Unsteady Incompressible Viscous Flow Research Articles

Mass transfer effects on vertical oscillating plate with heat flux

Theoretical solution of unsteady viscous incompressible flow past an infinite vertical oscillating plate with uniform heat flux and mass diffusion is presented here, taking into account of the homogeneous chemical reaction of first-order. The temperature from the plate to the fluid at an uniform rate and the mass is diffused uniformly. The dimensionless governing equations has been obtained by the Laplace transform method, when the plate is oscillating harmonically in its own plane. The effects of velocity and concentration are studied for different parameters like phase angle chemical reaction parameter, thermal Grashof number, mass Grashof number Schmidt number and time are studied. The so?lutions are valid only for small values of time t. It is observed that the velocity increases with decreasing phase angle ?t or chemical reaction parameter.

The development of a Cartesian cut cell method for incompressible viscous flows

AbstractThis paper describes the extension of the Cartesian cut cell method to applications involving unsteady incompressible viscous fluid flow. The underlying scheme is based on the solution of the full Navier–Stokes equations for a variable density fluid system using the artificial compressibility technique together with a Jameson‐type dual time iteration. The computational domain encompasses two fluid regions and the interface between them is treated as a contact discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures. The Cartesian cut cell technique is used for fitting the complex geometry of solid boundaries across a stationary background Cartesian grid which is located inside the computational domain. A time accurate solution is achieved by using an implicit dual‐time iteration technique based on a slope‐limited, high‐order, Godunov‐type scheme for the inviscid fluxes, while the viscous fluxes are estimated using central differencing. Validation of the new technique is by modelling the unsteady Couette flow and the Rayleigh–Taylor instability problems. Finally, a test case for wave run‐up and overtopping over an impermeable sea dike is performed. Copyright © 2006 John Wiley & Sons, Ltd.

Applications of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397–420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder.

A Modified Streamline Fractional Step Finite Element Method for Solving Unsteady Incompressible Viscous Flow

This paper proposes a numerical method for solving the unsteady incompressible Navier-Stokes equation at high Reynolds number by a modified fractional step finite element method. This method is based upon the velocity correction method and uses two concepts to conduct intermediate velocity. The first is the streamline method, which provides an accurate multidimensional generalization, and the second is the balancing tensor diffusivity, which is used as the artificial diffusion for the stabilization techniques. The accuracy of this method for the advection-diffusion equation is demonstrated for the rotating cone problem. The unsteady incompressible viscous flows, such as square cavity flow at Re ≤10000 and flow past a circlar cylinder at Re ≤2000, are simulated without any numerical instability.

Skin-friction for unsteady free convection MHD flow between two heated vertical parallel plates

Unsteady viscous incompressible free convection flow of an electrically conducting fluid between two heated vertical parallel plates is considered in the presence of a uniform magnetic field applied transversely to the flow. The induce field along the lines of motion varies transversely to the flow and the fluid temperature changing with time. An analytical solution for velocity, induced field and the temperature distributions are obtained for small and large magnetic Reynolds numbers. The skin-friction at the two plates is obtained. Velocity distribution, induced field and skin-friction are plotted against the distance from the plates. It has been observed that with the increase in Rm, the magnetic Reynolds number, at constant M, the Hartmann number, leads to an increase in the skin-friction gradually. But with the increase in M, at constant Rm, the skin-friction decreases.

A consistent splitting scheme for unsteady incompressible viscous flows I. Dirichlet boundary condition and applications

AbstractA well‐recognized approach for handling the incompressibility constraint by operating directly on the discretized Navier–Stokes equations is used to obtain the decoupling of the pressure from the velocity field. By following the current developments by Guermond and Shen, the possibilities of obtaining accurate pressure and reducing boundary‐layer effect for the pressure are analysed. The present study mainly reports the numerical solutions of an unsteady Navier–Stokes problem based on the so‐called consistent splitting scheme (J. Comput. Phys. 2003; 192:262–276). At the same time the Dirichlet boundary value conditions are considered. The accuracy of the method is carefully examined against the exact solution for an unsteady flow physics problem in a simply connected domain. The effectiveness is illustrated viz. several computations of 2D double lid‐driven cavity problems. Copyright © 2005 John Wiley & Sons, Ltd.

NUMERICAL SIMULATION OF FLOWS PAST A ROTATIONAL CIRCULAR CYLINDER BY TAYLOR-SERIES-EXPANSION AND LEAST SQUARES-BASED LATTICE BOLTZMANN METHOD

An explicit Taylor series expansion and least square-based lattice Boltzmann method (TLLBM) is used to simulate the two-dimensional unsteady viscous incompressible flows. TLLBM is based on the well-known Taylor series expansion and the least square optimization. It has no limitation on mesh structure and lattice model. Its marching in time is accurate. Therefore, it is very suitable for simulation of time dependent problems. Numerical experiments are performed for simulation of flows past a rotational circular cylinder. Good agreement is achieved between the present results and available data in the literature.

Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape

In this paper, a Cartesian grid method with cut cell approach has been developed to simulate two dimensional unsteady viscous incompressible flows with rigid bodies of arbitrary shape. A collocated finite volume method with nominally second-order accurate schemes in space is used for discretization. A pressure-free projection method is used to solve the equations governing incompressible flows. For fixed-body problems, the Adams–Bashforth scheme is employed for the advection terms and the Crank–Nicholson scheme for the diffusion terms. For moving-body problems, the fully implicit scheme is employed for both terms. The present cut cell approach with cell merging process ensures global mass/momentum conservation and avoid exceptionally small size of control volume which causes impractical time step size. The cell merging process not only keeps the shape resolution as good as before merging, but also makes both the location of cut face center and the construction of interpolation stencil easy and systematic, hence enables the straightforward extension to three dimensional space in the future. Various test examples, including a moving-body problem, were computed and validated against previous simulations or experiments to prove the accuracy and effectiveness of the present method. The observed order of accuracy in the spatial discretization is superlinear.

Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids

A novel immersed object method is developed for simulating two-dimensional unsteady incompressible viscous flows around arbitrarily moving rigid bodies. It has been implemented in a parallel unstructured finite volume incompressible Navier–Stokes solver, based on the artificial compressibility (AC) approach using a higher-order characteristics-based upwind scheme and matrix-free implicit dual time-stepping. In the immersed object method, an object is immersed in the flow field, and it is supposed to contain frozen fluid, which moves like a solid body. This is realized by introducing source terms in the momentum equations during the AC sub-iterations. An internal mesh within the object is employed to search and locate all the Eulerian nodes within the object in every time step for imposing the source terms. Unlike many existing methods, this method does not require complex searching, extrapolation and interpolation to find the intersections of the object boundary with the unstructured background mesh and assign flow condition onto the object boundary. If it is necessary to capture the boundary layer accurately, then a dense overlapping grid can then be constructed around the object for further refined calculation. The immersed object method has been used to simulate steady and unsteady incompressible viscous flows over a stationary circular cylinder, rotating square cylinder and moving disk in cavity. The results agree well with published numerical solutions and experimental measurements.

An efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids

This paper presents the development and validation of a parallel unstructured-grid fluid–structure interaction (FSI) solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries. The Navier–Stokes solver on unstructured moving grid using the arbitrary Lagrangian Eulerian formulation is based on the artificial compressibility approach and a high-order characteristics-based finite-volume scheme. Both unsteady flow and FSI are calculated with a matrix-free implicit dual time-stepping scheme. A membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions. This model can be employed to calculate arbitrary wall movement and variable tension along the membrane, together with a dynamic mesh method for large deformation of the flow field. The parallelization of the fluid–structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data. The parallel solver is used to simulate fluid flow in a two-dimensional channel with and without moving membrane for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by this method are excellent, using up to 16 processors on a SGI Origin 2000 parallel computer. A maximum speedup of 23.14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver. The parallel results obtained are compared with those using serial code and they are found to be identical. Copyright © 2005 John Wiley & Sons, Ltd.

Parallel-multigrid computation of unsteady incompressible viscous flows using a matrix-free implicit method and high-resolution characteristics-based scheme

A three-dimensional parallel unstructured non-nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated. The finite-volume Navier–Stokes solver is based on the artificial compressibility approach with a high-resolution method of characteristics-based scheme for handling convection terms. The unsteady flow is calculated with a matrix-free implicit dual time stepping scheme. The parallelization of the multigrid solver is achieved by multigrid domain decomposition approach (MG-DD), using single program multiple data (SPMD) and multiple instruction multiple data (MIMD) programming paradigm. There are two parallelization strategies proposed in this work, first strategy is a one-level parallelization strategy using geometric domain decomposition technique alone, second strategy is a two-level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques. Message-passing interface (MPI) and OpenMP standard are used to communicate data between processors and decompose loop iterations arrays, respectively. The parallel-multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid-driven cavity flow. A maximum speedup of 22.5 could be achieved on 32 processors, for instance, the lid-driven cavity flow of Re = 1000. The results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements. A detailed study of the time step size and number of pseudo-sub-iterations per time step required for simulating unsteady flow are presented in this paper.

Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method

The development and validation of a parallel unstructured tetrahedral non-nested multigrid (MG) method for simulation of unsteady 3D incompressible viscous flow is presented. The Navier–Stokes solver is based on the artificial compressibility method (ACM) and a higher-order characteristics-based finite-volume scheme on unstructured MG. Unsteady flow is calculated with an implicit dual time stepping scheme. The parallelization of the solver is achieved by a MG domain decomposition approach (MG-DD), using the Single Program Multiple Data (SPMD) programming paradigm. The Message-Passing Interface (MPI) Library is used for communication of data and loop arrays are decomposed using the OpenMP standard. The parallel codes using single grid and MG are used to simulate steady and unsteady incompressible viscous flows for a 3D lid-driven cavity flow for validation and performance evaluation purposes. The speedups and efficiencies obtained by both the parallel single grid and MG solvers are reasonably good for all test cases, using up to 32 processors on the SGI Origin 3400. The parallel results obtained agree well with those of serial solvers and with numerical solutions obtained by other researchers, as well as experimental measurements.

