A comparison of second-order and fourth-order pressure gradient algorithms in a σ-co-ordinate ocean model

In stratified three-dimensional models the use of a boundary-fitted vertical co-ordinate is known to produce errors in the horizontal pressure gradient calculation near steep topography. The error is due to the splitting of the horizontal pressure gradient term in each of the momentum equations into two parts and the subsequent incomplete cancellation of the truncation errors of those parts. In order to minimize these pressure gradient errors, a fourth-order-accurate pressure gradient calculation has been implemented and installed in SPEM, a three-dimensional primitive equation ocean model. The stability and accuracy of the new scheme are compared with those of the original second-order-accurate model in a series of calculations of unforced flow in the vicinity of an isolated seamount. The new scheme is shown to have much smaller pressure gradient errors over a wide range of parameter space as well as a greater parametric domain of numerical stability.     

A control volume finite difference method for buoyant flow in three-dimensional curvilinear non-orthogonal co-ordinates

This paper presents a control-volume-based finite difference method in non-orthogonal curvilinear coordinates on a local basis in which the vectors and tensors are all based on the general curvilinear coordinates for buoyant flow calculations in arbitrary three-dimensional geometries. The governing equations are transformed from Cartesian co-ordinates into generalized curvilinear co-ordinates. After integrating the set of equations for the control volumes, the finite difference equations are then formulated by a proper treatment of the heat flux and stress tensors and by incorporating the QUICK scheme for the convective terms. The solution procedure then follows the one for three-dimensional Cartesian co-ordinates. Examples are given in problems of natural convection in such three-dimensional enclosures as parallelepiped enclosures and horizontal closed cylinders with differentially heated ends. In the latter case, important applications have been found in crystal growth by means of chemical vapour deposition in a cylindrical ampoule, in which uniform heat fluxes along the two ends are required in order to produce high-quality crystals. Special attention is given to the insertion of baffles in the cylinder to improve the recirculating flow patterns near the two ends.     

On identification of flow separation and vortices in internal periodic flows

To permit simplified analysis of complex time-dependent flows, possible relationship between the near-wall flow, flow separation and vortices are studied numerically for a flow in a constricted two-dimensional channel. The pulsating incoming wave-form consists of a steady flow, followed by a half-sinus flow superimposed on the steady component. One pair of vortices is created in each cycle, one vortex near each wall. The vortices propagate downstream in the next cycles, promoting flow separation as they move. Existing flow separation criteria were not found to be uniformly valid. A relation between the near-wall flow and the vortical system exists only during the steady incoming flow phase of the cycle. It seems that local criteria of flow separation cannot be found for complex internal pulsating flow fields. However, the vorticity field can be utilized, even in complex time-periodic flows, for identifying vortices that have been formed by the roll-up of shear layers.     

Particle dispersion in a single-sided backward-facing step flow

The paper describes the particle dispersion in a single-sided backward-facing step flow. Particles of well-known sizes in the diameter range from 1 to 70 μm were suspended in an air flow and the particle motion over a step was measured by mean of a laser-Doppler anemometer. Thus, the local and integral flow quantities, i.e. the mean and turbulent velocity data could be measured precisely. In the experiments, monodispersed particle size distributions were used to exclude particle size related information ambiguity, known as triggering effects or size bias. The results of this study show qualitatively and quantitatively the difference in time-averaged particle dynamics for selected particle sizes in a backward-facing step flow. The experiments show, for different sizes, the changes in the particle velocity field in comparison with the velocity field of the continuous phase deduced from the 1 μm particles, and also imply the strong influences which different particle sizes have on flow data evaluation when size effects are not taken into account with particle-related optical measuring techniques.     

An exact solution for flow past an accelerated horizontal plate in a rotating fluid with the generalized Oldroyd-B model

In this paper, the generalized Oldroyd-B with fractional calculus approach is used. An exact solution in terms of Fox-H function for flow past an accelerated horizontal plate in a rotating fluid is obtained by using discrete Laplace transform method. A comparison among the influence of various parameters in the Oldroyd-B model and the angular velocity of the fluid on the velocity profiles is made through numerical method in graphic form.     

The numerical solution of the flow in a general bifurcating channel at moderately high Reynolds number using boundary-fitted co-ordinates,primitive variables and Newton iteration

A numerical code has been implemented for the numerical solution of the steady, incompressible Navier–Stokes equations using primitive variables in a bifurcating channel. A boundary-fitted, numerically generated grid is placed onto the domain of the channel which is transformed into either a rectilinear C- or T-shaped region. The differenced equations are solved using Newton's iteration which makes upwinding at high Reynolds number unnecessary. Practical implications of inverting the huge Jacobian matrix of Newton's method are discussed. The results have relative error of 2–3 × 10^?3 at Reynolds number 100, with T-geometry being marginally but significantly more accurate than C-geometry. Results have been obtained for Reynolds numbers up to 1000 for three bifurcations one of which models the carotid arterial bifurcation in the human head. For this latter bifurcation the wall shear stress is calculated in connection with the onset of atherosclerosis. Finally, the results of flows having different daughter tube end pressures are presented.     

Numerical study of magnetohydrodynamic laminar flow separation in a channel with smooth expansion

The flow of an electrically conducting incompressible viscous fluid in a plane channel with smooth expansion in the presence of a uniform transverse magnetic field has been analysed. A solution technique for the governing magnetohydrodynamic equations in primitive variable formulation has been developed. A co‐ordinate transformation has been employed to map the infinite irregular domain into a finite regular computational domain. The governing equations are discretized using finite‐difference approximations in staggered grid. Pressure Poisson equation and pressure correction formulae are derived and solved numerically. It is found that with increase in the magnetic field, the size of the flow separation zone diminishes and for sufficiently large magnetic field, the separation zone disappears completely. The peak u‐velocity decreases with increase in the magnetic field. It is also found that the asymmetric flow in a symmetric geometry, which occurs at moderate Reynolds numbers, becomes symmetric with sufficient increase in the transverse magnetic field. Thus, a transverse magnetic field of suitable strength has a stabilizing effect in controlling flow separation, as also in delaying the transition to turbulence. Copyright © 2008 John Wiley & Sons, Ltd.     

Supersonic flow separation in planar nozzles

We present experimental results on separation of supersonic flow inside a convergent–divergent (CD) nozzle. The study is motivated by the occurrence of mixing enhancement outside CD nozzles operated at low pressure ratio. A novel apparatus allows investigation of many nozzle geometries with large optical access and measurement of wall and centerline pressures. The nozzle area ratio ranged from 1.0 to 1.6 and the pressure ratio ranged from 1.2 to 1.8. At the low end of these ranges, the shock is nearly straight. As the area ratio and pressure ratio increase, the shock acquires two lambda feet. Towards the high end of the ranges, one lambda foot is consistently larger than the other and flow separation occurs asymmetrically. Downstream of the shock, flow accelerates to supersonic speed and then recompresses. The shock is unsteady, however, there is no evidence of resonant tones. The separation shear layer on the side of the large lambda foot exhibits intense instability that grows into large eddies near the nozzle exit. Time-resolved wall pressure measurements indicate that the shock oscillates in a piston-like manner and most of the energy of the oscillations is at low frequency.      

Transient oscillatory and constantly accelerated non-Newtonian flow in a porous medium

This paper looks at the magnetohydrodynamic (MHD) analysis for transient flow of an Oldroyd-B fluid in a porous medium. The presented analysis takes into account the modified Darcy's law. The flow is induced due to constantly accelerated and oscillating plate. Expressions for the corresponding velocity field and the adequate tangential stress are determined by means of the Fourier sine transform. The influence of various parameters of interest on the velocity and tangential stress has been shown and discussed. A comparison for different kinds of fluids is also provided.     

GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

A model for pressure loss, film thickness, and entrained fraction for gas-liquid annular flow

A two-component (air-water) annular flow model is presented requiring only flow rates, absolute pressure, temperature, and tube diameter. Film thicknesses (base film and wave height) are calculated from a critical film thickness model. Modeled pressure gradient is weighted by wave intermittency to compute average pressure gradient. Film flow rate and wave velocity are estimated using the universal velocity profile in the waves and a piecewise linear profile in the base film. For vertical flow, mean absolute errors for film thickness, wave velocity, and pressure gradient are 9%, 9%, and 19%, respectively. In horizontal flow, mean absolute errors for pressure gradient, base film thickness, and disturbance wave velocity are 17%, 10%, and 14%, respectively, on par with those from single-behavior models that require additional film thickness or other data as inputs.     

Adiabatic and diabatic two-phase venting flow in a microchannel

The growth and advection of the vapor phase in two-phase microchannel heat exchangers increase the system pressure and cause flow instabilities. One solution is to locally vent the vapor formed by capping the microchannels with a porous, hydrophobic membrane. In this paper we visualize this venting process in a single 124 μm by 98 μm copper microchannel with a 65 μm thick, 220 nm pore diameter hydrophobic Teflon membrane wall to determine the impact of varying flow conditions on the flow structures and venting process during adiabatic and diabatic operation. We characterize liquid velocities of 0.14, 0.36 and 0.65 m/s with superficial air velocities varying from 0.3 to 8 m/s. Wavy-stratified and stratified flow dominated low liquid velocities while annular type flows dominated at the higher velocities. Gas/vapor venting can be improved by increasing the venting area, increasing the trans-membrane pressure or using thinner, high permeability membranes. Diabatic experiments with mass flux velocities of 140 and 340 kg/s/m² and exit qualities up to 20% found that stratified type flows dominate at lower mass fluxes while churn-annular flow became more prevalent at the higher mass-flux and quality. The diabatic flow regimes are believed to significantly influence the pressure-drop and heat transfer coefficient in vapor venting heat exchangers.     

An investigation of pressure transients in pipelines with two-phase bubbly flow

This paper presents a two-dimensional model for the analysis of the pressure transient of a two-phase homogeneous bubbly mixture flowing in a pipeline and the numerical integration using the centre implicit method (CIM). Experiments were conducted to confirm the proposed sonic speed equation of an air–water mixture for an air concentration of less than 1%. The 2D CIM model is compared with the method of characteristics (MoC) for a two-phase bubbly flow in a pipeline. The comparisons show that the proposed 2D CIM model generally gives good agreement with the method of characteristics.     

Calculation of pressure on an airfoil contour in an unsteady separated flow

Simple formulas for calculating the pressure and the total hydrodynamic reactions acting on an arbitrarily moving airfoil are derived within the framework of the model of plane unsteady motion of an ideal incompressible fluid. Several vortex wakes may be shed from the airfoil owing to changes in velocity circulation around the airfoil contour. Cases with nonclosed and closed contours are considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 109–113, May–June, 2008.     

On the receptivity of pipe poiseuille flow with a bump on the wall under the periodical pressure

Asymptotic method was adopted to obtain a receptivity model for a pipe PoiseuiUe flow under periodical pressure, the wall of the pipe with a bump. Bi-orthogonal eigen-function systems and Chebyshev collocation method were used to resolve the problem.Various spatial modes and the receptivity coefficients were obtained. The results show that different modes dominate the flow in different stages, which is comparable with the phenomena observed in experiments.     

