Laminar co-rotating Batchelor vortex merging

The dynamics of laminar co-rotating vortex pairs without axial flow have been recently thoroughly studied through theoretical, experimental and numerical studies, which revealed different instabilities contributing to the decay of the vortices. In this paper, the objective is to extend the analysis to the case of co-rotating vortices with axial flow at low Reynolds numbers. A high-order incompressible Navier–Stokes flow solver is used. The momentum equations are spatially discretized on a staggered mesh by finite differences and all derivatives are evaluated with 10th order compact finite difference schemes with RK-4 temporal discretization. The initial condition is a linear superposition of two co-rotating circular Batchelor vortices with q = 1. It is found that there is an initial evolution that resembles the evolution that single q = 1 vortices go through. Azimuthal disturbances grow and result in the appearance of large-scale helical sheets of vorticity. With the development of these instability waves, the axial velocity deficit is weakened. The redistribution of both angular and axial momentum between the core and the surroundings drives the vortex core to a more stable configuration, with a higher q value. After these processes, the evolution is somewhat similar to a pair of co-rotating Lamb–Oseen vortices. A three-dimensional instability develops, with a large band of unstable modes, with the most amplified mode corresponding scaling with the vortex initial separation distance. P. J. S. A. Ferreira de Sousa wishes to acknowledge the support of FCT—SFRH/BD/1129/2000 and SFRH/BPD/21778/2005.     

Numerical study of eccentric Couette–Taylor flows and effect of eccentricity on flow patterns

In this study, the differential quadrature (DQ) method was used to simulate the eccentric Couette–Taylor vortex flow in an annulus between two eccentric cylinders with rotating inner cylinder and stationary outer cylinder. An approach combining the SIMPLE (semi-implicit method for pressure-linked equations) and DQ discretization on a non-staggered mesh was proposed to solve the time-dependent, three-dimensional incompressible Navier–Stokes equations in the primitive variable form. The eccentric steady Couette–Taylor flow patterns were obtained from the solution of three-dimensional Navier–Stokes equations. The reported numerical results for steady Couette flow were compared with those from Chou [1], and San and Szeri [2]. Very good agreement was achieved. For steady eccentric Taylor vortex flow, detailed flow patterns were obtained and analyzed. The effect of eccentricity on the eccentric Taylor vortex flow pattern was also studied.     

Fast numerical evaluation of flow fields with vortex cells

A vortex cell (in this paper) is an aerodynamically shaped cavity in the surface of a body, for example a wing, designed specially to trap the separated vortex within it, thus preventing large-scale unsteady vortex shedding from the wing. Vortex stabilisation can be achieved either by the special geometry, as has already been done experimentally, or by a system of active control. In realistic conditions the boundary and mixing layers in the vortex cell are always turbulent. In the present study a model for calculating the flow in a vortex cell was obtained by replacing the laminar viscosity with the turbulent viscosity in the known high-Reynolds-number asymptotic theory of steady laminar flows in vortex cells. The model was implemented numerically and was shown to be faster than solving the Reynolds-averaged Navier–Stokes equations. An experimental facility with a vortex cell was built and experiments performed. Comparisons of the experimental results with the predictions of the model are reasonably satisfactory. The results also indicate that at least for flows in near-circular vortex cells it is sufficient to have accurate turbulence models only in thin viscous layers, while outside the viscosity should only be small enough to make the flow effectively inviscid.     

Formula for the Flow Resistance Factor in a Pipe with a Sudden Expansion at Small Reynolds Numbers

A formula for the flow resistance factors in a pipe with a sudden expansion of the cross section at Reynolds numbers of 0.2 to 10 is obtained by numerical solution of the complete Navier–Stokes equations for incompressible fluids. The flow resistance factors obtained using the derived formula are compared to those found by numerical solution of the Navier–Stokes equations.     

Pollution dispersion analysis using the puff model with numerical flow field data

The computations of the flowfield and pollutant dispersion over a flat plate and the Russian hills of various slopes are described. The Gaussian plume and the puff model have been used to calculate concentration of pollutant. The Reynolds-averaged unsteady incompressible Navier–Stokes equation with low Reynolds k– model has been used to calculate the flowfield. The flow data of a flat plate and the Russian hills from Navier–Stokes equation solutions has been used as the input data for the puff model. The computational results of flowfield agree well with experimental results of both a flat plate and Russian hills. The concentration prediction by the Gaussian plume model and the Gaussian puff model also agrees fairly well with experiments.     

