Numerical solution of laminar recirculating flow between shrouded rotating disks

In this paper we develop a method to obtain the numerical solution of the problem of recirculating flow between shrouded rotating disks. The major difficulty of this type of problem is in obtaining convergence at high Reynolds numbers. With the technique developed in this paper we have obtained convergence for Reynolds numbers up to 10,000. The procedure can be extended to higher Reynolds numbers if desired. The contours of the stream function, vorticity function and angular velocity are presented for Reynolds numbers of 500, 2000, 5000 and 10000. The method is applicable to any problem which has similar equations of motion.     

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.     

Lattice Boltzmann simulation of lid-driven flow in deep cavities

In this paper, the lattice Boltzmann equation (LBE) method is applied for simulation of lid-driven flow in a two-dimensional, rectangular, deep cavity. First, the code is validated for the standard square cavity, and then the results of a deep cavity are presented. Steady results are presented for deep cavities with aspect ratios of 1.5–4, and Reynolds numbers of 50–3200. Several features of the flow, such as the location and strength of the primary vortex, and the corner-eddy dynamics are investigated and compared with previous findings from experiments and theory. Steady results for deep cavities show the existence of corner eddies at the bottom, which coalesce to form a second primary-eddy as the cavity aspect-ratio is increased above a critical value. However, at relatively high Reynolds numbers, the second primary-eddy is formed via a rapid transition of an unsteady wall-eddy. The predicted results from LBE simulations are shown to be consistent with experiments and theory.     

Finite difference methods for viscous incompressible global stability analysis

In the present article some high-order finite-difference schemes and in particularly dispersion-relation-preserving (DRP) family schemes, initially developed by Tam and Webb [Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys. 107 (1993) 262-281.] for computational aeroacoustic problems, are used for global stability issue. (The term global is not used in weakly-non-parallel framework but rather for fully non-parallel flows. Some authors like Theofilis [Advances in global linear instability analysis of non-parallel and three-dimensional flows, Progress in Aerospace Sciences 39 (2003) 249-315] refer to this approach as “BiGlobal”.) These DRP schemes are compared with different classical schemes as second and fourth-order finite-difference schemes, seven-order compact schemes and spectral collocation scheme which is usually employed in such stability problems. A detailed comparative study of these schemes for incompressible flows over two academic configurations (square lid-driven cavity and separated boundary layer at different Reynolds numbers) is presented, and we intend to show that these schemes are sufficiently accurate to perform global stability analyses.     

Incompressible moving boundary flows with the finite volume particle method

Mesh-free methods offer the potential for greatly simplified modeling of flow with moving walls and phase interfaces. The finite volume particle method (FVPM) is a mesh-free technique based on interparticle fluxes which are exactly analogous to intercell fluxes in the mesh-based finite volume method. Consequently, the method inherits many of the desirable properties of the classical finite volume method, including implicit conservation and a natural introduction of boundary conditions via appropriate flux terms. In this paper, we describe the extension of FVPM to incompressible viscous flow with moving boundaries. An arbitrary Lagrangian–Eulerian approach is used, in conjunction with the mesh-free discretisation, to facilitate a straightforward treatment of moving bodies. Non-uniform particle distribution is used to concentrate computational effort in regions of high gradients. The underlying method for viscous incompressible flow is validated for a lid-driven cavity problem at Reynolds numbers of 100 and 1000. To validate the simulation of moving boundaries, flow around a translating cylinder at Reynolds numbers of 20, 40 and 100 is modeled. Results for pressure distribution, surface forces and vortex shedding frequency are in good agreement with reference data from the literature and with FVPM results for an equivalent flow around a stationary cylinder. These results establish the capability of FVPM to simulate large wall motions accurately in an entirely mesh-free framework.     

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.     

