Lattice Boltzmann simulation of lid-driven swirling flow in confined cylindrical cavity

Lid-driven swirling flow in a confined cylindrical cavity is investigated using lattice Boltzmann equation (LBE) method. The steady, 3-dimensional flow is examined at different aspect (height-to-radius) ratios and Reynolds numbers. The LBE simulations are carried out using the multiple-relaxation-time method. The LBE simulation results are compared with the results of a finite volume solution of Navier-Stokes equations and with published experimental data. Numerical results are presented for cylindrical cavities with two aspect ratios of 1.5 and 2.5, and three Reynolds numbers of 990, 1010 and 1290. Effects of the aspect ratio and Reynolds number on the size, position and breakdown of the central recirculation bubble, together with the flow pattern in the cavity, are determined. Detailed topological features of the flow, such as, (1) structure and breakdown of the vortex along the axis, (2) azimuthal component of vorticity, and (3) circulation strength of flow about the axis are investigated and compared with previous findings from experiments and theory.The predicted results from LBE simulations are consistent with experiments and theory. Steady results reveal the occurrence of a breakdown bubble in agreement with the regime diagram due to Escudier. The vortex breakdown around a region may be characterized by a change in sign of the azimuthal vorticity near such locations. Investigations are carried out on the characteristics of angular momentum when the vortex breakdown occurs. The theoretical criterion for vortex breakdown to occur, as proposed by Brown and Lopez is verified using the numerical data obtained from the simulations.     

Multi relaxation time lattice Boltzmann simulations of deep lid driven cavity flows at different aspect ratios

In this paper, the multi relaxation time (MRT) lattice Boltzmann equation (LBE) was used to compute lid driven cavity flows at different Reynolds numbers (100–7500) and cavity aspect ratios (1–4 cavity width depth). Steady solutions were obtained for square cavity flows, however for deep cavity flows at 1.5 and 4 cavity width depth, unsteady solutions prevail at Re = 7500, where periodic flow exists manifested by the rapid changes of the shapes and locations of the corner vortices in strong contrast of the stationary primary vortex. The merger of the bottom corner vortices into a primary vortex and the reemergence of the corner vortices as the Reynolds number increases are more evident for the deep cavity flows. For the four cavity width depth cavity, four primary vortices were predicted by MRT model for Reynolds number beyond 1000, which were not predicted by previous single relaxation time (SRT) BGK LBE model, and this was verified by complementary Navier–Stokes simulations. Also, MRT model is more suitable for parallel computations than its BGK counterpart, due to the more intense local computations of the multi relaxation time procedure.     

Numerical study of the flow of a fluid of second grade in a square cavity

The flow of a fluid of second grade in a square cavity is studied with the help of finite difference equations. Convergence is obtained by seeking after the dominance of the main diagonal of the system. Solutions are obtained for a Reynolds number equal to 100, and Weissenberg numbers ranging from −0·5 to 1. Interesting effects are found when the Weissenberg number increases.     

Multiplicity of states for two-sided and four-sided lid driven cavity flows

Numerical simulations for incompressible flow in two-sided and four-sided lid driven cavities are reported in the present study. For the two-sided driven cavity, the upper wall is moved to the right and the left wall to the bottom with equal speeds. For the four-sided driven cavity, the upper wall is moved to the right, the lower wall to the left, while the left wall is moved downwards and the right wall upwards, with all four walls moving with equal speeds. At low Reynolds numbers, the resulting flow field is symmetric with respect to one of the cavity diagonals for the two-sided driven cavity, while it is symmetric with respect to both cavity diagonals for the four-sided driven cavity. At a critical Reynolds number of 1073 for the two-sided driven cavity and 129 for the four-sided driven cavity, the flow field bifurcates from a stable symmetric state to a stable asymmetric state. Three possible flow solutions exist above the critical Reynolds number, an unstable symmetric solution and two stable asymmetric solutions. All three possible solutions are recovered in the present study and flow bifurcation diagrams are constructed. Moreover, it is shown that the marching direction of the iterative solver determines which of the two asymmetric solutions is recovered.     

Vortex structure of steady flow in a rectangular cavity

The vortex structure of the two-dimensional steady flow in a lid-driven rectangular cavity at different depth-to-width ratios and Reynolds numbers is investigated using a lattice Boltzmann method. The aspect ratio varies from 0.1 to 7 and the Reynolds number ranges from 0.01 to 5000. The effects of the aspect ratio and Reynolds number on the size, center position and number of vortices are determined together with the flow pattern in the cavity. The present results not only confirm the vortex structure of Stokes flow reported by previous researchers, but also reveal some new evolution features of the vortices and their structure with the Reynolds number. When the Reynolds number approaches 0, the flow shows a characteristic feature of symmetric vortex structure. On the other hand, as the Reynolds number increases, the sizes and center positions of the vortices in the near-lid region appear to be strongly affected by the inertia force, resulting in an asymmetric vortex structure in this region. The influence of the inertia force decreases along the depth for the deep cavity flow. It is found that there is a critical value of the aspect ratio, which depends on the Reynolds number. When the critical value is exceeded, flow pattern in a certain region of cavity becomes symmetric again. These large symmetric vortices are similar in shape, and their sizes approach a constant.     

