Hydrodynamics of a particle impact on a wall

The problem of a particle impacting on a wall, a common phenomenon in particle-laden flows in the minerals and process industries, is investigated computationally using a spectral-element method with the grid adjusting to the movement of the particle towards the wall. Remeshing is required at regular intervals to avoid problems associated with mesh distortion and the constantly reducing maximum time-step associated with integration of the non-linear convective terms of the Navier–Stokes equations. Accurate interpolation between meshes is achieved using the same high-order interpolation employed by the spectral-element flow solver. This approach allows the full flow evolution to be followed from the initial approach, through impact and afterwards as the flow relaxes. The method is applied to the generic two-dimensional and three-dimensional bluff body geometries, the circular cylinder and the sphere. The principal case reported here is that of a particle colliding normally with a wall and sticking. For the circular cylinder, non-normal collisions are also considered. The impacts are studied for moderate Reynolds numbers up to approximately 1200. A cylindrical body impacting on a wall produces two vortices from its wake that convect away from the cylinder along the wall before stalling while lifting induced wall vorticity into the main flow. The situation for a sphere impact is similar, except in this case a vortex ring is formed from the wake vorticity. Again, secondary vorticity from the wall and particle plays a role. At higher Reynolds number, the secondary vorticity tends to form a semi-annular structure encircling the primary vortex core. At even higher Reynolds numbers, the secondary annular structure fragments into semi-discrete structures, which again encircle and orbit the primary core. Vorticity fields and passive tracer particles are used to characterize the interaction of the vortical structures. The evolution of the pressure and viscous drag coefficients during a collision are provided for a typical sphere impact. For a Reynolds number greater than approximately 1000 for a sphere and 400 for a cylinder, the primary vortex core produced by the impacting body undergoes a short-wavelength instability in the azimuthal/spanwise direction. Experimental visualisation using dye carried out in water is presented to validate the predictions.     

Homostrophic vortex interaction under external strain,in a coupled QG-SQG model

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.     

An application of a boundary element method to natural convection

The direct boundary element method is applied to the numerical modelling of thermal fluid flow in a transient state. The Navier-Stokes equations are considered under the Boussinesq approximation and the viscous thermal flow equations are expressed in terms of stream function, vorticity, and temperature in two dimensions. Boundary integral equations are derived using logarithmic potential and time-dependent heat potential as fundamental solutions. Boundary unknowns are discretized by linear boundary elements and flow domains are divided into a series of triangular cells. Charged points are translated upstream in the numerical evaluation of convective terms. Unknown stream function, vorticity, and temperature are staggered in the computational scheme.

Simple iteration is found to converge to the quasi steady-state flow. Boundary solutions for two-dimensional examples at a Reynolds number 100 and Grashoff number 10⁷ are obtained.     

Transport of rarefied gases inside a vortex tube induced by a surface wave

We derive the entrainment of rarefied gases induced by a surface wave (SW) propagating along the wall or interface of the vortex core in a rather long straight vortex-tube (vortex or vorticity filament) by using the perturbation method. The values of the critical reflux pressure-gradient for time-averaged macroscopic flows of rarefied gases decrease as the Knudsen number increases from zero. Nonlinear coupling due to the boundary effect and the streaming flow of the rarefied gases is observed as the Reynolds number (based on the wave speed) increases to 50 and the wave number of the surface wave is larger. We also address the zero-flux states of our results. Received: November 22, 2004; revised: May 17, 2005     

Conforming Chebyshev Spectral Collocation Methods for the Solution of Laminar flow in a Constricted Channel

The numerical simulation of steady planar two-dimensional laminarflow of an incompressible fluid through an abruptly contractingchannel using spectral domain decomposition methods is described.The key features of the method are the decomposition of theflow region into a number of rectangular subregions and spectralapproximations that are pointwise C¹ continuous across subregioninterfaces. Spectral approximations to the solution are obtainedfor Reynolds numbers in the range [0, 500]. The size of thesalient corner vortex decreases as the Reynolds number increasesfrom 0 to around 45. As the Reynolds number is increased further,the vortex grows slowly. A vortex is detected downstream ofthe contraction at a Reynolds number of around 175 that continuesto grow as the Reynolds number is increased further.     

