Flows In Irregular Domains Research Articles

Modified diffuse interface fluid model and its consistent energy-stable computation in arbitrary domains

Binary fluid flows in irregular domains are common phenomena in natural and industrial fields. To describe these physical processes, we herein present a novel fluid flow-coupled two-material model reflecting different wetting conditions on an arbitrary fluid-solid interface. A modified ternary Cahn–Hilliard (CH) diffuse interface model is adopted to capture the fluid-fluid interface. One of three components is fixed for all time to represent the profile of complex domains. By considering the contact angle condition on solid-fluid interface and equilibrium interface assumption, an extra wetting term is derived. To treat the fluid flow in domains with solid obstacles, we consider the entire system as a porous medium with variable permeability and add a large penalty term into the incompressible Navier–Stokes (NS) equations. It can be proven that the proposed modified Cahn–Hilliard–Navier–Stokes (CHNS) system satisfies the energy dissipation law (energy stability). Based on the scalar auxiliary variable (SAV) approach with appropriate correction techniques, the linear and consistent energy-stable scheme is developed. We introduce the numerical implementation and estimate the discrete version of energy stability in detail. The proposed method can be implemented on regular Cartesian grids with the absence of explicit boundary treatment. Moreover, the calculations are totally decoupled in each time step. Extensive numerical experiments not only indicate the expected accuracy and stability but also show the superior performance in arbitrary domains with different wetting conditions.

A coupled volume of fluid and level set method based on analytic PLIC for unstructured quadrilateral grids

We develop a coupled volume of fluid and level set (VOSET) method for unstructured quadrilateral grids to simulate incompressible two-phase interfacial flows in irregular domains. In the method, an analytic piecewise linear interface calculation (PLIC) is first proposed and its solution speed is much faster (about five times) than that of Brent's iterative method; then an iterative geometric operation by which the level set function near interfaces can be calculated, is extended to unstructured quadrilateral grids; moreover, the volume fraction advection is solved by a Lagrangian-Eulerian advection scheme. Finally, the present method is validated by the single vortex flow problem, bubble rising problem and falling droplet problem. Our simulation results show good agreement with those in previous studies.

On two-phase flow solvers in irregular domains with contact line

We present numerical methods that enable the direct numerical simulation of two-phase flows in irregular domains. A method is presented to account for surface tension effects in a mesh cell containing a triple line between the liquid, gas and solid phases. Our numerical method is based on the level-set method to capture the liquid–gas interface and on the single-phase Navier–Stokes solver in irregular domain proposed in [35] to impose the solid boundary in an Eulerian framework. We also present a strategy for the implicit treatment of the viscous term and how to impose both a Neumann boundary condition and a jump condition when solving for the pressure field. Special care is given on how to take into account the contact angle, the no-slip boundary condition for the velocity field and the volume forces. Finally, we present numerical results in two and three spatial dimensions evaluating our simulations with several benchmarks.

A Barycentric Interpolation Collocation Method for Darcy Flow in Two-Dimension

A meshless, barycentric interpolation collocation method for numerical approximation of Darcy flows is proposed. The barycentric Lagrange interpolation and its differentiation matrices are basic tool to discretize governing equations, Dirichlet and Neumann boundary conditions. For Darcy flows in irregular domains, embedding the irregular domain into a rectangular, the barycentric interpolation collocation method can be directly applied. The resultant saddle-point systems come from combining the discretized governing equations and boundary conditions, such that we can deal easy with all kinds of boundary condition either regular or irregular domains. Some numerical examples are given to illustrate the accuracy, stability and robust of presented method.

Domain Decomposition for Stokes-Darcy Flows with Curved Interfaces

A non-overlapping domain decomposition method is developed for coupled Stokes-Darcy flows in irregular domains. The Stokes region is discretized by standard Stokes finite elements while the Darcy region is discretized by the multipoint flux mixed finite element method. The subdomain grids may not match on the interfaces and mortar finite elements are employed to impose weakly interface continuity conditions. The interfaces can be curved and matching conditions are imposed via appropriate mappings from physical grids to reference grids with flat interfaces. The global problem is reduced to a mortar interface problem, which is solved by the conjugate gradient method. Each iteration involves solving subdomain problems of either Stokes or Darcy type, which is done in parallel. Computational experiments are presented to illustrate the convergence of the discretization and the condition number of the interface operator.

A novel Cartesian cut cell solver for incompressible viscous flow in irregular domains

A Cartesian cut cell solver with solution-based adaptive mesh refinement is developed for simulating viscous, incompressible flows with arbitrary complex geometries. The cut cells are automatically generated using Volume CAD (VCAD), a framework for storing geometric and material attribute data. Unlike earlier cut cell methods, this solver organizes the cutting patterns into only six categories and further subdivides the resulting pentagon into two quadrilaterals, such that mesh data can be stored by uniform data structure and the post-processing of flow data can be handled conveniently. A novel method is proposed to treat minuscule cut cells without the process of cell merging. A collocated finite volume method, which can be used even when multiple cell shapes and orthogonal and non-orthogonal grids exist in the decomposition, is employed to discretize the Navier–Stokes equations. A modified SIMPLE-based smoothing pressure correction scheme is applied in this cut cell method to suppress checkerboard pressure oscillations caused by collocated arrangement. The solver is first used to simulate a channel flow to demonstrate its calculation accuracy expressed with L1 and L∞ norm errors and then the method is utilized to solve three benchmark problems of flow and heat transfer within irregular domains to verify its feasibility, efficiency, accuracy and potential in engineering applications. Copyright © 2010 John Wiley & Sons, Ltd.

