Optimal shape design for the time‐dependent Navier–Stokes flow

This paper is concerned with the problem of shape optimization of two‐dimensional flows governed by the time‐dependent Navier–Stokes equations. We derive the structures of shape gradients for time‐dependent cost functionals by using the state derivative and its associated adjoint state. Finally, we apply a gradient‐type algorithm to our problem, and numerical examples show that our theory is useful for practical purposes and the proposed algorithm is feasible in low Reynolds number flows. Copyright © 2007 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Projection‐based variational multiscale method for incompressible Navier–Stokes equations in time‐dependent domains

A variational multiscale method for computations of incompressible Navier–Stokes equations in time‐dependent domains is presented. The proposed scheme is a three‐scale variational multiscale method with a projection‐based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh‐dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Colocated schemes for the incompressible Navier–Stokes equations on non‐smooth grids for two‐dimensional problems

The accuracy of colocated finite volume schemes for the incompressible Navier–Stokes equations on non‐smooth curvilinear grids is investigated. A frequently used scheme is found to be quite inaccurate on non‐smooth grids. In an attempt to improve the accuracy on such grids, three other schemes are described and tested. Two of these are found to give satisfactory results. Copyright © 2000 John Wiley & Sons, Ltd.     

Multigrid solution of the incompressible Navier–Stokes equations for three‐dimensional recirculating flow: coupled and decoupled smoothers compared

A comparison of multigrid methods for solving the incompressible Navier–Stokes equations in three dimensions is presented. The continuous equations are discretised on staggered grids using a second‐order monotonic scheme for the convective terms and implemented in defect correction form. The convergence characteristics of a decoupled method (SIMPLE) are compared with those of the cellwise coupled method (SCGS). The convergence rates obtained for computations of the three‐dimensional lid‐driven cavity problem are found to be very similar to those obtained for computations of the corresponding two‐dimensional problem with comparable grid density. Although the convergence rate of SCGS is thus superior to that of SIMPLE, the decoupled method is found to be more efficient computationally and requires less computing time for a given level of convergence. The linewise implementation of the coupled method (CLGS) is also investigated and shown to be more efficient than SCGS, although the convergence rate and computing time required per cycle are both found to depend on the direction of sweep. The optimal implementation of CLGS is found to be only marginally more effective than SIMPLE, but a change to the structure of the data storage would increase the advantage. Copyright © 1999 John Wiley & Sons, Ltd.     

Effect of magnetic Reynolds number on the two‐dimensional hydromagnetic flow around a cylinder

Numerical experiments have been conducted to study the effect of magnetic Reynolds number on the steady, two‐dimensional, viscous, incompressible and electrically conducting flow around a circular cylinder. Besides usual Reynolds number Re, the flow is governed by the magnetic Reynolds number R_m and Alfvén number β. The flow and magnetic field are uniform and parallel at large distances from the cylinder. The pressure Poisson equation is solved to find the pressure fields in the entire flow region. The effects of the magnetic field and electrical conductivity on the recirculation bubble, drag coefficient, standing vortex and pressure are presented and discussed. For low interaction parameter (N<1), the suppression of the flow‐separation is nearly independent of the conductivity of the fluid, whereas for large interaction parameters, the conductivity of the fluid strongly influences the control of flow‐separation. Copyright © 2008 John Wiley & Sons, Ltd.     

Anisotropic mesh adaption for time‐dependent problems

We propose a space–time adaptive procedure for a model parabolic problem based on a theoretically sound anisotropic a posteriori error analysis. A space–time finite element scheme (continuous in space but discontinuous in time) is employed to discretize this problem, thus allowing for non‐matching meshes at different time levels. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of liquid crystalline polymer flow into two‐dimensional thin cavity moulds

In this paper, flows of liquid crystalline polymers into two‐dimensional thin cavity moulds are simulated. The flows are modelled by Ericksen–Leslie equations of motion in the high viscosity limit. An elliptic pressure equation is derived under Hele–Shaw approximations, and the non‐isothermal natures of the flow are modelled. The equations are solved using the finite‐difference technique. A new boundary‐mapping technique is developed in this study to solve the difficulty in the finite‐difference treatment of arbitrarily shaped boundaries, which possess no natural coordinate system. This new method avoids the difficult mesh control in the body‐fitted mapping process and makes the mapping process easy to implement. It can also solve the problems caused by the uneven distribution of grid nodes in the traditional body‐fitted mapping technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of high‐Reynolds number flow around circular cylinders by a three‐step FEM–BEM model

An innovative computational model, developed to simulate high‐Reynolds number flow past circular cylinders in two‐dimensional incompressible viscous flows in external flow fields is described in this paper. The model, based on transient Navier–Stokes equations, can solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The pressure is assumed to be zero at infinite boundary and the external flow field is simulated using a direct boundary element method (BEM) by solving a pressure Poisson equation. A three‐step finite element method (FEM) is used to solve the momentum equations of the flow. The present model is applied to simulate high‐Reynolds number flow past a single circular cylinder and flow past two cylinders in which one acts as a control cylinder. The simulation results are compared with experimental data and other numerical models and are found to be feasible and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A nodal‐based implementation of a stabilized finite element method for incompressible flow problems

The objective of this paper is twofold. First, a stabilized finite element method (FEM) for the incompressible Navier–Stokes is presented and several numerical experiments are conducted to check its performance. This method is capable of dealing with all the instabilities that the standard Galerkin method presents, namely the pressure instability, the instability arising in convection‐dominated situations and the less popular instabilities found when the Navier–Stokes equations have a dominant Coriolis force or when there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal‐based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements, thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this gives rise to questions regarding the consistency and the conservation properties of the final scheme, which are addressed in this paper. Copyright © 2000 John Wiley & Sons, Ltd.     