Steady laminar flow in a 90 degree planar branch

The flow characteristics of a Newtonian fluid in a two-dimensional, planar, right angled Tee branch are studied over a range of inlet Reynolds number of 10–800 by solving the Navier-Stokes equations using a finite element discretization. The effects of the branch length and the grid size on the interior flow field are examined to assess the accuracy of the solutions. In one case the computed velocity field is compared with the Laser Doppler anemometry measurements available in the literature and excellent agreement has been obtained. The computed velocity field is believed to be accurate within about 5%. Results are presented for two types of experimentally realizable boundary conditions—viz. equal exit pressure at the outlet of each branch and specified flow split between the branches. For the case of equal exit pressures the fractional flow in the main duct increases with increasing Reynolds number and the flow characteristics in the side branch become akin to that in a cavity. For the case of specified flow split, the number, size and strength of the recirculation zones increase as more fluid is forced to go into the side branch. The length of the side branch appears to have very little influence on the interior flow field, particularly at higher Reynolds number. This observation is rationalized as being due to the parabolized approximation becoming more valid at higher Reynolds numbers. The critical Reynolds number at which the first recirculation zone appears in the side branch increases with increasing fractional flow in the side branch and with decreasing side branch width.     

Numerical studies of the flow around a circular cylinder by a finite element method

Numerical solutions of the steady, incompressible, viscous flow past a circular cylinder are presented for Reynolds numbers R ranging from 1 to 100. The governing Navier-Stokes equations in the form of a single, fourth order differential equation for stream function and the boundary conditions are replaced by an equivalent variational principle. The numerical method is based on a finite element approximation of this principle. The resulting non-linear system is solved by the Newton-Raphson process. The pressure field is obtained from a finite element solution of the Poisson equation once the stream function is known. The results are compared with those determined by other numerical techniques and experiments. In particular, the discussion is concerned with the development of the closed wake with Reynolds number, and the tendency of R ≥ 40 flow toward instability.     

Solutions of the two-dimensional Navier-Stokes equations by Chebyshev expansion methods

The steady two-dimensional Navier-Stokes equations in both the vorticity-stream function and the vorticity-velocity formulation are solved by Chebyshev expansion methods. Numerical experiments for the driven flow in a rectangular cavity and the developing flow in a circular tube at low Reynolds numbers are described.     

A finite element numerical solution of natural convection in enclosed cavities

Steady state free convective flow enclosed within a cavity and subjected to a temperature gradient is predicted using the finite element method. The matrix equations resulting from the finite element discretisation and formulation are solved using both an iterative and a modified Newton-Raphson scheme. An assessment of the variation in the characteristics of the flow regime is made in association with the dimensionless Prandtl and Rayleigh numbers. A further parameter of interest in such problems is the cavity aspect ratio. The upper limit for the Rayleigh number (based on cavity width) presented in the present paper is 10⁷. The flow patterns are obtained for Prandtl numbers in the range 10^?2 ? Pr ? 10³ and for aspect ratios 1, 10, 20. Where possible the results are compared with existing solutions obtained using the finite difference method. A satisfactory correlation exists where such comparisons can be made. The results complement and extend those obtained during previous theoretical and numerical investigations.     

A comparison of the method of lines to finite difference techniques in solving time-dependent partial differential equations

Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (i) Burger's equation over a finite space domain by a forward time—central space explicit method, and (ii) the stream function—vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to “set up” time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.     

Fluid flows in microchannels with cavities

Pressure-driven gas and liquid flows through microchannels with cavities have been studied using both experimental measurements and numerical computations. Several microchannels with cavities varying in shape, number and dimensions have been fabricated. One set of microdevices, integrated with sensors on a silicon wafer, is used for flow rate and pressure distribution measurements in gas flows. Another set of microdevices, fabricated using glass-to-silicon wafer bonding, is utilized for visualization of liquid flow patterns. The cavity effect on the flow in the microchannel is found to be very small, with the mass flow rate increasing slightly with increasing number of cavities. The flow pattern in the cavity depends on two control parameters; it is fully attached only if both the reduced Reynolds number and the cavity number are small. A flow regime map has been constructed, where the critical values for the transition from attached to separated flow are determined. The numerical computations reveal another control parameter, the cavity aspect ratio. The flow in the cavity is similar only if all three control parameters are the same. Finally, the vorticity distribution and related circulation in the cavity are analyzed. [1546].     

