Multiple vortex formation in steady laminar axisymmetric impinging flow

Steady confined laminar axisymmetric impinging flow of a Newtonian fluid is relevant in many situations, an important application being heat and mass transfer from a solid surface to an impinging jet. This paper focuses on the evolution of the structure of the radial flow field in the channel region beyond the impingement zone. We employ an upwind scheme with an established numerical technique to solve the stream function and vorticity equations for a range of Reynolds numbers Re and geometrical aspect ratios e. Our results show the progressive complexity in the radial flow due to multiple points of flow separation and reattachment, and we provide a detailed demarcation of the Re-e plane based on flow separation behavior. In addition to the primary and secondary vortices anchored on the confining and impinging surfaces, respectively, we describe the formation and properties of a tertiary vortex which is wholly enclosed within the primary vortex. At a fixed Reynolds number, the tertiary vortex is observed only for a specific range of the aspect ratio, and we catalog its birth, growth and demise as the aspect ratio is varied. The range of aspect ratios over which the tertiary vortex exists is seen to increase with the Reynolds number. These results show that the fine structure of the radial flow at high Reynolds number continues to be dependent on the aspect ratio in a complex manner. At a given aspect ratio, the sizes of the vortices increases with Reynolds number, scaling as ∼Re^1/3, and for sufficiently large Re, the length of the tertiary vortex can exceed that of the secondary vortex. The primary and secondary vortex lengths satisfy an asymptotic relationship independent of Re and e, the numerically computed value of α being ∼2. Similarly, the locations of these vortices bear simple linear relationships independent of Re and e. Furthermore, despite the complex fine structure of the flow field, macroscopic flow properties such as vortex circulation and excess pressure loss continue to exhibit relatively simple dependence on Re and e, in accordance with previous results at much lower Reynolds numbers. Finally, some comments are made regarding the possibility of additional cascaded or isolated vortices occurring at even higher Reynolds numbers and aspect ratios.     

Early induction of secondary vortices for micromixing enhancement

In the present work a new type of micromixer has been proposed and its mixing characteristic has been analyzed. The micromixer can be viewed as a U-tube with a side inlet. Here micromixing is enhanced by the secondary vortex generation induced by the curvature of the tube. The flow in the mixer geometry is investigated theoretically to understand micro-mixing using computational fluid dynamics (CFD). For this we use the Navier–Stokes equations coupled with species transport. Mixing is quantified using mixing quality which is a measure of the uniformity of the concentration in a given geometry. Special attention is paid to the occurrence of the secondary vortices close to the mid point of the outer wall and its role in mixing. Simulations are also done to study the flow in U shaped channels. The simulation results show that the new design leads to an early introduction of secondary vortices than a simple U tube. Thus in the new design the secondary vortices are induced at a Re = 120 as opposed to the classical value of Re = 400 (when there is no side inlet) reported in the literature. Mixing is studied for different diffusivities and combination of inlet velocities. We also compare the performance of our design with the classical T and Y mixers. The early induction implies that we can have good mixing at low Re. Consequently, when used as a micro-reactor we can combine good mixing with high residence times to obtain good conversions in our system.     

Inertial migration of single particle in a square microchannel over wide ranges of <Emphasis Type="Italic">Re</Emphasis> and particle sizes

Inertial migration of particles has been widely used in inertial microfluidic systems to passively manipulate cells/particles. However, the migration behaviors and the underlying mechanisms, especially in a square microchannel, are still not very clear. In this paper, the immersed boundary-lattice Boltzmann method (IB-LBM) was introduced and validated to explore the migration characteristics and the underlying mechanisms of an inertial focusing single particle in a square microchannel. The grid-independence analysis was made first to highlight that the grid number across the thin liquid film (between a particle and its neighboring channel wall) was of significant importance in accurately capturing the migrating particle’s dynamics. Then, the inertial migration of a single particle was numerically investigated over wide ranges of Reynolds number (Re, from 10 to 500) and particle sizes (diameter-to-height ratio a/H, from 0.16 to 0.5). It was interesting to find that as Re increased, the channel face equilibrium (CFE) position moved outward to channel walls at first, and then inflected inwards to the channel center at high Re (Re?>?200). To account for the physical mechanisms behind this behavior, the secondary flow induced by the inertial focusing single particle was further investigated. It was found that as Re increased, two vortices appeared around the particle and grew gradually, which pushed the particle away from the channel wall at high Re. Finally, a correlation was proposed based on the numerical data to predict the critical length L_c (defined to describe the size of fluid domain that was strongly influenced by the particle) according to the particle size a/H and Re.     

