# Numerical method

COSA solves the compressible Euler and Navier-Stokes equations on structured multi-block grids both in two and three dimensions. For turbulent flows, the code presently uses either the k-ω turbulence model of Wilcox or the shear stress transport turbulence model of Menter. The code features a steady solver, an implicit and an explicit time-domain (TD) solver, and a frequency-domain harmonic balance (HB) solver for the rapid and accurate calculation of strongly nonlinear periodic flow fields typically arising in rotary fluid machinery.

COSA can solve unsteady rotor flows and other fluid machinery problems either in the absolute or relative frame of reference. In the case of unsteady periodic flows, a multi-frequency periodicity boundary condition (MFPBC) enables a cost reduction of the HB simulation proportional to the number of rotor blades, because such MFPBC enables the simulation of a single blade sector rather than the entire rotor.

The explicit integration of the steady equations is based on a multi-stage Runhe-Kutta (RK) smoother, and is accelerated by local time-stepping, central variable-coefficient implicit residual smoothing, and full approximation scheme multigrid. The implicit time-domain solver uses the dual-time-stepping approach, and the system of equations arising at each new physical time is solved iteratively using the same procedure adopted for solving the steady equations. The explicit time-domain solver uses instead time-accurate RK time-marching for problems requiring very high temporal resolutions. The harmonic balance solver of COSA and its efficient numerical implementation are one of the unique features of this CFD code. The HB flow equations are solved with a procedure similar to that used to solve the steady equations.

Another crucially important numerical and modeling feature of COSA is its low-speed preconditioning (LSP) strategy for complex steady and unsteady fluid problems featuring regions of incompressible flow, that is characterized by very low Mach number. The use of LSP allows a single code to be used for high-speed flows, such as those encountered in transonic wing aerodynamics and the flow field past gas turbine blades, low-speed flows, such as those associated with the flow field past an automobile, and mixed-speed problems, such as the flow field past the blades of new large multimegawatt horizontal axis wind turbines. In these machines, the relatively high relative flow speed in the blade tip region induces non-negligible compressibility effects, whereas the substantially lower relative speeds at the inboard part of the blade result in a practically incompressible flow regime.