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Motivated by applications to fluid flows with conjugate heat transfer and electrokinetic effects, we propose a direct forcing immersed boundary method for simulating general, discontinuous, Dirichlet and Robin conditions at the interface between two materials. In comparison to existing methods, our approach uses smaller stencils and accommodates complex geometries with sharp corners. The method is built on the concept of a “forcing pair,” defined as two grid points that are adjacent to each other, but on opposite sides of an interface. For 2D problems this approach can simultaneously enforce discontinuous Dirichlet and Robin conditions using a six-point stencil at one of the forcing points, and a 12-point stencil at the other. In comparison, prior work requires up to 14-point stencils at both points. We also propose two methods of accommodating surfaces with sharp corners. The first locally reduces stencils in sharp corners. The second uses the signed distance function to globally smooth all corners on a surface. The smoothing is defined to recover the actual corners as the grid is refined. We verify second-order spatial accuracy of our proposed methods by comparing to manufactured solutions to the Poisson equation with challenging dis- continuous fields across immersed surfaces. Next, to explore the performance of our method for simulating fluid flows with conjugate heat transport, we couple our method to the incompressible Navier–Stokes and continuity equations using a finite-volume projection method. We verify the spatial-temporal accuracy of the solver using manufactured solutions and an analytical solution for circular Couette flow with conjugate heat transfer. Finally, to demonstrate that our method can model moving surfaces, we simulate fluid flow and conjugate heat transport between a stationary cylinder and a rotating ellipse or square.
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Minimal surfaces on mirror-symmetric frames: a fluid dynamics analogy
Chaplygin’s hodograph method of classical fluid mechanics is applied to explicitly solve the Plateau problem of finding minimal surfaces. The minimal surfaces are formed between two mirror-symmetric polygonal frames having a common axis of symmetry. Two classes of minimal surfaces are found: the class of regular surfaces continuously connecting the supporting frames forming a tube with complex shape; and the class of singular surfaces which have a partitioning film closing the tube in between. As an illustration of the general solution, minimal surfaces supported by triangular frames are fully described. The theory is experimentally validated using soap films. The general solution is compared with the known particular solutions obtained by the Weierstrass inverse method.
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- Award ID(s):
- 1655740
- PAR ID:
- 10230372
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 897
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.391
