A fast immersed interface method for solving Stokes flows on irregular domains

We present a fast immersed interface method for solving the steady Stokes flows involving the rigid boundaries. The immersed rigid boundary is represented by a set of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundary, singular forces at the rigid boundary are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are approximated using the cubic splines. The strength of singular forces is determined by solving a small system of equations via the GMRES method. The Stokes equations are discretized using finite difference method with the incorporation of jump conditions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate Stokes flows on irregular domains.     

Finite Difference Solutions of Incompressible Flow Problems with Corner Singularities

Two problems that include corner singularities are considered. The first concerns the flow of a viscous fluid in a channel driven by a constant pressure gradient, when the velocity satisfies a two-dimensional Poisson equation. The second is Stokes flow in a two-dimensional region when the stream-function satisfies the biharmonic equation. For both problems the boundaries of the domains contain corners. For corner angles greater than some critical value, the stress or the vorticity is singular. Using both a formal analysis and numerical results, we show that numerical approximations for the stream-function and velocity, obtained by using standard second-order finite difference methods, still converge to the exact solutions despite the corner singularities. However, the convergence rate is lower than second-order and the deterioration in the accuracy is not local, i.e., not confined to the corner. On the other hand, even though the vorticity solution of the Stokes problem does not converge, it diverges only locally. At a finite distance from the corner, the vorticity converges with the same rate as the stream-function. Adaptive methods for improving the accuracy are also discussed.     

A simple implementation of the immersed interface methods for Stokes flows with singular forces

In this paper, we introduce a simple version of the immersed interface method (IIM) for Stokes flows with singular forces along an interface. The numerical method is based on applying the Taylor’s expansions along the normal direction and incorporating the solution and its normal derivative jumps into the finite difference approximations. The fluid variables are solved in a staggered grid, and a new accurate interpolating scheme for the non-smooth velocity has been developed. The numerical results show that the scheme is second-order accurate.     

Development of a meshless Galerkin boundary node method for viscous fluid flows

In this paper, a meshless Galerkin boundary node method is developed for boundary-only analysis of the interior and exterior incompressible viscous fluid flows, governed by the Stokes equations, in biharmonic stream function formulation. This method combines scattered points and boundary integral equations. Some of the novel features of this meshless scheme are boundary conditions can be enforced directly and easily despite the meshless shape functions lack the delta function property, and system matrices are symmetric and positive definite. The error analysis and convergence study of both velocity and pressure are presented in Sobolev spaces. The performance of this approach is illustrated and assessed through some numerical examples.     

First vorticity–velocity–pressure numerical scheme for the Stokes problem

We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.     

An augmented approach for Stokes equations with a discontinuous viscosity and singular forces

For Stokes equations with a discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, it is known that the pressure is discontinuous and the velocity is non-smooth. It has been shown that these discontinuities are coupled together, which makes it difficult to obtain accurate numerical solutions. In this paper, a new numerical method that decouples the jump conditions of the fluid variables through two augmented variables has been developed. The GMRES iterative method is used to solve the Schur complement system for the augmented variables that are only defined on the interface. The augmented approach also rescales the Stokes equations in such a way that a fast Poisson solver can be used in each iteration. Numerical tests using examples that have analytic solutions show that the new method has average second order accuracy for the velocity in the infinity norm. An example of a moving interface problem is also presented.     

Pressure condition for lattice Boltzmann methods on domains with curved boundaries

We propose a lattice Boltzmann algorithm for an average pressure boundary condition at outlets in pipe flow systems. The advantage of this boundary condition is that only the average pressure is used to recover the non-trivial flow fields. The asymptotic analysis shows that this algorithm works for general curved boundaries and renders a second order accurate velocity and a first order accurate pressure approximation of the incompressible Navier–Stokes solution. Here, we verify the accuracy by numerical simulations of a Poiseuille flow and a less symmetric flow with non-trivial pressure field in channels inclined with arbitrary angle, and flows in a pipe with three outlets.     

An Adaptive SDG Method for the Stokes System

Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance.     

Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.     

Analytical solution of gas lubricated slider microbearing

The analytical solution of a two-dimensional, isothermal, compressible gas flow in a slider microbearing is presented. A higher order accuracy of the solution is achieved by applying the boundary condition of Kn ² order for the velocity slip on the wall, together with the momentum equation of the same order (known as the Burnett equation). The analytical solution is obtained by the perturbation analysis. The order of all terms in continuum and momentum equations and in boundary conditions is evaluated by incorporating the exact relation between the Mach, Reynolds and Knudsen numbers in the modelling procedure. Low Mach number flows in microbearing with slowly varying cross-sections are considered, and it is shown that under these conditions the Burnett equation has the same form as the Navier–Stokes equation. Obtained analytical results for pressure distribution, load capacity and velocity field are compared with numerical solutions of the Boltzmann equation and some semi-analytical results, and excellent agreement is achieved. The model presented in this paper is a useful tool for the prediction of flow conditions in the microbearings. Also, its results are the benchmark test for the verifications of various numerical procedures.     