Derivation and characteristics analysis of an acoustics–convection upstream resolution algorithm for the two‐dimensional Euler and Navier–Stokes equations

The first of a two‐paper series, this paper introduces a new decomposition not of the hyperbolic flux vector but of the flux vector Jacobian. The paper then details for the Euler and Navier–Stokes equations an intrinsically infinite directional upstream‐bias formulation that rests on the mathematics and physics of multi‐dimensional acoustics and convection. Based upon characteristic velocities, this formulation introduces the upstream bias directly at the differential equation level, before the spatial discretization, within a characteristics‐bias governing system. Through a decomposition of the Euler flux divergence into multi‐dimensional acoustics and convection–acoustics components, this characteristics‐bias system induces consistent upstream bias along all directions of spatial wave propagation, with anisotropic variable‐strength upstreaming that correlates with the spatial distribution of characteristic velocities. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel solution of three‐dimensional Marangoni flow in liquid bridges

This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three‐dimensional and time‐dependent Navier–Stokes equations on distributed‐memory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non‐linear Navier–Stokes problem, governing the Marangoni flows behavior, can effectively be parallelized on a distributed memory parallel machine by remapping the distributed data structure. The numerical code is based on a three‐dimensional Simplified Marker and Cell (SMAC) primitive variable method applied to a staggered finite difference grid. Using this method, the problem is split into two problems, one parabolic and the other elliptic A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo pressure elliptic equation, representing the most time‐consuming part of the algorithm. A grid‐partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed‐Up is ever smaller than the ideal linear Speed‐Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier–Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency. Copyright © 1999 John Wiley & Sons, Ltd.     

SIMPLE‐type preconditioners for cell‐centered,colocated finite volume discretization of incompressible Reynolds‐averaged Navier–Stokes equations

This paper contains a comparison of four SIMPLE‐type methods used as solver and as preconditioner for the iterative solution of the (Reynolds‐averaged) Navier–Stokes equations, discretized with a finite volume method for cell‐centered, colocated variables on unstructured grids. A matrix‐free implementation is presented, and special attention is given to the treatment of the stabilization matrix to maintain a compact stencil suitable for unstructured grids. We find SIMPLER preconditioning to be robust and efficient for academic test cases and industrial test cases. Compared with the classical SIMPLE solver, SIMPLER preconditioning reduces the number of nonlinear iterations by a factor 5–20 and the CPU time by a factor 2–5 depending on the case. The flow around a ship hull at Reynolds number 2E9, for example, on a grid with cell aspect ratio up to 1:1E6, can be computed in 3 instead of 15 h.Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulation of flapping‐wing insect hovering flight at unsteady flow

A computational fluid dynamics (CFD) analysis was conducted to study the unsteady aerodynamics of a virtual flying bumblebee during hovering flight. The integrated geometry of bumblebee was established to define the shape of a three‐dimensional virtual bumblebee model with beating its wings, accurately mimicking the three‐dimensional movements of wings during hovering flight. The kinematics data of wings documented from the measurement to the bumblebee in normal hovering flight aided by the high‐speed video. The Navier–Stokes equations are solved numerically. The solution provides the flow and pressure fields, from which the aerodynamic forces and vorticity wake structure are obtained. Insights into the unsteady aerodynamic force generation process are gained from the force and flow‐structure information. The CFD analysis has established an overall understanding of the viscous and unsteady flow around the virtual flying bumblebee and of the time course of instantaneous force production, which reveals that hovering flight is dominated by the unsteady aerodynamics of both the instantaneous dynamics and also the past history of the wing. A coherent leading‐edge vortex with axial flow and the attached wingtip vortex and trailing edge vortex were detected. The leading edge vortex, wing tip vortex and trailing edge vortex, which caused by the pressure difference between the upper and the lower surface of wings. The axial flow, which include the spanwise flow and chordwise flow, is derived from the spanwise pressure gradient and chordwise pressure gradient, will stabilize the vortex and gives it a characteristic spiral conical shape. Copyright © 2006 John Wiley & Sons, Ltd.     

A locally DIV‐free projection scheme for incompressible flows based on non‐conforming finite elements

We present a projection scheme whose end‐of‐step velocity is locally pointwise divergence free, using a continuous ?₁ approximation for the velocity in the momentum equation, a first‐order Crouzeix–Raviart approximation at the projection step, and a ?₀ approximation for the pressure in both steps. The analysis of the scheme is done only for grids that guarantee the existence of a divergence free conforming ?₁ interpolant for the velocity. Optimal estimates for the velocity error in L²‐ and H¹‐norms are deduced. The numerical results demonstrate that these estimates should also hold on grids on which the continuous ?₁ approximation for the velocity locks. Since the end‐of‐step velocity is locally solenoidal, the scheme is recommendable for problems requiring good mass conservation. Copyright © 2005 John Wiley & Sons, Ltd.     

Two‐level consistent splitting methods based on three corrections for the time‐dependent Navier–Stokes equations

Three kinds of two‐level consistent splitting algorithms for the time‐dependent Navier–Stokes equations are discussed. The basic technique of two‐level type methods for solving the nonlinear problem is first to solve a nonlinear problem in a coarse‐level subspace, then to solve a linear equation in a fine‐level subspace. Hence, the two‐level methods can save a lot of work compared with the one‐level methods. The approaches to linearization are considered based on Stokes, Newton, and Oseen corrections. The stability and convergence demonstrate that the two‐level methods can acquire the optimal accuracy with the proper choice of the coarse and fine mesh scales. Numerical examples show that Stokes correction is the simplest, Newton correction has the best accuracy, while Oseen correction is preferable for the large Reynolds number problems and the long‐time simulations among the three methods. Copyright © 2015 John Wiley & Sons, Ltd.