Accurate Steady-State Solutions for Unstable Problems
Renan de S. Teixeira1; Leonardo S. de B. Alves2
1 Instituto Nacional de Metrologia, Qualidade e Tecnologia - INMETRO; 2 Universidade Federal Fluminense - UFF
doi:10.20906/CPS/COB-2015-1512
Resumo
Several fluid mechanics problems require accurate reference solutions for their analysis and computational studies. Thermal and hydrodynamic stability analysis are such examples, where the reference solution should be a representative base flow. Numerical simulations of unstable problems are yet another example where accurate reference solutions are important to minimizing the unwanted oscillations due ill-posed initial and boundary conditions. Steady-state provide the best alternative to be used as reference solutions. Selective frequency damping (SFD) was developed to provide such steady-state solutions. It works by introducing a source term in the governing equations that forces the time marching scheme to converge towards a reference solution, which is the filtered version of the unsteady solution marching in time. Since this source term disappears at steady-state, the reference solution becomes the steady-state in this limit. However, this technique has been applied to globally unstable problems only. Recently, physical-time damping (PTD) was developed to generate steady-states for any type of unstable flows. The method works with the dissipative properties of the Euler implicit scheme. But this method is not able to introduce enough dissipation overcome absolute instabilities with large temporal growth rates. This is caused by nonlinear effects, which limit the dissipative properties of the implicit Euler scheme. The present paper describes a robust technique to generate steady-states for unstable problems, called Minimal Gain Marching (MGM) schemes. This method is based on constructing time marching schemes that achieve the minimal gain of the linear stability spectrum at the smallest possible time step. This paper extends previous studies, showing that such a technique works quite well for several different flows with different stability characteristics.
Palavras-chave: Initial and Boundary Conditions; Steady-State Solutions; Numerical Stability Analysis; Marching Schemes