Analysis of Sound Radiated by Axisymmetric Vortex Ring

A theoretical and numerical study using the unsteady, 3D Navier-Stokes equations to generate axisymmetric vortex rings is reported. Increasing the vorticity, the vortex ring transition to a turbulent state are analyzed. After transition to a turbulent stage, the self-similarity of the temporal evolution of the flow is observed. Then we can compare sound radiated by the vortex ring to jet noise, at similar Reynolds number and low Mach number. The agreement between the simulation results and the simplified model is good. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A computational study of the error in the vortex method associated with discontinuous vorticity

We investigate computationally the error computed by the vortex method for a discontinuous patch of vorticity. Specifically, the computed velocity and vorticity of an elliptical path of constant vorticity, known as the Kirchhoff ellipse, are compared to the analytic velocity and vorticity. The error in the velocity and the vorticity for the Kirchhoff ellipse as computed by the vortex method is presented. This error is studied as a function of the aspect ratio of the ellipse, the blob function, the spacing between the centers of the computational elements, and the blob radius. Both the error at the initial time and the error after three revolutions of the ellipse are discussed. © 1996 John Wiley & Sons, Inc.     

Convergence of vortex methods for weak solutions to the 2-D euler equations with vortex sheet data

We prove the convergence of vortex blob methods to classical weak solutions for the two-dimensional incompressible Euler equations with initial data satisfying the conditions that the vorticity is a finite Radon measure of distinguished sign and the kinetic energy is locally bounded. This includes the important example of vortex sheets. The result is valid as long as the computational grid size h does not exceed the smoothing blob size ε, i.e., h/ε ≦ C.. ©1995 John Wiley & Sons, Inc.     

Algorithm for solving the Navier–Stokes equations for the modeling of creeping flows

We study a coupled algorithm for solving the two-dimensional Navier–Stokes equations in the stream function–vorticity variables. The algorithm is based on a finite-difference scheme in which the inertial terms in the vortex transport equation are taken from the lower time layer and the dissipative terms, from the upper time layer. In the linear approximation, we study the stability of this algorithm and use test computations to show its advantages when modeling flows at moderate Reynolds numbers.     

First-order accurate finite difference schemes for boundary vorticity approximations in curvilinear geometries

In this work we develop first-order accurate, forward finite difference schemes for the first derivative on both a uniform and a non-uniform grid. The schemes are applied to the calculation of vorticity on a solid wall of a curvilinear, two-dimensional channel. The von Mises coordinates are used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the derived finite difference schemes are used to calculate the vorticity at the computational boundary grid points using combinations of up to five computational domain grid points. This work extends previous work (Awartani et al., 2005) [3] in which higher-order schemes were obtained for the first derivative using up to four computational domain grid points. The aim here is to shed further light onto the use of first-order accurate non-uniform finite difference schemes that are essential when the von Mises transformation is used. Results show that the best schemes are those that use a natural sequence of non-uniform grid points. It is further shown that for non-uniform grid with clustering near the boundary, solution deteriorates with increasing number of grid points used. By contrast, when a uniform grid is used, solution improves with increasing number of grid points used.     

Formulation of boundary conditions for vorticity in viscous incompressible flow problems

For the stream function-vorticity formulation of the Navier-Stokes equations, vorticity boundary conditions are required on the body surface and the far-field boundary. A two-parameter approximating formula is derived that relates the velocity and vorticity on the outer boundary of the computational domain. The formula is used to construct an algorithm for correcting the conventional far-field boundary conditions. Specifically, a soft boundary condition is set for the vorticity and a uniform flux is specified for the transversal velocity. A third-order accurate three-parameter formula for the vorticity on the wall is derived. The use of the formula does not degrade the convergence of the iterative process of finding the vorticity as compared with a previously derived and tested two-parameter formula.     

Adaptive Wavelet Simulation of a Flow around an Impulsively Started Cylinder Using Penalisation

We present an adaptive wavelet method to integrate the velocity–vorticity formulation of the two-dimensional Navier–Stokes equations coupled with a penalisation technique to handle efficiently solid boundaries of arbitrary shape. We demonstrate the validity of this method, called coherent vortex simulation (CVS), to compute the flow around an impulsively started cylinder at high Reynolds number and compare the results with a classical vortex method.     

On Prandtl-Batchelor theory of a cylindrical eddy: Existence and uniqueness

For a steady laminar two-dimensional flow, Prandtl and Batchelor proposed a property in the case of a region of nested closed streamlines. This Prandtl-Batchelor(PB) theory claims the constancy of the vorticity in the limit of infinite Reynolds number R ( or vanishing viscosity n\nu ) within such a region. To establish this result rigorously, as a first step we here show that a boundary layer corresponding to the PB theory exists and is unique for the circular eddy under relatively small perturbations of the Euler limit wall velocity.     

