Free Surface Algorithms for Immiscible Fluids

Return to Clarence Burg’s home page

Immiscible fluids are fluids that do not mix, so that the interface between the two fluids is well defined throughout the simulation.  The example of interest is the air/water interface for flows past moving ships of interest to naval hydrodynamics applications.  Because the fluids do not mix, the simulation algorithm should maintain the crisp interface as a crisp interface.  Various methods have been employed for the solution of this free surface problem including the following

*                  Surface Tracking Methods

*                  Volume of Fluid / Level Set Methods

*                  Ghost-Cell / Level Set Methods

In the Surface Tracking Method, only the water side of the domain is simulated, by making the assumption that the air side does not influence the location of the free surface.  Thus, only the water side is gridded, and this grid must move with the motion of the free surface, while conforming to the hull.  Complicated grid movement algorithms are required, especially for unstructured grids and due to the need to conform to the hull. 

For mild free surface simulations (typically, upto Fr = 0.3), this approach works well, since the free surface interface is captured crisply.  But, for complicated free surface patterns, such as cresting and re-entrant waves and for topological changes due to the free surface detaching or attaching to various points along the hull, this algorithm has severe difficulties. 

Inspite of these difficulties, this approach has been the most popular approach for most structured flow solvers, such as CFD-Ship-Iowa, FINFlow, Parnassos, and others.  I have also implemented this surface tracking approach within Mississippi State University’s unstructured flow solver U2NCLE, using the torsional spring method to move the grid.

Burg, C. O. E., Marcum, D. L., "Moving Towards High-Fidelity RANS Calculations of Maneuvering Surface Vessels Using Unstructured Grids", Proceedings of 8th International Conference on Numerical Ship Hydrodynamics, Busan, Korea, Sept. 2003.  PDF

Burg, C. O. E., Sreenivas, K., Hyams, D. G., and Mitchell, B., "Unstructured Nonlinear Free Surface Simulations for the Fully-Appended DTMB Model 5415 Series Hull Including Rotating Propulsors", Proceedings of 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, July, 2002.  PDF

Burg, C. O. E., Sreenivas, K., Hyams, D. G., and Mitchell, B., "Unstructured Nonlinear Free Surface Flow Simulations: Validation and Verification", AIAA Paper 2202-2977, 32nd AIAA Fluid Dynamics Conference, St. Louis, June, 2002.

 

In the Volume of Fluid / Level Set Method, both the water and air side of the interface is simulated, typically on a fixed grid.  An auxiliary equation, the level set equation, is solved to determine the location of the free surface.  A wide variety of approaches have been used, with subtle differences, but the main idea is that the level set function varies from 0 to 1 and specifies the percentage of air and water in each element or cell.  For water, the function is 1, and for air, the function is 0.  Numerically, this function is smeared across the interface, so that the cells near the interface have both air and water.  This smearing of the interface prevents the simulation of a crisp interface.  Researchers typically use an interface sharpening approach after each iteration to re-establish a narrow interface, and they use a locally-refined grid near the interface so that this smearing occurs over a small region.  Many unstructured flow solvers, including the commercial solvers, COMET and CFX-5, use this volume of fluid approach. 

 

In the Ghost Cell / Level Set Method, only the water side of the interface is simulated, but both sides are gridded.  Using the assumption that the air side of the interface has little influence on the location of the free surface, the level set equation evolves based on the velocities on the water side.  Typically, the level set equation represents a signed distance function to the free surface, with 0 being the location of the free surface.  To obtain the flow quantities on the air side, the velocity is extrapolated via the Ghost-Cell method to the air side, and the pressure is determined via the hydrostatic pressure assumption.  Because the air side is not simulated, the numerical stability difficulties associated with the jump in density across the interface are avoided, so that this interface can be reasonably captured on a relatively coarse grid.  This methodology requires the re-initialization of the level set function to maintain a signed distance function, and the process of extrapolating the velocities is up to interpretation, as each implementation does this differently.  I have implemented this approach within Mississippi State University’s unstructured flow solver U2NCLE, obtaining some intriguing results. 

Burg, C. O. E., “Single-Phase Level Set Simulations for Unstructured Incompressible Flows”, AIAA Paper 2005-5350, 17th AIAA Computational Fluid Dynamics Conference, Toronto, Ontario, June, 2005.  PDF

These free surface algorithms typically suffer from numerical instabilities, loss of physical realism due to the smearing of the interface, and loss of mass conservation, since no effort is made within the algorithm to ensure mass conservation.  Thus, time-accurate simulations are problematic for these algorithms. 

 

To correct for these problems, I am studying algorithms to combine the Ghost-Cell approach with a level set representation consistent with the Volume-of-Fluid approach, which will guarantee mass conservation.