VOF-Technology in

STAR-CCM+

Samir Muzaferija & Milovan Perić

Contents

Introduction to multiphase flows

Theoretical background for VOF-method

High-Resolution Interface-Capturing (HRIC) scheme

Accounting for surface tension effects

Extensions of VOF-method

Waves: generation, propagation, damping…

Free surface flows: application examples

Future development

Introduction to Multiphase Flows

VOF-approach is suitable, when

the grid is fine enough to resolve

the interface between two

immiscible fluids.

Sometimes not all parts of the flow

are suited for VOF-treatment…

Examples: Atomization nozzle

flow and jet break-up (right) and

flow around a hydrofoil (below)

Interface Conditions, I

• Conditions at an interface between two immiscibe fluids:

Kinematic condition: No flow through interface.

Dynamic conditions: Balance of normal and tangential stresses (surface tension forces):

VOF: Theory, I

• VOF considers a single effective fluid whose properties vary according to volume fraction of individual fluids:

• The mass conservation equation for fluid i reads:

• It can be rearranged into an equation in integral form:

This equation is used to compute the transport of volume fraction αi.

VOF: Theory, II

• The mass conservation equation for the effective fluid is obtained by summing up all component equations and using the condition:

• The integral form of mass conservation equation (used to compute pressure correction) reads:

• The properties of effective fluid are computed according to volume fractions:

VOF: Theory, III

• All fluids (liquids and gases) can be compressible.

• If density is a function of pressure and temperature, we have:

• For an ideal gas, the following relations hold:

• The source term due to compressibility is then:

Interface-Capturing Method, I

• For sharp interfaces, special discretization for convective terms in the equation for volume fraction αi is needed (to avoid excessive spreading).

• The method must produce bounded solutions, i.e. each volume fraction must lie between 0 and 1 and the sum of all volume fractions must be 1 at each control volume.

• Bounded schemes must fall within a certain region of the normalized variable diagram; the normalized variables are defined as:

Interface-Capturing Method, II

• The boundedness requirement:

The normalized variable

diagram and the proposed

high-resolution interface-

capturing (HRIC) scheme

HRIC-Scheme, I

• The HRIC-scheme defines the face value of the normalized variable as follows:

• This value is corrected by the local Courant-number (CFLl and CFLu are scheme parameters – default 0.5 and 1):

HRIC-Scheme, II

• Another correction is introduced to account for the orientation of interface relative to cell face:

• This correction reduces the tendency of interface to align with the grid…

• Cθ is the scheme parameter (default value: 0.05)

HRIC-Scheme, III

• The convected cell-face value of volume fraction is finally determined as:

• The face value can also be expressed as a blend of upwind and downwind values:

• The blending factor is a function of normalized face variable and volume fraction values at U, C and D nodes:

Surface Tension Effects, I

• The kinematic interface condition is implicitly accounted for by the transport equation for volume fraction.

• The dynamic interface conditions require additional forces in the momentum equations in cells containing free surface…

• Surface tension forces are converted to volume forces:

Since the gradient of volume

fraction is zero away from

interface, these terms are

equal to zero everywhere

except along interface…

Surface Tension Effects, II

• The unit vector normal to interface is obtained from the gradient of volume fraction:

• The curvature of free surface is obtained from the divergence of the unit vector normal to interface:

• The volume fraction field needs to be smoothed before the curvature is computed (sharp interface leads to a non-smooth curvature field).

Surface Tension Effects, III

• The so called „parasitic currents“ can develop, if the fluid moves only slowly or not at all, and the surface tension effects dominate (high curvature or surface tension coefficient)...

• The reason: pressure and surface tension forces must be in equilibrium when fluid is at rest – but the numerical approximations do not guarantee that (one term is linear and the other is non-linear):

• There are many partial solutions to this problem in literature, but none works in all situations…

Surface Tension Effects, IV

• Where free surface is in contact with wall, contact angle needs to be prescribed.

Surface Tension Effects, IV

• One can distinguish between:

Static contact angle

Dynamic advancing contact angle on dry surface

Dynamic advancing contact angle on wet surface

Dynamic receding contact angle

• The contact angle is enforced as:

nfs = - n

w cos θ

w + t

w sin θ

w

Surface Tension Effects, V

• Contact angle and dynamic contact line at a moving wall (e.g. in a coating process)...

Extensions of VOF-Method

• One can add additional models in the equation for volume fraction (diffusion, sources) in order to model effects like non-sharp interfaces, phase change etc.

• This is the main advantage of this approach compared to level-set and similar schemes...

• VOF-framework is already used in STAR-CCM+ for the following models:

Cavitation

Boiling

Evaporation and condensation at free surface

Melting and solidification

• STAR-CCM+ provides several wave models:

– For initialization of volume fraction, velocity and pressure fields;

– For a transient inlet boundary condition.

