Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models

Abstract Large eddy simulation reduced order models (LES-ROMs) are ROMs that leverage LES ideas (e.g., filtering and closure modeling) to construct accurate and efficient ROMs for convection-dominated (e.g., turbulent) flows. Eddy viscosity (EV) ROMs (e.g., Smagorinsky ROM (S-ROM)) are LES-ROMs whose closure model consists of a diffusion-like operator in which the viscosity depends on the ROM velocity. We propose the Ladyzhenskaya ROM (L-ROM), which is a generalization of the S-ROM. Furthermore, we prove two fundamental numerical analysis results for the new L-ROM and the classical S-ROM: (i) We prove the verifiability of the L-ROM and S-ROM, i.e., that the ROM error is bounded (up to a constant) by the ROM closure error. (ii) We introduce the concept of ROM limit consistency (in a discrete sense), and prove that the L-ROM and S-ROM are limit consistent, i.e., that as the ROM dimension approaches the rank of the snapshot matrix, d , and the ROM lengthscale goes to zero, the ROM solution converges to the “true solution" , i.e., the solution of the d -dimensional ROM. Finally, we illustrate numerically the verifiability and limit consistency of the new L-ROM and S-ROM in two under-resolved convection-dominated problems that display sharp gradients: the 1D Burgers equation with a small diffusion coefficient; and the 2D lid-driven cavity flow at Reynolds number $$\textrm{Re}=15,000$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Re</mml:mtext> <mml:mo>=</mml:mo> <mml:mn>15</mml:mn> <mml:mo>,</mml:mo> <mml:mn>000</mml:mn> </mml:mrow> </mml:math> .