Numerical simulation of the flow around rows of cylinders

Two-dimensional, laminar, unsteady, water flow around cylinder arrays of unequal sizes was simulated using FLUENT™ at Reynolds numbers below 150 (based on the free-stream velocity and first row cylinder diameter). The flow pattern through two rows of inline cylinders showed incomplete vortex shedding behind the first row at a separation distance less than 2d. Karman vortices were not formed and a near-stagnant separated flow region appeared between the aligned cylinders. Cylinders in staggered arrangements shed Karman vortices regardless of the separation between the two rows. This research has shed light on the detailed flow through paper machine forming fabrics.     

Numerical modeling of interaction of a current with a circular cylinder near a rigid bed

The numerical modeling of 2D turbulent flow around a smooth horizontal circular cylinder near a rigid bed with gap ratio G/D = 0.3 at Reynolds number Re_D = 9500 is investigated. Ansys^® 10.0-FLOTRAN program package is used to solve the governing equations by FEM, and the performance of the standard k ? ε, standard k ? ω, and SST turbulence models are examined. A sensitivity study for the three turbulence models is carried out on three computational meshes with different densities near the cylinder surface. The computational velocity fields and the Strouhal numbers from the present simulations are compared with those obtained from the PIV measurement. It is found that the time-averaged velocity field of the flow in the proximity of the cylinder is closely affected by the mesh resolution near the cylinder surface, and the mesh refinement in radial direction improves the results of present simulations. The shedding of vortices in the cylinder wake is not predicted by k ? ε model on all the three meshes. The results for the time-averaged velocity field show that the numerical modeling using either of k ? ω and SST turbulence models on the finest mesh used on the cylinder surface is reasonably successful.     

Influence of adjusting the inlet channel confluence angle on mixing behaviour in inertial microfluidic mixers

Recent drive for high-throughput microfluidic systems has triggered tremendous research effort to develop efficient, high-throughput microfluidic mixers. In particular, inducing a fluid–fluid collision at high flow rate in microfluidic channel has been suggested as an effective strategy to enhance mixing. However, previous studies using T-shaped microfluidic mixers showed that, in addition to fluid–fluid collision, the confluence angle of fluid stream in microfluidic channel also has a dramatic effect on mixing. This study suggests the possibility to enhance mixing by simply changing the inlet confluence angle of the streams. In this work, we assess the mixing behaviour of microfluidic mixers with variable inlet confluence angle with the Reynolds number (Re) range of 2.83–566. It is shown that the increase in inlet confluence angle enables the reduction of Re required for complete mixing. Simulation results demonstrate that increasing the confluence angle facilitates the interaction of vortices in mixers to induce an enhanced mixing. We further demonstrate that the increased interaction of vortices also prompts the turbulent emulsification where a significant reduction in emulsion size is observed for each mixer with increased inlet confluence angle at same Re.     

Variational multiscale large-eddy simulations of the flow past a circular cylinder: Reynolds number effects

Variational multiscale large-eddy simulations (VMS–LES) of the flow around a circular cylinder are carried out at different Reynolds numbers in the subcritical regime, viz. Re = 3900, 10,000 and 20,000, based on the cylinder diameter. A mixed finite-element/finite-volume discretization on unstructured grids is used. The separation between the largest and the smallest resolved scales is obtained through a variational projection operator and finite-volume cell agglomeration. The WALE subgrid scale model is used to account for the effects of the unresolved scales; in the VMS approach, it is only added to the smallest resolved ones. The capability of this methodology to accurately predict the aerodynamic forces acting on the cylinder and in capturing the flow features are evaluated for the different Reynolds numbers considered. The sensitivity of the results to different simulation parameters, viz. agglomeration level and numerical viscosity, is also investigated at Re = 20,000.     

