Steady Incompressible Viscous Flow Research Articles

Stagnation point flow of a viscous incompressible fluid

A steady two-dimensional incompressible viscous fluid flow in the vicinity of a stagnation point has been discussed. The reduced third order nonlinear ordinary differential equation of velocity has been solved using fourth order Runge-Kutta method with shooting technique. The qualitative (graphical) as well as quantitative (tabular) analysis of the velocity distribution has been made. Some of the important findings are (i) The viscous effect are confined in the boundary layer for fluids with very small viscosity. (ii) Outside the boundary layer, the viscous effect is negligible and the fluid is considered as inviscid. (iii) The difference in velocity between adjacent layers is decreasing for less viscous fluids.

V-type multigrid method in a lid-driven cavity

The study of steady incompressible viscous flow in a lid-driven cavity is done using V-type multigrid method. The second order accurate finite difference form of the Navier-Stoke’s equation is solved up to Reynolds number 10,000 using grids of maximum size 257 × 257. The validation of the result is ensured by the flow properties of primary eddy which includes the location and strength of the centres at different Reynolds numbers with the literature. Stream function, vorticity contours in the lid-driven cavity and velocity profiles at mid-section of the cavity also provide additional information for the study of flow.

Convective heat transfer in a tube filled with homogeneous and inhomogeneous porous medium

The aim of this study is to analyze the flow and heat transfer inside a straight tube filled with porous medium analytically and numerically. Steady incompressible viscous fluid flow through the porous medium is modeled using Darcy equation as well as Brinkman equation and the results are compared. Uniform heat flux boundary condition along with homogeneous/heterogeneous porous medium is considered. Closed-form solutions for velocity profile, temperature profile, and Nusselt number as a function of shape parameter (σ) are obtained. In the limiting case of infinite permeability, the results from Brinkman model reduce to that of Hagen-Poiseuille flow and in the limiting case of zero permeability, the results approach that of Darcy model. For both homogeneous and heterogeneous porous media, Darcy model is shown to provide an upper bound for Nusselt number. The theoretical results are validated using numerical simulations on COMSOL Multiphysics.

Unsteady Preconditioned Characteristic Boundary Conditions for Direct Numerical Simulation of Incompressible Flows

The unsteady preconditioned characteristic boundary conditions (UPCBCs) based on the artificial compressibility (AC) method are formulated and applied at artificial boundaries for the direct numerical simulation (DNS) of incompressible flows. The compatibility equations including the unsteady terms are mathematically derived in the generalized curvilinear coordinates and then incorporated as boundary conditions (BCs) in a high-order accurate incompressible flow solver. The spatial derivative terms of the system of equations are discretized using the fourth-order compact finite difference (FD) scheme, consistent with the high-order accuracy required for the DNS. The time integration is carried out using an implicit dual time-stepping method, and the robustness and performance of the numerical algorithm are enhanced by applying the preconditioning technique and calculating the automated AC parameter based on the local flow conditions. At first, the steady viscous incompressible flow for a two-dimensional plane jet is computed to assess the accuracy and performance of the solution method by applying the UPCBCs in comparison with the simplified BCs. Then, the simulation of the unsteady incompressible viscous flow around a two-dimensional NACA0012 airfoil at the Reynolds number 800 and the angle of attack 20° is performed, and it is indicated that the computational cost of the solution is considerably decreased by implementing the UPCBCs at the artificial far-field boundary compared with the simplified BCs, which is remarkably valuable for the DNSs. Finally, the computation of the transition to unsteady for the incompressible flow around a three-dimensional NACA0012 wing at the same conditions of the two-dimensional case is successfully performed, and the results obtained are compared with those of previous studies that exhibit good agreement. Indications are that the present solution method by applying the UPCBCs at the artificial far-field boundary can be effectively used for the DNSs of the incompressible flows.

