Block-Structured
AMR for Free Surface Flows
In this study, the multi-moment
finite volume method (FVM) is extended for directly numerical simulation (DNS)
of two-phase flow problems. A CIP (constraint interpolation profile) – CSL
(semi-Lagrangian) scheme is used to solve the advection part of the momentum
equation. A new implementation of VOF (volume of fluid method) -APPLIC
(approximated piecewise linear calculation) method is presented for capturing
the moving interface. For accurately modeling of the surface tension, the
original height function (HF) method is improved to exclude isolated volumetric
fractions in curvature estimation. Besides, we propose an extrapolation method
to push the curvature value of the interfacial cells to its neighbors. One of
the highlights of this research lies in the application of parallel
block-structured adaptive mesh refinement (BAMR) strategy, by which the demand
for computational resources can be reduced significantly.
High computational efficiency and ease
of implementation are always of our primary consideration. Based on the
principle, the combination of BAMR, VOF/APPLIC, height function method and multi-moment
FVM is very suitable for large scale parallel computation of interfacial flows under
the effect of surface tension. Numerical tests reveal the accuracy of present approach
in predicting the capillary wave instability, droplet collisions. BAMR solver shows
remarkable efficiency in simulating fuel jet injection problems. As discussed
in Section 3.6, BAMR solver saves over 5/6 of total unknowns with finer
resolution and larger computational domain, compared with conventional
Cartesian method.
Curvature
Estimation
Fig. 1 
and
error for curvature estimation of a
circular interface (only interfacial cells are involved), △: CV (convolved
VOF method), ◇: RDF
(reconstructed distance function method),
□: HF
(height function) of S.J. Cummins et al., ●: present HF, – – –: second order, – ∙ – ∙ –:
first order.
Capillary Wave
Capillary
Waves of the Same Density
Fig. 2 Time
evolution of the amplitude of the capillary wave as a function of
non-dimensional time
, 
, 
, 
. –: A. Prosperetti’s
theoretical solution, ○:
results of present BAMR solver.
Air-Water
Capillary Wave
Fig. 3 Time
evolution of the amplitude of the capillary wave as a function of
non-dimensional time, , 
, 
. –: A. Prosperetti’s
theoretical solution, ○:
results of present BAMR solver.
Capillary
Breakup of Liquid Jet
Fig. 4 Comparison of relative deformations of the interface. (a). Maximum radius variation
,
(b). Minimum radius variation
. – – –: L. Rayleigh [45] (
),
∙∙∙∙∙: C. Weber [46] (
), ∙–∙–∙:
C. Weber [46] (
),
○:
present BAMR (), △: present BAMR (), □: present BAMR ().
Fig. 5 Interface
deformation of capillary jet instability when
.
Droplet
Collisions
Table. 1 Fluid
properties for present droplet collisions.
|
Fluid |
Density (kg/m3) |
Dynamic viscosity (kg/m∙s) |
Surface tension
coefficient (kg/s) |
|
Tetradecane (C14H30) |
758 |
2.128×10-3 |
0.026 |
|
Nitrogen |
1.138 |
1.787×10-5 |
Table. 2 Parameters
for different cases of the droplet collisions.
|
Case |
Diameter ( |
|
|
|
|
|
I |
354 |
1.05 |
0.71 |
64.9 |
312.8 |
|
II |
336 |
1.25 |
0.06 |
61.4 |
296.5 |
|
III |
356 |
1.05 |
0.25 |
70.8 |
327.7 |
|
IV |
356 |
1.05 |
0.39 |
48.1 |
270.1 |
Fig. 6 Off-center collisions of hydrocarbon droplet (case-I), the predictions of VOF/PLIC and
VOF/APPLIC are presented for comparison.
Fig. 7 Off-center collisions of hydrocarbon droplet (case-II), only the predictions of VOF/APPLIC are
given.
Fig. 8 Off-center collisions of hydrocarbon droplet (case-III), only the predictions of VOF/APPLIC are
given.
Fig. 9 Off-center collisions of hydrocarbon droplet (case-IV), only the predictions of VOF/APPLIC are
given.
Straight Liquid
Jet Spray
Table. 3 
numbers and flow properties
for jet spray simulation.
|
Descriptions |
We |
Re |
(m/s) |
(mm) |
Ambient
(MPa) |
(Kg/m3) |
(Kg/m3) |
(Pa∙s) |
(Pa∙s) |
(N/m) |
|
Case-I |
1270 |
440 |
30 |
0.1 |
3 |
34.5 |
848 |
2.87E-6 |
1.97E-5 |
3.0E-2 |
|
Case-II |
3530 |
740 |
50 |
Fig. 10 Breaking-up of the liquid film.
Fig. 11 Instantaneous
free surface of liquid jet spray simulations. Left, case-I; right, case-II.
Fig. 12 BAMR
blocks of liquid jet spray test (Case-II), each block contains 8×8×8 uniform
mesh.
Reference
C. Liu*, C.
Hu, Blocked Adaptive Method for Directly Numerical Simulation of Interfacial
Flows, submitted.
C. Hu*, C. Liu, Simulation of Violent Free Surface Flow by AMR Method, Journal of Hydrodynamics, (2018) accepted.