A numerical investigation of wall effects in three‐dimensional,laminar flow over a backward facing step with a constant aspect and expansion ratio

The wall effects are investigated in the three‐dimensional laminar flow over a backward‐facing step. For this purpose, a numerical experiment is designed under actual laboratory conditions. The aspect ratio of the computational domain is 1:40 and the expansion ratio is 1:2. The Reynolds number ranges from 100 to 950. The governing equations are the steady state, isothermal and incompressible Navier–Stokes equations for Newtonian fluids. They are solved with a homemade Galerkin finite element code. The computations are validated with data from available laboratory and numerical experiments. The results focus on the variation of both velocity profiles and lengths of eddies along the lower and upper wall in the spanwise direction. Calculated streamlines in the streamwise and transverse direction show how the flow is distorted near the lateral wall and how it develops up to the plane of symmetry. The study of skin friction lines along the top and bottom wall of the domain reveals a flow that takes place in the spanwise direction. This spanwise component of the flow becomes more dominant with increasing Reynolds number and is impossible to be sustained at steady state for Reynolds numbers higher than 950 for this particular geometry. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical investigation of the influence of the aspect ratio in three‐dimensional separated flows

The influence of aspect ratio in three‐dimensional, numerical experiments of separated flows is studied in the case of the backward‐facing step at Reynolds numbers 600, 800, and 950. The computational domain is designed as an actual laboratory experiment. The governing equations are the steady state, isothermal, and incompressible Navier–Stokes equations. The expansion ratio of the computational domain is 1:2. The aspect ratio varies from 1:10 to 1:40. The results of the computations focus on the spanwise variations of the length and the strength of the two eddies along the lower and upper wall. It is concluded that both numerical and laboratory experiments should be designed with an aspect ratio of at least 1:20, if only the accuracy of the position of the detachment and the re‐attachment points matters. If the accuracy of the shear‐stress distributions is also taken into account, then an aspect ratio of at least 1:30 should be chosen. Finally, if the magnitudes of the vortex centers are also considered, then only the aspect ratio of 1:40 qualifies for a realization of two‐dimensional flow conditions in the plane of symmetry. This is contrary to the common practice in the field, at least from the side of laboratory experiments, where an aspect ratio of 1:10 is still considered adequate for this purpose. Copyright © 2011 John Wiley & Sons, Ltd.     

Side‐wall effects on the bifurcation of the flow through a sudden expansion

Three‐dimensional computations have been performed to study the flow through a symmetric sudden expansion with an expansion ratio of 3 at low Reynolds numbers. The aspect ratio of the flow channel is allowed to vary within a wide range to examine its influence on the flow which bifurcates from a symmetric state to an asymmetric state. The results reveal that the critical Reynolds number of the symmetry‐breaking bifurcation increases while the aspect ratio is reduced. The flow behaviour near the side walls is illustrated by using limiting streamlines. The origin of the singular points identifiable on the side wall can be traced back to the recirculating flows and the relevant reattachment/separation points in the core of the channel. It is seen that the determination of the exact critical Reynolds number is not trivial because it depends on how to define asymmetric flow. Computations have also been conducted to show that a slight asymmetry in the channel geometry causes a smooth transition from symmetric to non‐symmetric states. Copyright © 2007 John Wiley & Sons, Ltd.     

A local domain‐free discretization method to simulate three‐dimensional compressible inviscid flows

In this paper, the domain‐free discretization method (DFD) is extended to simulate the three‐dimensional compressible inviscid flows governed by Euler equations. The discretization strategy of DFD is that the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior‐dependent points are updated at each time step by extrapolation along the wall normal direction in conjunction with the wall boundary conditions and the simplified momentum equation in the vicinity of the wall. Spatial discretization is achieved with the help of the finite element Galerkin approximation. The concept of ‘osculating plane’ is adopted, with which the local DFD can be easily implemented for the three‐dimensional case. Geometry‐adaptive tetrahedral mesh is employed for three‐dimensional calculations. Finally, we validate the DFD method for three‐dimensional compressible inviscid flow simulations by computing transonic flows over the ONERA M6 wing. Comparison with the reference experimental data and numerical results on boundary‐conforming grid was displayed and the results show that the present DFD results compare very well with the reference data. Copyright © 2009 John Wiley & Sons, Ltd.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Steady and unsteady laminar flows of Newtonian and generalized Newtonian fluids in a planar T‐junction

An investigation of laminar steady and unsteady flows in a two‐dimensional T‐junction was carried out for Newtonian and a non‐Newtonian fluid analogue to blood. The flow conditions considered are of relevance to hemodynamical applications and the localization of coronary diseases, and the main objective was to quantify the accuracy of the predictions and to provide benchmark data that are missing for this prototypical geometry. Under steady flow, calculations were performed for a wide range of Reynolds numbers and extraction flow rate ratios, and accurate data for the recirculation sizes were obtained and are tabulated. The two recirculation zones increased with Reynolds number, but the behaviour was non‐monotonic with the flow rate ratio. For the pulsating flows a periodic instability was found, which manifests itself by the breakdown of the main vortex into two pieces and the subsequent advection of one of them, while the secondary vortex in the main duct was absent for a sixth of the oscillating period. Shear stress maxima were found on the walls opposite the recirculations, where the main fluid streams impinge onto the walls. For the blood analogue fluid, the recirculations were found to be 10% longer but also short lived than the corresponding Newtonian eddies, and the wall shear stresses are also significantly different especially in the branch duct. Copyright © 2007 John Wiley & Sons, Ltd.     

A two‐step Taylor‐characteristic‐based Galerkin method for incompressible flows and its application to flow over triangular cylinder with different incidence angles

An alternative characteristic‐based scheme, the two‐step Taylor‐characteristic‐based Galerkin method is developed based on the introduction of multi‐step temporal Taylor series expansion up to second order along the characteristic of the momentum equation. Contrary to the classical characteristic‐based split (CBS) method, the current characteristic‐based method does not require splitting the momentum equation, and segregate the calculation of the pressure from that of the velocity by using the momentum–pressure Poisson equation method. Some benchmark problems are used to examine the effectiveness of the proposed algorithm and to compare with the original CBS method, and the results show that the proposed method has preferable accuracy with less numerical dissipation. We further applied the method to the numerical simulation of flow around equilateral triangular cylinder with different incidence angles in free stream. In this numerical investigation, the flow simulations are carried out in the low Reynolds number range. Instantaneous streamlines around the cylinder are used as a means to visualize the wake region behind, and they clearly show the flow pattern around the cylinder in time. The influence of incidence angle on flow characteristic parameters such as Strouhal number, Drag and Lift coefficients are discussed quantitatively. Copyright © 2009 John Wiley & Sons, Ltd.     

A comparative study of inflow conditions for two‐ and three‐dimensional spatially developing mixing layers using large eddy simulation

A series of spatially developing mixing layers are simulated using the large eddy simulation (LES) technique. A hyperbolic tangent function and data derived from boundary layer simulations are used to generate the inflow condition, and their effects on the flow are compared. The simulations are performed in both two and three dimensions. In two‐dimensional simulations, both types of inflow conditions produce a layer that grows through successive pairings of Kelvin–Helmholtz (K–H) vortices, but the composition ratio is lower for the hyperbolic tangent inflow simulations. The two‐dimensional simulations do not undergo a transition to turbulence. The three‐dimensional simulations produce a transition to turbulence, and coherent structures are found in the post‐transition region of the flow. The composition ratio of the three‐dimensional layers is reduced in comparison to the counterpart two‐dimensional runs. The mechanisms of growth are investigated in each type of simulation, and amalgamative pairing interactions are found in the pre‐transition region of the three‐dimensional simulations, and throughout the entire computational domain of those carried out in two‐dimensions. The structures beyond the post‐transition region of the three‐dimensional simulations appear to behave in a much different manner to their pre‐transition cousins, with no pairing‐type interactions observed in the turbulent flow. In order to accurately simulate spatially developing mixing layers, it is postulated that the inflow conditions must closely correspond to the conditions present in the reference experiment. Copyright © 2007 John Wiley & Sons, Ltd.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

A two‐scale low‐Reynolds number turbulence model

In this study, a two‐scale low‐Reynolds number turbulence model is proposed. The Kolmogorov turbulence time scale, based on fluid kinematic viscosity and the dissipation rate of turbulent kinetic energy (ν, ε), is adopted to address the viscous effects and the rapid increasing of dissipation rate in the near‐wall region. As a wall is approached, the turbulence time scale transits smoothly from a turbulent kinetic energy based (k, ε) scale to a (ν, ε) scale. The damping functions of the low‐Reynolds number models can thus be simplified and the near‐wall turbulence characteristics, such as the ε distribution, are correctly reproduced. The proposed two‐scale low‐Reynolds number turbulence model is first examined in detail by predicting a two‐dimensional channel flow, and then it is applied to predict a backward‐facing step flow. Numerical results are compared with the direct numerical simulation (DNS) budgets, experimental data and the model results of Chien, and Lam and Bremhorst respectively. It is proved that the proposed two‐scale model indeed improves the predictions of the turbulent flows considered. Copyright © 2000 John Wiley & Sons, Ltd.     

An implicit two‐dimensional non‐hydrostatic model for free‐surface flows

An implicit finite volume model in sigma coordinate system is developed to simulate two‐dimensional (2D) vertical free surface flows, deploying a non‐hydrostatic pressure distribution. The algorithm is based on a projection method which solves the complete 2D Navier–Stokes equations in two steps. First the pressure term in the momentum equations is excluded and the resultant advection–diffusion equations are solved. In the second step the continuity and the momentum equation with only the pressure terms are solved to give a block tri‐diagonal system of equation with pressure as the unknown. This system can be solved by a direct matrix solver without iteration. A new implicit treatment of non‐hydrostatic pressure, similar to the lower layers is applied to the top layer which makes the model free of any hydrostatic pressure assumption all through the water column. This treatment enables the model to evaluate both free surface elevation and wave celerity more accurately. A series of numerical tests including free‐surface flows with significant vertical accelerations and nonlinear behaviour in shoaling zone are performed. Comparison between numerical results, analytical solutions and experimental data demonstrates a satisfactory performance. Copyright © 2006 John Wiley & Sons, Ltd.     

计算离心泵叶轮流场的扫描流束有限元方法

本文提出一种求解离心式叶轮流场的数值方法，将流动求解区域离散为有限个由流线构成其边界的单元，采用伽辽金法建立的单元方程在一条流束上集合为方程组，流线上的节点坐标亦作为未知量包含在有限元方程中，通过扫描计算，逐步解得流线位置及流动参数。本文应用叶轮的通流理论流动模型，采用扫描流速有限元方法对离心泵叶轮流场进行了计算，并与有关文献作了比较。     

High‐resolution,monotone solution of the adjoint shallow‐water equations

A monotone, second‐order accurate numerical scheme is presented for solving the differential form of the adjoint shallow‐water equations in generalized two‐dimensional coordinates. Fluctuation‐splitting is utilized to achieve a high‐resolution solution of the equations in primitive form. One‐step and two‐step schemes are presented and shown to achieve solutions of similarly high accuracy in one dimension. However, the two‐step method is shown to yield more accurate solutions to problems in which unsteady wave speeds are present. In two dimensions, the two‐step scheme is tested in the context of two parameter identification problems, and it is shown to accurately transmit the information needed to identify unknown forcing parameters based on measurements of the system response. The first problem involves the identification of an upstream flood hydrograph based on downstream depth measurements. The second problem involves the identification of a long wave state in the far‐field based on near‐field depth measurements. Copyright © 2002 John Wiley & Sons, Ltd.     

