A novel implicit numerical treatment for non‐linear turbulence models using high and low Reynolds number formulations

Non‐linear turbulence models can be seen as an improvement of the classical eddy‐viscosity concept due to their better capacity to simulate characteristics of important flows. However, application of non‐linear models demand robustness of the numerical method applied, requiring a stable discretization scheme for convergence of all variables involved. Usually, non‐linear terms are handled in an explicit manner leading to possible numerical instabilities. Thus, the present work shows the steps taken to adapt a general non‐linear constitutive equation using a new semi‐implicit numerical treatment for the non‐linear diffusion terms. The objective is to increase the degree of implicitness of the solution algorithm to enhance convergence characteristics. Flow over a backward‐facing step was computed using the control volume method applied to a boundary‐fitted coordinate system. The SIMPLE algorithm was used to relax the algebraic equations. Classical wall function and a low Reynolds number model were employed to describe the flow near the wall. The results showed that for certain combination of relaxation parameters, the semi‐implicit treatment proposed here was the sole successful treatment in order to achieve solution convergence. Also, application of the implicit method described here shows that the stability of the solution either increases (high Reynolds with non‐orthogonal mesh) or preserves the same (low Reynolds number applications). Additional advantages of the procedure proposed here lie in the possibility of testing different non‐linear expressions if one considers the enhanced robustness and stability obtained for the entire numerical algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

The stability of two concentric non-Newtonian fluids in circular pipe flow

The flow of two concentric non-Newtonian fluids, under constant pressure gradient in a circular tube, is studied by linear stability analysis. The viscosities of the two fluids are different and their dependence on shear stress is described by the Ellis model. It is found that the steady state flow can be unstable, depending on certain combinations of the values of physical parameters, to infinitesimal axisymmetric disturbances of large wavelengths, for any Reynolds number however small. The flow is predominantly stable if the inner fluid is more viscous and predominantly unstable if the outer fluid is more viscous. Stronger dependence of viscosity on shear stress can both stabilize and destabilize the flow. Interfacial tension is also destabilizing when the Weber number is small than about 10⁴.     

Sensitivity computations of eddy viscosity models with an application in drag computation

This paper presents a numerical study of the sensitivity of an eddy viscosity model with respect to the variation of the eddy viscosity parameter for the two‐dimensional driven cavity problem and flow around a cylinder. The main objective is to provide a comparison between computing the sensitivity using sensitivity equation and computing the sensitivity using finite difference methods and also numerically illustrate the application of the sensitivity computations in improving drag flow functional. Copyright © 2006 John Wiley & Sons, Ltd.     

An implicit time‐marching algorithm for shallow water models based on the generalized wave continuity equation

Wave equation models currently discretize the generalized wave continuity equation with a three‐time‐level scheme centered at k and the momentum equation with a two‐time‐level scheme centered at k+1/2; non‐linear terms are evaluated explicitly. However in highly non‐linear applications, the algorithm becomes unstable at even moderate Courant numbers. This paper examines an implicit treatment of the non‐linear terms using an iterative time‐marching algorithm. Depending on the domain, results from one‐dimensional experiments show up to a tenfold increase in stability and temporal accuracy. The sensitivity of stability to variations in the G‐parameter (a numerical weighting parameter in the generalized wave continuity equation) was examined; results show that the greatest increase in stability occurs with G/τ=2–50. In the one‐dimensional experiments, three different types of node spacing techniques—constant, variable, and LTEA (Localized Truncation Error Analysis)—were examined; stability is positively correlated to the uniformity of the node spacing. Lastly, a scaling analysis demonstrates that the magnitudes of the non‐linear terms are positively correlated to those that most influence stability, particularly the term containing the G‐parameter. It is evident that the new algorithm improves stability and temporal accuracy in a cost‐effective manner. Copyright © 2001 John Wiley & Sons, Ltd.     

Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot

A Fourier–Galerkin spectral technique for solving coupled higher‐order initial‐boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so‐called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived. The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth‐order algebraic for both linear and nonlinear problems. Though algebraic, the fifth‐order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems. An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement. The new technique is applied to the G‐jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters. Copyright © 2008 John Wiley & Sons, Ltd.     

