}^{C o m p r e s s i v e}$
Diagram
Function $F (θ)$	${\| \cos (θ) \|}^{C_{θ}} w h e r e C_{θ} i s t h e a n g l e f a c t o r$ The default value is 0.05.	$\sqrt{\| \cos (θ) \|}$ The angle factor for this scheme is constant at 0.5. This means that for a relatively large range of angles, a significant amount of QUICK contributions are blended in.
Function $F (θ)$

$ξ_{D}$ is a specified downwind limit above which only downwind differencing is applied to the compressive portion ( $ξ_{f}^{C o m p r e s s i v e}$ ) of the modified HRIC scheme. This property governs the level of compressiveness of the scheme and, therefore, the fluid interface sharpness. Reducing the value of $ξ_{D}$ leads to a more precise capture of fluid interfaces and sharpness in appearance, whilst $ξ_{D} = 1$ represents pure upwind contributions for the modified HRIC scheme.

The calculated value of $ξ_{f}$ is further corrected according to the local Courant number:

Figure 4. EQUATION_DISPLAY

Co = \frac{v \cdot a_{f}}{V_{P_{C}}} δ t

(2593)

This correction is to take into account the availability criterion. This criterion says that the amount of one fluid that is convected across a cell face during a time-step is always less than or equal to the amount available in the donor cell. This correction is made according to the expressions:

Figure 5. EQUATION_DISPLAY

{ξ_{f}}^{*} = {\begin{matrix} ξ_{f} & if & C o < {C o}_{l} \\ ξ_{C} + (ξ_{f} - ξ_{C}) \frac{{C o}_{u} - C o}{{C o}_{u} - {C o}_{l}} & if & {C o}_{l} \leq C o < {C o}_{u} \\ ξ_{C} & if & {C o}_{u} \leq C o \end{matrix}

(2594)

and it plays a role only in transient simulations. Default values for ${C o}_{l}$ and ${C o}_{u}$ are 0.5 and 1. They are introduced to control blending of HRIC and FOU schemes depending on the Courant number. For values of $C o < {C o}_{l}$ HRIC schemes are used, for ${C o}_{l} < C o < {C o}_{u}$ a blend of HRIC and FOU is used, and for ${C o}_{u} < C o$ FOU is used.

The blending is introduced in the case when a large time variation of the free surface shape is present, and the time-step is too large to resolve details of it. It brings stability and robustness to the scheme. In this case, smaller values of ${C o}_{u}$ and ${C o}_{l}$ help to promote convergence. Smaller values activate FOU sooner, and the calculation is more stable. At the same time, the interface is more “smudged” in regions with large $C o$ numbers.

For problems which have a steady-state solution, the values of ${C o}_{u}$ and ${C o}_{l}$ must be large, such that HRIC is used irrespective of the time-step selected. Any number larger than the maximum CFL is appropriate.

The cell-face value $ξ_{f}$ is now calculated as:

Figure 6. EQUATION_DISPLAY

α_{f}^{H R I C} = ξ_{f}^{*} (α_{D} - α_{U}) + α_{U}

(2595)

Interface Smoothing

In some situations, particularly when the mesh has a large aspect ratio, spurious oscillations can appear in the volume fraction field. These oscillations are caused by the volume fraction gradient that is used in extrapolating the value of the volume fraction in the upwind cell. To help suppress the oscillations, a smoothed version of the volume fraction gradient can be used to predict the upwind value.

The normalized cell value is computed as:

Figure 7. EQUATION_DISPLAY

ξ_{C} = \frac{α_{C} - α_{U}}{α_{D} - α_{U}}

(2596)

where $α_{C}$ is the value in the central cell (upwind of the acceptor cell), $α_{D}$ is the downwind value in the acceptor cell, and $α_{U}$ is the upwind value (in the upwind cell). See the normalized variable diagram (NVD).

$α_{U}$ is extrapolated using the volume fraction gradient $\nabla α_{f, U}$ and the centroid-to-centroid distance vector $d x$ as:

Figure 8. EQUATION_DISPLAY

α_{U} = α_{f, D} - 2 d x \cdot \nabla α_{f, U}

(2597)

The normalized cell value $ξ_{C}$ then dictates the blending ratio of upwind- and downwind values to define the volume fraction value at the cell faces. To reduce the mesh dependency of the volume fraction gradient in Eqn. (2597), the gradient $\nabla α_{f, U}$ is replaced by a smoothed version. This leads to a less mesh dependent extrapolation of the upwind value $α_{U}$ and thus a more stable behavior of the HRIC scheme on anisotropic meshes.

This feature is only available for the HRIC convection scheme. The smoothing process is controlled by the user parameters [Min, Max] Gradient Smoothing Steps and Reference Length Scale Factor. See HRIC Gradient Smoothing Properties.

Sharp Interface Reconstruction with Adaptive Mesh Refinement

When Adaptive Mesh Refinement (AMR) refines cells based on the Free Surface Mesh Refinement and Lsi Mesh Refinement criteria, solution quantities are automatically interpolated to the adapted mesh. When AMR refines cells that contain the interface between phases, a reconstruction based interpolation technique can be used to provide a sharp representation of the interface on the refined cells. This interpolation strategy requires the interface to be sufficiently sharp on the coarse mesh (cells to be refined), and obtains the volume fractions on the refined cells by calculating induced cut volumes.

The Free Surface Mesh Refinement feature is only available for the HRIC convection scheme.

For an interface cell with volume $V$ and a phase volume fraction $α$ , the phase volume that needs to be conserved is given by:

Figure 9. EQUATION_DISPLAY

V_{α} = α V

(2598)

The induced cut volume for the interface cell is calculated as follows:

A planar interface $i$ with interface normal $n_{i}$ is constructed and iteratively positioned inside the parent cell until the corresponding cut volume $V_{i}$ equals the phase volume:

Figure 10. EQUATION_DISPLAY

$V_{i} = V_{α}$

(2599)

This planar reconstruction is assumed to separate the given phase from the remaining phases:
Given the interface position calculated in Step 1, the cut volumes of $i$ with each child cell and the corresponding volume fractions are derived: