The flow induced in a three layer stratified fluid by a submerged sink or source with stagnation points on the free surfaces

A boundary integral technique is developed to study the free surface flow of a steady, two-dimensional, incompressible, irrotational and inviscid fluid which is induced in both two and three layer stratified fluids in the presence of gravity by a submerged sink or source with stagnation points on the free surfaces. A special form of the Riemann–Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing non-linear boundary integral–differential equations which have been solved for the fluid velocity on the free surfaces and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various Froude numbers and sink heights in both two and three layer fluids. Further, we have also studied the critical Froude numbers for which no convergent solutions are possible for any larger values of the Froude number. We have found that the free surfaces are dependent on two parameters, namely the Froude number and the ratio of sink height to the thickness of either the middle layer in a three layer system and the bottom layer in a two layer system.     

Diffusion of a chemically reactive species of a power-law fluid past a stretching surface

A numerical solution for the steady magnetohydrodynamic (MHD) non-Newtonian power-law fluid flow over a continuously moving surface with species concentration and chemical reaction has been obtained. The viscous flow is driven solely by the linearly stretching sheet, and the reactive species emitted from this sheet undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. Using a similarity transformation, the governing non-linear partial differential equations are transformed into coupled nonlinear ordinary differential equations. The governing equations of the mathematical model show that the flow and mass transfer characteristics depend on six parameters, namely, the power-law index, the magnetic parameter, the local Grashof number with respect to species diffusion, the modified Schmidt number, the reaction rate parameter, and the wall concentration parameter. Numerical solutions for these coupled equations are obtained by the Keller-Box method, and the solutions obtained are presented through graphs and tables. The numerical results obtained reveal that the magnetic field significantly increases the magnitude of the skin friction, but slightly reduces the mass transfer rate. However, the surface mass transfer strongly depends on the modified Schmidt number and the reaction rate parameter; it increases with increasing values of these parameters. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Numerical errors of explicit finite difference approximation for two-dimensional solute transport equation with linear sorption

The numerical errors associated with explicit upstream finite difference solutions of two-dimensional advection—Dispersion equation with linear sorption are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation. The numerical truncation errors are defined using Peclet and Courant numbers in the X and Y direction, a sink/source dimensionless number and new Peclet and Courant numbers in the XY plane. The effects of these truncation errors on the explicit solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution in uniform flow field. The results show that these errors are not negligible and correcting the finite difference scheme for them results in a more accurate solution.     

Peristaltic transport of a third order fluid under the effect of a magnetic field

The peristaltic transport of a third order fluid in a planar channel is considered. The fluid is electrically conducting by a transverse magnetic field. The perturbation solution is obtained using small Deborah number. Expressions of stream function, longitudinal velocity and pressure gradient valid for long wavelength are developed. Numerical integration is performed to analyze the effect of Hartman number on the pressure rise and frictional force. It is noted that both Hartman and Deborah numbers suppress the flow.     

Bio-inspired computational heuristics to study the boundary layer flow of the Falkner-Scan system with mass transfer and wall stretching

In the present study, a bio-inspired computational intelligence technique is developed for finding the solutions of the celebrated Falkner-Skan equation arising in fluid mechanics problems using feed-forward Artificial Neural Networks (ANNs), Genetic Algorithms (GA), the Active-Set (AS) method and their combination namely a GA-AS approach. The differential equations based ANNs modeling of the Falkner-Skan system is constructed by defining an unsupervised error function. The training of the design parameters of ANNs is carried out with the help of viable global search through GAs and fine tuning of the results is achieved with an efficient local search using the AS method. The proposed scheme is applied to a number of scenarios for the Falkner-Skan system based on boundary layer flow over a moving wall with mass transfer in the presence of a free stream with power-law velocity distributions. The dynamics of the system are investigated for different cases of mass transfer and wall stretching. The proposed results are compared with analytical and numerical solutions to verify the correctness of the approach. The accuracy and convergence of the proposed solver are validated through sufficient large numbers of independent runs in terms of different performance indices based on mean absolute error, Thail’s inequality coefficient and Nash-Suitcliff efficiency. These solutions greatly enrich possible approaches for stochastic numerical solution of the celebrated Falkner-Skan system.     

