VOF-Technology in STAR-CCM+ Samir Muzaferija & Milovan Peri ć
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 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:
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 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
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 (CFL l and CFL u 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… • C θ 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 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 • 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: n fs = - 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
Wave Models • STAR-CCM+ provides several wave models: – For initialization of volume fraction, velocity and pressure fields; – For a transient inlet boundary condition. • 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)
Wave Damping • 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+: w w x sd – Starting point for wave damping (propagation in x -direction) x ed – End point for wave damping (boundary) f 1 , f 2 and n d – Parameters of the damping model w – Vertical velocity component
Time-Accurate Wave Propagation, I • 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… • Test case: – Stokes 5 th -order wave – Wavelength 102.7 m – Wave height 5.8 m – Wave period 8 s – Solution domain 4 wavelengths long…
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 ( Δ x = 2.5 m, Δ z = 0.5 m) 1st-order scheme , 100 Δ t/T (CFL = 0.41), after 4 periods 5 cells 2nd-order scheme , 100 Δ t/T (CFL = 0.41), after 4 periods 10 cells
Application examples • 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
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 = ρu 2 D/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)
Drop Impact on a Wall, II Animation showing droplet impact on the wall and rebound due to non-wetting contact angle...
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 Web speed: 0.5 m/s Coarse grid Vacuum: 2000 Pa On a coarse mesh, outlet flux oscillates strongly, on a fine mesh much less… Refined grid
Micro-Gravity Free Surface Shape, I Silicon oil in a cylindrical container subjected to a sudden reduction in gravity (to 1e-6 m/s^2) changes free surface shape to s pherical… Fluid is at rest both initially and at the end of simulation – parasitic currents require reduced CFL- Wall limits for HRIC… Symmetry axis
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