Flow patterns past two circular cylinders in proximity

Flow patterns past two nearby circular cylinders of equal diameter immersed in the cross-flow at low Reynolds numbers (Re ? 160), were numerically studied using an immersed boundary method. We considered all possible arrangements of the two cylinders in terms of the distance between the two cylinders and the inclination angle of the line connecting the cylinder centers with respect to the direction of the main flow. Ten distinct flow patterns were identified in total based on vorticity contours and streamlines, which are Steady, Near-Steady, Base-Bleed, Biased-Base-Bleed, Shear-Layer-Reattachment, Induced-Separation, Vortex-Impingement, Flip-Flopping, Modulated Periodic, and Synchronized-Vortex-Shedding. Collecting all the numerical results obtained, we propose a general flow-pattern diagram for each Re, and a contour diagram on vortex-shedding frequency for each cylinder at Re = 100. The perfect symmetry implied in the geometrical configuration allows one to use these diagrams to identify flow pattern and vortex-shedding frequencies in the presence of two circular cylinders of equal diameter arbitrarily positioned in physical space with respect to the main-flow direction.     

Transition in a 2-D lid-driven cavity flow

Direct numerical simulations about the transition process from laminar to chaotic flow in square lid-driven cavity flows are considered in this paper. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasi-periodic, inverse period-doubling, period-doubling, and chaotic self-sustained flow regimes. The numerical experiments are conducted by solving the 2-D incompressible Navier-Stokes equations with increasing Reynolds numbers (Re). The spatial discretization consists of a seventh-order upwind-biased method for the convection term and a sixth-order central method for the diffusive term. The numerical experiments reveal that the first Hopf bifurcation takes place at Re equal to 7402±4%, and a consequent periodic flow with the frequency equal to 0.59 is obtained. As Re is increased to 10,300, a new fundamental frequency (FF) is added to the velocity spectrum and a quasi-periodic flow regime is reached. For slightly higher Re (10,325), the new FF disappears and the flow returns to a periodic regime. Furthermore, the flow experiences an inverse period doubling at 10,325 <Re< 10,700 and a period-doubling regime at 10,600 <Re< 10,900. Eventually, for flows with Re greater than 11,000, a scenario for the onset of chaotic flow is obtained. The transition processes are illustrated by increasing Re using time-velocity histories, Fourier power spectra, and the phase-space trajectories. In view of the conducted grid independent study, the values of the critical Re presented above are estimated to be accurate within ±4%.     

Thermohydraulics of square-section microchannels following a serpentine path

Fully developed laminar flow and heat transfer behaviour in serpentine channels with a square cross-section has been studied using computational fluid dynamics. Studies were performed up to Re=200, beyond which the flow became unsteady. The effect of geometric configuration was examined in detail for Re=110, 0.525<R _c/d<2 and 3.6<L/d<12 (where d is the side length of the square section, R _c is radius of curvature of the serpentine bends, and L is the half-wavelength of the serpentine path). Simulations were carried out at (Pr=0.7, 6.13 and 100) constant wall heat flux (H2 boundary condition) and constant wall temperature (T boundary condition). Dean vortices formed at the bends promote fluid mixing transverse to the main flow direction. This leads to significant heat transfer enhancement (up to a factor of 8 at high Pr and Re) with relatively small pressure-drop penalty (factor of 1.8 at high Re). Increasing R _c/d mitigates these effects while the effect of increasing L/d decreases the frictional penalty without greatly affecting the heat transfer enhancement.     

