scholarly journals Penalization model for Navier–Stokes–Darcy equations with application to porosity-oriented topology optimization

2018 ◽  
Vol 28 (08) ◽  
pp. 1481-1512 ◽  
Author(s):  
Alain Bastide ◽  
Pierre-Henri Cocquet ◽  
Delphine Ramalingom
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Topology Optimization ◽  
Stokes Equations ◽  
Optimal Solution ◽  
Navier Stokes ◽  
Necessary Condition ◽  
Navier Stokes Equations ◽  
Topology Optimization Problem ◽  
Darcy Equations

Topology optimization for fluid flow aims at finding the location of a porous medium minimizing a cost functional under constraints given by the Navier–Stokes equations. The location of the porous media is usually taken into account by adding a penalization term [Formula: see text], where [Formula: see text] is a kinematic viscosity divided by a permeability and [Formula: see text] is the velocity of the fluid. The fluid part is obtained when [Formula: see text] while the porous (solid) part is defined for large enough [Formula: see text] since this formally yields [Formula: see text]. The main drawback of this method is that only solid that does not let the fluid to enter, that is perfect solid, can be considered. In this paper, we propose to use the porosity of the media as optimization parameter hence to minimize some cost function by finding the location of a porous media. The latter is taken into account through a singular perturbation of the Navier–Stokes equations for which we prove that its weak-limit corresponds to an interface fluid-porous medium problem modeled by the Navier–Stokes–Darcy equations. This model is then used as constraint for a topology optimization problem. We give necessary condition for such problem to have at least an optimal solution and derive first order necessary optimality condition. This paper ends with some numerical simulations, for Stokes flow, to show the interest of this approach.

Fluid flow over and through a regular bundle of rigid fibres

2016 ◽  
Vol 792 ◽  
pp. 5-35 ◽  
Author(s):  
Giuseppe A. Zampogna ◽  
Alessandro Bottaro
Keyword(s):  
Porous Medium ◽  
Fluid Flow ◽  
Flow Field ◽  
Stokes Equations ◽  
Transversely Isotropic ◽  
Navier Stokes ◽  
Leading Order ◽  
Macroscopic Scale ◽  
Navier Stokes Equations ◽  
Homogenized Model

The interaction between a fluid flow and a transversely isotropic porous medium is described. A homogenized model is used to treat the flow field in the porous region, and different interface conditions, needed to match solutions at the boundary between the pure fluid and the porous regions, are evaluated. Two problems in different flow regimes (laminar and turbulent) are considered to validate the system, which includes inertia in the leading-order equations for the permeability tensor through a Oseen approximation. The components of the permeability, which characterize microscopically the porous medium and determine the flow field at the macroscopic scale, are reasonably well estimated by the theory, both in the laminar and the turbulent case. This is demonstrated by comparing the model’s results to both experimental measurements and direct numerical simulations of the Navier–Stokes equations which resolve the flow also through the pores of the medium.

Modelling Turbulent Flow in Deformable Highly Porous Seabed and Structures

2018 ◽  
Author(s):  
Hisham Elsafti ◽  
Hocine Oumeraci
Keyword(s):  
Porous Media ◽  
Transient Flow ◽  
Stokes Equations ◽  
Pore Fluid ◽  
Navier Stokes ◽  
Model Parameters ◽  
Navier Stokes Equations ◽  
Deformable Porous Media ◽  
Physical Experiments ◽  
Fluid Coupling

In this study, the fully-coupled and fully-dynamic Biot governing equations in the open-source geotechFoam solver are extended to account for pore fluid viscous stresses. Additionally, turbulent pore fluid flow in deformable porous media is modeled by means of the conventional eddy viscosity concept without the need to resolve all turbulence scales. A new approach is presented to account for porous media resistance to flow (solid-to-fluid coupling) by means of an effective viscosity, which accounts for tortuosity, grain shape and local turbulences induced by flow through porous media. The new model is compared to an implemented extended Darcy-Forchheimer model in the Navier-Stokes equations, which accounts for laminar, transitional, turbulent and transient flow regimes. Further, to account for skeleton deformation, the porosity and other model parameters are updated with regard to strain of geomaterials. The presented model is calibrated by means of available results of physical experiments of unidirectional and oscillatory flows.

scholarly journals Development of An Integrated Numerical Model for Simulating Wave Interaction with Permeable Submerged Breakwaters Using Extended Navier–Stokes Equations

2020 ◽  
Vol 8 (2) ◽  
pp. 87 ◽  
Author(s):  
Paran Pourteimouri ◽  
Kourosh Hejazi
Keyword(s):  
Porous Medium ◽  
Wave Propagation ◽  
Numerical Model ◽  
Analytical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Stokes Wave ◽  
Navier Stokes Equations ◽  
Numerical Predictions ◽  
The Impact

