Technology
Vorticity Confinement (VC)
 |
Vorticity Isosurfaces
for forward flight |
Vorticity Confinement (VC) is a revolutionary, new physics based turbulence model invented and developed by Dr. John Steinhoff, a professor at the University of Tennessee Space Institute and Consultant to Flow Analysis Solutions, Inc., in the late 1980s to solve vortex dominated flows. It was first formulated in 1988, to capture concentrated vortices shed from the wings and helicopter rotors and later became popular in a wide range of research areas. A “carrier field” and the rotationally invariant vorticity field itself, was used to “guide” and confine the vorticity. At high Reynolds numbers, turbulence is typically characterized by concentrated vorticity and conversely for most incompressible flows, vorticity is confined to the turbulent regions, which are typically thin.. Discretizing and solving partial differential equations in these thin regions is typically very time consuming and so we have developed a combined turbulence model, which includes discretization and can automatically treat thin regions without grid refinement.
During the 1990’s and 2000’s it became widely used in the field of engineering. There is a basic similarity to the solitary wave approach which is extensively used in many condensed physics applications.
VC
VC transports vorticity along the gradient of the carrier field. This satisfied a partial differential equation which effectively results in solitary wave solutions. The vortex structures, treated as solitary waves remain stable, convect with local velocity.
 |
| Jet Plume without confinement |
VC involves an added dissipation (
) to the partial differential equation, which automatically balance with gradient transport (
), produce stable solutions for a wide range of conditions. The transport is effected by adding acceleration along the cross product of the local vorticity and the unit vector formed by its gradient. This added acceleration only transports the vorticity and not the fluid itself. Zero divergence for incompressible flows can then be easily enforced as if this term were not present. This can be thought as a tangential acceleration about the vortex center. This invention solved the problem of unwanted numerical diffusion and resulted in vortices that maintained a stable, fixed radius for indefinite transport distances.
 |
| Jet Plume with confinement |
One feature is that since the fluid is rotating around the vortex centre, almost all of this added momentum will cancel and integrate to a small value. In this way, momentum is closely conserved. This has proven to be a very robust technique because we are combining an inward transport with an outward dissipation. It was initially used to transport concentrated vortices. However, it was soon realized that the same method could serve where vorticity transport is along the normal vector directed towards the surface. This served as an efficient model for boundary layers which like shed vortices, do not spread in a normal direction. The tangential dissipation along the surface prevented the irregularities such as “stair-casing” effects. It proved to be a good model for turbulent boundary layers. Since the inner boundary layer is approximately constant, it performs better than more complex models on a much coarser grid at much lower cost. In addition, this was not a fixed boundary layer but was only created by added VC1 type acceleration. This meant that the VC1 boundary layer could separate if it approached the corner or region of strong adverse pressure gradient. In this way, this technique served as a very efficient model for the boundary which as a first approximation did not require the points to be specified, unlike panel methods or some conventional models. This is extremely convenient for immersing complex shapes without requiring body fitted adaptive grids. One important advantage of this approach is that exactly the same computational grid is used to solve for the flow, irrespective of the body shape. Due to the inward propagation of vorticity, the vortex structure was stable against spreading, due to numerical diffusion.
WC
A new numerical method, “Wave Confinement” (WC), is developed to efficiently solve the linear wave equation. This is similar to the originally developed “Vorticity Confinement” (VC) method for fluid mechanics problems. It involves modification of the discrete wave equation by adding an extra nonlinear term that can accurately propagate the pulses for long distances without numerical dispersion/diffusion.
 |
Wave propagation with
reflective boundaries |
These pulses are propagated as stable surfaces and do not suffer phase shift or amplitude exchange in spite of nonlinearity. The pulses remain thin, unlike conventional higher order numerical schemes, which only converge as N (the number of grid cells across the pulse) becomes large. The additional term does not interfere with conservation of the important integral quantities such as total amplitude, motion of the centroid, etc. Properties like varying index of refraction, diffraction, multiple reflections are included, and accuracy has been validated.
The generated short pulses can be best described as solitary waves, which can recover the shape after a collision due to nondestructive interaction between the pulses. Within the pulse, the dissipative effects due to the numerical errors are balanced with those of nonlinearity and the pulse will retain its original form and speed even after many collisions. Scattering over complex bodies can be modeled with no use of complicated adaptive grid generation schemes around the bodies. The confinement term smoothes the boundary and prevents stair casing effects but the boundary remains thin. Validation studies have been performed for a number of real flow models and compared to the exact solutions. It is observed that the solutions match quite well with the exact solution. This approximation has a number of advantages over the existing paraxial approximation used to simulate radar and radio wave propagation.
Immersed boundary Layer Model
 |
| Immersed boundary |
To enforce no-slip boundary conditions on immersed surfaces, first, the surface is represented implicitly by a smooth “level set” function, “F”, defined at each grid point. This is just the (signed) distance from each grid point to the nearest point on the surface of an object – positive outside, negative inside. Then, at each time step during the solution, velocities in the interior are simply set to zero. In a computation using VC, this results in thin vortical region along the surface, which is smooth in the tangential direction, with no “staircase” effects. The important point is that no special logic is required in the “cut” cells, unlike many conventional schemes: only the same VC equations are applied, as in the rest of the grid, but with a different form for F. Also, unlike many conventional immersed surface schemes, which are inviscid because of cell size constraints, there is effectively a no-slip boundary condition, which results in a boundary layer with well-defined total vorticity and which, because of VC, remains thin, even after separation. The method is especially effective for complex configurations with separation from sharp corners. Also, even with constant coefficients, it can approximately treat separation from smooth surfaces. General blunt bodies, which typically shed turbulent vorticity induce a velocity around an upstream body. It is inconsistent to use body fitted grids as the vorticity convects through a non fitted grid.
