Numerical Evidence of Multiple Solutions for the Reynolds-Averaged Navier-Stokes Equations for High-Lift Configurations

Prediction of Lift Cells for Stalling Wings by Lifting-Line Theory

“…Also unfortunately, the challenge of multiple solutions has recently added itself to the scene. Specifically, RANS studies on grids of defendable density and with full convergence of the iterations (to "machine zero") are producing multiple solutions: in the case of Kamenetskiy et al, as many as eight [1]. This is seen both on simple infinite airfoils and on a finite wing of moderate complexity.…”

“…A SIMPLE theory is formulated for infinite and finite wings, aimed at the stall regime with the purpose of explaining the spontaneous formation of lift cells, which have been observed in experiments and in a recent computational fluid dynamics (CFD) study [1]. A periodic version of lifting-line theory, applied to infinite airfoils, appears to be new.…”

“…It is then often considered "normal" for the search for a steady state to fail, and users switch to time-accurate simulations, although more algorithmically powerful solvers such as the one used in [1] do find fully converged solutions, even under extreme conditions such as an airfoil at a 90 deg angle of attack. Such unsteady RANS (URANS) simulations enter vortex-shedding modes, which are plausible and can have value for vibration prediction but have an uncertain accuracy and a debatable relationship with the "third" direction of infinite geometries (sometimes called "extruded airfoils") [5].…”

Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Towards CFD 2030

“…Incipient or large scale unsteadiness may require the use of time-accurate simulation or continuation to obtain the desired solution when multiple solutions or hysteresis are observed. 15 For example, Nishikawa et al 84 demonstrate this behavior for the Drag Prediction Workshop configuration in Fig. 7.…”

“…An adaptive unstructured grid viscous simulation tool with the same robustness and throughput as the adaptive Euler method would have great impact for building aerodynamic databases. Dalle et al 12 may show some evidence of nonunique solutions 14,15 where the output-based technique is refining the solution related to the closest attractor or trying to extract a steady solution out of an inherently unsteady problem. These issues may be hard to discern without proper physics, discretization error controls, and strong nonlinear solvers.…”