Simulation of coating flows with slip effects

This work is concerned with the numerical prediction of wire coating flows. Both annular tube‐tooling and pressure‐tooling type extrusion–drag flows are investigated for viscous fluids. The effects of slip at die walls are analysed and free surfaces are computed. Flow conditions around the die exit are considered, contrasting imposition of no‐slip and various instances of slip models for die wall conditions. Numerical solutions are computed by means of a time marching Taylor–Galerkin/pressure–correction finite element scheme, that demonstrates how slip conditions on die walls mitigate stress singularities at the die exit. For pressure‐tooling and with appropriate handling of slip, reduction in shear rate at the die exit may be achieved. Maximum shear rates for tube‐tooling are about one quarter of those encountered in pressure‐tooling. Equivalently, extension rates peak at land entry, and tube‐tooling values are one third of those observed for pressure‐tooling. With slip and tube‐tooling, peak shear values at die exit may be almost completely eliminated. Nevertheless, in contrast to the pressure‐tooling scenario, this produces larger peak shear rates upstream within the land region than would otherwise be the case for no‐slip. Copyright © 2000 John Wiley & Sons, Ltd.     

Viscoelastic computations for reverse roll coating with dynamic wetting lines and the Phan-Thien-Tanner models

The computational modelling of reverse roll coating with dynamic wetting line has been analysed for various non-Newtonian viscoelastic materials appealing to the Phan-Thien-Tanner (PTT) network class of models suitable for typical polymer solutions, with properties of shear thinning and strain hardening/softening. The numerical technique utilizes a hybrid finite element-sub-cell finite volume algorithm with a dynamic free-surface location, drawing upon a fractional-staged predictor-corrector semi-implicit time-stepping procedure of an incremental pressure-correction form. The numerical solution is investigated following a systematic study which allows for parametric variation in elasticity (We-variation), extensional hardening-softening (ε), and solvent fraction (β). Under incompressible flow conditions, linear PTT (LPTT) and exponential PTT (EPTT) models were used to solve the paint strip coatings, under reverse roll-coating configuration. This involves two-dimensional planar reverse roll-coating domains, considering a range of Weissenberg numbers (We) up to critical levels, addressing velocity fields and vortex development, pressure and lift profiles, shear rate, and stress fields. Various differences are observed when comparing solutions for these constitutive models. Concerning the effects of elasticity, increase in We stimulates vortex structures, which are visible at both the downstream meniscus and upstream narrowest nip region, whilst decreasing the peak pressure and lift values at the nip constriction. At low values (ε > 0.5, β = 0.1) of extensional viscosity, the LPTT flow fields were much easier to extract, attaining critical We levels up to unity, in contrast to critical We levels of 0.4 for EPTT solutions. This finding is reversed at higher extensional viscosity levels (ε < 0.5). This trend reveals qualitative agreement with theoretical studies. Noting flow behaviour under EPTT solution, increasing the peak level of strain hardening/softening is found to stimulate vortex activity around the nip region, with a corresponding increase in peak pressure and lift values.     

Effects of Reynolds number on physiological‐type pulsatile flows in a pipe with ring‐type constrictions

The effects of Reynolds number on the physiological‐type of laminar pulsatile flow fields within the vicinity of mechanical ring‐type constriction in small pipes were studied numerically. The parameters considered are: the Reynolds number (Re) in the range of 50–1500; Strouhal number (St) in the range of 0.00156–3.98; Womersley number (Nw) from 0.0 to 50.0. The pulsatile flows considered were physiological‐type of simulated flows. Within a pulsating cycle, detailed flow characteristics were studied through the pulsating contours of streamline (ψ), vorticity (Ω), shear stress (τ) and isobar. The relations between the instantaneous flow rate (Q) and instantaneous pressure gradients (dp/dz) are observed to be elliptic. The relations between the instantaneous flow rate (Q) and pressure loss (P_loss) are quadratic. Linear relations were observed between the instantaneous flow rate (Q) and the maximum velocity, maximum vorticity and maximum shear stress. The Reynolds number of the flow in a pulsating cycle was found to have significant effects on the recirculation length and the pressure gradient within the pulsatile flow regime. Copyright © 1999 John Wiley & Sons, Ltd.     

