A Vanka-based parameter-robust multigrid relaxation for the Stokes–Darcy Brinkman problems

We consider a block-structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element-based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka-based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two-grid and V(1,1)-cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter.     

Hydrodynamic forces and moments due to interaction of linear water waves with truncated partial-porous cylinders in finite depth

An exact analytical method is employed for studying the diffraction problems in an ocean due to the presence of a specific type of cylinders. In this current work, two models are studied: (i) a floating surface-piercing truncated partial-porous cylinder, (ii) a surface-piercing truncated partial-porous cylinder placed at the bottom. In both cases, the configuration of the composite cylinder is such that it consists of an impermeable inner cylinder rising above the free surface and a coaxial truncated porous cylinder around the lower part of the inner cylinder with the top of the porous cylinder being impermeable. By using linear water wave theory, a three-dimensional representation of the problem is developed based on eigenfunction expansion method. The condition on the porous boundary is defined by applying Darcy’s law. Pressure and velocity satisfy continuity conditions across the linear interface between the adjacent fluid domains. Hydrodynamic force, moment and wave run-up are calculated by using the velocity potentials. Comparisons are carried out with results of wave diffraction by a floating and bottom-mounted compound cylinder, i.e., when the whole cylinder is non-porous. Handy agreements are observed from these comparisons. Through numerical tests, various experiments are carried out to investigate the impact of various parameters, such as porous coefficients, draft ratio, the ratio of inner and outer radii, the water depth etc., on hydrodynamic force, moment and wave run-up. The results clearly indicate that an appropriate optimal ratio for various parameters may be considered in designing practical ocean structures with minimum adverse hydrodynamic effect. The appearance of resonance in the results and role of porosity in mitigating resonance effect are explained. Proposal to select various appropriate parameters for the best possible effect is put forward.     

Physics-Informed Neural Networks for Solving Coupled Stokes–Darcy Equation

In this paper, a grid-free deep learning method based on a physics-informed neural network is proposed for solving coupled Stokes–Darcy equations with Bever–Joseph–Saffman interface conditions. This method has the advantage of avoiding grid generation and can greatly reduce the amount of computation when solving complex problems. Although original physical neural network algorithms have been used to solve many differential equations, we find that the direct use of physical neural networks to solve coupled Stokes–Darcy equations does not provide accurate solutions in some cases, such as rigid terms due to small parameters and interface discontinuity problems. In order to improve the approximation ability of a physics-informed neural network, we propose a loss-function-weighted function strategy, a parallel network structure strategy, and a local adaptive activation function strategy. In addition, the physical information neural network with an added strategy provides inspiration for solving other more complicated problems of multi-physical field coupling. Finally, the effectiveness of the proposed strategy is verified by numerical experiments.     

Studying the Darcy-Stefan problem on phase transition in a saturated porous soil

The Cauchy problem for the Darcy-Stefan model, which describes the process of freezing (thawing) of a saturated porous soil with allowance for liquid-phase filtration, is considered. The model includes the Darcy law, the equation of liquid-phase incompressibility, the equation of absence of solid-phase motion, the equation of energy balance in the porous soil-saturating continuous medium system, and also the Stefan condition and the condition of continuity of the normal components of the velocity field at the interface boundary. The existence of generalized solutions of the problem satisfying an additional condition of entropy nondecreasing in a thermomechanical system (i.e., the second law of thermodynamics) is proved by the method of the kinetic equation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 81–93, July–August, 2008.     

Convergence order estimates of the local discontinuous Galerkin method for instationary Darcy flow

We present a new a priori stability and convergence analysis for the local discontinuous Galerkin method applied to the instationary Darcy problem formulated on a d‐dimensional polytope with nonhomogeneous Neumann and Dirichlet boundary conditions. In addition to including a spatially and temporally varying permeability tensor into all estimates, the utilized analysis technique produces a convergence order result for the primary and the flux variables. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown, and our analysis provides some insights into the role played by this particular choice of stabilization. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1374–1394, 2017     

On the properties of compressible gas flow in a porous media

The flow of an adiabatic gas through a porous media is treated analytically for steady one- and two-dimensional flows. The effect on a compressible Darcy flow by inertia and Forchheimer terms is studied. Finally, wave solutions are found which exhibit a cut-off frequency and a phase shift between pressure and velocity of the gas, with the velocity lagging behind the pressure.Nomenclature A area of tube for one-dimensional flow - B drag coefficient associated with Forchheimer term - c speed of sound - M Mach number - p ^* gas pressure - p dimensionless gas pressure - s coordinate along the axis of tube - t ^* time variable - t dimensionless time variable - V^* gas velocity in the porous media - V dimensionless gas velocity Greek Letters ratio of specific heat capacities - phase angle between gas pressure and velocity for linear waves - parameter indicating the importance of the inertia term - viscosity - _p natural frequency of the porous media - ^* gas density - dimensionless gas density - parameter indicating the importance of the Forchheimer term - porosity of porous media - velocity potential - stream function     

