Separation-Induced Transition to Turbulence: Second-Moment Closure Modelling

The paper presents some results of application of a low-Re-number second-moment closure (SMC) to modelling the laminar-to-turbulent transition induced by a separation bubble. The same model, tested earlier in a number of low and high-Re-number flows, was found also to reproduce reasonably well several cases of bypass transition, as well as cyclic sequence of laminarization and turbulence revival in oscillating flows at transitional Re numbers, without any artificial transition triggering. The focus of the paper is on separation-induced transition in flow over a flat plate with a circular leading edge, and on a plane surface on which a laminar separation bubble was generated by imposed suction on the wall-opposite boundary. The results show acceptable agreement with available experimental data, large-eddy and direct numerical simulations (LES, DNS). The importance of applying higher-order discretization schemes for reproducing both the bubble and the transition is also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Note on the Numerical Treatment of the k-epsilon Turbulence Model*

Numerical solution of the equations arising from the κ mdash; ε turbulence model has difficulties inherent to nonlinear convection-reaction-diffusion equations with strong reaction terms, resulting in that numerical schemes easily become unstable. We present a formulation that stresses on the robustness of the solution method, tackling common problems that produce instability. The main contribution concerns the loss of positivity of κ and ε, which is addressed by acting on the coefficients of the reaction and diffusion terms rather than on the turbulent variables themselves. In addition, a linearized implicit, non-iterative, treatment of the wall law is proposed.     

An upwind scheme for the three-dimensional boundary layer equations

The development of a calculation method to solve the compressible, three-dimensional, turbulent boundary layer equations is described. An implicit finite difference solution procedure is adopted involving local upwinding of convective transport terms. A consistent approach to discretization and linearization is taken by casting all equations in a similar form. The implementation of algebraic, one-equation and two-equation turbulence models is described. An initial validation of the method is made by comparing prediction with measurements in two quasi-three-dimensional boundary layer flows. Some of the more obvious deficiencies in current turbulence-modelling standards for three-dimensional flows are discussed.     

Experience Gained Using Second-Moment Closure Modeling for Transitional Flows Due to Boundary Layer Separation

This paper presents detailed information on the experience gained during the attempts to model a set of transitional flows due to boundary layer separation. These flows are developed on a flat plate with a semi-circular leading edge and they have been coded by the ERCOFTAC Special Interest Group on Transition, as T3L flows. Different freestream velocities and turbulence intensities configure these transitional flows and, by consequence, govern the transition mechanism, resulting in larger or smaller transitional regions. The modeling of the T3L flows has been performed by adopting a low-Reynolds number second-moment closure turbulence model. The results showed satisfactory agreement with the experimental measurements, although some difficulties regarding successful convergence have been faced. The final conclusion is that turbulence modeling can present quite accurate results for transitional flows without any additional use of ad-hoc modifications or additional equations based on various transition models and intermittency transport modeling.     

Turbulence from 1870 to 1920: The birth of a noun and of a concept

We consider here the works of French, British, and German researchers in fluid mechanics from 1870 to the beginning of the twentieth century. Our aim is to understand how the term “turbulence” introduced by William Thomson in 1887, which was not used by the main researchers of the time, including Joseph Boussinesq, Osborne Reynolds, Lord Rayleigh, Horace Lamb in the first editions of his book, became classical in the 1920s. We trace the first introductions of the terms “turbulence”, “turbulent flow” in the works of relatively unknown researchers between 1889 and 1903, until it reaches the vocabulary of mainstream researchers in fluid mechanics and physics. Our result is that the shift was in 1906–1908, when the term was used in the 1906 edition of the book of Horace Lamb, and in Lanchester's book, followed by a series of papers of German researchers before the First World War.The use of the word “turbulence”, a word used for a long time for crowds or for children, in a scientific context, corresponds to the introduction of a new concept, a new understanding of a scientific phenomenon clearly identified as being different from laminar motion. The study of the use of this term is also the study of the diffusion of a new concept among researchers of the time.     

A Unifying Explanation for the Damping of Turbulence by Additives or External Forces

Recent calculations for the turbulent flow of a suspension of solid spheres in a gas were carried out by solving a Navier Stokes equation for the fluid that recognizes the presence of particles as external point forces. These show that a strong damping of the fluid turbulence can be realized at remarkably small volume fractions. The suggestion is made that the presence of point forces, pseudo-point forces, or added point shear stresses could provide a general explanation for turbulence suppression caused by additives or by the acceleration of a turbulent boundary-layer.     

A numerical study of the turbulent flow around the stern of ship models

The present work is concerned with the numerical calculation of the turbulent flow field around the stern of ship models. The finite volume approximation is employed to solve the Reynolds equations in the physical domain using a body-fitted, locally orthogonal curvilinear co-ordinate system. The Reynolds stresses are modelled according to the standard k-ε turbulence model. Various numerical schemes (i.e. hybrid, skew upwind and central differencing) are examined and grid dependence tests have been performed to compare calculated with experimental results. Moreover, a direct solution of the momentum equations within the near-wall region is tried to avoid the disadvantages of the wall function approach. Comparisons between calculations and measurements are made for two ship models, i.e. the SSPA and HSVA model.     