Viscoelastic simulations of stick‐slip and die‐swell flows

Numerical solutions of viscoelastic flows are demonstrated for a time marching, semi‐implicit Taylor–Galerkin/pressure‐correction algorithm. Steady solutions are sought for free boundary problems involving combinations of die‐swell and stick‐slip conditions. Flows with and without drag flow are investigated comparatively, so that the influence of the additional component of the drag flow may be analysed effectively. The influence of die‐swell is considered that has application to various industrial processes, such as wire coating. Solutions for two‐dimensional axisymmetric flows with an Oldroyd‐B model are presented that compare favourably with the literature. The study advances our prior fixed domain formulation with this algorithm, into the realm of free‐surface viscoelastic flows. The work involves streamline‐upwind/Petrov–Galerkin weighting and velocity gradient recovery techniques that are applied upon the constitutive equation. Free surface solution reprojection and a new pressure‐drop/mass balance scheme are proposed. Copyright © 2001 John Wiley & Sons, Ltd.     

A non‐linear k–ε model with realizability for prediction of flows around bluff bodies

The incompressible flow around bluff bodies (a square cylinder and a cube) is investigated numerically using turbulence models. A non‐linear k–ε model, which can take into account the anisotropy of turbulence with less CPU time and computer memory then RSM or LES, is adopted as a turbulence model. In tuning of the model coefficients of the non‐linear terms are adjusted through the examination of previous experimental studies in simple shear flows. For the tuning of the coefficient in the eddy viscosity (=C_μ), the realizability constraints are derived in three types of basic 2D flow patterns, namely, a simple shear flow, flow around a saddle and a focal point. C_μ is then determined as a function of the strain and rotation parameters to satisfy the realizability. The turbulence model is first applied to a 2D flow around a square cylinder and the model performance for unsteady flows is examined focussing on the period and the amplitude of the flow oscillation induced by Karman vortex shedding. The applicability of the model to 3D flows is examined through the computation of the flow around a surface‐mounted cubic obstacle. The numerical results show that the present model performs satisfactorily to reproduce complex turbulent flows around bluff bodies. Copyright © 2003 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Computations of strongly swirling flows with second‐moment closures

The present study is concerned with simulating turbulent, strongly swirling flows by eddy viscosity model and Reynolds stress transport model variants adopting linear and quadratic form of the pressure–strain models. Flows with different inlet swirl numbers, 2.25 and 0.85, were investigated. Detailed comparisons of the predicted results and measurements were presented to assess the merits of model variants. For the swirl number 2.25 case, due to the inherent capability of the Reynolds stress models to capture the strong swirl and turbulence interaction, both the linear and quadratic form of the pressure–strain models predict the flow adequately. In strong contrast, the k–ϵ model predicts an excessively diffusive flow fields. For the swirl number 0.85 case, both the k–ϵ and Reynolds stress model with linear pressure–strain process, show an excessive diffusive transport of the flow fields. The quadratic pressure–strain model, on the other hand, mimics the correct flow development with the recirculating region being correctly predicted. Copyright © 1999 John Wiley & Sons, Ltd.     

Computation of two‐dimensional flows past ram‐air parachutes

Computational results for flow past a two‐dimensional model of a ram‐air parachute with leading edge cut are presented. Both laminar (Re=10⁴) and turbulent (Re=10⁶) flows are computed. A well‐proven stabilized finite element method (FEM), which has been applied to various flow problems earlier, is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The Baldwin–Lomax model is employed for turbulence closure. Turbulent flow computations past a Clarck‐Y airfoil without a leading edge cut, for α=7.5°, result in an attached flow. The leading edge cut causes the flow to become unsteady and leads to a significant loss in lift and an increase in drag. The flow inside the parafoil cell remains almost stagnant, resulting in a high value of pressure, which is responsible for giving the parafoil its shape. The value of the lift‐to‐drag ratio obtained with the present computations is in good agreement with those reported in the literature. The effect of the size and location of the leading edge cut is studied. It is found that the flow on the upper surface of the parafoil is fairly insensitive to the configuration of the cut. However, the flow quality on the lower surface improves as the leading edge cut becomes smaller. The lift‐to‐drag ratio for various configurations of the leading edge cut varies between 3.4 and 5.8. It is observed that even though the time histories of the aerodynamic coefficients from the laminar and turbulent flow computations are quite different, their time‐averaged values are quite similar. Copyright © 2001 John Wiley & Sons, Ltd.     