Mixed finite element methods for generalized Forchheimer flow in porous media

Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single‐phase fluid in a porous medium in ?^d, d ≤ 3, subject to Forchhheimer's law—a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L^∞(J; L²(Ω)) and in L^∞(J; H(div; Ω)) are demonstrated for the pressure and momentum, respectively. Error estimates are also derived in L^∞(J; L^∞(Ω)) for the pressure. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Lattice gas simulation of Darcy flow with memory-free collisions

A lattice gas algorithm is proposed for the simulation of water flow in the unsaturated zone. Microscopic dynamics of a two-dimensional model system are defined. Up to four fluid particles occupy the sites of a square lattice. At each time step, the particles are sent to neighbouring sites according to probabilistic rules which depend on the permeability and the potential but not on the input velocities of the particles. On the macroscopic scale, the flow is described by a diffusion term and a Darcy term. Several extensions including higher dimension are discussed.List of Symbols c ⁽ⁿ⁾ constant in the definition of the rejection probabilityP forn = 1,2,3 particles at a site 0 c ⁽ⁿ⁾ 1 - D diffusion constant - D vertical extent of the system, measured in cells - E _i vector connecting a site to its neighbour in directioni - i direction of a nearest neighbour site,i = 1,..., 4 - j direction of a nearest neighbour site,j = 1,..., 4 - j mass transport (fluid flow),j = v - j ^x x-component of the flowj - k(x) spatial dependence of the permeability, user defined under the constraint 0 k 1 - k () the part of the permeability which depends on the degree of saturation (seek) - k ⁽ⁿ⁾ (x) effective permeability at a sitex that holdsn particles - L horizontal extent of the system, measured in cells - l _mac macroscopic length scale, e.g. one meter - l _mic microscopic length scale (one lattice constant) - m integer number of time steps - n (x) number of particles at the lattice sitex - N _A total number of particles on all A-sites - P probability for rejection of a randomly selected direction or set of directions - p arithmetic mean of the probability for a site to receive a particle from a particular neighbour (the average is taken over the four neighbours) - p _i ⁽ⁿ⁾ probability that one out ofn particles at a site is sent in directioni - p _ij ⁽²⁾ probability that the two particles at a site are sent in directionsi andj - t time - t _mac macroscopic time scale, e.g. one day - t _mic microscopic time scale (one time step) - v fluid velocity - x space vector, mostly two-dimensional:x = (x, y) - x horizontal component ofx - y vertical component ofx - quotient of microscopic and macroscopic time scales,t _mic /t _mac - quotient of microscopic and macroscopic length scales,l _mic /l _mac - _i p + _i is the probability that a particle is received from the neighbour atx +E _i - K(X, ) effective permeability,k =k(x)k () - correlation length - degree of saturation, used synonymously with density (homogeneous porosity) - ₀ value of a homogeneous particle density - ø(x) external potential (user defined), ø = ^gr + ^mat - ø(x) arithmetic mean of the external potential at the four sites surroundingx - ø _i external potential at the sitex +E _i - total potential, = ø + ^den - ^gr(x) gravitational potential - ^mat(x) matrix potential - ^den() density-dependent potential - _n potential depending on the occupation number - ⁽ⁿ⁾ (x) probability that sitex is occupied byn particles - ₀ ^{(n) (n)} in a system with homogeneous particle density - mac macroscopic - mic microscopic     

Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force

Effect of Lorentz forces on natural convection in a complex shaped cavity filled with nanoliquid immersed in porous medium is investigated by means of Control volume based finite element method (CVFEM). Non Darcy model is taken into account for porous media. The working fluid is Fe₃O₄ –water and its viscosity considered as function of magnetic field. Figures are illustrated for different values of Darcy number (Da), Fe₃O₄ -water volume fraction (?), Rayleigh (Ra) and Hartmann (Ha) numbers. Results depict that enhancing in Lorentz forces results in reduce in nanofluid motion and increase the thickness of thermal boundary. Convective heat transfer enhances with rise of Darcy number.