Numerical simulation of 2D square driven cavity using fourth-order compact finite difference schemes

Fourth-order compact finite difference schemes are employed with multigrid techniques to simulate the two-dimensional square driven cavity flow with small to large Reynolds numbers. The governing Navier-Stokes equation is linearized in streamfunction and vorticity formulation. The fourth-order compact approximation schemes are coupled with fourth-order approximations for velocities and vorticity boundaries. Numerical solutions are obtained for square driven cavity flow at high Reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods.     

Flow over a bluff body from moderate to high Reynolds numbers using large eddy simulation

Large eddy simulation (LES) is performed to study the uniform approach flow over a square-section cylinder with different Reynolds numbers, ranging from 10³ to 5 × 10⁶. Two different sub-grid scale models, the Smagorinsky and a dynamic one-equation model, are employed. An incompressible finite-volume code, based on a non-staggered grid arrangement and an implicit fractional step method with second-order accuracy in space and time, is used.

The structure of the flow is studied with the instantaneous and the mean quantities such as pressure, turbulent stresses, turbulent kinetic energy, vorticity, the second invariant of velocity gradient and streamlines. The Strouhal number, the mean and RMS values of the lift and drag are computed for various Reynolds numbers, which show a good agreement with the available experimental results. It is found that the effect of Reynolds number on the global quantities, the mean and the large scale instantaneous flow-structures is not much at the higher Reynolds numbers, i.e. Re > 2 × 10⁴. In this range of Reynolds numbers, the small scales of the instantaneous structures are more complex and chaotic as they compare with the larger ones.     

The 2D lid-driven cavity problem revisited

Numerical simulations of the 2D lid-driven cavity flow are performed for a wide range of Reynolds numbers. Accurate benchmark results are provided for steady solutions as well as for periodic solutions around the critical Reynolds number. Numerous comparisons with the results available in the literature are given. The first Hopf bifurcation is localized by a study of the linearized problem.     

Numerical solution of the incompressible Navier–Stokes equations with exponential type schemes

A new exponential type finite-difference scheme of second-order accuracy for solving the unsteady incompressible Navier–Stokes equation is presented. The driven flow in a square cavity is used as the model problem. Numerical results for various Reynolds numbers are given, and are in good agreement with those presented by Ghia et al. (Ghia, U., Ghia, K.N. and Shin, C.T., 1982, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multi-grid method. Journal of Computational Physics, 48, 387–411.).     

Initiation and structure of axisymmetric eddies in a rotating stream

Axisymmetric flow of a rotating stream is examined numerically to determine conditions under which an isolated eddy will form on the axis of rotation. An explicit finite difference procedure is used to integrate the time dependent transport equations. Solutions provide details of the flow structure and are presented for a range of Reynolds numbers and swirl ratios. Calculated results are interpreted in terms of recent physical experiments insofar as is possible.     

A finite-difference scheme for the solution of the steady Navier-Stokes equations

A new finite-difference scheme of second-order accuracy, free of artificial diffusion, for solving the steady-state incompressible Navier-Stokes equations is presented. The scheme uses a false transient approach with a combination of variable time step size and over-relaxation for the convection and diffusion terms. The driven flow in a square cavity is used as the model problem. The convergence is much better than the conventional Gauss-Seidel type iterative process. Results for Reynolds numbers up to 5000 are presented.     

Surface effects on transient three-dimensional flows around rotating spheres at moderate Reynolds numbers

Transient wake flow patterns and dynamic forces acting on a rotating spherical particle with non-uniform surface blowing are studied numerically for Reynolds numbers up to 300 and dimensionless angular velocities up to Ω=1. This range of Reynolds numbers includes the three distinct wake regimes i.e., the steady axisymmetric, the steady non-symmetrical and the unsteady with vortex shedding. The Navier–Stokes equations for an incompressible viscous flow are solved by a finite volume method in a three-dimensional, time accurate manner. An interesting feature associated with particle rotation and surface blowing is that they can affect the near wake structure in such a way that unsteady three-dimensional wake flow with vortex shedding develops at lower Reynolds numbers as compared to flow over a solid sphere in the absence of these effects and thus, vortex shedding occurs even at Re=200. Global properties, such as the lift and drag coefficients, and the Strouhal number are also significantly affected. It is shown that the present data for the average lift and drag coefficients correlate well with:

C_L/(1+Ω)^3.6=0.11

C_D(1+20V_S)^0.2/(1+Ω)^Re/1000=24(1+Re^2/3/6)/Re

where V_S is the average surface blowing velocity normalized by the free stream velocity.     

The effects of inertia on the structure of viscous corner eddies

Viscous eddies in the region close to a sharp corner are examined. The asymmetry in their structure that is apparent in numerical solutions for moderate values of the Reynolds number is derived analytically. A comparison is given with previous numerical studies and the agreement is found to be good. Some numerical verification of the analytical results is obtained from a study of the driven cavity flow problem for Reynolds numbers in the range 0–1000.