Turbulence in a skewed three‐dimensional wall‐bounded shear flow: effect of mean vorticity on structure modification

Mean‐flow three‐dimensionalities affect both the turbulence level and the coherent flow structures in wall‐bounded shear flows. A tailor‐made flow configuration was designed to enable a thorough investigation of moderately and severely skewed channel flows. A unidirectional shear‐driven plane Couette flow was skewed by means of an imposed spanwise pressure gradient. Three different cases with 8°, 34°and 52°skewing were simulated numerically and the results compared with data from a purely two‐dimensional plane Couette flow. The resulting three‐dimensional flow field became statistically stationary and homogeneous in the streamwise and spanwise directions while the mean velocity vector V and the mean vorticity vector Ω remained parallel with the walls. Mean flow profiles were presented together with all components of the Reynolds stress tensor. The mean shear rate in the core region gradually increased with increasing skewing whereas the velocity fluctuations were enhanced in the spanwise direction and reduced in the streamwise direction. The Reynolds shear stress is known to be closely related to the coherent flow structures in the near‐wall region. The instantaneous and ensemble‐averaged flow structures were turned by the skewed mean flow. We demonstrated for the medium‐skewed case that the coherent structures should be examined in a coordinate system aligned with V to enable a sound interpretation of 3D effects. The conventional symmetry between Case 1 and Case 2 vortices was broken and Case 1 vortices turned out to be stronger than Case 2. This observation is in conflict with the common understanding on the basis of the spanwise (secondary) mean shear rate. A refined model was proposed to interpret the structure modifications in three‐dimensional wall‐flows. What matters is the orientation of the mean vorticity vector Ω relative to the vortex vorticity vector ω _v, that is, the sign of Ω · ω _v. In the present situation, Ω · ω _v > 0 for the Case 1 vortices causing a strengthening relative to the Case 2 vortices. Copyright © 2011 John Wiley & Sons, Ltd.     

Turbulence in a three‐dimensional wall‐bounded shear flow

A new turbulent flow with distinct three‐dimensional characteristics has been designed in order to study the impact of mean‐flow skewing on the turbulent coherent vortices and Reynolds‐averaged statistics. The skewing of a unidirectional plane Couette flow was achieved by means of a spanwise pressure gradient. Direct numerical simulations of the statistically steady Couette–Poiseuille flow enabled in‐depth explorations of the turbulence field in the skewed flow. The imposition of a modest spanwise gradient turned the mean flow about 8° away from the original Couette flow direction and this turning angle remained nearly the same over the entire cross section. Nevertheless, a substantial non‐alignment between the turbulent shear stress angle and the mean velocity gradient angle was observed. The structure parameter turned out to slightly exceed that in the pure Couette flow, contrary to the observations made in some other three‐dimensional shear flows. Coherent flow structures, which are known to be associated with the Reynolds shear stress in near‐wall regions, were identified by the λ₂‐criterion. Instantaneous and ensemble‐averaged vortices resembled those found in the unidirectional Couette flow. In the skewed flow, however, the vortex structures were turned to align with the local mean‐flow direction. The conventional symmetry between Case 1 and Case 2 vortices was broken due to the mean‐flow three‐dimensionality. The turning of the coherent vortices and the accompanying symmetry‐breaking gave rise to secondary and tertiary turbulent shear stress components. By averaging the already ensemble‐averaged shear stresses associated with Case 1 and Case 2 vortices in the homogeneous directions, a direct link between the educed near‐wall structures and the Reynolds‐averaged turbulent stresses was established. These observations provide evidence in support of the hypothesis that the structural model proposed for two‐dimensional turbulent boundary layers remains valid also in flows with moderate mean three‐dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

Evaluation of one‐ and two‐equation low‐Re turbulence models. Part I—Axisymmetric separating and swirling flows

This first segment of the two‐part paper systematically examines several turbulence models in the context of three flows, namely a simple flat‐plate turbulent boundary layer, an axisymmetric separating flow, and a swirling flow. The test cases are chosen on the basis of availability of high‐quality and detailed experimental data. The tested turbulence models are integrated to solid surfaces and consist of: Rodi's two‐layer k–ε model, Chien's low‐Reynolds number k–ε model, Wilcox's k–ω model, Menter's two‐equation shear‐stress‐transport model, and the one‐equation model of Spalart and Allmaras. The objective of the study is to establish the prediction accuracy of these turbulence models with respect to axisymmetric separating flows, and flows of high streamline curvature. At the same time, the study establishes the minimum spatial resolution requirements for each of these turbulence closures, and identifies the proper low‐Mach‐number preconditioning and artificial diffusion settings of a Reynolds‐averaged Navier–Stokes algorithm for optimum rate of convergence and minimum adverse impact on prediction accuracy. Copyright © 2003 John Wiley & Sons, Ltd.     