3D Steady and Unsteady Bifurcations in a Shock-wave/Laminar Boundary Layer Interaction: A Numerical Study

The principal objective of this paper is to study some unsteady characteristics of an interaction between an incident oblique shock wave impinging a laminar boundary layer developing on a plate plane. More precisely, this paper shows that some unsteadiness, in particular the low frequency unsteadiness, originate in a supercritical Hopf bifurcation related to the dynamics of the separated boundary layer and not necessarily to the coherent structures resulting from the turbulent character of the boundary layer crossing the shock wave. Numerical computations of a shock-wave/laminar boundary-layer interaction (SWBLI) have been compared with a classical test case (Degrez test case) and both two-dimensional and three-dimensional (3D) unsteady Navier–Stokes equations are numerically solved with an implicit dual time stepping for the temporal algorithm and high order AUSM+ scheme for the spatial discretization. A parametric study on the oblique shock-wave angle has been performed to characterize the unsteady behaviour onset. Finally, discussions and assumptions are made about the origin of the 3D low frequency unsteadiness.     

Unsteady PTV velocity field past an airfoil solved with DNS: Part 1. Algorithm of hybrid simulation and hybrid velocity field at <Emphasis Type="Italic">Re</Emphasis> ≈ 10<Superscript>3</Superscript>

We develop a hybrid unsteady-flow simulation technique combining direct numerical simulation (DNS) and particle tracking velocimetry (PTV) and demonstrate its capabilities by investigating flows past an airfoil. We rectify instantaneous PTV velocity fields in a least-squares sense so that they satisfy the equation of continuity, and feed them to the DNS by equating the computational time step with the frame rate of the time-resolved PTV system. As a result, we can reconstruct unsteady velocity fields that satisfy the governing equations based on experimental data, with the resolution comparable to numerical simulation. In addition, unsteady pressure distribution can be solved simultaneously. In this study, particle velocities are acquired on a laser-light sheet in a water tunnel, and unsteady flow fields are reconstructed with the hybrid algorithm solving the incompressible Navier–Stokes equations in two dimensions. By performing the hybrid simulation, we investigate nominally two-dimensional flows past the NACA0012 airfoil at low Reynolds numbers. In part 1, we introduce the algorithm of the proposed technique and discuss the characteristics of hybrid velocity fields. In particular, we focus on a vortex shedding phenomenon under a deep stall condition (α = 15°) at Reynolds numbers of Re = 1000 and 1300, and compare the hybrid velocity fields with those computed with two-dimensional DNS. In part 2, the extension to higher Reynolds numbers is considered. The accuracy of the hybrid simulation is evaluated by comparing with independent experimental results at various angles of attack and Reynolds numbers up to Re = 10⁴. The capabilities of the hybrid simulation are also compared with two-dimensional unsteady Reynolds-Averaged Navier–Stokes (URANS) solutions in part 2. In the first part of these twin papers, we demonstrate that the hybrid velocity field approaches the PTV velocity field over time. We find that intensive alternate vortex shedding past the airfoil, which is predicted by the two-dimensional DNS, is substantially suppressed in the hybrid simulation and the resultant flow field is similar to the PTV velocity field, which is projection of the three-dimensional velocity field on the streamwise plane. We attempt to identify the motion that originates three-dimensional flow patterns by highlighting the deviation of the PTV velocity field from the two-dimensional governing equations at each snapshot. The results indicate that the intensive spots of the deviation appear in the regions in which three-dimensional instabilities are induced in the shear layer separated from the pressure side.     

The merging of two co-rotating vortices: a numerical study

The merging of two-dimensional co-rotating vortices is analysed through direct numerical simulations at large Reynolds numbers. It is shown how the Reynolds number affects each of the three phases that characterise this phenomenon. In the first phase, we examine the merging onset and focus on its definition. During the second rapid phase, the contributions of various flow regions upon the dynamics of a vortex are quantitatively studied. These regions are respectively the companion vortex, the filaments and an intermediate zone between vortices and filaments. The third phase is interpreted in terms of an advection diffusion process. Finally the final profile and circulation of the merged vortex is determined: the two thirds of the total circulation of the two initial vortices is contained in the newly formed vortex.     

