Simulation of Free-Surface Flows

With STAR-CCM+

Samir Muzaferija and Milovan Perić

CD-adapco

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 and propagation

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

• 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:

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

(details available in STAR-

CCM+ documentation)

HRIC-Scheme, IV

Simulation of sloshing in a tank due to sinusoidal sway motion:

one-cell sharp interface before wave overturns (left) and smeared

Interface after splashing (right), when the interface is in reality not sharp…

Interface Sharpening

• In order to prevent dilution, one can activate “interface sharpening” by setting “Sharpening factor” to a value >0.

• The sharpening model is based on “anti-diffusion” and acts only in cells at the interface…

• This is usually required only for violent sloshing and similar phenomena…

Local Grid Refinement, I

• One should, when possible, align grid with free surface where it is flat…

• One should, when possible, avoid vertical grid coarsening in free-surface zone where its deformation is small…

• The reason: volume fraction is convected into finer cells and leads to smeared interface…

Flow around a vertical cylinder – two grids for the same initial free surface position

Local Grid Refinement, II

Initial value from this cell feeds into next two, from there into next four – the smeared

interface does not get sharper by refining time step (only “Sharpening Factor” helps –

but it is better to adapt the grid to free surface that to use artificial anti-diffusion…)

Impulsively started flow around a vertical cylinder

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

• Recently, a new model called “Interface Momentum Dissipation” was introduced in STAR-CCM+ to reduce the effects of parasitic currents…

• The momentum dissipation term is added to the momentum equations only in the vicinity of the interface…

• It acts similarly as an increased fluid viscosity near interface (more on the gas side): µint grad(v)

• Interface Momentum Dissipation decreases rapidly with distance from interface…

Surface Tension Effects, V

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

Surface Tension Effects, VI

• 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

Interface Momentum Dissipation:

Ink Jet Droplet, I

Without IMD

With IMD

Without IMD, parasitic currents are strong (maximum velocity 35.88 m/s);

With IMD, parasitic currents are hardly visible (maximum velocity 8.98 m/s)

Without IMD, the interface is smeared behind secondary droplet and at nozzle exit;

With IMD, the interface is sharp almost everywhere…

Without IMD

With IMD

Interface Momentum Dissipation:

Ink Jet Droplet, II

Without IMD:

Strong parasitic

currents, maximum

velocity 4.97 m/s

(10x web speed)

With IMD:

Very weak parasitic

currents, maximum

velocity 0.506 m/s (1%

above web speed)

Interface Momentum Dissipation:

Flow in a Slot Coater, I

Interface Momentum Dissipation:

Flow in a Slot Coater, II

Without IMD:

Front meniscus has

irregular shape due

to high parasitic

velocities

With IMD:

Smooth front

meniscus

Interface Momentum Dissipation:

Flow in a Slot Coater, II

Without IMD:

Flow rate at outlet fluctuates due to

high parasitic velocities

With IMD:

Flow rate at outlet fluctuates less

Interface Momentum Dissipation:

Flow in and Around a Rising Bubble

Left: Without IMD

Strong parasitic currents, maximum

velocity 11.68 m/s, interface smeared

through high velocity normal to it, the

flow inside bubble cannot be

recognized…

Right: With IMD

Hardly visible parasitic currents,

maximum velocity 0.39 m/s (30

times lower than before), interface is

sharp (resolved by one cell) and one

can clearly see the flow inside

bubble…

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:

Evaporation and condensation

Melting and solidification

Cavitation

Boiling

• STAR-CCM+ provides several wave models:

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

– For transient inlet boundary conditions.

• 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

• 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…

Time-Accurate Wave Propagation

Scaled 10 times in vertical direction…

Stokes 5th-order wave after 11 periods (8.977 s), resolved by 80 cells per wave-

length (125 m) and 20 cells per wave height (5 m); damping over the last 300 m

Internal Wave Generation

• The source term in equation for volume fraction can be used to simulate injection and suction…

• … which can be used to create waves at free surface…

• By a suitable choice of the position and shape of the “source zone” and an appropriate source term function, one can generate waves of desired shape…

• The advantage of this approach: waves radiated by a solid structure can pass over the source region without reflection (which happens when waves are created by inlet boundary conditions)

• Improvements to the treatment of contact angle (better

recognition of contact line, distinguishing direction of

motion etc.)

• 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: VoF

Simulation of Pouring