Development of a second order approximation for the Navier-Stokes equations

The purpose of this paper is the development of a 2nd order finite difference approximation to the steady state Navier-Stokes equations governing flow of an incompressible fluid in a closed cavity. The approximation leads to a system of equations that has proved to be very stable. In fact, numerical convergence was obtained for Reynolds numbers up to 20,000. However, it is shown that extremely small mesh sizes are needed for excellent accuracy with a Reynolds number of this magnitude. The method uses a nine point finite difference approximation to the convection term of the vorticity equation. At the same time it is capable of avoiding values at corner nodes where discontinuities in the boundary conditions occur. Figures include level curves of the stream and vorticity functions for an assortment of grid sizes and Reynolds numbers.     

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.     

Multiplicity of steady solutions in two-dimensional lid-driven cavity flows by Lattice Boltzmann Method

This work is concerned with the computation of two- and four-sided lid-driven square cavity flows and also two-sided rectangular cavity flows with parallel wall motion by the Lattice Boltzmann Method (LBM) to obtain multiple stable solutions. In the two-sided square cavity two of the adjacent walls move with equal velocity and in the four-sided square cavity all the four walls move in such a way that parallel walls move in opposite directions with the same velocity; in the two-sided rectangular lid-driven cavity flow the longer facing walls move in the same direction with equal velocity. Conventional numerical solutions show that the symmetric solutions exist for all Reynolds numbers for all the geometries, whereas multiplicity of stable states exist only above certain critical Reynolds numbers. Here we demonstrate that Lattice Boltzmann method can be effectively used to capture multiple steady solutions for all the aforesaid geometries. The strategy employed to obtain these solutions is also described.     

Multidimensional upwind scheme of diagonal Cartesian method for an advection-diffusion problem

The natural calculation region in fluid dynamics involves complex boundaries. When using the Cartesian grid to approximate complex boundaries, two difficulties develop: the boundary zigzag effect and disagreement of direction of grid line and velocity. The multidimensional upwind scheme of the diagonal Cartesian method (DCM), using both Cartesian grid lines and diagonal lines segments, is presented in this paper to simulate the complex boundaries of the multiple-layer quasi 3D model equations. The DCM improves the simulation accuracy for the boundaries and calculation time increases only slightly compared to the Cartesian method. In order to verify the new scheme, a test case is presented which rotates the cavity flow at 45° to compare the numerical calculation results at different Reynolds numbers. The test case shows that the scheme is accurate and efficient in improving the simulation results. Then the three-dimensional advection-diffusion processes in the tidal water of the Hongyanhe Power Plant are simulated using this model. Numerical results show that the scheme is not only efficient on an experiment basis, but also efficient and reliable when applied to a large scale natural water area.     

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.).     

Passive mixing in a three-dimensional serpentine microchannel

A three-dimensional serpentine microchannel design with a “C shaped” repeating unit is presented in this paper as a means of implementing chaotic advection to passively enhance fluid mixing. The device is fabricated in a silicon wafer using a double-sided KOH wet-etching technique to realize a three-dimensional channel geometry. Experiments using phenolphthalein and sodium hydroxide solutions demonstrate the ability of flow in this channel to mix faster and more uniformly than either pure molecular diffusion or flow in a “square-wave” channel for Reynolds numbers from 6 to 70. The mixing capability of the channel increases with increasing Reynolds number. At least 98% of the maximum intensity of reacted phenolphthalein is observed in the channel after five mixing segments for Reynolds numbers greater than 25. At a Reynolds number of 70, the serpentine channel produces 16 times more reacted phenolphthalein than a straight channel and 1.6 times more than the square-wave channel. Mixing rates in the serpentine channel at the higher Reynolds numbers are consistent with the occurrence of chaotic advection. Visualization of the interface formed in the channel between streams of water and ethyl alcohol indicates that the mixing is due to both diffusion and fluid stirring     

Transitional cylindrical swirling flow in presence of a flat free surface

This article is devoted to the study of an incompressible viscous flow of a fluid partly enclosed in a cylindrical container with an open top surface and driven by the constant rotation of the bottom wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry and of the prescribed boundary conditions of Dirichlet type—no-slip on the cavity walls. The top surface of the cylindrical cavity is left open with an imposed stress-free boundary condition, while a no-slip condition with a prescribed rotational velocity is imposed on the bottom wall. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new flow states that were investigated based on a fully three-dimensional solution of the Navier-Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a Legendre spectral element method.     