A finite volume method to solve the 3D Navier–Stokes equations on unstructured collocated meshes

A new method to solve the Navier–Stokes equations for incompressible viscous flows and the transport of a scalar quantity is proposed. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circumcenters. Theoretical results and numerical properties of the scheme are provided. Predictions of lid-driven cavity flow, flows past a cylinder and heat transport in a cylinder are performed to validate the method.     

On the Development of a Nonprimitive Navier–Stokes Formulation Subject to Rigorous Implementation of a New Vorticity Integral Condition

In this paper, a new integral vorticity boundary condition has been developed and implemented to compute solution of nonprimitive Navier–Stokes equation. Global integral vorticity condition which is of primitive character can be considered to be of entirely different kind compared to other vorticity conditions that are used for computation in literature. The procedure realized as explicit boundary vorticity conditions imitates the original integral equation. The main purpose of this paper is to design an algorithm which is easy to implement and versatile. This algorithm based on the new vorticity integral condition captures accurate vorticity distribution on the boundary of computational flow field and can be used for both wall bounded flows as well as flows in open domain. The approach has been arrived at without utilizing any ghost grid point outside of the computational domain. Convergence analysis of this alternative vorticity integral condition in combination with semi-discrete centered difference approximation of linear Stokes equation has been carried out. We have also computed correct pressure field near the wall, for both attached and separated boundary layer flows, by using streamfunction and vorticity field variables. The competency of the proposed boundary methodology vis-a-vis other popular vorticity boundary conditions has been amply appraised by its use in a model problem that embodies the essential features of the incompressibility and viscosity. Subsequently the proposed methodology has been further validated by computing analytical solution of steady Stokes equation. Finally, it has been applied to three benchmark problems governed by the incompressible Navier–Stokes equations, viz. lid driven cavity, backward facing step and flow past a circular cylinder. The results obtained are in excellent agreement with computational and experimental results available in literature, thereby establishing efficiency and accuracy of the proposed algorithm. We were able to accurately predict both vorticity and pressure fields.     

A multi-level boundary element method for Stokes flows in irregular two-dimensional domains

Recently, we have developed multi-level boundary element methods (MLBEM) for the solution of the Laplace and Helmholtz equations that involve asymptotically decaying non-oscillatory and oscillatory singular kernels, respectively. The accuracy and efficiency of the fast boundary element methods for steady-state heat diffusion and acoustics problems have been investigated for square domains. The current work extends the MLBEM methodology to the solution of Stokes equation in more complex two-dimensional domains. The performance of the fast boundary element method for the Stokes flows is first investigated for a model problem in a unit square. Then, we consider an example problem possessing an analytical solution in a rectangular domain with 5:1 aspect ratio, and finally, we study the performance of the MLBEM algorithm in a C-shaped domain.     

Performances of upwind methods in predicting shear-like flows

Upwind methods are considered as the most appropriate numerical tools for predicting high speed flows. However, disturbing problems may arise in dealing with shear-like flows such as boundary layers. Here, the fluid dynamics is dominated by the diffusion processes and the convective terms, which are estimated through an upwind method, have to play a secondary and almost negligible role. Unfortunately, the presence of density and velocity gradients inside boundary layers introduces, in some upwind method, purely numerical effects. In some methods, the boundary layer region may grow, unphysically, beyond the correct thickness because of a spurious dissipation generated by an incorrect evaluation of the convective terms. In other methods, pressure oscillations may appear inside the boundary layer. Such situations, well known in the scientific community, can be properly analyzed by developing a linearized form of the first order scheme for the Euler conservation laws in a particular problem, with the evaluation of the fluxes made as dictated by the formulation of each upwind method. This analysis provides suggestions and hints useful to understand and predict the behavior of an upwind method in simulating shear-like flows.     