On the numerical solution of the turbulence energy equations for wave and tidal flows

This paper deals with the numerical solution, using finite difference methods, of the hydrodynamic and turbulence energy equations which describe wind wave and tidally induced flow. Calculations are performed using staggered and non-staggered finite difference grids in the vertical, with various time discretizations of the production and dissipation terms in the turbulence energy equations. It is shown that the time discretization of these terms can significantly influence the stability of the solution. The effect of time filtering on the numerical stability of the solution is also considered. The form of the mixing length is shown to significantly influence the bed stress in wind wave problems. A no-slip condition is applied at the sea bed, and the associated high-shear bottom boundary layer is resolved by transforming the equations onto a logarithmic or log-linear co-ordinate system before applying the finite difference scheme. A computationally economic method is developed which remains stable even when a very fine vertical grid (over 200 points) is used with a time step of up to 30 min.     

Multifronts and transputer networks for solving fluid mechanical finite element systems

This paper is concerned with the implementation of a multifrontal solver on a network of transputers. We briefly discuss the transputer, outline the frontal and multifrontal schemes and consider the implementation of these schemes on the network. Results are presented for two test problems in fluid mechanics showing that the solver displays close to linear speed-up.     

A variational model for fracture mechanics: Numerical experiments

In the variational model for brittle fracture proposed in Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319-1342], the minimum problem is formulated as a free discontinuity problem for the energy functional of a linear elastic body. A family of approximating regularized problems is then defined, each of which can be solved numerically by a finite element procedure. Here we re-formulate the minimum problem within the context of finite elasticity. The main change is the introduction of the dependence of the strain energy density on the determinant of the deformation gradient. This change requires new, more general existence and Γ-convergence results. The results of some two-dimensional numerical simulations are presented, and compared with corresponding simulations made in Bourdin et al. [2000. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797-826] for the linear elastic model.     

A velocity formulation for flow past a symmetric profile with a wake

A model having velocity components as basic unknowns is presented for calculation of two-dimensional flow past a symmetric profile with a wake in a channel. A modified least squares functional is used for the finite element solution of velocities. The determination of the position of the free streamline is treated as an optimum design problem. The concepts of cost function, geometry parameter and sensitivity derivative are employed. Numerical results are compared with published results obtained with streamfunction formulations.     

Finite difference methods for modeling porous media flows

The purpose of this study is to determine what finite-difference algorithms are best used in numerical simulation of two-dimensional single-phase saturated porous media flows when the models have a nondiagonal symmetric tensor for the mobility (or hydraulic conductivity) that has nontrivial jump discontinuities along lines that are not aligned with the coordinate axes. Such problems arise naturally in many modeling situations and, in addition, when simpler problems are studied using adaptive grids.The answer is surprising, the simplest finite-difference method, called the MAC Scheme with Linear Averaging, performs nearly as well as most other algorithms over a wide range of problems. A new algorithm, called the Full Harmonic Averaged Scheme, is significantly more costly to use, but does perform better than the simplest scheme in certain interesting cases. The simplest finite-difference method is compared to some finite-element simulations taken from the literature; the finite-difference algorithm performs better.Many of the conclusions of the paper rest on testing the algorithms on a new class of problems with analytic solutions. The problems have a nondiagonal mobility tensor and can have a jump discontinuity of arbitrary height.The research on which this report is based was financed in part by the U.S. Department of Energy through the New Mexico Waste-management Education and Research Consortium (WERC) and Sandia National Laboratories and the Department of Energy under Contract DE-AC04-76DP00789, Dr. M. G. Marietta, Technical Monitor, and the Schlumberger Foundation.     

A net inflow method for incompressible viscous flow with moving free surface

A new finite element procedure called the net inflow method has been developed to simulate time-dependent incompressible viscous flow including moving free surfaces and inertial effects. As a fixed mesh approach with triangular element, the net inflow method can be used to analyse the free surface flow in both regular and irregular domains. Most of the empty elements are excluded from the computational domain, which is adjusted successively to cover the entire region occupied by the liquid. The volume of liquid in a control volume is updated by integrating the net inflow of liquid during each iteration. No additional kinetic equation or material marker needs to be considered. The pressure on the free surface and in the liquid region can be solved explicitly with the continuity equation or implicitly by using the penalty function method. The radial planar free surface flow near a 2D point source and the dam-breaking problem on either a dry bed or a still liquid have been analysed and presented in this paper. The predictions agree very well with available analytical solutions, experimental measurements and/or other numerical results.     

A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

A finite difference technique for solving the Newtonian jet swell problem

A finite difference technique that incorporates a numerical mapping has been successfully applied to analyse both planar and axisymmetric Newtonian jets. A pressure gradient equation and a free-surface slope equation have been derived for free-surface iteration. The computation of pressure inside the jet surface using the pressure gradient equation is stable and accurate at high Reynolds numbers. The free-surface slope equation is needed for updating the free surface and is applicable for jets with strong surface tension effects. The present development can simulate the Newtonian jets for Reynolds numbers as high as 2000 and capillary number as low as 10^?5. Numerical predictions by the present technique are close to the results of previous finite element simulations.     

Sur la fermeture des termes rapides pour des écoulements moyens rotationnelsOn the closure of rapid terms for rotational mean flows

The modelling of homogeneous turbulent flows with mean rotatio, considered in a previous Note, is handled under the form $\text{M}{}^1\text{=}\text{M}\text{[}\text{b}\text{,}\text{y}\text{,}\text{Q}{}^1\text{]}$ in terms of componental and dimensional anisotropies, and of the symmetrized stropholysis. A systematic technique of expansion is proposed. The necessary realisability conditions are then applied. It is shown that there exists no realisable functional $\text{M}$ which would be isotropic with respect to its arguments. To cite this article: J. Piquet, C. R. Mecanique 331 (2003).