NUMERICAL MODELLING OF THREE-DIMENSIONAL FLY-ASH FLOW IN POWER UTILITY BOILERS

A two-fluid Eulerian model in combination with a particle–wall collision model and generalized Eulerian boundary conditions for the particulate phase is employed to predict complex three- dimensional fly-ash flows which often cause severe erosion to boiler tubes located in power utility boilers. Mean momentum and mass conservation equations are solved for each phase using a finite volume scheme with two-way coupling and a modified renormalization group (RNG)-based k –ϵ turbulence model. Comparison of predicted particle concentration with measured data is made and excellent agreement is obtained. The detailed character of the particulate velocity field and concentration just downstream of the 180° bend shows a marked dependence on the Stokes number not previously reported. © 1997 by John Wiley & Sons, Ltd.     

Acoustic Lagrangian velocity measurement in a turbulent air jet

Measuring Lagrangian velocities in a turbulent flow is of a great interest for turbulence modeling. We report measurements made in an axisymmetric turbulent air jet at Reynolds number R _λ ≃ 320, using acoustical Doppler scattering. Helium-filled soap bubbles are used as Lagrangian tracers. We describe an experimental setup which allows the simultaneous measurement of the full three-component Lagrangian velocity and the longitudinal Eulerian one. Lagrangian velocity probability density functions (PDF) are found Gaussian, close to Eulerian ones. Velocity correlations are analysed as well as the statistical dependence between components.     

Heat transfer enhancement in grooved channels due to flow bifurcations

The flow bifurcation scenario and heat transfer characteristics in grooved channels, are investigated by direct numerical simulations of the mass, momentum and energy equations, using the spectral element methods for increasing Reynolds numbers in the laminar and transitional regimes. The Eulerian flow characteristics show a transition scenario of two Hopf bifurcations when the flow evolves from a laminar to a time-dependent periodic and then to a quasi-periodic flow. The first Hopf bifurcation occurs to a critical Reynolds number Rec₁ that is significantly lower than the critical Reynolds number for a plane-channel flow. The periodic and quasi-periodic flows are characterized by fundamental frequencies ω₁ and m· ω₁+n·ω₂, respectively, with m and n integers. Friction factor and pumping power evaluations demonstrate that these parameters are much higher than the plane channel values. The time-average mean Nusselt number remains mostly constant in the laminar regime and continuously increases in the transitional regime. The rate of increase of this Nusselt number is higher for a quasi-periodic than for a periodic flow regime. This higher rate originates because better flow mixing develops in quasi-periodic flow regimes. The flow bifurcation scenario occurring in grooved channels is similar to the Ruelle-Takens-Newhouse transition scenario of Eulerian chaos, observed in symmetric and asymmetric wavy channels.     

Instantaneous planar pressure determination from PIV in turbulent flow

This paper deals with the determination of instantaneous planar pressure fields from velocity data obtained by particle image velocimetry (PIV) in turbulent flow. The operating principles of pressure determination using a Eulerian or a Lagrangian approach are described together with theoretical considerations on its expected performance. These considerations are verified by a performance assessment on a synthetic flow field. Based on these results, guidelines regarding the temporal and spatial resolution required are proposed. The interrogation window size needs to be 5 times smaller than the flow structures and the acquisition frequency needs to be 10 times higher than the corresponding flow frequency (e.g. Eulerian time scales for the Eulerian approach). To further assess the experimental viability of the pressure evaluation methods, stereoscopic PIV and tomographic PIV experiments on a square cylinder flow (Re _D  = 9,500) were performed, employing surface pressure data for validation. The experimental results were found to support the proposed guidelines.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonreacting chemical transport in two-phase reservoirs — Factoring diffusive and wave properties

The vertical transport of mass, energy andn unreacting chemical species in a two-phase reservoir is analysed when capillarity can be ignored. This results in a singular system of equations, comprising mixed parabolic and hyperbolic equations. We derive a natural factorisation of these equations into diffusive and wave equations. If diffusive or conductive effects are important for onlyN–1 of the chemical species, then there areN parabolic equations, andn+2–N wave equations. Each wave equation allows for the corresponding variable to be discontinous, or equivalently, for shock propagation to occur. Steady flows were shown to allow for more than two (vapour and liquid dominated) saturations for a given mass, energy and chemical flux, but when thermal conduction and chemical diffusion are unimportant, only the vapour and liquid dominated cases appear likely to occur. For infinitesimal shocks there is a continuous flux vector for each diffusive variable. The existence of these continuous flux vectors results in significant simplifications of the corresponding wave equations. It remains an open question if continuous flux vectors exist for finite shocks. General expressions are obtained for the diffusion and wave matrices, which require no knowledge of continuous flux vectors.     