Die swell measurements of second‐order fluids: numerical experiments

An analysis of the flow of a second‐order fluid is presented. Reference values for some variables are defined, and with these a non‐dimensional formulation of the governing equations. From this formulation, three dimensionless numbers appear; one is the Reynolds number, and two numbers that are called the first‐ and second‐dimensionless normal stress (NSD) coefficients. The equations of motion are solved by a finite element method using a commercially available program (Fidap), and the steady state converged solution was used to measure the die swell. The factors that influence die swell and that are studied in this work include: the die geometry for circular cross sectional dies, including tubular, converging, diverging, half‐converging/half‐tubular shapes; fluid characteristics such as Reynolds number and first‐ and second‐DNS coefficients (both positive and negative values); and flow rates, as determined by the maximum velocity in a parabolic velocity profile at the entrance to the die. The results suggest that shear and deformation histories of the fluid directly influence not only swell characteristics, but also convergence characteristics of the numerical simulation. © 1999 John Wiley & Sons, Ltd.     

A computational methodology for two‐dimensional fluid flows

A weighted residual collocation methodology for simulating two‐dimensional shear‐driven and natural convection flows has been presented. Using a dyadic mesh refinement, the methodology generates a basis and a multiresolution scheme to approximate a fluid flow. To extend the benefits of the dyadic mesh refinement approach to the field of computational fluid dynamics, this article has studied an iterative interpolation scheme for the construction and differentiation of a basis function in a two‐dimensional mesh that is a finite collection of rectangular elements. We have verified that, on a given mesh, the discretization error is controlled by the order of the basis function. The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation. We have also verified the technique with a dynamical core of a two‐dimensional flow in primitive variables. An excellent result has been observed—on resolving a shear layer and on the conservation of the potential and the kinetic energies—with respect to previously reported benchmark simulations. In particular, the shear‐driven simulation at CFL = 2.5 (Courant–Friedrichs–Lewy) and (Reynolds number) exhibits a linear speed up of CPU time with an increase of the time step, Δt. For the natural convection flow, the conversion of the potential energy to the kinetic energy and the conservation of total energy is resolved by the proposed method. The computed streamlines and the velocity fields have been demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.     

Time‐dependent turbulent cavitating flow computations with interfacial transport and filter‐based models

Turbulent cavitating flow computations need to address both cavitation and turbulence modelling issues. A recently developed interfacial dynamics‐based cavitation model (IDCM) incorporates the interfacial transport into the computational modelling of cavitation dynamics. For time‐dependent flows, it is known that the engineering turbulence closure such as the original k–ε model often over‐predicts the eddy viscosity values reducing the unsteadiness. A recently proposed filter‐based modification has shown that it can effectively modulate the eddy viscosity, rendering better simulation capabilities for time‐dependent flow computations in term of the unsteady characteristics. In the present study, the IDCM along with the filter‐based k–ε turbulence model is adopted to simulate 2‐D cavitating flows over the Clark‐Y airfoil. The chord Reynolds number is Re=7.0 × 10⁵. Two angles‐of‐attack of 5 and 8° associated with several cavitation numbers covering different flow regimes are conducted. The simulation results are assessed with the experimental data including lift, drag and velocity profiles. The interplay between cavitation and turbulence models reveals substantial differences in time‐dependent flow results even though the time‐averaged characteristics are similar. Copyright © 2005 John Wiley & Sons, Ltd.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

A new second‐moment closure approach for turbulent swirling confined flows

An improved anisotropic model for the dissipation rate—ε—of the turbulent kinetic energy (k), to be used together with a non‐linear pressure‐strain correlations model, is proposed. Experimental data from the open literature for two confined turbulent swirling flows are used to assess the performance of the proposed model in comparison to the standard ε transport equation and to a linear approach to model the pressure‐strain term that appears in the exact equations for the Reynolds‐stress tensor. For the less strongly swirling flow the predictions show much more sensitivity to the εtransport equation than to the pressure‐strain model. In opposition, for the more strongly swirling flow, the results show that the predictions are much sensitive to the pressure‐strain model. Nevertheless, the improved εtransport equation together with the non‐linear pressure strain model yield predictions in good agreement with experiments in both studied cases. Copyright © 2003 John Wiley & Sons, Ltd.     

