A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models

In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches

Non-Darcy flow through porous media

Since its introduction, Darcy's law has been implemented as a mathematical tool that allows simple calculation and prediction of low velocity subsurface flows. However, turbulence, non-isothermal conditions, as well as other factors can create conditions where Darcy's law does not accurately describe the head and velocity distributions within a given porous matrix. Darcy's law has been widely applied to analytical and numerical modeling of fluid flow through porous matrix, regardless of the hydrogeologic setting. This study attempts to quantify the error incurred by these models through simultaneous numerical modeling of the mass continuity equation using Darcy's law as well as Forchheimer's relation. To this end, results from steady-state and transient Darcy-based and Forchheimer-based numerical models are presented in this study.Geological Science

Efficient prediction of broadband trailing edge noise and application to porous edge treatment

Trailing edge noise generated by turbulent flow traveling past an edge of an airfoil is one of the most essential aeroacoustic sound generation mechanisms. It is of great interest for noise problems in various areas of industrial application. First principle based CAA with short response time are needed in the industrial design process for reliable prediction of spectral differences in turbulent-boundary-layer trailing-edge noise due to design modifications. In this paper, an aeroacoustic method is studied, resting on a hybrid CFD/CAA procedure. In a first step RANS simulation provides a time-averaged solution, including the mean-flow and turbulence statistics such as length-scale, time-scale and turbulence kinetic energy. Based on these, fluctuating sound sources are then stochastically generated by the Fast Random Particle-Mesh Method to simulate in a second CAA step broadband aeroacoustic sound. From experimental findings it is well known that porous trailing edges significantly lower trailing edge noise level over a large range of frequencies reaching up to 8dB reduction. Furthermore, sound reduction depends on the porous material parameters, e.g. geometry, porosity, permeability and pore size. The paper presents first results for an extended hybrid CFD/CAA method including porous materials with prescribed parameters. To incorporate the effect of porosity, an extended formulation of the Acoustic Perturbation Equations with source terms is derived based on a reformulation of the volume averaged Navier-Stokes equations into perturbation form. Proper implementation of the Darcy and Forchheimer terms is verified for sound propagation in homogeneous and anisotropic porous medium. Sound generation is studied for a generic symmetric NACA0012 airfoil without lift to separate secondary effects of lift and camber on sound from those of the basic edge noise treatments.Comment: 37 page