DeHNSSo: The delft harmonic Navier-Stokes solver for nonlinear stability problems with complex geometric features
Description
A nonlinear Harmonic Navier-Stokes (HNS) framework is introduced for simulating instabilities in laminar spanwise-invariant shear layers, featuring sharp and smooth wall surface protuberances. While such cases play a critical role in the process of laminar-to-turbulent transition, classical stability theory analyses such as parabolized or local stability methods fail to provide (accurate) results, due to their underlying assumptions. The generalized incompressible Navier-Stokes (NS) equations are expanded in perturbed form, using a spanwise and temporal Fourier ansatz for flow perturbations. The resulting equations are discretized using spectral collocation in the wall-normal direction and finite-difference methods in the streamwise direction. The equations are then solved using a direct sparse-matrix solver. The nonlinear mode interaction terms are converged iteratively. The solution implementation makes use of a generalized domain transformation to account for geometrical smooth surface features, such as humps. No-slip conditions can be embedded in the interior domain to account for the presence of sharp surface features such as forward- or backward-facing steps. Common difficulties with Navier-Stokes solvers, such as the treatment of the outflow boundary and convergence of nonlinear terms, are considered in detail. The performance of the developed solver is evaluated against several cases of representative boundary layer instability growth, including linear and nonlinear growth of Tollmien-Schlichting waves in a Blasius boundary layer and stationary crossflow instabilities in a swept flat-plate boundary layer. The latter problem is also treated in the presence of a geometrical smooth hump and a sharp forward-facing step at the wall. HNS simulation results, such as perturbation amplitudes, growth rates, and shape functions, are compared to benchmark flow stability analysis methods such as Parabolized Stability Equations (PSE), Adaptive Harmonic Linearized Navier-Stokes (AHLNS), or Direct Numerical Simulations (DNS). Good agreement is observed in all cases. The HNS solver is subjected to a grid convergence study and a simple performance benchmark, namely memory usage and computational cost. The computational cost is found to be considerably lower than high-fidelity DNS at comparable grid resolutions.