Electroosmotic flow control in complex microgeometries

Numerical simulation results for pure electroosmotic and combined electroosmotic/pressure driven Stokes flows are presented in the cross-flow and Y-split junctions. The numerical algorithm is based on a mixed structured/unstructured spectral element formulation, which results in high-order accurate resolution of thin electric double layers with discretization flexibility for complex engineering geometries. The results for pure electroosmotic flows in cross-flow junctions under multiple electric fields show similarities between the electric and velocity fields. The combined electroosmotic/pressure driven flows are also simulated by regulating the flowrate in different branches of the cross-flow junctions. Flow control in the Stokes flow regime is shown to have linear dependence on the magnitude of the externally applied electric field, both for pure electroosmotic and combined flows. This linear behavior enables utilization of electroosmotic forces as nonmechanical means of flow control for microfluidic applications     

Hybrid atomistic�Ccontinuum simulations of fluid flows involving interfaces

In this work, we use an hybrid atomistic–continuum (HAC) simulation method to study transient and steady isothermal flows of Lennard-Jones fluids near interfaces. Our hybrid method is based on a domain decomposition algorithm. The flow domain is composed of two overlapping regions: an atomistic region described by molecular dynamics, and a continuum region described by a finite volume discretization of the incompressible Navier–Stokes equations. To show the interest of such an hybrid method to compute flows near fluid/solid interface, we first applied our hybrid scheme to the classical Couette flow, where the moving wall is modelled at the atomistic scale. In addition, we also studied an oscillatory shear flow. Then, to compute flows near fluid/fluid interface, we applied our method to a two-phase Couette flow (liquid/gas), where the interface is modelled at the molecular scale. We show that hybrid results can sometimes differ from those provided by analytical solutions deduced from continuum mechanics equations combined with usual boundary/interface relations. For the Couette and oscillatory shear flows, a good agreement is found between hybrid simulations and macroscopic analytical solutions, however, we noticed that the fluid in contact with the wall can be more entailed than what expected. For the liquid/gas Couette flow, the hybrid simulation exhibits an unexpected jump of the velocity in the interfacial region, corresponding to a partial slip between the two fluid phases. Those interesting results highlight the interest of using an HAC method to deal with systems for which surfaces/interfaces effects are important.     

Gravity capillary waves generated by a moving pressure distribution in the presence of constant vorticity and dissipation

Two-dimensional nonlinear free-surface flows due to a pressure distribution moving at a constant velocity at the surface of a fluid of infinite depth are considered. The effects of the gravity and of the surface tension are included in the dynamic boundary condition. The vorticity in the fluid is assumed to be constant. The dissipation is modelled by a quasi potential approximation. The problem is solved numerically by a boundary integral equation method and numerical solutions are presented. The results unify previous findings for irrotational gravity capillary waves, waves in the presence of constant vorticity and free surface flows with dissipation.     

Study of subsonic�Csupersonic gas flow through micro/nanoscale nozzles using unstructured DSMC solver

We use an extended direct simulation Monte Carlo (DSMC) method, applicable to unstructured meshes, to numerically simulate a wide range of rarefaction regimes from subsonic to supersonic flows through micro/nanoscale converging–diverging nozzles. Our unstructured DSMC method considers a uniform distribution of particles, employs proper subcell geometry, and follows an appropriate particle tracking algorithm. Using the unstructured DSMC, we study the effects of back pressure, gas/surface interactions (diffuse/specular reflections), and Knudsen number on the flow field in micro/nanoscale nozzles. If we apply the back pressure at the nozzle outlet, a boundary layer separation occurs before the outlet and a region with reverse flow appears inside the boundary layer. Meanwhile, the core region of inviscid flow experiences multiple shock-expansion waves. In order to accurately simulate the outflow, we extend a buffer zone at the nozzle outlet. We show that a high viscous force creation in the wall boundary layer prevents any supersonic flow formation in the divergent part of the nozzle if the Knudsen number exceeds a moderate magnitude. We also show that the wall boundary layer prevents forming any normal shock in the divergent part. In reality, Mach cores would appear at the nozzle center followed by bow shocks and expansion region. We compare the current DSMC results with the solution of the Navier–Stokes equations subject to the velocity slip and temperature jump boundary conditions. We use OpenFOAM as a compressible flow solver to treat the Navier–Stokes equations.     

Smoothed surface gradients for panel methods

A weak form to compute the dipolar and monopolar surface gradients, related to a low-order panel method, is shown. The flow problem is formulated by means of a three-dimensional potential model and the discretization is based on Morino's formulation for the perturbation velocity potential. On the body surface, this representation reduces to a boundary integral equation with the source (or monopolar) and the doublet (or dipolar) densities. The first of the two is found by application of the boundary flow condition, and the second one is the unknown over the body surface. A lower panel method is used for the analytic integrations of both the monopolar and dipolar influence coefficients. The surface velocity field is computed after solving the linear system, with a strong and a weak form of the Stokes theorem, which is oriented to fairly non-structured panel meshes. The proposed method is validated by comparing the numerical results with analytical ones for an isolated sphere and includes a prediction over a car-like configuration.