Large Eddy Simulations: How to evaluate resolution

The present paper gives an analysis of fully developed channel flow at Reynolds number of Re=u_τδ/ν=4000 based on the friction velocity, u_τ, and half the channel height, δ. Since the Reynolds number is high, the LES is coupled to a URANS model near the wall (hybrid LES–RANS) which acts as a wall model. It it found that the energy spectra is not a good measure of LES resolution; neither is the ratio of the resolved turbulent kinetic energy to the total one (i.e. resolved plus modelled turbulent kinetic energy). It is suggested that two-point correlations are the best measures for estimating LES resolution. It is commonly assumed that SGS dissipation takes place at high wavenumbers. Energy spectra of the fluctuating velocity gradients show that this is not true; the major part of the SGS dissipation takes place at low to midrange wavenumbers. Furthermore, the energy spectra of the fluctuating velocity gradients reveals that the accuracy of the predicted velocity gradients at the highest resolved wavenumbers is very poor.     

Numerical solution scheme for inert,disperse, and dilute gas-particle flows

A multi-velocity formulation is proposed for the solution of an Eulerian representation of an inert, disperse, and dilute particle-phase of a gas-particle flow. Single-velocity formulations are capable of predicting regions of zero particle concentration but are problematic with crossing particle trajectories or compression waves. The multi-velocity formulation described here can account for crossing particle trajectories by splitting the particle-phase into distinct velocity families which are transported separately in the flow. Switching of the particle families at solid boundaries and due to momentum transfer with the gas-phase is conducted in a manner that enforces conservation of mass, momentum, and energy. This numerical method is combined with a parallel block-based adaptive mesh refinement algorithm that is very effective in treating problems with disparate length scales. The block-based data structure lends itself naturally to domain decomposition and thereby enables efficient and scalable implementations of the algorithm on distributed-memory multi-processor architectures. Numerical results are described to demonstrate the capabilities of the approach for predicting gas-particle flows.     

Evaluation of erosion in a two‐way coupled fluid–particle system

In this study, the effects of flow turbulence intensity, temperature, particle sizes and impinging velocity on erosion by particle impact are demonstrated numerically. Underlying turbulent flow on an Eulerian frame is described by the compressible Reynolds averaged Navier–Stokes equations with a RNG k–ε turbulence model. The particle trajectories and particle–wall interactions are evaluated by a Eulerian–Lagrangian approach in a two‐way coupling system. An erosion model considering material weight removal from surfaces is used to predict erosive wear. Computational validation against measured data is demonstrated satisfactorily. The analysis of erosion shows that the prevention of erosion is enhanced by increasing the effects of flow temperature and turbulence intensity and reducing particle inertial momentum. Copyright © 2001 John Wiley & Sons, Ltd.     

An advection-based model to increase the temporal resolution of PIV time series

A numerical implementation of the advection equation is proposed to increase the temporal resolution of PIV time series. The method is based on the principle that velocity fluctuations are transported passively, similar to Taylor’s hypothesis of frozen turbulence. In the present work, the advection model is extended to unsteady three-dimensional flows. The main objective of the method is that of lowering the requirement on the PIV repetition rate from the Eulerian frequency toward the Lagrangian one. The local trajectory of the fluid parcel is obtained by forward projection of the instantaneous velocity at the preceding time instant and backward projection from the subsequent time step. The trajectories are approximated by the instantaneous streamlines, which yields accurate results when the amplitude of velocity fluctuations is small with respect to the convective motion. The verification is performed with two experiments conducted at temporal resolutions significantly higher than that dictated by Nyquist criterion. The flow past the trailing edge of a NACA0012 airfoil closely approximates frozen turbulence, where the largest ratio between the Lagrangian and Eulerian temporal scales is expected. An order of magnitude reduction of the needed acquisition frequency is demonstrated by the velocity spectra of super-sampled series. The application to three-dimensional data is made with time-resolved tomographic PIV measurements of a transitional jet. Here, the 3D advection equation is implemented to estimate the fluid trajectories. The reduction in the minimum sampling rate by the use of super-sampling in this case is less, due to the fact that vortices occurring in the jet shear layer are not well approximated by sole advection at large time separation. Both cases reveal that the current requirements for time-resolved PIV experiments can be revised when information is poured from space to time. An additional favorable effect is observed by the analysis in the frequency domain whereby the spectrum becomes significantly less prone to aliasing error for the super-sampled data series.     

A direct numerical simulation of axisymmetric liquid–gas two‐phase laminar flows in a pipe

An accurate finite‐volume‐based numerical method for the simulation of an isothermal two‐phase flow, consisting of a liquid slug translating in a non‐reacting gas in a circular pipe is presented. This method is built on a sharp interface concept and developed on an Eulerian Cartesian fixed‐grid with a cut‐cell scheme and marker points to track the moving interface. The unsteady, axisymmetric Navier–Stokes equations in both liquid and gas phases are solved separately. The mass continuity and momentum flux conditions are explicitly matched at the true surface phase boundary to determine the interface shape and movement. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures. It is uniquely demonstrated for the first time with the current method that conservation of mass is strictly enforced for continuous infusion of flow into the domain of computation. The method has been used to compute the velocity and pressure fields and the deformation of the liquid core. It is also shown that the current method is capable of producing accurate results for a wide range of Reynolds number, Re, Weber number, We, and large property jumps at the interface. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparison of the Eulerian and Lagrangian approaches in studying the flow pattern and heat transfer in a separated axisymmetric turbulent gas-droplet flow