A level set‐based immersed interface method for solving incompressible viscous flows with the prescribed velocity at the boundary

AbstractA second‐order accurate immersed interface method (IIM) is presented for solving the incompressible Navier–Stokes equations with the prescribed velocity at the boundary, which is an extension of the IIM of Le et al. (J. Comput. Phys. 2006; 220:109–138) to a level set representation of the boundary in place of the Lagrangian representation of the boundary using control points on a uniform Cartesian grid. In order to enforce the prescribed velocity boundary condition, the singular forces at the immersed boundary are applied on the fluid. These forces are related to the jump in pressure and the jumps in the derivatives of both the pressure and velocity, and are approximated via using the local Hermite cubic spline interpolation. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier–Stokes equations are discretized via using finite difference method with the incorporation of jump conditions on a staggered Cartesian grid and solved by a second‐order accurate projection method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate the viscous flows in irregular domains. Copyright © 2009 John Wiley & Sons, Ltd.

An implicit-forcing immersed boundary method for simulating viscous flows in irregular domains

We present a method for solving the incompressible Navier–Stokes equations in irregular domains. These equations are discretized using finite difference method in a uniform Cartesian grid. Stationary rigid boundaries are embedded in the Cartesian grid and singular forces are applied at the rigid boundaries to impose the no-slip conditions. The singular forces are then distributed to the nearby Cartesian grid points using the immersed boundary method. In the present work, the singular forces are computed implicitly by solving a small system of equations at each time step. This system of equations is derived from a second order projection method. The main advantage of this method is that it imposes the no-slip boundary condition exactly and avoids the need for small time step to maintain stability. The ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2-dimensional flows past a circular cylinder, multiple rigid obstacles and 3-dimensional flow past a sphere.

Dense particulate flow model on unstructured mesh

A numerical method based on two-dimensional (2D) unstructured meshes is developed to solve the discrete particle model (DPM). Inter-particle interactions are taken into account for dense particulate flows, which are described by binary collisions in a hard-sphere model. The particle volume fraction is calculated accurately and physical scalars from the Eulerian grid to the Lagrangian particle positions are mapped through gradient interpolations. The governing equations for the continuous phase are discretized using a finite volume method on an unstructured grid and solved by the algebraic multi-grid (AMG) method. The SIMPLE algorithm employed to solve single-phase flows on unstructured meshes is extended to the pressure–velocity equations. Momentum coupling between the two phases is strongly implicit resulting in a very robust convergence of the AMG solver. Data structuring and mapping techniques for further enhancement of the flexibility and computational efficiency of the numerical model are introduced. Several test cases confirm that the numerical method can be applied to gas–solid and gas–liquid flows in irregular domains without regard to element types of the mesh. The numerical model presented in this paper partly overcomes the difficulties in simulating dense particulate flows using the DPM in 2D irregular domains.

Dynamic Subgrid-Scale Modeling for Large-Eddy Simulations in Complex Topologies

The dynamic eddy-viscosity relationship is a suitable choice for modeling the subgrid-scales (SGS) in a large-eddy simulation (LES) of complex turbulent flows in irregular domains. This algebraic relationship is easy to implement and its dynamic coefficient will give negligible turbulent viscosity contributions in the flow regions that are irrotational or laminar. Its fine-scale turbulence predictions can be qualitatively reasonable if the local grid resolution maintains the SGS field predominantly within the equilibrium range of turbulent energy spectra. This performance is given herein by two curvilinear coordinate forms of the dynamic Smagorinsky model that are formally derived and a-priori tested using the resolved physics of the cylinder wake. The conservative form evaluates the dynamic coefficient in the computational (transformed) space whereas its non-conservative counterpart operates in the physical domain. Although both forms equally captured the real normal SGS stress reasonably well, the real shear stress and dissipation rates were severely under-predicted. Mixing the eddy-viscosity choice with a scale-similarity model can ease this latter deficiency.

Simulation of non-hydrostatic, density-stratified flow in irregular domains

A numerical model has been developed for simulating density-stratified flow in domains with irregular but simple topography. The model was designed for simulating strong interactions between internal gravity waves and topography, e.g. exchange flows in contracting channels, tidally or convectively driven flow over two-dimensional sills or waves propagating onto a shoaling bed. The model is based on the non-hydrostatic, Boussinesq equations of motion for a continuously stratified fluid in a rotating frame, subject to user-configurable boundary conditions. An orthogonal boundary fitting co-ordinate system is used for the numerical computations, which rely on a fourth-order compact differentiation scheme, a third-order explicit time stepping and a multi-grid based pressure projection algorithm. The numerical techniques are described and a suite of validation studies are presented. The validation studies include a pointwise comparison of numerical simulations with both analytical solutions and laboratory measurements of non-linear solitary wave propagation. Simulation results for flows lacking analytical or laboratory data are analysed a posteriori to demonstrate satisfaction of the potential energy balance. Computational results are compared with two-layer hydraulic predictions in the case of exchange flow through a contracting channel. Finally, a simulation of circulation driven by spatially non-uniform surface buoyancy flux in an irregular basin is discussed.