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# Methods and Applications of Analysis

## Volume 21 (2014)

### Number 1

### Simulation of high re boundary layer flows using vorticity confinement

Pages: 123 – 150

DOI: http://dx.doi.org/10.4310/MAA.2014.v21.n1.a6

#### Authors

#### Abstract

We describe how Vorticity Confinement (VC) can be regarded as a new pde formulation of the slightly viscous incompressible flow equations. These equations, when discretizedand solved, generate nonlinear solitary waves that can be used to efficiently approximate a largeclass of external flow problems, including the effects of separating turbulent boundary layers. Theseproblems can involve subsonic flow over complex structures such as ships, buildings, and realistictopography such as hills. These problems typically involve the simulation of a large ensemble of flowconditions for each configuration to be designed or analyzed. One of the most difficult aspects ofthese simulations is that often the main effects of the dynamics of thin evolving vortical structuresmust be solved for.

The VC method appears to be effective for many of these problems, since it requires much lesscomputing and setup time than current Navier Stokes “RANS” approximations. The method involvestreating the flow over a solid body as a two-scale problem: The first component is an “outer” smoothlyvarying, mainly irrotational flow with perhaps large scale vortical components where standard CFDtechniques can be used. The second component is composed of thin vortical regions. These vorticalparts consist of mostly thin attached boundary layers, thin separating vortex sheets, which roll upand thin vortex filaments, which result from roll up. The VC method involves treating these regionswith a single equation that has three equilibrium states corresponding to these regions. The equationallows transition between these equilibrium states so that for example, boundary layers can separateand roll up into vortex filaments, and vortex filaments can join and reconnect with other filaments.These properties survive discretization and require no extra logic.

VC can be used to treat the entire flow in a locally-Cartesian computational grid with the solidsurfaces “immersed” in the grid so that they can be quickly generated for many configurations.Adaptive or conforming fine scale grid cells are then not required to approximate the thin vorticalboundary layers, or thin separating vortex sheets. Instead, vortical structures created with Vorticity Confinement, which are essentially thin, non-diffusing, or confined “Nonlinear Solitary Waves”(NSW’s) are used to “carry” the vorticity in these regions. The VC method has the efficiency ofpanel methods, but the generality and ease of use of fixed grid Euler equation methods. In thispaper we concentrate on attached and separating boundary layers; there are already in the literaturea large number of papers describing the use of VC for free, convecting vortices.

#### Keywords

vorticity confinement, vorticity, vortex dominated flow, computational fluid dynamics, boundary layer separation, solitary waves, permanent wave

#### 2010 Mathematics Subject Classification

65-xx