Sensitivity analysis of low Reynolds number channel flow using the finite volume method

An unsteady finite volume‐based fractional step algorithm solved on a staggered grid has been developed for computing design sensitivity parameters in two‐dimensional flows. Verification of the numerical code is performed for the case of low Reynolds number, pressure‐driven flow through a straight channel, which has an exact steady‐state solution to the Navier–Stokes equations. Sensitivity of the flow to the channel height, fluid viscosity, and imposed pressure gradient is considered. Three different numerical techniques for computing the design sensitivity parameters: finite difference, complex‐step differentiation, and sensitivity equation method (SEM), are compared in terms of numerical error (relative to the exact solution), computational expense, and ease of implementation. Results indicate that, of all the three methods, complex step is the most accurate and requires the least computational time. In addition, treatment of the boundary conditions in SEM is addressed, within the framework of the present finite volume approach, with special attention given to parameter dependence in the boundary conditions. Error estimation based on the Grid Convergence Index provides a good indication of the exact error in the SEM solutions. An example of application of the use of sensitivity parameters to estimate the propagation of input uncertainty through the numerical simulation is also provided. Copyright © 2007 John Wiley & Sons, Ltd.     

On stability of strong and weak reflected shocks

A time-realistic adaptive unstructured Euler code is used to demonstrate the numerical existence and investigate the stability of both weak and strong reflected shocks in regular reflection. For supersonic parallel flow in a channel, impinging on two symmetrical opposing wedges, the weak reflected configuration is insensitive to downstream pressure disturbances and therefore stable. The strong reflected shock configuration is unstable to positive perturbations in back-pressure and neutrally stable to negative perturbations. A unique -shock structure provides the transition mechanism between weak and strong reflected shock configurations. Received 6 September 1999 / Accepted 10 August 2000     

Stability of the crossflow pattern around a slender and influence of disturbance

Topological structure and stability of a slender cross flow is discussed by the stability theory of dynamic system. The inner boundary of flow field was limiting streamline and it was proved that the topological structure connected saddles by limiting streamline is stable. It is proved that the development of slender vortices leads to the change of topological structure about cross flow. And it is the change from stable and symmetrical vortices flow pattern to unstable and symmetrical vortices flow pattern, and then to stable and asymmetrical vortices flow pattern due to little disturbance which leads to the development of asymmetrical slender vortices. The influence of disturbance to flowfield structure was discussed by unfolding theory too.     

Numerical investigation of the flow front behaviour in the co‐injection moulding process

This paper presents a three‐dimensional (3D) solution algorithm for solving the sequential co‐injection moulding process. The flow of skin and core materials inside a rectangular cavity is investigated both numerically and experimentally. A 3D finite element flow analysis code is used to solve the governing equations of the non‐isothermal sequential co‐injection moulding. The predicted flow front behaviour is compared to the experimental observations for various skin/core volume ratio, injection speed, injection temperature, and core injection delay. Simulation results are in good agreement with experimental data and indicate correctly the trends in solution change when processing parameters are changing. Solutions are also shown for the filling of a spiral‐flow mould. The numerical approach is shown to predict the core expansion phase during which the flow front of core and skin materials advance together without breakthrough. Breakthrough phenomena is also predicted and the numerical solution is in good agreement with the experiment. Copyright © 2005 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.     

Onset of flow instability in a heated capillary tube

We study the stability of flow in a heated capillary tube with an evaporating meniscus. The behavior of the vapor/liquid system, which undergoes small perturbations, is analyzed by linear approximation, in the frame of a one-dimensional model of capillary flow, with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes, on the flow stability is studied. The velocity, pressure and temperature oscillations in a capillary tube with a constant wall heat flux or a constant wall temperature are determined. A scenario of a possible process at small and moderate Peclet numbers corresponding to the flow in capillaries is considered. The boundaries of stability, subdividing the domains of stable and unstable flows, are outlined, and the values of geometrical and operating parameters corresponding to the transition from stable to unstable flow are estimated. It is shown that the stable capillary flow occurs at relatively small wall heat fluxes, whereas at high ones, the flow is unstable, with continuously growing velocity, pressure and temperature oscillations.     