Space-time discretizing optimal DRP schemes for flow and wave propagation problems

The present paper reports constrained optimization of explicit Runge–Kutta (RK) schemes, coupled with optimal upwind compact scheme to achieve dispersion relation preservation (DRP) property for high performance computing. Essential ideas of optimization employed in arriving at the proposed time integration scheme are extension of the earlier work reported in Rajpoot et al. (J Comput Phys 2010;229:3623–51). This is in turn an application of the correct error evolution equation in Sengupta et al. (J Comput Phys 2007;226:1211–8). Resultant DRP scheme demonstrated the idea for explicit spatial central difference schemes. Present work is similar, extending it for near-spectral accuracy compact schemes. Practical utility of the developed method is demonstrated by solution of model problems and for flow problems by solving Navier–Stokes equation, some of which cannot be solved by conventional schemes, as the problem of rotary oscillation of cylinder.Developed method is calibrated with: (i) flow past a circular cylinder performing rotary oscillation at Re = 150 and (ii) flow inside a 2D lid-driven cavity (LDC) at Reynolds numbers of Re = 1000 and Re = 10,000. Quantitative and qualitative comparisons show excellent match for rotary oscillation cylinder cases with the experimental results of Thiria et al. (J Fluid Mech 2006;560:123–47). Results for LDC for Re = 1000 are compared with that in Botella & Peyret (Comp Fluids 1998;27:421–33) and results for Re = 10,000 are compared with recent published ones showing triangular vortex in the core.     

Numerical simulation of the flow field in a model of the nasal cavity

Results of a numerical simulation of the flow in a model of the human nasal cavity using an AUSM-based method of second-order accuracy on a multi-block structured grid are presented and compared with experimental data. Computations are performed for inspiration and expiration at rest with Reynolds numbers Re=1560 and Re=1230 at the nostril, respectively. The comparison shows good agreement with experimental findings.     

A Parallel Overset Grid High-Order Flow Solver for Large Eddy Simulation

This work describes the development and validation of a parallel high-order compact finite difference Navier–Stokes solver for application to large-eddy simulation (LES) and direct numerical simulation. The implicit solver can employ up to sixth-order spatial formulations and tenth-order filtering. The parallelization of the solver is founded on the overset grid technique. LES were then performed for turbulent channel flow with Reynolds numbers ranging from Re _τ=180 to 590, and flow past a circular cylinder with a transitional wake at Re _D =3900. The channel flow solutions were obtained using both an implicit LES (ILES) approach and a dynamic sub-grid scale model. The ILES method obtained virtually identical solutions at half the computational cost. The original vector and new parallel solvers produce indistinguishable mean flow solutions for the circular cylinder. Repeating the cylinder simulation on a much finer mesh resulted in significantly better agreement with experimental data in the near wake than the coarse grid solution and other previous numerical studies.     

DNS of a plane mixing layer for the investigation of sound generation mechanisms

Direct numerical simulations in two and three dimensions have been performed to investigate the sound generation by vortex pairing in a compressible plane mixing layer with Ma₁ = 0.5 being the upper and Ma₂ = 0.25 being the lower stream Mach number. The Reynolds number based on the vorticity thickness at the inflow and the velocity of the upper stream is Re=500. The flow is forced at the inflow with eigenfunctions obtained from viscous linear stability theory including three-dimensional disturbances. The results are verified with linear stability theory and the two-dimensional simulations performed by Colonius et al. [Colonius T, Lele SK, Moin P. Sound generation in a mixing layer. J Fluid Mech 1997;330:375-409]. The excitation of a steady longitudinal vortex mode leads to an early three-dimensional deformation of the travelling spanwise vortices and reduced sound emission to the slower fluid stream side.     

