Variational multiscale approximation of the one‐dimensional forced Burgers equation: The role of orthogonal subgrid scales in turbulence modeling

• Published in: International journal for numerical methods in fluids 2018-02, Vol.86 (5), p.313-328
• Main Authors: Bayona, Camilo, Baiges, Joan, Codina, Ramon
• Format: Article
• Language: English
• Subjects: Anàlisi numèrica; Approximation; Burgers equation; Computer simulation; Direct numerical simulation; Energy spectra; Large eddy simulation; Matemàtiques i estadística; Mathematical analysis; Mathematical models; Modelling; Models matemàtics; Multiscale analysis; orthogonal subgrid scales; Simulation; Turbulence; Turbulència; variational multiscale method; Vortices; Àrees temàtiques de la UPC
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4420

Summary A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite‐dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS‐VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS‐VMS method without any modification of the continuous problem. We apply the orthogonal subgrid scales‐variational multiscale (OSGS‐VMS) method to the one‐dimensional Burgers equation. The space where the orthogonal subscales are defined is described in terms of the finite‐dimensional resolved space, and its properties are analyzed. Results obtained with the proposed method are compared with the solution of a DNS of a turbulent problem. Good accuracy is obtained, and it is shown that OSGS‐VMS results improve as the grid is refined. This allows us to conclude that OSGS‐VMS works properly in the DNS limit. The scale dependence of the OSGS‐VMS added dissipation is studied, and it is shown that it behaves as a turbulence model. The OSGS‐VMS results are compared with the classical Smagorinsky LES method. We conclude that OSGS‐VMS is preferred rather than Smagorinsky LES method, when the grid size is equal to the filter width.