Two-mode Galerkin projection captures Prandtl secondary flow in square ducts without turbulence statistics
Minimal Galerkin models derived from linear eigenmodes predict secondary mean flows in square ducts. The approach requires no prior turbulent statistics and yields analytical relations between mean coefficients and Reynolds stresses. Agreement with DNS improves with added modes, offering a low-cost route to engineering flow predictions.
Researchers projected the Navier-Stokes equations onto leading eigenmodes of the linearized operator around the laminar base flow, then integrated the resulting low-dimensional ODE systems with added zero-mean stochastic forcing. The secondary mean flow emerges solely from the nonzero time-averaged coefficient of one symmetry-compatible mode whose amplitude is set by the joint second moments of the other coefficients, which can be evaluated analytically. Validation against streamwise-constant DNS with identical forcing shows matching vortex topology and direction; Reynolds-stress distributions improve systematically as model dimension increases from two to higher truncations.
El-Nadi et al.: Five-mode models will reproduce DNS secondary-flow kinetic energy within 15% in square ducts at Re_tau=180 by end of 2027.
Sources (2)
- [1]Primary Source(https://arxiv.org/abs/2608.13771)
- [2]Supporting Source(https://doi.org/10.1017/jfm.2018.456)