A POD-Galerkin reduced-order model for isotropic viscoelastic turbulent flow

Abstract

In this article, a proper orthogonal decomposition (POD) reduced-order model for isotropic turbulent flow of viscoelastic fluid is established for the first time. Particularly, since the present studies about viscoleastic fluid are mainly for revealing the mechanism of turbulence, we try to establish the reduced order model for momentum equations and constitutive equations, finally get both velocities and deformation rates calculated. Through decomposing the sampling matrices which are obtained by direct numerical simulation (DNS) using finite volume method (FVM), the velocity basis functions and deformation rate basis functions are generated respectively. According to the Galerkin projection method, the equations for velocity spectrum coefficients and deformation rate spectrum coefficients are deducted, which are coupled pluralistic nonlinear equations and solved iteratively by the Newton-Raphson method. To illustrate the performance of the proposed model for the viscoelastic fluid flow, a two-dimensional decaying isotropic turbulence testing case is designed in Example 1. It is found that the established reduced-order model obtains good accuracy when the decaying flow is at its early stage, but the errors get considerable when the flow steps into transition flow. In addition, a three-dimensional forced isotropic viscoelastic turbulence testing case is designed in Example 2. It is indicated that the errors of viscoelastic forced isotropic turbulent flow are acceptable. Finally, the calculation speed of the established reduced-order model is found to be much faster than that of DNS.

Uncertainty Quantification for Numerical Solutions of the Nonlinear Partial Differential Equations by Using the Multi-Fidelity Monte Carlo Method

“…The POD basis vector V ∈ R N×n of the matrix V is the n largest left singular vectors of the matrix Y. The POD reduced-order system is obtained by Galerkin projection [22,26]:…”

“…Here, for the NLS equation, inspired by Ref. [19], the proper orthogonal decomposition (POD) method [22,25,26] coupled with discrete empirical interpolation method (DEIM) [27][28][29][30] is used to construct an optimal low fidelity model, which can deal with the complexity of higher order nonlinear terms of the NLS equation. Otherwise, for Burgers' equation, we attempt to construct another kind of the low-fidelity model by using the operator reduction (OR) method [31][32][33], combined with DEIM for the parameterized Burgers' equation.…”

A Galerkin‐free/equation‐free model reduction method for single‐phase flow in fractured porous media

“…The proper orthogonal decomposition reduced‐order model (POD‐ROM) can reduce the number of degrees of freedom for original high‐dimensional physical problems by combining the POD and the Galerkin projection method . It has been applied in many fields, such as turbulence flow, heat conduction and convective heat transfer, two‐phase flow, and gas flow . Recently, the POD method has been extended to simulate flow in porous media .…”