Three-dimensional numerical study of jet-in-crossflow characteristics at low Reynolds number

A numerical study of a square jet in a cross flow is carried out at a Reynolds number of 100. The flow field and heat transfer characteristic downstream of the jet have been explored by solving three-dimensional unsteady Navier–Stokes equations and energy equation using higher order spatial and temporal discretization. The projection of vortical structure on a plane is seen to give the component of vortex normal to the plane. Four combinations of velocity profile namely (1) uniform crossflow and uniform jet, (2) laminar boundary layer crossflow and uniform jet, (3) uniform crossflow and parabolic jet profile, and (4) laminar boundary layer crossflow and parabolic jet are compared at same phase to see their effect on the flow field and heat transfer characteristic. All the four cases are seen to exhibit unsteadiness but the jet with parabolic profile is seen to show stronger unsteadiness. The instantaneous vortical structures of all the cases at the same phase show that the structures are more complex for the jet with parabolic velocity profile. The temperature field is seen to be correlated with the vortical structures. Comparison of the time averaged flow field reveals that the jet penetration is the highest for the jet having parabolic profile and boundary layer crossflow. The adiabatic effectiveness is observed to be more for the jet with uniform velocity profile and uniform crossflow and was least for the jet with parabolic velocity profile and boundary layer crossflow.     

Phase-resolved PIV measurements in a transitional plane wall jet: a numerical comparison

The transitional process of a forced plane wall jet is studied both experimentally and numerically. Experimentally, Particle Image Velocimetry (PIV) and laser-sheet/smoke flow-visualization techniques are implemented to provide an overall understanding of the flow features. Numerically, time-accurate computational results are obtained by solving the two-dimensional, unsteady Navier–Stokes equations. Comparison of PIV data and two-dimensional computed results shows excellent agreement in the early stages of transition, demonstrating that the numerical study can be used to complement the experimental one. The results show that, under the influence of external excitation, linear-instability growth is bypassed and a discrete shear-layer vortex is formed in the immediate vicinity of the nozzle exit. This vortex interacts with the boundary-layer vorticity, leading to the formation of another vortex in the inner layer. These two vortices form a vortex couple that for high forcing convects downstream in a stable manner. By adoption of either a no-slip or a slip boundary condition in the numerical computation, it is determined that the flow development is relatively insensitive to the imposed wall-boundary condition. This seems to suggest that the physical mechanism leading to the formation of the boundary-layer vortex is an inviscid rotational one. Received: 14 February 1998/Accepted: 11 August 1998     

Simulations of the unsteady separated flow past a normal flat plate

Well-resolved two-dimensional numerical simulations of the unsteady separated flow past a normal flat plate at low Reynolds numbers have been performed using a fractional step procedure with high-order spatial discretization. A fifth-order upwind-biased scheme is used for the convective terms and the diffusive terms are represented by a fourth-order central difference scheme. The pressure Poisson equation is solved using a direct method based on eigenvalue decomposition of the coefficient matrix. A systematic study of the flow has been conducted with high temporal and spatial resolutions for a series of Reynolds numbers. The interactions of the vortices shed form the shear layers in the near-and far-wake regions are studied. For Reynolds numbers less than 250 the vortices are observed to convect parallel to the freestream. However, at higher Reynolds numbers (500 and 1000), complex interactions including vortex pairing, tearing and deformations are seen to occur in the far-wake region. Values of the drag coefficient and the wake closure length are presented and compared with previous experimental and numerical studies.     

Numerical study of the unsteady flow and heat transfer in channels with periodically mounted square bars

 Numerical investigations of unsteady laminar flow and heat transfer in a channel of height H with periodically mounted square bars of height d = 0.2H arranged side by side to the approaching flow have been conducted for different transverse separation distances of the bars. Five cases with transverse separation distance of 0, 0.5, 1, 1.5 and 2d for a Reynolds number of 300 in a channel with a periodicity length of 2H were studied. The unsteady Navier–Stokes equations and the energy equation have been solved by a finite volume code with staggered grids combined with the SIMPLEC algorithm and a fine grid resolution. Due to the arrangement of bars detached from the channel walls the flow is unsteady with vortex shedding from the bars. The amplitude and mean values of the drag coefficients, skin friction coefficients, friction factor and Nusselt numbers have a strong dependence of the transverse separation distance of the bars. Received on 28 January 2000     