The Eulerian and Lagrangian approaches are used to perform a numerical study of the disperse phase dynamics, turbulence, and heat transfer in a turbulent gas-droplet flow in a tube with sudden expansion with the following ranges of two-phase flow parameters: initial droplet size d ₁ = 0–200 µm and mass fraction of droplets M _L1 = 0–0.1. The main difference between the Eulerian and Lagrangian approaches is the difference in the predictions of the droplet mass fraction: the Eulerian approach predicts a smaller value of M _L both in the recirculation region and in the flow core (the difference reaches 15–20%). It is demonstrated that the disperse phase mass fraction calculated by the Lagrangian approach agrees better with measured data than the corresponding value predicted by the Eulerian approach.     

Effects of chemical reaction on mixed convection flow of a polar fluid through a porous medium in the presence of internal heat generation

This paper reports a detailed numerical investigation on mixed convection flow of a polar fluid through a porous medium due to the combined effects of thermal and mass diffusion. The energy equation accounts for heat generation or absorption, while the nth order homogeneous chemical reaction between the fluid and the diffusing species is included in the mass diffusion equation. The governing equations of the linear momentum, angular momentum, energy and concentration are obtained in a non-similar form by introducing a suitable group of transformations. The final set of non-similar coupled non-linear partial differential equations is solved using an implicit finite-difference scheme in combination with quasi-linearization technique. The effects of various parameters on the velocity, angular velocity, temperature and concentration fields are investigated. Numerical results for the skin friction coefficient, wall stress of angular velocity, Nusselt number and Sherwood number are also presented.     

A lagrangian lattice boltzmann method for euler equations

A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time stepsn andn+1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.     

Laminar Mixed Convection Adjacent to Vertical Continuously Stretching Sheets With Variable Viscosity and Variable Thermal Diffusivity

The paper presents a study of the laminar mixed convection adjacent to vertical continuously stretching sheets, taking into account the effects of variable viscosity and variable thermal diffusivity. The similarity solutions are reported for isothermal sheet moving with a velocity of the form u_w=Bx^0.5 and a continuous linearly stretching sheet with a linear surface temperature distribution. The equations of conservation of mass, momentum and energy, which govern the flow and heat transfer, are solved numerically by using the shooting method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. The numerical results show that, variable viscosity, variable thermal diffusivity, the velocity exponent parameter, the temperature exponent parameter and the buoyancy force parameter have significant influences on the velocity and temperature profiles, shear stress and Nusselt number in two cases air and water.     

Measuring turbulence energy with PIV in a backward-facing step flow

Turbulence energy is estimated in a backward-facing step flow with three-component (3C, stereo) particle image velocimetry (PIV). Estimates of turbulence energy transport equation for convection, turbulence transport, turbulence production, viscous diffusion, and viscous dissipation in addition to Reynolds stresses are computed directly from PIV data. Almost all the turbulence energy terms in the backward-facing step case can be measured with 3C PIV, except the pressure-transport term, which is obtained by difference of the other turbulence energy terms. The effect of the velocity spatial sampling resolution in derivative estimations is investigated with four two-dimensional PIV measurement sets. This sampling resolution information is used to calibrate the turbulence energies estimated by 3C PIV measurements. The focus of this study is on the separated shear layer of the backward-facing step. The measurements with 3C PIV are carried out in a turbulent water flow at Reynolds number of about 15,000, based on the step height h and the inlet streamwise maximum mean velocity U₀. The expansion ratio (ER) is 1.5. Turbulence energy budget profiles in locations x/h=4, x/h=6, and x/h=10 are compared with DNS data of a turbulent flow. The shapes of profiles agree well with each other. Different ERs between the PIV case (1.5) and the DNS case (1.2) cause higher values for the turbulence energies measured by PIV than the energies by DNS when x/h=10 is approached. PIV results also show that the turbulence energy level in these experiments is generally higher than that of the DNS data.     

Velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium

In this paper the velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium are obtained for the governing equations (Kaviany [7]) following the technique adopted by Chandrashekara [2] which are concerned with the interesting situations of the existence of transverse, velocity and thermal boundary layers. Here the pressure gradient is just balanced by the first and second order solid matrix resistances for small permeability and observed that by increasing of the flow resistance the asymptotic value for the heat transfer rate increases. Further we concluded that the transverse boundary layers are thicker than that of axial boundary layers. Hence we evaluated the expressions for the boundary layer thickness, the shear stress at the semi-infinite plate and _T (the ratio of the thicknesses of the thermal boundary layer and momentum boundary layer). The variations of these quantities for different values of the porous parameterB and the flow resistanceF have been discussed in detail with the help of tables. The curves for velocity and temperature distributions have been plotted for different values ofB andF.Lastly we have evaluated the heat fluxq(x) and found that it depends entirely upon the Reynolds numberRe, Prandtl numberPr,B andF.