1. CFX Slide serial no 1 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Lecture 5 Free Surface Flow

2. CFX Slide serial no 2 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Outline Introduction to Free Surface Flow Homogeneous Multiphase Implementation and Examples Surface Tension Advanced Topics  Inhomogeneous Free Surface Flow

3. CFX Slide serial no 3 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface What is Free Surface Flow Free surface flow  separated multiphase flow  fluids separated by distinct resolvable interface  examples: open channel flow, flow around ship hulls, water jet in air (Pelton wheel), tank filling, etc.

4. CFX Slide serial no 4 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Classification Froude number  L=h (water depth) for shallow water flow  L= /2  (wavelength) for sinusoidal wave train in deep water  for flow around ship hulls, there is not a single wave velocity, but we can still define a Froude number based on the ship geometry Analogies with Mach number  flow can be subcritical, transcritical, or supercritical  hydraulic jump is a ‘shock’

5. CFX Slide serial no 5 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Homogeneous MPF: Concept Homogeneous MPF model  Air and water are separated by a distinct free surface interface (may be smeared by numerics)  Only one velocity at each point in space: bulk velocity  Sufficient to solve for this bulk velocity field  Also valid when interphase drag is very large and body forces are absent

6. CFX Slide serial no 6 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Homogeneous MPF: Momentum Phasic momentum equation: Sum over phases: Essentially a single-phase momentum equation with mixture density and viscosity

7. CFX Slide serial no 7 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Homogeneous MPF: Continuity Phasic continuity: If homogeneous: Neglect compressibility  equation is symmetric for all phases  solve for (N-1) volume fractions and treat the other as a ballast Volume continuity: Incompressibility assumed for clarity only (not fundamental limitation)

8. CFX Slide serial no 8 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Homogeneous MPF: Other equations Bulk equations solved for other homogeneous field variables also  turbulence  additional variables Energy is a special case  temperature is homogeneous  enthalpy is a principal variable (not shared among fluids)  solution: solve two phasic equations with a large interphase heat transfer term to force temperatures to be the same

9. CFX Slide serial no 9 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Implementation MPF Model  usually homogeneous MPF model Advection and transient terms  compressive discretization  interface typically smeared over 2-3 elements Pressure-velocity coupling (Rhie-Chow)  special treatment of buoyancy force to keep flow well-behaved at interface

10. CFX Slide serial no 10 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Maxwell’s Expt. 2-D transient problem Solved with hex and prismatic meshes Hex mesh (~10000 nodes on plane)

11. CFX Slide serial no 11 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Maxwell’s Expt.

12. CFX Slide serial no 12 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Wigley Hull

13. CFX Slide serial no 13 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Wigley Hull

14. CFX Slide serial no 14 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Jet with Adaption No adaption One stepTwo steps

15. CFX Slide serial no 15 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Free Surface Flow: Gear Box

16. CFX Slide serial no 16 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Surface Tension An attractive force at the free surface interface Normal component  smooths regions of high curvature  induces pressure rise within droplet: Tangential component  moves fluid along interface toward region of high   often called Marangoni effect (  decreases with temperature) F

17. CFX Slide serial no 17 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Surface Tension: Wall Adhesion Non-wettingWetting Wall adhesion is responsible for capillary rise in tubes

18. CFX Slide serial no 18 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Dimensionless Groups Weber number: Capillary number: Marangoni number:

19. CFX Slide serial no 19 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Surface Tension: Modelling Conceptually a surface force at interface  awkward to deal with interface topology Reformulate as a continuum force  Brackbill, Kothe, Zemach 1992  wall contact angle specifies direction of normal at wall

20. CFX Slide serial no 20 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Surface Tension: Extreme angle

21. CFX Slide serial no 21 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Surface Tension: Colliding drops

22. CFX Slide serial no 22 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Surface Tension: Colliding drops

23. CFX Slide serial no 23 © 2003 ANSYS CFX CFX-5.6 Multiphase Course Free Surface Inhomogeneous Free Surface Splashing  air entrained in water; behaves as disperse fluid  combine free surface discretization with full MPF model  use a drag law appropriate for the motion of the disperse fluid