Melting Heat Transfer and MHD Boundary Layer Flow of Eyring-Powell Nanofluid Over a Nonlinear Stretching Sheet with Slip

The steady laminar incompressible viscous magneto hydrodynamic boundary layer flow of an Eyring- Powell fluid over a nonlinear stretching flat surface in a nanofluid with slip condition and heat transfer through melting effect has been investigated numerically. The resulting nonlinear governing partial differential equations with associated boundary conditions of the problem have been formulated and transformed into a non-similar form. The resultant equations are then solved numerically using the Runge-Kutta fourth order method along with the shooting technique. The physical significance of different parameters on the velocity, temperature and nanoparticle volume fraction profiles is discussed through graphical illustrations. The impact of physical parameters on the local skin friction coefficient and rate of heat transfer is shown in tabulated form.

Numerical Solution of the Problem of Incompressible Fluid Flow in a Plane Channel with a Backward-Facing Step at High Reynolds Numbers

Numerical solutions of the problem of steady incompressible viscous fluid flow in a plane channel with a backward-facing step have been obtained by the grid method. The fluid motion is described by the Navier–Stokes equations in velocity-pressure variables. The main computations were performed on a uniform 6001 × 301 grid. The control-volume method of the second order in space was used for the difference approximation of the original equations. The results were validated for the range of Reynolds numbers (100 ≤ Re ≤ 3000) by comparing them with the experimental and theoretical data found in the literature. The stability of the computational algorithm at high Reynolds numbers was achieved by using a fine difference grid (a small grid step). The study has been carried out for a short channel at Reynolds numbers from 1000 to 10 000 with a step of 1000. A nonstandard structure of the primary vortex behind the step—the presence of numerous centers of rotation both inside the vortex and in the near-wall region under it—has been revealed. The number of centers of rotation in the primary recirculation zone is shown to grow with increasing Reynolds number. The profiles of the coefficients of friction and hydrodynamic resistance to the flow as a function of Reynolds number have also been analyzed. The results obtained can be useful for comparison and validation of the solutions of problems of such a type.

Isogeometric Boundary Element Analysis of steady incompressible viscous flow, Part 2: 3-D problems

This is a sequel to a previous paper (Beer et al., 2017) [1] where a novel approach was presented to the 2-D Boundary Element analysis of steady incompressible viscous flow. Here the method is extended to three dimensions. NURBS basis functions are used for describing the geometry of the problem and for approximating the unknowns. In addition, the arising volume integrals are treated differently to published work and volumes are described by bounding NURBS surfaces instead of cells and only one mapping is used. The advantage of the present approach is that complex boundary shapes can be described with very few parameters and that no generation of cells is required. For the solution of the non-linear equations full and modified Newton–Raphson methods are used. A comparison of the two methods is made on the classical example of a forced cavity flow, where accurate two-dimensional solutions are available in the literature. Finally, it is shown on a practical example of an airfoil how more complex boundary shapes can be approximated with few parameters and a solution obtained with a small number of unknowns.

SIMULATION OF THE LID-DRIVEN CAVITY FLOW AT REYNOLDS NUMBERS BETWEEN 100 AND 1000 USING THE MULTI-RELAXATION-TIME LATTICE BOLTZMANN METHOD

The Multi-Relaxation-Time Lattice Boltzmann Method (MRT LBM) is used to numerically simulate the steady viscous incompressible flow in a lid-driven cavity. The simulations are performed for a range of Reynolds numbers between 100 and 1000. The simulation results include the flow streamlines which clearly show the location of the primary main vortex and the two side vortices, the horizontal and vertical velocity component profiles at different sections inside the cavity and the location of the vortices centers. The numerical results are compared against published results and show a perfect agreement. The simulation results are given for a range of Reynolds numbers not reported in the published literature. The presented results can be used for benchmarking other numerical methods in the reported range of the Reynolds number.

Application of a preconditioned high‐order accurate artificial compressibility‐based incompressible flow solver in wide range of Reynolds numbers

SummaryIn the present study, the preconditioned incompressible Navier‐Stokes equations with the artificial compressibility method formulated in the generalized curvilinear coordinates are numerically solved by using a high‐order compact finite‐difference scheme for accurately and efficiently computing the incompressible flows in a wide range of Reynolds numbers. A fourth‐order compact finite‐difference scheme is utilized to accurately discretize the spatial derivative terms of the governing equations, and the time integration is carried out based on the dual time‐stepping method. The capability of the proposed solution methodology for the computations of the steady and unsteady incompressible viscous flows from very low to high Reynolds numbers is investigated through the simulation of different 2‐dimensional benchmark problems, and the results obtained are compared with the existing analytical, numerical, and experimental data. A sensitivity analysis is also performed to evaluate the effects of the size of the computational domain and other numerical parameters on the accuracy and performance of the solution algorithm. The present solution procedure is also extended to 3 dimensions and applied for computing the incompressible flow over a sphere. Indications are that the application of the preconditioning in the solution algorithm together with the high‐order discretization method in the generalized curvilinear coordinates provides an accurate and robust solution method for simulating the incompressible flows over practical geometries in a wide range of Reynolds numbers including the creeping flows.