An experimental and numerical study of the structure and stability of laminar opposed-jet flows

Experiments, simulations, and numerical bifurcation analysis are used to study the incompressible flow between two opposed tubes with disks mounted at their exits. The experiments in this axisymmetric geometry show that for low and equal Reynolds numbers, Re, at both nozzles, the flow remains symmetric about the plane halfway through the nozzle exits and the stagnation plane is located halfway between the two jets. When Re is increased past a critical value, asymmetric flow fields are obtained even when the momentum fluxes of the two opposed streams are equal. For unequal Re at the jet exits, when the fixed velocity (and the corresponding Reynolds number, Re₁) of one stream is low, the stagnation plane location, SPL, changes smoothly with the Re₂. For high enough Re₁, a hysteretic jump of SPL is observed. Particle Image Velocimetry and flow visualization demonstrate that within the hysteretic range, the two stable flow fields are anti-symmetric. The experimental setup is also studied with transient incompressible flow simulations using a spectral element solver. It is found that to accurately model the flow, we either need to extend the domain into the nozzles, or impose experimental velocity profiles at the nozzle exits. As in the experiments asymmetric flows are obtained past a critical Re. Finally, bifurcation analysis using a Newton-Picard method shows that the transition from symmetric to asymmetric flows results from the loss of stability of the symmetric flows at a pitchfork bifurcation.     

Efficient treatment of complex geometries for large eddy simulations of turbulent flows

Incompressible turbulent flow over a backward facing step at Re_h=5100 is investigated by large eddy simulations (LES). The ratio of the oncoming boundary layer thickness δ to the step height h was set to 1.2. Additionally channel flows at various Re_τ numbers are presented for the validation of the numerical code. The results are compared with existing DNS and experimental databases. The present study focuses on different procedures for LES of engineering problems in complex geometries using structured rectangular grids. Two different methods that are able to treat complex geometrical configurations are implemented, examined and compared; namely the domain decomposition approach based on Schur’s complement and the immersed boundary method. In the present study both methods make use of a fast direct Poisson’s pressure solver based on a heavily modified version of the public domain package FISHPAK. The latter was optimised and fully parallelised for shared memory architectures, for solutions on rectangular grids stretched in one or two directions. The resulting code reaches performances of 1.0 μs/node/iter, allowing low cost computations on grids of the order of million points. The main objective of the present study was to investigate the potential of different methods for LES in complex geometrical configurations like bluff body flows and wakes. One of the main findings is that careful selection of numerical methods and implementation techniques can lead to accurate and very efficient codes, where the geometric complexity does not lead to algorithmic or numerical complexity.     

Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations

In this paper, we propose a new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier-Stokes equations. This new family of numerical methods has first been proposed in [16] for purely hyperbolic systems and has been called P_NP_M schemes, where N indicates the polynomial degree of the test functions and M is the degree of the polynomials used for flux and source computation. A particular feature of the general P_NP_M schemes is that they contain classical high order accurate finite volume schemes (N=0) as well as standard discontinuous Galerkin methods (M=N) just as special cases, which therefore allows for a direct efficiency comparison.In the application section of this paper we first show numerical convergence results on unstructured meshes obtained for the compressible Navier-Stokes equations with Sutherland’s viscosity law, comparing all third to sixth order accurate P_NP_M schemes with each other. In order to validate the method also in practice we show several classical steady and unsteady CFD applications, such as the laminar boundary layer flow over a flat plate at high Reynolds numbers, flow past a NACA0012 airfoil, the unsteady flows past a circular cylinder and a sphere, the unsteady flows of a compressible mixing layer in two space dimensions and finally we also show applications to supersonic flows with shock Mach numbers up to M_s=10.     