A simplified model for the viscous crossflow in a slotted test section

A simplified physical model is constructed which simulates the viscous crossflow in a fluid layer near the slots at a fixed streamwise location in a slotted wind tunnel. For low to moderate Reynolds numbers, numerical solutions of the two-dimensional, incompressible Navier-Stokes equation in stream function and vorticity, which govern the model flow, are obtained. Fairly general slot geometry is incorporated by means of the Thompson-Thames-Mastin transformation. An approximate factorization scheme with cyclic acceleration parameters is employed to solve a finite difference analog of the stream function equation. The vorticity equation is numerically solved with a modified version of the classical alternating direction implicit (ADI) scheme. Although no quantitative assessment of solution accuracy can be made, numerical results for variation in incremental wall pressure around the slot are at least qualitatively similar to some experimental results of Berndt and Sorenson[21].     

Effect of slip boundary condition on flow computation in the presence of rotational body forces

The slip boundary condition often used in numerical computation of flow problems is examined to find its influence on the solution when there is a rotational body force field. Two specific examples are considered having different force distributions that are generated electromagnetically. The numerical study shows that in a fluid region weakly affected by the body force, a slip wall can be used to approximate a realistic no-slip solid surface if the reynolds number is reasonably large. On the other hand, in a region where the body force has a strong effect, a no-slip wall may never be replaced by a slip wall even if the Reynolds number is extremely high.     

Temporal/spatial simulation of the stratified far wake of a sphere

Far wakes are generally studied on the basis of a temporal development study, i.e. the flow is assumed periodic in streamwise direction and one focuses on its evolution in time. It is then required to set up an appropriate initial condition to start the calculation. Like in some previous works, we study the far wake of a sphere in a stratified fluid, but rather than starting from an ad hoc initial condition, we first carry out a spatial development simulation in order to define a relevant initialization. The control parameters of the flow are the Prandtl, Reynolds and internal Froude numbers, that we take equal to Pr = 7, Re = 10,000 and F = 25. The numerical method is based on a spectral multi-domain Fourier–Chebyshev approximation, stabilized by using a spectral vanishing viscosity technique, that may be interpreted as a spectral large-eddy simulation closure restricted to the high frequencies. Very realistic results are obtained, especially showing the so-called three-dimensional (3D), non-equilibrium (NEQ) and quasi two-dimensional (Q2D) regimes, with satisfactory comparisons with the experiments.     

Numerical computation of momentum jets and forced plums

A numerical study of vertical momentum jets and forced plumes issuing to similar receiving media is presented. The complete partial differential equations governing steady, incompressible, turbulent flow are solved in axisymmetric coordinates using finite-difference techniques. Solutions were based on the stream function-vorticity transport approach for a Boussinesq fluid. Buoyant driving forces were coupled to the vorticity equation by a buoyancy transport equation wich included effects of temperature and other constituents. Turbulent transport coefficients were computed iteratively using the Prandt eddy diffusivitiy model. Results for the momentum jet, axial and radial distributions of velocity and concentration, show excellent agreement with published data. Forced plum computations are presented which include similar results for densimetric Froude numbers ranging from 1 to 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.     

Lift and multiple equilibrium positions of a single particle in Newtonian and Oldroyd-B fluids

A heavy particle is lifted from the bottom of a channel in a plane Poiseuille flow when the Reynolds number is larger than a critical value. In this paper we obtain correlations for lift-off of particles in Oldroyd-B fluids. The fluid elasticity reduces the critical shear Reynolds number for lift-off. The effect of the gap size between the particle and the wall, on the lift force, is also studied. A particle lifted from the channel wall attains an equilibrium height at which its buoyant weight is balanced by the hydrodynamic lift force. Choi and Joseph [Choi HG, Joseph DD. Fluidization by lift of 300 circular particles in plane Poiseuille flow by direct numerical simulation. J Fluid Mech 2001;438:101-128] first observed multiple equilibrium positions for a particle in Newtonian fluids. We report several new results for the Newtonian fluid case based on a detailed study of the multiple equilibrium solutions, e.g. we find that at a given Reynolds number there are regions inside the channel where no particle, irrespective of its weight, can attain a stable equilibrium position. This would result in particle-depleted zones in channels with Poiseuille flows of a dilute suspension of particles of varying densities. Multiple equilibrium positions of particles are also found in Oldroyd-B fluids. All the results in this paper are based on 2D direct numerical simulations.     