Linear Forcing により定常化された周期的粘性流解析解と適用

Linear Forcing により定常化された周期的粘性流解析解と適用

Numerical solutions of 2D steady incompressible backward-facing step flow at high Reynolds numbers

Numerical solutions of the test problem of 2D steady incompressible viscous flow over backward-facing step are presented in the article. The Navier-Stokes equations in a velocity-pressure formulation are numerically solved using a uniform grid of 6001´301 points. The control-volume technique for second-order difference approximation for spatial derivatives is used. The obtained numerical solutions were validated for a wide range of Reynolds numbers (100 £ Re £ 3000), comparing them to numerous experimental and numerical results found in the literature. The use of an original technology for building a solution at the exit from the channel (an open boundary of the solution domain) has made it possible to drastically reduce the length of the channel and, as a consequence, to decrease a size of the grid step. As a result, the increase in the stability of the computational algorithm took place, and it was possible to obtain solutions of the problem for large Reynolds numbers. Main research calculations have been performed for a short channel at Reynolds numbers from 1000 to 10000 with increment 1000. The obtained results have revealed a non-standard structure of primary vortex behind the step - numerous centers of rotation are present both in the vortex and under it in the area adjacent to the wall. It is shown that the number of the centers of rotation of the primary recirculation zone grows in accordance with the increase of the Reynoldsnumber. Also, the behavior of profiles of friction coefficients and hydrodynamic resistance to the flow depending on the Reynolds number are analyzed in the work. The obtained results may be useful for comparison and verification of solutions of this type of problems.

On the equatorial Ekman layer

The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the small Ekman number limit $E\ll 1$, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the $E^{1/2}$ Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder $\mathscr{C}$, is geostrophic. Stewartson identified a complicated nested layer structure on $\mathscr{C}$, which comprises relatively thick quasigeostrophic $E^{2/7}$- (inside $\mathscr{C}$) and $E^{1/4}$- (outside $\mathscr{C}$) layers. They embed a thinner ageostrophic $E^{1/3}$ shear layer (on $\mathscr{C}$), which merges with the inner sphere Ekman layer to form the $E^{2/5}$-equatorial Ekman layer of axial length $E^{1/5}$. Under appropriate scaling, this $E^{2/5}$-layer problem may be formulated, correct to leading order, independent of $E$. Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate $z$) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using a non-local integral boundary condition at finite $z$ to account for the far-field behaviour. Adopting $z^{-1}$ as a small parameter we extend Stewartson’s similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.

Level set-based topology optimization of steady state incompressible viscous flows under outflow rate inequality constraint

Historically, fluidic devices such as switches, amplifiers, and oscillators, have an advantage, compared with electronic devices, in terms of maintenance-free operation and operating life. Therefore, prior to the great progress in electronic technologies that has occurred during the past several decades, the structure and function of fluidic devices were the subject of extensive research. In particular, the structures of these fluidic devices are often composed of complicated flow channel layouts. Recently, fluidic devices are again attracting significant attention, stimulated by progress in the development of MEMS technologies. In this study, to develop an energy-efficient structure for a MEMS-scale fluidic device, we apply a topology optimization method to an optimal design problem for a steady state incompressible viscous flow field. We use a level set-based topology optimization method incorporating the concept of the phase field method for the topology optimization so that clear boundaries between the solid and fluid domains are expressed in the optimal configurations. To define the topology optimization problem for a fluid regime, the expressions of the primary and adjoint problems are formulated concretely, to minimize viscous energy dissipation. Moreover, to ensure the intended design outflow rate at a designated outlet, the optimization problem includes an outflow rate inequality constraint in this paper. Following the concept of the standard adjoint variable method, a stable optimization process that satisfies the outflow rate inequality constraint is achieved. Two numerical examples, one for two-terminal and the other for multi-terminal flow, are provided to demonstrate the usefulness of the proposed level set-based topology optimization method.