Boundary Element Modeling of Dynamics of a Bubble in Contact with a Solid Surface at Low Reynolds Numbers

In the present study, the dynamics of a bubble attached to the surface and driven by the acoustic field at low Reynolds numbers are considered. The approach is based on the boundary element method (BEM) for Stokes flows, which is especially effective for the numerical solution of problems in the three-dimensional case. However, the dynamics of computing compressible bubbles are difficult to formulate due to the degeneration of the conventional BEM for Stokes equations. In the present approach, an additional relation based on the Lorenz reciprocity principle is used to resolve the problem. To describe the contact line dynamics a semiempirical law of motion is used. Such an approach allows us to bypass the known issue of nonintegrability stresses in the moving triple point. The behavior of a bubble attached to the surface in the cases of a pinned or moving contact line is studied. The developed method can be used for the detailed study of bubble dynamics in contact with a solid wall in order to determine the optimal conditions and parameters of surface cleaning processes.     

An immersed-boundary method for compressible viscous flows

This paper combines a state-of-the-art method for solving the preconditioned compressible Navier-Stokes equations accurately and efficiently for a wide range of the Mach number with an immersed-boundary approach which allows one to use Cartesian grids for arbitrarily complex geometries. The method is validated versus well documented test problems for a wide range of the Reynolds and Mach numbers. The numerical results demonstrate the efficiency and versatility of the proposed approach as well as its accuracy, from incompressible to supersonic flow conditions, for moderate values of the Reynolds number. Further improvements, obtained via local grid refinement or non-linear wall functions, can render the proposed approach a formidable tool for studying complex three-dimensional flows of industrial interest.     

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.     

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.     

Topology optimization of internal partitions in a flow-reversing chamber muffler for noise reduction

A topology-optimization-based design method for a flow-reversing chamber muffler is suggested to maximize the transmission loss value at a target frequency considering flow power dissipation. Rigid partitions for high noise reduction should be carefully placed inside the muffler to avoid extreme flow power dissipation due to a 180° change in flow direction from an inlet to an outlet. The optimal flow path for minimum flow power dissipation is well known to change depending on the Reynolds number, which is a function of the inlet flow velocity. To optimize the partition layout with an optimal flow path in an expansion chamber at a given Reynolds number, a flow-reversing chamber muffler design problem is formulated by topology optimization. The formulated topology optimization problem is implemented using the finite element method with a gradient-based optimization algorithm and is solved for various design conditions such as the target frequencies, rigid partition volumes, Reynolds numbers, non-design domain settings, and allowed amounts of flow power dissipation. The effectiveness of our suggested approach is verified by comparing the optimized partition layouts obtained by the suggested method and previous methods.     

Explicit algebraic Reynolds stress model of incompressible turbulent flow in rotating square duct

An explicit algebraic Reynolds stress model (EARSM) is proposed for the simulation of the incompressible three-dimensional Reynolds averaged Navier-Stokes (RANS) equations. The spatial discretization of the RANS equations is performed by a finite volume method with nonstaggered variable arrangement.The EARSM model which accounts for rotational effects is used to compute the turbulent flows in rotating straight square duct. The Reynolds number of 48,000 is based on the bulk velocity and the hydraulic diameter of the duct and is kept constant in the range of the rotational numbers. The second order closure (EARSM) yields an asymmetric mean velocity profile as well as turbulence properties. Effects of rotation near the cyclonic (suction side) and anticyclonic (pressure side) walls are well observed. Direct numerical simulation and large eddy simulation data are available for this case. The comparison of EARSM results with these accurate simulations shows a very good agreement.