Innovative Grid-Based Vorticity–Velocity Solver for Analysis of Vorticity-Dominated Flows

Abstract

Unsteady vorticity-dominated flows are prevalent in a wide variety of engineering applications and are particularly challenging to predict efficiently. Perhaps in no other application is this more apparent than rotorcraft, where prediction is complicated by a wide variety of interactional aerodynamics between strong vortical wakes shed from rotating components and various bodies and wings immersed in the flow. While conventional computational fluid dynamics methods are equipped with the modeling capabilities to support these flow behaviors, they are subject to significant numerical diffusion whose practical import is that high computational costs involving large grids, complex adaptation methods, and high interpolation orders are required to produce sufficiently accurate simulations and prevent the artificial diffusion of vortical structures. Hybrid computational fluid dynamics methods that couple a body-fitted solver to a vorticity-velocity method hold considerable promise in alleviating these costs while effectively preserving vorticity. However, several fundamental and long-standing issues identified with vorticity-velocity methods have precluded their wider adoption. Here, the four most prominent concerns are addressed: 1) maintaining a solenoidal vorticity field; 2) supporting a low-dissipation/dispersion scheme on a general multiresolution grid; 3) providing a consistent and efficient implementation of the viscous terms; and 4) properly including the unsteady pressure contributions. Nomenclature A = angular impulse B = unsteady pressure B = normalized unsteady pressure E = energy f = flux across a surface GR; r = Green's function H = helicity I = linear impulse n = unit normal vector P = static pressure P = normalized steady pressure R = position vector with components x; y; z R = centroid of vorticity Re Γ = Reynolds number with respect to vortex ring circulation; Γ∕v r = radial location S = initial distance between vortex rings S = surface S = vorticity source term t = time U = freestream velocity u = velocity u = centroid of velocity V = volume Γ = circulation Δs = cell size ε = enstrophy η = divergence-free correction term θ = initial vortex ring inclination υ = viscosity ρ = density σ = core radius σ = pressure source term ψ = stream function Ω = integrated vorticity ω = vorticity

Dual-solver hybrid computational approaches for design and analysis of vertical lift vehicles

“…Further time savings can be achieved if the mesh extent expands at the convection speed. The most well-known VVTM solvers are VTM [36], the seminal version of this approach, and VorTran-M/M2 [37,38]. Like with Lagrangian formulations, FMMs can also be applied to grid-based schemes [39] to reduce the computational cost.…”