Basic idea of time-linearized methods

The time-linearized solver in TRACE [23], [11] , called linearTRACE, provides an efficient way to solve the unsteady RANS equations for small time-harmonic perturbations of a steady flow \(\bar q\). Based on the assumption that the nonlinear terms in the Navier-Stokes equations behave essentially linearly for fluctuations of small amplitudes, one replaces the flux functions with their Jacobians. More precisely, instead of the unsteady RANS equations,

\begin{equation} \frac{\partial q}{\partial t} + R(q ) = 0 \end{equation}

for the flow

\begin{equation} q(t,x) = \bar q(x) + \tilde q (t,x), \end{equation}

one considers the unsteady linear equations

\begin{equation} \frac{\partial \tilde q}{\partial t} + \frac{\partial R}{\partial q}\Big|_{\bar q} \cdot \tilde q = 0. \end{equation}

The advantage of the linearized equations is that fluctuations of different frequencies are decoupled. In fact, Fourier transformation in time yields the time-linearized equations

\begin{equation} \Big( i \omega + \frac{\partial R}{\partial q}\Big|_{\bar q} \Big) \cdot \widehat q_\omega = 0 \end{equation}

for each angular frequency \(\omega > 0\). Roughly speaking, the time-linearized equations are derived from the unsteady nonlinear equations in two steps:

Approximation of the URANS equations by its linearization.
Transfer from time-domain to frequency domain using the Fourier transformation.

Application

Time-linearized techniques may be used for two applications:

Aeroelasticity, i.e. flutter and forced-response analysis [23] .
Prediction of tonal noise generation, transmission and propagation [11] .

Solution technique

The time-linearized system is solver using a parallelized preconditioned GMRES method. Since memory requirements and CPU-time per iteration increase with the number of iterations performed, GMRES is usually restarted after a fixed interval of iterations. There are two preconditioners available:

Symmetric successive over-relaxation, SSOR( \(\omega, k\)), possibly using \(k\) SSOR iterations as a preconditioner for one GMRES step. SSOR(0.7,0) is the default preconditioner.
Incomplete LU decomposition with a given level of fill \(p\).

Boundary conditions and periodic boundaries

Apart from the linearization of the wall boundary conditions, linearTRACE offers non-reflecting 1D and 2D boundary conditions for inlets and outlets. Moreover, at periodic boundaries a phase-lag boundary condition may be used.