A meshless method for numerical simulation of depth‐averaged turbulence flows using a k‐ϵ model

A three‐dimensional numerical model is developed to analyze free surface flows and water impact problems. The flow of an incompressible viscous fluid is solved using the unsteady Navier–Stokes equations. Pseudo‐time derivatives are introduced into the equations to improve computational efficiency. The interface between the two phases is tracked using a volume‐of‐fluid interface tracking algorithm developed in a generalized curvilinear coordinate system. The accuracy of the volume‐of‐fluid method is first evaluated by the multiple numerical benchmark tests, including two‐dimensional and three‐dimensional deformation cases on curvilinear grids. The performance and capability of the numerical model for water impact problems are demonstrated by simulations of water entries of the free‐falling hemisphere and cone, based on comparisons of water impact loadings, velocities, and penetrations of the body with experimental data. For further validation, computations of the dam‐break flows are presented, based on an analysis of the wave front propagation, water level, and the dynamic pressure impact of the waves on the downstream walls, on a specific container, and on a tall structure. Extensive comparisons between the obtained solutions, the experimental data, and the results of other numerical simulations in the literature are presented and show a good agreement. Copyright © 2015 John Wiley & Sons, Ltd.     

Natural convection flow under a magnetic field in an inclined square enclosure differentialy heated on adjacent walls

Steady, laminar, natural-convection flow in the presence of a magnetic field in an inclined square enclosure differentially heated along the bottom and left vertical walls while the other walls are kept isothermal was considered. The governing equations were solved numerically for the stream function, vorticity and temperature ratio using the differential quadrature method for various Grashof and Hartmann numbers, inclination angle of the enclosure and direction of the magnetic field. The orientation of the enclosure changes the temperature gradient inside and has a significant effect on the flow pattern. Magnetic field suppresses the convective flow and its direction also influences the flow pattern, causing the appearance of inner loops and multiple eddies. The surface heat flux along the bottom wall is slightly increased by clockwise inclination and reduced by half by the counterclockwise inclination. The surface heat flux along the upper portion of the left side wall is reversed by the rise of warmer fluids due to the convection currents for no inclination and clockwise inclination of the enclosure.     

A two‐dimensional numerical scheme of dry/wet fronts for the Saint‐Venant system of shallow water equations

We propose a new two‐dimensional numerical scheme to solve the Saint‐Venant system of shallow water equations in the presence of partially flooded cells. Our method is well balanced, positivity preserving, and handles dry states. The latter is ensured by using the draining time step technique in the time integration process, which guarantees non‐negative water depths. Unlike previous schemes, our technique does not generate high velocities at the dry/wet boundaries, which are responsible for small time step sizes and slow simulation runs. We prove that the new scheme preserves ‘lake at rest’ steady states and guarantees the positivity of the computed fluid depth in the partially flooded cells. We test the new scheme, along with another recent scheme from the literature, against the analytical solution for a parabolic basin and show the improved simulation performance of the new scheme for two real‐world scenarios. Copyright © 2014 John Wiley & Sons, Ltd.     

Time‐dependent flow across a step: the slip with friction boundary condition

The paper studies numerically the slip with friction boundary condition in the time‐dependent incompressible Navier–Stokes equations. Numerical tests on two‐ and three‐dimensional channel flows across a step using this boundary condition on the bottom wall are performed. The influence of the friction parameter on the flow field is studied and the results are explained according to the physics of the flow. Due to the stretching and tilting of vortices, the three‐dimensional results differ in many respects from the two‐dimensional ones. Copyright © 2005 John Wiley & Sons, Ltd.     

Turbulent mixed convection flow over a backward-facing step––the effect of the step heights

Effect of the backward-facing step heights on turbulent mixed convection flow along a vertical flat plate is examined experimentally. The step geometry consists of an adiabatic backward-facing step, an upstream wall and a downstream wall. Both the upstream and downstream walls are heated to a uniform and constant temperature. Laser–Doppler velocimeter and cold wire anemometer were used, respectively, to measure simultaneously the time-mean velocity and temperature distributions and their turbulent fluctuations. The experiment was carried out for step heights of 0, 11, and 22 mm, at a free stream air velocity, u_∞, of 0.41 m/s, and a temperature difference, ΔT, of 30 °C between the heated walls and the free stream air. The present results reveal that the turbulence intensity of the streamwise and transverse velocity fluctuations and the intensity of temperature fluctuations downstream of the step increase as the step height increases. Also, it was found that both the reattachment length and the heat transfer rate from the downstream heated wall increase with increasing step height.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.