Simulation of dynamic stall for a NACA 0012 airfoil using a vortex method

The unsteady, incompressible, viscous laminar flow over a NACA 0012 airfoil is simulated, and the effects of several parameters investigated. A vortex method is used to solve the two-dimensional Navier–Stokes equations in the vorticity/stream-function form. By applying an operator-splitting method, the “convection” and “diffusion” equations are solved sequentially at each time step. The convection equation is solved using the vortex-in-cell method, and the diffusion equation using a second-order ADI finite difference scheme. The airfoil profile is obtained by mapping a circle in the computational domain into the physical domain through a Joukowski transformation. The effects of several parameters are investigated, such as the reduced frequency, mean angle of attack, location of pitch axis, and the Reynolds number. It is observed that the reduced frequency has the most influence on the flow field.     

Forced convection heat transfer from an equilateral triangular cylinder at low Reynolds numbers

An unsteady two-dimensional numerical simulation is performed to investigate the forced convection heat transfer for flow past a long heated equilateral triangular cylinder in an unconfined medium for the low Reynolds number laminar regime. The Reynolds number considered in this study ranges from 50 to 250 with three different values of Prandtl number (Pr?=?0.71, 7 and 100). Fictitious confining boundaries are chosen on the lateral sides of the computational domain that makes the blockage ratio β?=?5?% in order to make the problem computationally feasible. An unstructured triangular mesh is used for the computational domain discretization and the simulation is carried out with the commercial CFD solver Fluent. The flow and heat transfer characteristics are analyzed with the streamline and isotherm patterns at various Reynolds numbers. The dimensionless frequency of vortex shedding (Strouhal number), drag coefficient and Nusselt numbers are presented and discussed. The results obtained are in good agreement with the available results in the literature.     

Existence of Weak Solutions for the Unsteady Interaction of a Viscous Fluid with an Elastic Plate

The purpose of this work is to study the existence of solutions for an unsteady fluid-structure interaction problem. We consider a three-dimensional viscous incompressible fluid governed by the Navier–Stokes equations, interacting with a flexible elastic plate located on one part of the fluid boundary. The fluid domain evolves according to the structure’s displacement, itself resulting from the fluid force. We prove the existence of at least one weak solution as long as the structure does not touch the fixed part of the fluid boundary. The same result holds also for a two-dimensional fluid interacting with a one-dimensional membrane.     

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.     

Flow past a sphere in density-stratified fluid

Stratified flow past a three-dimensional obstacle such as a sphere has been a long-lasting subject of geophysical, environmental and engineering fluid dynamics. In order to investigate the effect of the stratification on the near wake, in particular, the unsteady vortex formation behind a sphere, numerical simulations of stratified flows past a sphere are conducted. The time-dependent Navier–Stokes equations are solved using a three-dimensional finite element method and a modified explicit time integration scheme. Laminar flow regime is considered, and linear stratification of density is assumed under Boussinesq approximation. The effects of stratification is implemented by density transport without diffusion. The computed results include the characteristics of the near wake as well as the effects of stratification on the separation angle. Under increased stratification, the separation on the sphere is suppressed and the wake structure behind the sphere becomes planar, resembling that behind a vertical cylinder. With further increase in stratification, the wake becomes unsteady, and consists of planar vortex shedding similar to von Karman vortex streets.     

Stability of the boundary layer of a non-Newtonian liquid obeying a power-type rheological law

The stability of the stationary (steady-state) laminar boundary layer of a non-Newtonian liquid obeying a power-type rheological law at a semiinfinite plate situated in a longitudinal flow is analyzed. An approximate formula is derived for estimating the minimum Reynolds number at which the flow loses stability with respect to slight two-dimensional perturbations. Calculations of the point of stability loss for aqueous solutions of carboxyl methyl cellulose are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–124, March–April, 1971.     

Experimental and numerical POD study of the Coanda effect used to reduce self-sustained tones

The two-dimensional flow along an inclined plate may be detached or reattached to the plate by Coanda effect. Experimentally, we explore the influence of the inclination angle and of the Reynolds number on the attachment and detachment phenomena, and on the hysteresis loop. A proper orthogonal decomposition (POD) of the flow is applied to a LES simulation resulting data. A low-dimensional dynamical model is obtained using by Galerkin projection of the Navier–Stokes equations upon the POD basis functions. We show that this model represents qualitatively the characteristics of the flow.