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

AbstractAn unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.

3D Visualization of the separated fluid flows

For the detailed investigation of the 3D unsteady incompressible viscous separated fluid flows around a sphere (for 200 ≤ Re ≤ 700) and a circular cylinder (for 200 ≤ Re ≤ 400) the direct numerical simulation and 3D visualization are used. For 3D visualization of the fluid flows around a sphere the definition of vortex core as a connected region containing two negative eigenvalues of the S^2 + Ω^2 tensor is used (where S_{i,j} and Ω_{i,j} are the rate of strain and the rate of rotation tensors). The formation mechanism of vortices in the sphere wake for Re = 500 is described in detail. For 3D visualization of the fluid flows around a circular cylinder the 3D isosurfaces of the streamwise component of vorticity ω_x are used.

Solution of three-dimensional unsteady external flow using a coupled arbitrary Lagrangian FEM–BEM model

A computational model has been developed in this paper to solve three-dimensional unsteady incompressible viscous flow problems in external flow fields. The model is based on primitive variables in Navier–Stokes equations under transient conditions. The model can be used to solve infinite boundary value problems by extracting the boundary effects on a specified finite computational domain using projection method of the Navier–Stokes equations. The momentum equation of fluid motion is solved using the three-step finite element method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation by considering the pressure to be zero at the infinite boundary. Arbitrary Lagrangian–Eulerian method is incorporated in the present model to solve the moving boundary problems. The model has been applied initially to simulate a cubic cavity flow problem for verification purpose and further used to simulate the flow past a square cylinder in two dimensions. Finally, the external flow problem of flow induced by the movement of a sphere inside a viscous flow field in three dimensions has also been considered. The simulation results are found to be reasonable and satisfactory.

Parallel unsteady incompressible viscous flow computations using an unstructured multigrid method

The development and validation of a parallel unstructured non-nested multigrid method for simulation of unsteady incompressible viscous flow is presented. The Navier–Stokes solver is based on the artificial compressibility approach and a higher-order characteristics-based finite-volume scheme on an unstructured multigrid. Unsteady flow is calculated with an implicit dual time-stepping scheme. The parallelization of the solver is achieved by multigrid domain decomposition approach (MG-DD), using the single program multiple data (SPMD) programming paradigm and message-passing interface (MPI) for communication of data. The parallel codes using single grid and multigrid are used to simulate steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by both the parallel single grid and multigrid solvers are reasonably good for both test cases, using up to 32 processors on the SGI Origin 2000. A maximum speedup of 12 could be achieved on 16 processors for high-Reynolds number unsteady viscous flow. The parallel results obtained were compared with those using serial single grid and multigrid codes and it remains the same as those obtained by serial solvers and agrees well with numerical solutions obtained by other researchers as well as experimental measurements.

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

AbstractTime‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.

SIMULATION OF UNSTEADY INCOMPRESSIBLE FLOWS BY USING TAYLOR SERIES EXPANSION- AND LEAST SQUARE-BASED LATTICE BOLTZMANN METHOD

In this work, an explicit Taylor series expansion- and least square-based lattice Boltzmann method (LBM) is used to simulate two-dimensional unsteady incompressible viscous flows. The new method is based on the standard LBM with introduction of the Taylor series expansion and the least squares approach. The final equation is an explicit form and essentially has no limitation on mesh structure and lattice model. Since the Taylor series expansion is only applied in the spatial direction, the time accuracy of the new method is kept the same as the standard LBM, which seems to benefit for unsteady flow simulation. To validate the new method, two test problems, that is, the vortex shedding behind a circular cylinder at low Reynolds numbers and the oscillating flow in a lid driven cavity, were considered in this work. Numerical results obtained by the new method agree very well with available data in the literature.

A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

AbstractA new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.

Exact solutions of non-Newtonian fluid flows with prescribed vorticity

The equations of motion of a non-Newtonian second-grade fluid flow are highly nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions. Due to the complexity of the equations, inverse methods described by Nemenyi [1] have become attractive in the study of non-Newtonian fluids. In these methods, certain physical or geometrical properties of the flow field are assumed a priori. Lin and Tobak [2] studied steady plane viscous incompressible flows for a chosen vorticity function by decomposing the nonlinear fourth-order partial differential equation in the streamfunction. This excellent approach yielded two second-order linear partial differential equations in the streamfunction. Hui [3] used this approach to study unsteady plane viscous incompressible flows. During the past decade, there has been substantial interest in flows of viscoelastic liquids due to the occurrence of these flows in industrial processes. In this paper, we study the steady and unsteady incompressible viscous non-Newtonian second-grade fluid flows in which the vorticity is proportional to the streamfunction perturbed by a uniform stream. The solutions obtained are exact solutions and represent various non-parallel flows of second-grade fluids. The plan of this paper is as follows: In Sect. 2, the equations of motion of an unsteady plane incompressible second-grade fluid are given, and the vorticity function is assumed to be∇ 2 ψ=A(ψ−Ux−BUy 2). In Sect. 3, solutions for the steady flow are obtained. In Sect. 4, solutions for unsteady flow are obtained.