Steady two-dimensional flow past a normal flat plate

A vorticity/stream function formulation is used to obtain a numerical simulation of steady two-dimensional flow of a viscous incompressible fluid past a normal flat plate for a range of Reynolds numbers. A method of Fornberg [J. Fluid Mech. 98, 819 (1980)] is used to determine upstream and downstream boundary conditions on the stream function. Special care is taken in the neighbourhood of the singularities in vorticity at the plate edges and this is very important because any errors introduced are swept downstream and severely affect such quantities as the length and width of the attached eddies. The computed results are compared with those of a laboratory experiment in which a plane strip is drawn through water and ethylene glycol for the range of Reynolds numbers for which the experimental flow is stable.     

On the existence of Burgers vortices for high Reynolds numbers

Axisymmetric or non-axisymmetric Burgers vortices have been studied numerically as a model of concentrated vorticity fields. Recently it has been rigorously proved that non-axisymmetric Burgers vortices exist for all values of the vortex Reynolds number if an asymmetry parameter is sufficiently small. On the other hand, several numerical results indicate that Burgers vortices have simpler structures as the vortex Reynolds number is increasing, even when the asymmetry parameter is not sufficiently small. In this paper we give a rigorous explanation for this numerical observation and extend the existence and stability results of Burgers vortices for high vortex Reynolds numbers.     

A Suggested Mechanism for Nonlinear Wall Roughness Effects on High Reynolds Number Flow Stability

The interactions between an uneven wall and free stream unsteadiness and their resultant nonlinear influence on flow stability are considered by means of a related model problem concerning the nonlinear stability of streaming flow past a moving wavy wall. The particular streaming flows studied are plane Poiseuille flow and attached boundary-layer flow, and the theory is presented for the high Reynolds number regime in each case. That regime can permit inter alia much more analytical and physical understanding to be obtained than the finite Reynolds number regime; this may be at the expense of some loss of real application, but not necessarily so, as the present study shows. The fundamental differences found between the forced nonlinear stability properties of the two cases are influenced to a large extent by the surprising contrasts existing even in the unforced situations. For the high Reynolds number effects of nonlinearity alone are destabilizing for plane Poiseuille flow, in contrast with both the initial suggestion of earlier numerical work (our prediction is shown to be consistent with these results nevertheless) and the corresponding high Reynolds number effects in boundary-layer stability. A small amplitude of unevenness at the wall can still have a significant impact on the bifurcation of disturbances to finite-amplitude periodic solutions, however, producing a destabilizing influence on plane Poiseuille flow but a stabilizing influence on boundary-layer flow.     

The Effect of the Backward-Facing Step Height on Flow in a Channel with Rectangular Cross Section

The investigation of a channel flow with a backward-facing step on one wall is described. The height of the incoming channel H is adjustable as to allow alter the ratio of step height h over H; h/H = 1 : 10, 1 : 4, 1.02 : 1(h + H = 0.275m). Mutual correlations between the extremes of static pressure distributions, limits of the separation zone and of the corner vortex structures and Reynolds number are discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigating Asymptotic Suction Boundary Layers using a One-Dimensional Stochastic Turbulence Model

The one-dimensional turbulence model (ODT) is applied to study turbulent asymptotic suction boundary layers for a Reynolds number of Re = u_∞/v₀ = 333, where u_∞ and v₀ are the free stream and suction velocity, respectively. In here we will demonstrate that a large eddy suppression mechanism may reduce the influence of ODT model parameters, such as the viscous cut-off parameter Z. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral Analysis of Large Scale Structures in Fully Developed Turbulent Pipe Flow

The present work focuses on spectral analysis of the streamwise velocity fluctuations obtained, utilizing the Cottbus Large Pipe (CoLaPipe) test facility for R⁺ ≤ 3500, where R⁺ is the Reynolds number based on the wall friction velocity, u_τ, and the pipe radius, R. Measurements of the streamwise spectra have been conducted using a single hot-wire probe. Few runs have been also performed using Particle Image Velocimetry (PIV) as a structure visualization evidence and for spatial correlation purposes. The spectral analysis is being carried out to reveal few insights into pipe flow structure, allowing to follow the foot prints of such structures as well as providing estimates of their energy contents. The cumulative energy is examined as a function of the streamwise wavelength, describing the most energetic motions found in spectral data at various wall normal locations. For the current Reynolds number range, the Very Large Scale Motions (VLSM) and the Large Scale Motions (LSM) were evident as localized peaks in pre-multiplied spectra, having mean wavelengths approximately of 12R, and 3R, respectively. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)