Fully iterative ILU preconditioning of the unsteady Navier–Stokes equations for GPGPU

In this work we investigate the numerical difficulties that arise in optimizing the efficiency of Newtonian fluids simulations on a massively parallel computing hardware like a GPU. In particular, we will concentrate on the resulting algebraic problem. We will present an approximate, fully-iterative, ILU preconditioner and we will show that this solution is very efficient on a GPU if compared with an intrinsic massively parallel preconditioner like the diagonal preconditioner, which indeed goes faster than more robust techniques, like ILU, despite their strong decrease in the number of iterations. We refer to GMRES as the iterative scheme used to solve the linear system. In particular, we will deal with the solution of incompressible flows with variable density and we will investigate the interplay between Reynolds and Atwood numbers. We will show that the numerical simulation at medium–high Reynolds numbers produces linear systems whose matrices can be reasonably preconditioned with the diagonal preconditioner, while at low Reynolds numbers the higher viscosity of the fluid makes the diagonal preconditioner ineffective in the solution time requested from GMRES and, decreasing the Reynolds number, unable to let GMRES converge at all. In this situation, we will show how an adequate iterative approach to the parallel solution of the triangular systems that result from the ILU preconditioning, turns out to be robust and efficient. We will show numerical results for variable-density fluids, discretized with the scheme described in Calgaro et al. (2008), in classical benchmarks and, in particular, in the well-known Rayleigh–Taylor instability.     

Effects of slip on sheet-driven flow and heat transfer of a non-Newtonian fluid past a stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHFcase) in presence of a uniform heat source or sink. Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field, heat source (sink) parameter and the third grade fluid parameters on the velocity, skin friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.     

Forced convection in electro-osmotic/Poiseuille micro-channel flows of viscoelastic fluids: fully developed flow with imposed wall heat flux

The analytical solution for heat transfer in a dynamic and thermally fully developed channel flow of the simplified Phan-Thien–Tanner fluid induced by combined electro-osmosis and pressure gradient was obtained assuming that material properties are independent of temperature. The flow forcing was quantified by an appropriate dimensionless parameter and its effect and that of all other relevant dimensionless numbers is presented and discussed. Specifically, the forced convection occurs under conditions of constant wall heat flux and the solution includes the effects of Weissenberg number, electric double layer (EDL) thickness, forcing ratio parameter, viscous dissipation as well as of Joule heating due to the electric currents and was obtained under the simplifying Debye–Hückel approximation. Generally speaking, the Joule effect is stronger than the viscous dissipation except in very narrow channels, but these fall outside the validity of the Debye–Hückel conditions. For pure electro-osmosis, viscous dissipation is restricted to the near-wall region and virtually nonexistent elsewhere, so it is irrelevant for thin electric double layers and Joule heating is more relevant. As the EDL thickens and/or the pressure gradient contribution increases, the role of viscous dissipation grows and shear-thinning effects also appear more clearly on the Nusselt number. Generally speaking, an increase in internal heating results in lower Nusselt numbers and this effect is stronger than the effect of shear-thinning, which is responsible for a slight increase in the Nusselt number.     

The influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport

The effects of both wall slip conditions and heat transfer on peristaltic flow of MHD Newtonian fluid in a porous channel with elastic wall properties have been studied under the assumptions of long-wavelength and low-Reynolds number. The analytical solution has been derived for the stream function and temperature. The results for velocity, temperature, stream function and heat transfer coefficient obtained in the analysis have been evaluated numerically and discussed briefly. The numerical result shows that more trapped bolus appears with increasing Knudsen number.     