A stabilized formulation for the advection–diffusion equation using the Generalized Finite Element Method

This paper presents a stable formulation for the advection–diffusion equation based on the Generalized (or eXtended) Finite Element Method, GFEM (or X‐FEM). Using enrichment functions that represent the exponential character of the exact solution, smooth numerical solutions are obtained for problems with steep gradients and high Péclet numbers in one‐ and two‐dimensions. In contrast with traditional stabilized methods that require the construction of stability parameters and stabilization terms, the present work avoids numerical instabilities by improving the classical Galerkin solution with enrichment functions (that need not be polynomials) using GFEM, which is an instance of the partition of unity framework. This work also presents a strategy for constructing enrichment functions for problems involving complex geometries by employing a global–local‐type approach. Representative numerical results are presented to illustrate the performance of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamic ship response integration for numerical prediction of squat in highly restricted waterways

In this paper we are interested in numerical modeling of ‘dynamic’ phenomenon of squat by finite elements analysis. It proposes a set of modular numerical tools; therefore, interchangeable. This model enables the study of the interaction between a two‐dimensional potential flow in highly restricted waterways on irregular shaped bottom with stationary free surface in taking into account the dynamic response of a ship. The proposed model has been used to validate a stability model based on the extension of the one‐dimensional theory of Schijf to the dynamic effects by pointing out stable and unstable squat positions for a ship. It is also shown that for two cases of depth change in shallow water (‘step’), unstable position may be reached. Copyright © 2009 John Wiley & Sons, Ltd.     

Transient two‐layer thin‐film flow

The transient two‐layer thin‐film planar flow is investigated theoretically in this study. The interplay among inertia, viscous and surface/interfacial tension is emphasized. It is found that the film and interface profiles, as well as the flow field, are strongly influenced by the viscosity ratio, velocity and film thickness ratios at inception, and the surface‐to‐interfacial tension ratio. The nonlinear stability of the steady state reveals the formation of a solitary wave after flow inception, which propagates in the form of a convective instability, with the steady state recovered only in the tail (upstream) region of the wave. In the presence of surface/interfacial tension, surface modulation appears, which grows in wavelength and amplitude with position. The flow is found to be particularly stable for higher viscosity of the lower film layer. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulations of interfacial instabilities on a rotating miscible droplet in a time‐dependent gap Hele–Shaw cell with significant Coriolis effects

Interfacial instability of a rotating miscible droplet with significant Coriolis force in a Hele–Shaw cell is simulated numerically. The influences of the relevant control parameters are first discussed qualitatively by fingering patterns. More vigorous fingerings are found at higher rotational effects, a lower viscosity contrast and a weaker effective surface tension (Korteweg constant). For a time‐dependent gap Hele–Shaw cell, a higher cell lifting rate makes the rotating droplet bear an inward straining flow, which leads to fingering enhancement. On the contrary, a higher pressing rate provides more stable effects by additional squeezing outward flow. A quantitative analysis between the Coriolis effects and tilting angles of fingers is addressed. For arbitrary combinations of all relevant control parameters, the values of tilting angles follow a nearly linear relationship with the Coriolis effects. We estimate the correlation between the relevant control parameters (dimensionless Coriolis factor Re, viscosity parameter R, cell lifting rate a) and tilting angles (θ) of fingers that can be approximated as for significant Korteweg stresses. Copyright © 2005 John Wiley & Sons, Ltd.     

On the construction of manufactured solutions for one and two‐equation eddy‐viscosity models

This paper presents manufactured solutions (MSs) for some well‐known eddy‐viscosity turbulence models, viz. the Spalart & Allmaras one‐equation model and the TNT and BSL versions of the two‐equation k–ω model. The manufactured flow solutions apply to two‐dimensional, steady, wall‐bounded, incompressible, turbulent flows. The two velocity components and the pressure are identical for all MSs, but various alternatives are considered for specifying the eddy‐viscosity and other turbulence quantities in the turbulence models. The results obtained for the proposed MSs with a second‐order accurate numerical method show that the MSs for turbulence quantities must be constructed carefully to avoid instabilities in the numerical solutions. This behaviour is model dependent: the performance of the Spalart & Allmaras and k–ω models is significantly affected by the type of MS. In one of the MSs tested, even the two versions of the k–ω model exhibit significant differences in the convergence properties. Copyright © 2006 John Wiley & Sons, Ltd.     