Relationship between vortex and sound in under‐expanded supersonic impinging jets

The study of an under‐expanded supersonic jet impinging on a flat plate by using large‐eddy simulation is reported. A third‐order upwind compact difference and a fourth‐order symmetric compact scheme are employed to discretize the nondimensional axisymmetric compressible Favre‐filtered Navier–Stokes equations in space, whereas the third‐order Runge–Kutta method with the total variation diminishing property is adopted to deal with the temporal discretization. The numerical simulation successfully captures the shock wave and vortex structures with different scales in the flow field. Waves with high and low frequencies traveling forward and reflecting back, and sound sources in different locations can be observed. By comparison with the frequency of the impinging tone from the experiment, it can be deduced that the change of pressure and swirling strength in the shear layer, pressure change on the impinging plate, and vortex merging in the jet shear layer are interdependent with the impinging tone. The effects of nozzle lip thickness on the impinging jet flow field have been investigated. The results show that the values of pressure fluctuation and vortex swirling strength in the shear layer near the nozzle have an extremum with the variation of the nozzle lip thickness. The results provide a theoretical foundation for the design of supersonic nozzles. Copyright © 2011 John Wiley & Sons, Ltd.     

A further work on multi‐phase two‐fluid approach for compressible multi‐phase flows

This paper is to continue our previous work Niu (Int. J. Numer. Meth. Fluids 2001; 36 :351–371) on solving a two‐fluid model for compressible liquid–gas flows using the AUSMDV scheme. We first propose a pressure–velocity‐based diffusion term originally derived from AUSMDV scheme Wada and Liou (SIAM J. Sci. Comput. 1997; 18 (3):633—657) to enhance its robustness. The scheme can be applied to gas and liquid fluids universally. We then employ the stratified flow model Chang and Liou (J. Comput. Physics 2007; 225 :240–873) for spatial discretization. By defining the fluids in different regions and introducing inter‐phasic force on cell boundary, the stratified flow model allows the conservation laws to be applied on each phase, and therefore, it is able to capture fluid discontinuities, such as the fluid interfaces and shock waves, accurately. Several benchmark tests are studied, including the Ransom's Faucet problem, 1D air–water shock tube problems, 2D shock‐water column and 2D shock‐bubble interaction problems. The results indicate that the incorporation of the new dissipation into AUSM⁺‐up scheme and the stratified flow model is simple, accurate and robust enough for the compressible multi‐phase flows. Copyright © 2008 John Wiley & Sons, Ltd.     

An application of the finite difference‐based lattice Boltzmann model to simulating flow‐induced noise

This paper presents a novel approach to simulate aerodynamically generated sounds by modifying the finite difference‐based lattice BGK compressible fluid model for the purpose of speeding up the calculation and also stabilizing the numerical scheme. With the model, aerodynamic sounds generated by a uniform flow around a two‐dimensional circular cylinder at Re = 150 are simulated. The third‐order‐accurate up‐wind scheme is used for the spatial derivatives, and the second‐order‐accurate Runge–Kutta method is applied for the time marching. The results show that we successively capture very small acoustic pressure fluctuations, with the same frequency of the Karman vortex street, much smaller than the whole pressure fluctuation around a circular cylinder. The propagation velocity of the acoustic waves shows that the points of peak pressure are biased upstream owing to the Doppler effect in the uniform flow. For the downstream, on the other hand, it is faster. It is also apparent that the amplitude of sound pressure is proportional to r^?1/2, r being the distance from the centre of the circular cylinder. Moreover, the edgetone generated by a two‐dimensional jet impinging on a wedge to predict the frequency characteristics of the discrete oscillations of a jet‐edge feedback cycle is investigated. The jet is chosen long enough to guarantee the parabolic velocity profile of the jet at the outlet, and the edge is of an angle of α = 23°. At a stand‐off distance w, the edge is inserted along the centreline of the jet, and a sinuous instability wave with real frequency is assumed to be created in the vicinity of the nozzle exit and to propagate towards the downstream. We have succeeded in capturing small pressure fluctuations resulting from periodic oscillation of jet around the edge. Copyright © 2006 John Wiley & Sons, Ltd.