Numerical simulations of particle migration in a viscoelastic fluid subjected to Poiseuille flow

In this work we present 2D numerical simulations on the migration of a particle suspended in a viscoelastic fluid under Poiseuille flow. A Giesekus model is chosen as constitutive equation of the suspending liquid. In order to study the sole effect of the fluid viscoelasticity, both fluid and particle inertia are neglected.The governing equations are solved through the finite element method with proper stabilization techniques to get convergent solutions at relatively large flow rates. An Arbitrary Lagrangian–Eulerian (ALE) formulation is adopted to manage the particle motion. The mesh grid is moved along the flow so as to limit particle motion only in the gradient direction to substantially reduce mesh distortion and remeshing.Viscoelasticity of the suspending fluid induces particle cross-streamline migration. Both large Deborah number and shear thinning speed up the migration velocity. When the particle is small compared to the gap (small confinement), the particle migrates towards the channel centerline or the wall depending on its initial position. Above a critical confinement (large particles), the channel centerline is no longer attracting, and the particle is predicted to migrate towards the closest wall when its initial position is not on the channel centerline. As the particle approaches the wall, the translational velocity in the flow direction is found to become equal to the linear velocity corresponding to the rolling motion over the wall without slip.     

Direct computation of perturbed impinging hot jets

The unsteady flow and temperature fields of an impinging hot jet at a Reynolds number of 1000 and a nozzle-to-plate distance of 6 jet diameters have been obtained by direct numerical solution of the compressible time-dependent three-dimensional Navier-Stokes equations using highly accurate numerical methods. Effects of an external perturbation on the flow and heat transfer characteristics of the transitional impinging jet have been examined. Oscillatory behaviour induced by the external perturbation has been observed for the impinging jet. The external perturbation leads to the large-scale vortical structures in the primary jet stream, which subsequently lead to the strong oscillatory behaviour of the impinging jet. The vortical structures lead to flow transitional behaviour that enhances mixing of the hot jet with the ambient fluid. It has been observed that the wall boundary layer of the impinging jet is thin. Both the instantaneous and time-averaged wall shear and normal stresses and Nusselt number are examined. Although the external perturbation strongly affects the flow structures in the primary jet stream, it does not have significant effects on the wall stresses and heat transfer characteristics of the impinging jet due to the re-laminarization effect of the wall.     

A numerical study of temporal shallow mixing layers using BGK-based schemes

A numerical study of the temporal shallow mixing layers is performed. The depth-averaged shallow water equations are solved by the finite volume method based on the Bhatnagar–Gross–Krook (BGK) equation. The filtering operation is applied to the governing equations and the well-known Smagorinsky model for the subgrid-scale (SGS) stress is employed in order to present a large eddy simulation (LES). The roll-up and pairing processes are clearly shown and the corresponding kinetic energy spectra are calculated. The effects of the Froude number and the bottom friction are numerically investigated. It is shown that the growth rate of the mixing layer decreases as the Froude number increases, which is very similar to the compressible mixing layers when considering the effects of the Mach number. The numerical results also indicate that the increase in bottom friction can enhance the stability of the flows, which is physically reasonable and consistent with the theoretical and experimental findings.     

Simulation of unsteady compressible flow in a channel with vibrating walls - Influence of the frequency

This study deals with the numerical solution of a 2D unsteady flow of a compressible viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract and the flow can represent a model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract.The flow is described by the system of Navier–Stokes equations for laminar flows. The numerical solution is implemented using the finite volume method (FVM) and the predictor–corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. Due to the motion of the grid, the basic system of conservation laws is considered in the Arbitrary Lagrangian–Eulerian (ALE) form.The authors present the numerical simulations of flow fields in the channel, acquired from a program developed exclusively for this purpose. The numerical results for unsteady flows in the channel are presented for inlet Mach number M_∞ = 0.012, Reynolds number Re_∞ = 4.5 × 10³ and the wall motion frequency 20 and 100 Hz.