• Currently available models:

– 1st-order linear wave theory

– Non-linear 5th-order Stokes wave theory (Fenton, 1985)

– Pierson-Moskowitz and JONSWAP long-crested wave spectra

– Superposition of linear waves with varying amplitude, period and direction of propagation (can be set-up via Excel-file)

Wave Models

w w

• Vertical motion is damped by introducing smoothly

increasing resistance…

• The method proposed by Choi and Yoon (Costal Engineering,

Vol. 56, pp. 1043-1060, 2009) has been implemented into

STAR-CCM+:

Wave Damping

xsd

– Starting point for wave damping (propagation in x-direction)

xed

– End point for wave damping (boundary)

f1 , f

2 and n

d – Parameters of the damping model

w – Vertical velocity component

• Accurate wave propagation requires 2nd-order time-

integration method.

• Second-order method (quadratic interpolation in time)

requires that the wave propagates less than half a cell

per time step.

• First-order scheme is always stable but less accurate…

• Test case:

– Stokes 5th-order wave

– Wavelength 102.7 m

– Wave height 5.8 m

– Wave period 8 s

– Solution domain 4 wavelengths long…

Time-Accurate Wave Propagation, I

Time-Accurate Wave Propagation, II

Wave damping was applied over the last 100 m before outlet... 41 cells

per wave length, 11.5 cells per wave height (Δx = 2.5 m, Δz = 0.5 m)

1st-order scheme, 100 Δt/T (CFL = 0.41), after 4 periods

2nd-order scheme, 100 Δt/T (CFL = 0.41), after 4 periods

5 cells

10 cells

• Droplet impact on a wall

• Flow in a slot coater

• Micro-gravity free surface re-orientation

• Flow around ships

• Wave impact on offshore structures

• Flow over a weir

• Simulation of pouring

Application examples

Drop Impact on a Wall, I

• A water droplet with a diameter D = 2.7 mm hits a wall with a speed of 4.551 m/s.

• Wall surface is waxed: contact angle is 105° for advancing interface and 95° for receding interface.

• Surface tension coefficient: σ = 0.073 N/m

• Weber number: We = ρu2D/σ = 763

• Mesh size at wall: 6 µm

• Time step: 0.2 µs

• Comparison with experiments by S. Sikalo and E. Ganic (Phenomena of droplet-surface interactions, Experimental Thermal and Fluid Science, 2006)

Animation showing droplet impact on the wall and rebound due to non-wetting contact angle...

Drop Impact on a Wall, II

Drop Impact on a Wall, III

Comparison of predicted and measured spreading of liquid droplet on the wall

Comparison of predicted and measured height of liquid above wall at the impingement location...

Simulation of Slot Coating, I

Prediction of stable operation window of a slot coater as a function of vacuum level

Stable region

predicted well on

very coarse grid

Simulation of Slot Coating, II

Effects of grid refinement (web speed 0.8 m/s,

under-pressure 500 Pa):

Coarse grid

Refined grid

Simulation of Slot Coating, III

Effects of grid refinement: Flow rates at inlet and outlet

Coarse grid

Refined grid

Web speed: 0.5 m/s

Vacuum: 2000 Pa

On a coarse mesh,

outlet flux oscillates

strongly, on a fine mesh

much less…

Micro-Gravity Free Surface Shape, I

Symmetry axis

Wall

Silicon oil in a cylindrical

container subjected to a

sudden reduction in gravity

(to 1e-6 m/s^2) changes

free surface shape to

spherical…

Fluid is at rest both

initially and at the end

of simulation –

parasitic currents

require reduced CFL-

limits for HRIC…

Micro-Gravity Free Surface Shape, II

Comparison of predicted and experimentally observed position of free surface

during transition process at symmetry axis and at wall (experiments by Michaelis

and Dreyer, in Multiphase Science and Technology, Vol. 16, pp. 219-238, 2004)

Symmetry axis

Wall

Flow Around Ships, I

Comparison of predicted and measured wave profiles around

container ship at Froude number 0.26

Flow Around Ships, II

Comparison of measured and predicted wave profiles

around a military vessel (destroyer DTMB 5415)…

Wave Impact on Offshore Structures, I

Simulation of wave impact on a

platform in shallow water by DNV

(published at OMAE2012

Conference)

Simulation of wave impact on a

jack-up platform in shallow water

by GL (published at OMAE2009

Conference)

Wave Impact on Offshore Structures, II

Coupled simulation of

flow using STAR-CCM+

and deformation of

platform structure using

ABAQUS.

Simulation by CD-adapco

Engineering Services for

Chevron. Published at

OMAE2012 Conference.

Evidence of damage on a

platform after it was hit by

a hurricane

Deformation in a

simulation: good

agreement with field

observation…

Flow Over a Weir

Simulation of Pouring, I

• Improvements to computation of free surface curvature

(to reduce the parasitic currents)

• Transition to other multiphase models:

– VOF to Lagrangian and vice-versa

– Fluid film to VOF and vice versa

• Eulerian or Lagrangian multiphase models within VOF

phases

Future Developments