An integrated two-dimensional vertical (2DV) model was developed to investigate wave interactions with permeable submerged breakwaters. The integrated model is capable of predicting the flow field in both surface water and porous media on the basis of the extended volume-averaged Reynolds-averaged Navier–Stokes equations (VARANS). The impact of porous medium was considered by the inclusion of the additional terms of drag and inertia forces into conventional Navier–Stokes equations. Finite volume method (FVM) in an arbitrary Lagrangian–Eulerian (ALE) formulation was adopted for discretization of the governing equations. Projection method was utilized to solve the unsteady incompressible extended Navier–Stokes equations. The time-dependent volume and surface porosities were calculated at each time step using the fraction of a grid open to water and the total porosity of porous medium. The numerical model was first verified against analytical solutions of small amplitude progressive Stokes wave and solitary wave propagation in the absence of a bottom-mounted barrier. Comparisons showed pleasing agreements between the numerical predictions and analytical solutions. The model was then further validated by comparing the numerical model results with the experimental measurements of wave propagation over a permeable submerged breakwater reported in the literature. Good agreements were obtained for the free surface elevations at various spatial and temporal scales, velocity fields around and inside the obstacle, as well as the velocity profiles.

scholarly journals Upscaling of Navier–Stokes equations in porous media: Theoretical, numerical and experimental approach

2009 ◽  
Vol 36 (7) ◽  
pp. 1200-1206 ◽  
Author(s):  
Guillermo A. Narsilio ◽  
Olivier Buzzi ◽  
Stephen Fityus ◽  
Tae Sup Yun ◽  
David W. Smith
Keyword(s):  
Porous Media ◽  
Experimental Approach ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

Cellular‐automaton fluids: A model for flow in porous media

Geophysics ◽  
1988 ◽  
Vol 53 (4) ◽  
pp. 509-518 ◽  
Author(s):  
Daniel H. Rothman
Keyword(s):  
Porous Media ◽  
Fluid Flow ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
Pressure Gradient ◽  
Cellular Automaton ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cellular Automaton Fluids

Numerical models of fluid flow through porous media can be developed from either microscopic or macroscopic properties. The large‐scale viewpoint is perhaps the most prevalent. Darcy’s law relates the chief macroscopic parameters of interest—flow rate, permeability, viscosity, and pressure gradient—and may be invoked to solve for any of these parameters when the others are known. In practical situations, however, this solution may not be possible. Attention is then typically focused on the estimation of permeability, and numerous numerical methods based on knowledge of the microscopic pore‐space geometry have been proposed. Because the intrinsic inhomogeneity of porous media makes the application of proper boundary conditions difficult, microscopic flow calculations have typically been achieved with idealized arrays of geometrically simple pores, throats, and cracks. I propose here an attractive alternative which can freely and accurately model fluid flow in grossly irregular geometries. This new method solves the Navier‐Stokes equations numerically using the cellular‐automaton fluid model introduced by Frisch, Hasslacher, and Pomeau. The cellular‐ automaton fluid is extraordinarily simple—particles of unit mass traveling with unit velocity reside on a triangular lattice and obey elementary collision rules—but is capable of modeling much of the rich complexity of real fluid flow. Cellular‐automaton fluids are applicable to the study of porous media. In particular, numerical methods can be used to apply the appropriate boundary conditions, create a pressure gradient, and measure the permeability. Scale of the cellular‐automaton lattice is an important issue; the linear dimension of a void region must be approximately twice the mean free path of a lattice gas particle. Finally, an example of flow in a 2-D porous medium demonstrates not only the numerical solution of the Navier‐Stokes equations in a highly irregular geometry, but also numerical estimation of permeability and a verification of Darcy’s law.

Oscillatory forcing of flow through porous media. Part 1. Steady flow

2002 ◽  
Vol 465 ◽  
pp. 213-235 ◽  
Author(s):  
D. R. GRAHAM ◽  
J. J. L. HIGDON
Keyword(s):  
Porous Media ◽  
Flow Rate ◽  
Stokes Equations ◽  
Nonlinear Flow ◽  
Navier Stokes ◽  
Full Time ◽  
Inertial Effects ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
The Mean

Oscillatory forcing of a porous medium may have a dramatic effect on the mean flow rate produced by a steady applied pressure gradient. The oscillatory forcing may excite nonlinear inertial effects leading to either enhancement or retardation of the mean flow. Here, in Part 1, we consider the effects of non-zero inertial forces on steady flows in porous media, and investigate the changes in the flow character arising from changes in both the strength of the inertial terms and the geometry of the medium. The steady-state Navier–Stokes equations are solved via a Galerkin finite element method to determine the velocity fields for simple two-dimensional models of porous media. Two geometric models are considered based on constricted channels and periodic arrays of circular cylinders. For both geometries, we observe solution multiplicity yielding both symmetric and asymmetric flow patterns. For the cylinder arrays, we demonstrate that inertial effects lead to anisotropy in the effective permeability, with the direction of minimum resistance dependent on the solid volume fraction. We identify nonlinear flow phenomena which might be exploited by oscillatory forcing to yield a net increase in the mean flow rate. In Part 2, we take up the subject of unsteady flows governed by the full time-dependent Navier–Stokes equations.

Sign in / Sign up

Export Citation Format

Share Document