ADI method based on [formula omitted]-continuous two-node integrated-RBF elements for viscous flows

We propose a C2-continuous alternating direction implicit (ADI) method for the solution of the streamfunction–vorticity equations governing steady 2D incompressible viscous fluid flows. Discretisation is simply achieved with Cartesian grids. Local two-node integrated radial basis function elements (IRBFEs) [D.-A. An-Vo, N. Mai-Duy, T. Tran-Cong, A C2-continuous control-volume technique based on Cartesian grids and two-node integrated-RBF elements for second-order elliptic problems, CMES: Comput. Model. Eng. Sci. 72 (2011) 299–334] are used for the discretisation of the diffusion terms, and then the convection terms are incorporated into system matrices by treating nodal derivatives as unknowns. ADI procedure is applied for the time integration. Following ADI factorisation, the two-dimensional problem becomes a sequence of one-dimensional problems. The solution strategy consists of multiple use of a one-dimensional sparse matrix algorithm that helps saving the computational cost. High levels of accuracy and efficiency of the present methods are demonstrated with solutions of several benchmark problems defined on rectangular and non-rectangular domains.

Von Kàrmàn problem for a rotating permeable disk

The von Karman problem of steady viscous incompressible flow in a half-space beneath a permeable, porous disk of infinite radius, uniformly rotating it its own plane, is considered. The Bevers-Joseph condition which takes account for viscous fluid slip relative to the plane disk is taken for the boundary condition for the tangential velocity component. This makes it possible to consider the flow only outside the porous disk. The disk permeability effect on the boundary layer is studied, in particular, its influence on the dependence of the moment of viscous forces distributed over the disk about the axis of rotation on the permeability and the pressure difference.

Modified alternating direction-implicit iteration method for linear systems from the incompressible Navier–Stokes equations

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver.

Solutions in Variably Inclined MHD Flows

We study the plane MHD flows when the velocity and magnetic fields are variably inclined and investigate the steady viscous incompressible flow problems of a fluid having infinite electrical conductivity in the presence of a magnetic field. Accounting for infinite electrical conductivity makes the flow problem realistic and attractive because the magnetic Reynolds number is very small for most liquid metals. Particular problems are discussed when magnetic lines are variably inclined but nowhere aligned with streamlines, when the fluid is viscous and non-viscous. Streamlines are parabolic as shown in the graphs.

On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, we present block SSOR and modified block SSOR iteration methods based on the special structures of the coefficient matrices. In each step of the block SSOR iteration, we employ the block LU factorization to solve the sub-systems of linear equations. We show that the block LU factorization is existent and stable when the coefficient matrices are block diagonally dominant of type-II by columns. Under suitable conditions, we establish convergence theorems for both block SSOR and modified block SSOR iteration methods. In addition, the block SSOR iteration and AF-ADI method are considered as preconditioners for the nonsymmetric systems of linear equations. Numerical experiments show that both block SSOR and modified block SSOR iterations are feasible iterative solvers and they are also effective for preconditioning Krylov subspace methods such as GMRES and BiCGSTAB when used to solve this class of systems of linear equations.

Numerical Solution of Non-Isothermal Fluid Flows Using Local Radial Basis Functions (LRBF) Interpolation and a Velocity-Correction Method

Meshfree point collocation method (MPCM) is developed, solving the velocity-vorticity formulation of Navier-Stokes equations, for two-dimensional, steady state incompressible viscous flow problems in the presence of heat transfer. Particular emphasis is placed on the application of the velocity-correction method, ensuring the continuity equation. The Gaussian Radial Basis Functions (GRBF) interpolation is employed to construct the shape functions in conjunction with the framework of the point collocation method. The cases of forced, natural and mixed convection in a 2D rectangular enclosure are examined. The accuracy and the sta- bility of the proposed scheme are demonstrated through three representative, well known and established benchmark problems. Results are presented for high values of the characteristics non-dimensional numbers of the flow, that is, the Reynolds, the Rayleigh and the Richardson number.

Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows

With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.