MHD flow past a circular cylinder using the immersed boundary method

The immersed boundary method (IB hereafter) is an efficient numerical methodology for treating purely hydrodynamic flows in geometrically complicated flow-domains. Recently Grigoriadis et als. [1] proposed an extension of the IB method that accounts for electromagnetic effects near non-conducting boundaries in magnetohydrodynamic (MHD) flows. The proposed extension (hereafter called MIB method) integrates naturally within the original IB concept and is suitable for magnetohydrodynamic (MHD) simulations of liquid metal flows. It is based on the proper definition of an externally applied current density field in order to satisfy the Maxwell equations in the presence of arbitrarily-shaped, non-conducting immersed boundaries. The efficiency of the proposed method is achieved by fast direct solutions of the two poisson equations for the hydrodynamic pressure and the electrostatic potential.The purpose of the present study is to establish the performance of the new MIB method in challenging configurations for which sufficient details are available in the literature. For this purpose, we have considered the classical MHD problem of a conducting fluid that is exposed to an external magnetic field while flowing across a circular cylinder with electrically insulated boundaries. Two- and three-dimensional, steady and unsteady, flow regimes were examined for Reynolds numbers Re_d ranging up to 200 based on the cylinder’s diameter. The intensity of the external magnetic field, as characterized by the magnetic interaction parameter N, varied from N=0 for the purely hydrodynamic cases up to N=5 for the MHD cases. For each simulation, a sufficiently fine Cartesian computational mesh was selected to ensure adequate resolution of the thin boundary layers developing due to the magnetic field, the so called Hartmann and sidewall layers. Results for a wide range of flow and magnetic field strength parameters show that the MIB method is capable of accurately reproducing integral parameters, such as the lift and drag coefficients, as well as the geometrical details of the recirculation zones. The results of the present study suggest that the proposed MIB methodology provides a powerful numerical tool for accurate MHD simulations, and that it can extend the applicability of existing Cartesian flow solvers as well as the range of computable MHD flows. Moreover, the new MIB method has been used to carrry out a series of accurate simulations allowing the determination of asymptotic laws for the lift and drag coefficients and the extent of the recirculation length as a function of the amplitude of the magnetic field. These results are reported herein.     

Cell pattern and stagnation ring of the flow driven by the counter-rotation in a fluid-filled cylinder

For the flow driven by the counter-rotation between the top and bottom endwalls in a fluid-filled cylinder, a stagnation ring can be observed on the slower rotating endwall in experiment. Its appearance corresponds to a two-cell flow pattern in the meridional plane, where a flow separation forms in the Ekman boundary layer. In this paper we numerically show that, in addition to the single-cell and two-cell patterns previously studied, there exist more complex cell patterns, namely, three-cell and merged-cell patterns, when the flow is driven under differently counter-rotating manner that is realized between the top and bottom endwalls as a whole against the sidewall. Such a counter-rotating flow makes the stagnation ring to appear simultaneously on both top and bottom endwalls rather than just on the slower rotating endwall. Moreover, the three-cell and merged-cell patterns, which are formed by a combination of the Ekman layer separation with the “vortex breakdown bubble”, are unique characteristics. The appearance of the cell pattern and stagnation ring is primarily decided by the counter rotation-rate ratio s, but is also affected by the Reynolds number Re and height-to-radius aspect ratio Λ, so a cell-pattern zone and a stagnation ring zone are proposed numerically as a function of s and Re for a given Λ.     

Lattice Boltzmann simulation of lid-driven flow in trapezoidal cavities

In this paper, an incompressible lattice Bhatnagar–Gross–Krook (LBGK) model proposed by Guo et al. is used to simulate lid-driven flow in a two-dimensional isosceles trapezoidal cavity. Due to the complex boundary of the trapezoidal cavity, here the extrapolation scheme proposed by Guo et al. is used to treat curved boundary. In our numerical simulations, the effects of the Reynolds number (Re) and the top angle θ on the strength, center position and number of vortices in the isosceles trapezoidal cavities are studied. Re is varied from 100 to 15,000, and the top angle θ ranges from 50 to 90. Numerical results show that, as Re increases, the phenomena in the cavity become more and more complex, and the number of the vortexes increases. We also found that the vortex near the bottom wall breaks up into two smaller vortices as θ increases up to a critical value. Furthermore, as Re is increased, the flow in the cavity undergoes a complex transition (from steady to the periodic flow, and finally to the chaotic flow). At last, the scope of critical Re for flow transition from steady to periodic state, and from periodic to chaotic state is presented for different top angles θ.