{"id":5291,"date":"2021-04-01T18:21:00","date_gmt":"2021-04-01T16:21:00","guid":{"rendered":"https:\/\/www.mi.uni-koeln.de\/NumSim\/?p=5291"},"modified":"2021-04-01T18:36:59","modified_gmt":"2021-04-01T16:36:59","slug":"snapshot-simulation-of-a-kelvin-helmholtz-instability-using-second-order-finite-volume-schemes-and-fourth-order-discontinuous-galerkin-methods","status":"publish","type":"post","link":"https:\/\/www.mi.uni-koeln.de\/NumSim\/2021\/04\/01\/snapshot-simulation-of-a-kelvin-helmholtz-instability-using-second-order-finite-volume-schemes-and-fourth-order-discontinuous-galerkin-methods\/","title":{"rendered":"Snapshot: Simulation of a Kelvin-Helmholtz instability using second order Finite Volume schemes and fourth order Discontinuous Galerkin methods"},"content":{"rendered":"

We present in-viscid and viscous simulations of a Kelvin-Helmholtz instability using second a order accurate monotoniced-central finite volume (FV) method and a fourth order accurate discontinuous Galerkin (DG) method. The initial condition is given by [1]:<\/p>\n

$$\\rho (t=0) = \\frac{1}{2}
\n+ \\frac{3}{4} B,
\n~~~~~~~~~
\np (t=0) = 1,~~~~~~~~~~
\n$$<\/p>\n

$$
\nv_1 (t=0) = \\frac{1}{2} \\left( B-1 \\right),
\n~~~~~~~
\nv_2 (t=0) = \\frac{1}{10} \\sin(2 \\pi x),
\n$$<\/p>\n

with $$B=\\tanh \\left( 15 y + 7.5 \\right) – \\tanh(15y-7.5).$$<\/p>\n

We first present the FV results at end time $t=3.7$, which are computed using a monotoniced-central second order discretization of the Euler equations of gas dynamics on uniform grids.<\/p>\n

\"\"<\/a>\"\"<\/a><\/p>\n

The next results use a fourth order DG discretization of the Navier-Stokes equations on uniform grids using $Re=320.000$ at end time $t=3.7$. The highest resolution (4096\u00b2 DOFs) is a direct numerical simulation (DNS) of the problem, where all scales are resolved.<\/p>\n

\"\"<\/a>\"\"<\/a><\/p>\n

It is remarkable that the numerical dissipation of the second order FV scheme causes the in-viscid simulation with 2048\u00b2 DOFs to look very similar to the viscous DNS solution at $Re=320.000$.<\/p>\n

[1] A.M. Rueda-Ram\u00edrez, G.J Gassner (2021). A Subcell Finite Volume Positivity-Preserving Limiter for DGSEM Discretizations of the Euler Equations<\/em>. https:\/\/arxiv.org\/pdf\/2102.06017.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

We present in-viscid and viscous simulations of a Kelvin-Helmholtz instability using second a order accurate monotoniced-central finite volume (FV) method and a fourth order accurate discontinuous Galerkin (DG) method. The initial condition is given by [1]: $$\\rho (t=0) = \\frac{1}{2} … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[49],"tags":[],"post_mailing_queue_ids":[],"_links":{"self":[{"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/posts\/5291"}],"collection":[{"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/comments?post=5291"}],"version-history":[{"count":20,"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/posts\/5291\/revisions"}],"predecessor-version":[{"id":5327,"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/posts\/5291\/revisions\/5327"}],"wp:attachment":[{"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/media?parent=5291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/categories?post=5291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-json\/wp\/v2\/tags?post=5291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}