The Krylov accelerated SIMPLE(R) method for flow problems in industrial furnaces

Numerical modeling of the melting and combustion process is an important tool in gaining understanding of the physical and chemical phenomena that occur in a gas‐ or oil‐fired glass‐melting furnace. The incompressible Navier–Stokes equations are used to model the gas flow in the furnace. The discrete Navier–Stokes equations are solved by the SIMPLE(R) pressure‐correction method. In these applications, many SIMPLE(R) iterations are necessary to obtain an accurate solution. In this paper, Krylov accelerated versions are proposed: GCR‐SIMPLE(R). The properties of these methods are investigated for a simple two‐dimensional flow. Thereafter, the efficiencies of the methods are compared for three‐dimensional flows in industrial glass‐melting furnaces. Copyright © 2000 John Wiley & Sons, Ltd.     

On pressure separation algorithms (PSepA) for improving the accuracy of incompressible flow simulations

We investigate a special technique called ‘pressure separation algorithm’ (PSepA) (see Applied Mathematics and Computation 2005; 165 :275–290 for an introduction) that is able to significantly improve the accuracy of incompressible flow simulations for problems with large pressure gradients. In our numerical studies with the computational fluid dynamics package FEATFLOW ( www.featflow.de ), we mainly focus on low‐order Stokes elements with nonconforming finite element approximations for the velocity and piecewise constant pressure functions. However, preliminary numerical tests show that this advantageous behavior can also be obtained for higher‐order discretizations, for instance, with Q₂/P₁ finite elements. We analyze the application of this simple, but very efficient, algorithm to several stationary and nonstationary benchmark configurations in 2D and 3D (driven cavity and flow around obstacles), and we also demonstrate its effect to spurious velocities in multiphase flow simulations (‘static bubble’ configuration) if combined with edge‐oriented, resp., interior penalty finite element method stabilization techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Determination of turbulent pressure loss of non-newtonian oil flow in rough pipes

Summary A detailed theoretical analysis is carried out of the turbulent flow of non-newtonian fluids in rough pipes in which approximate assumptions are of negligible importance. A new friction-factor equation (BNS-equation) is obtained which is valid for turbulent flow of power-law type non-newtonian fluids in the transition region between smooth and wholly rough wall turbulence.The accuracy of the deduced formula was checked by in situ measurements on a crude oil pipeline having a diameter of 305 mm and a length of 161 km. For this pipeline the Reynolds numbers ranged between 10⁴ and 10⁵. The computed and the measured friction factors showed good agreement. The mean of the absolute values of the relative error was 4.1%, while the standard deviation was 0.81%, these results appear to give better fitting than other similar equations published earlier.The BNS-equation is the first analytically deduced relationship for non-newtonian crude oils which includes the effects of both Reynolds number and relative roughness of the pipe.With 6 figures and 2 tables     

Numerical study on the effect of initial flow velocity on liquid film thickness of accelerated slug flow in a micro tube

Numerical simulation of air–water slug flows accelerated from steady states with different initial velocities in a micro tube is conducted. It is shown that the liquid film formed between the gas bubble and the wall in an accelerated flow is significantly thinner than that in a steady flow at the same instantaneous capillary number. Specifically, the liquid film thickness is kept almost unchanged just after the onset of acceleration, and then gradually increases and eventually converges to that of an accelerated flow from zero initial velocity. Due to the flow acceleration, the Stokes layer is generated from the wall, and the instant velocity profile can be given by superposition of the Stokes layer and the initial parabolic velocity profile of a steady flow. It is found that the velocity profile inside a liquid slug away from the bubble can be well predicted by the analytical solution of a single-phase flow with acceleration. The change of the velocity profile in an accelerated flow changes the balance between the inertia, surface tension and viscous forces around the meniscus region, and thus the resultant liquid film thickness. By introducing the displacement thickness, the existing correlation for liquid film thickness in a steady flow (Han and Shikazono, 2009) is extended so that it can be applied to a flow with acceleration from an arbitrary initial velocity. It is demonstrated that the proposed correlation can predict liquid film thickness at Re < 4600 within the range of ±10% accuracy.