A projection method for the spectral solution of non‐homogeneous and incompressible Navier–Stokes equations

This paper is devoted to the development of a parallel, spectral and second‐order time‐accurate method for solving the incompressible and variable density Navier–Stokes equations. The method is well suited for finite thickness density layers and is very efficient, especially for three‐dimensional computations. It is based on an exact projection technique. To enforce incompressibility, for a non‐homogeneous fluid, the pressure is computed using an iterative algorithm. A complete study of the convergence properties of this algorithm is done for different density variations. Numerical simulations showing, qualitatively, the capabilities of the developed Navier–Stokes solver for many realistic problems are presented. The numerical procedure is also validated quantitatively by reproducing growth rates from the linear instability theory in a three‐dimensional direct numerical simulation of an unstable, non‐homogeneous, flow configuration. It is also shown that, even in a turbulent flow, the spectral accuracy is recovered. Copyright © 2012 John Wiley & Sons, Ltd.     

钝锥三维粘性绕流背风面分离的数值模拟

本文将作者在文献[1]中提出的方法推广应用于求解三维可压缩 N-S 方程和简化 N-S 方程,并对近似因式分解法应用于三维问题的稳定性进行了分析。指出,对二维问题原无条件稳定的格式,经近似因式分解后仍是无条件稳定的;对于三维问题,原无条件稳定的格式经普通近似因式分解后所得到的格式可能是不稳定的或条件稳定的。利用系数矩阵分裂法所得到的近似因式分解格式可仍是无条件稳定的,只要适当加大分裂后的系数反差即可。 文中给出了钝锥超音速三维粘性绕流结果。得到了背风面分离的流动图像,物面压力值与实验值吻合。     

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re ₁ , Re ₂ ) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states. Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by M.Y. Hussaini     

A 1D-Averaged Model for Stable and Unstable Miscible Flows in Porous Media with Varying Péclet Numbers and Aspect Ratios

This paper deals with reducing the number of spatial dimensions of the models used to solve stable and unstable miscible flows in saturated and homogeneous porous media. Unstable miscible displacements occur when a fluid displaces another fluid of higher viscosity, with which it can fully mix. Stable flows occur if the displaced fluid is less viscous than the displacing one. First, a 1D-averaged model is identified, capable of accurately describing unstable flows at high Péclet numbers. Second, another 1D-averaged model is determined, capable of accurately predicting miscible displacements at low Péclet numbers. Third, a new model is proposed, for any Péclet number and for both stable and unstable flows, as a combination of the previous two models. This combination involves three parameters whose values depend on the dimensionless numbers of the problem, namely, the viscosity ratio M, the Péclet number P_e, the aspect ratio A, and the dispersion length ratio ε. These parameters are computed for several values of M, P_e, A with ε=1 by matching results from direct 2D simulations, obtained from a numerical model previously validated against experimental data. It is found that a 1D-averaged model combining an extended version of the Todd–Longstaff model and the diffusive term of the 1D-miscible model forms an accurate general model for miscible displacements in homogeneous porous media. This paper also provides a large set of data computed from high-resolution 2D simulations of unstable miscible displacements.     

An adaptive boundary element approach to transient free surface flow as applied to injection molding

An adaptive (Lagrangian) boundary element approach is proposed for the general two‐dimensional simulation of confined moving‐boundary flow of viscous incompressible fluids. Only the quasi‐steady creeping (Stokes) flow of a Newtonian fluid is examined. The method is stable as it includes remeshing capabilities of the deforming moving boundary, and thus it can handle large deformations. An algorithm is developed for mesh refinement of the deforming moving‐boundary mesh. Several flow problems are presented to illustrate the utility of the approach, with particular emphasis on cavity filling and viscous fingering, as applied to conventional and gas‐assisted injection molding. The accuracy of the method is assessed through the problem of jet flow and the transient fountain flow between two flat plates. Copyright © 2000 John Wiley & Sons, Ltd.