Simu Simula lation tion of of F Free ee-Su Surf rfac ace e Flo lows ws Wit ith h ST STAR AR-CC CCM+ M+ Samir Muzaferija and Milovan Peri ć 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 α i .
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 α i 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 Impulsively started flow around a vertical cylinder 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…)
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 e xcept 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): µ int 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: n fs = - 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)
Interface Momentum Dissipation: Ink Jet Droplet, II Without IMD With IMD Without IMD, the interface is smeared behind secondary droplet and at nozzle exit; With IMD, the interface is sharp almost everywhere…
Interface Momentum Dissipation: Flow in a Slot Coater, I 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, 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
Wave Models • STAR-CCM+ provides several wave models: – For initialization of volume fraction, velocity and pressure fields; – For transient inlet boundary conditions. • Currently available models: – 1 st -order linear wave theory – Non-linear 5 th -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)
Time-Accurate Wave Propagation • Accurate wave propagation requires 2 nd -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… Scaled 10 times in vertical direction… Stokes 5 th -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)
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