turbulence models for fluid-structure interaction …

Emirates Journal for Engineering Research, 11(2), 1-18(2006) (Review Paper)

1

TURBULENCE MODELS FOR FLUID-STRUCTURE INTERACTION APPLICATIONS

A. BALABEL and D. DINKLER

Institute for Structural Analysis, TU-Braunschweig, Germany

(Received March 2006 and accepted October 2006)

لرياضية وطرق المحاآاة العددية للسريان المضطرب والتي هذا البحث مراجعة في التطورات الحديثة للنماذج ايقدم هذا وقد تم أيضا في هذا البحث القيام ببحث متعمق قي . تستخدم في تطبيقات التأثير المتبادل بين الموائع والمنشأات

تطبيقات تحليل أداء وآفاءة العديد من المحاولات المختلفة في مجال السريان المضطرب والتي تم إستخدامها في الهذا ويعتبر الحافز الرئيسي لتقديم هذا العمل هو . المعقدة والمرتبطة بمجال التأثير المتبادل بين الموائع والمنشآت

الرغبة في الجمع بين مميزات الطرق العددية الحديثة والمستخدمة في مجال التأثير المتبادل بين الموائع والمنشآت لمستخدمة في توصيف السريان المضطرب والتي تتميز بالتعقيد الرياضي وآثرة والأداء المتميز للنماذج الحديثة ا

وتتمثل الأهداف الرئيسية لهذا العمل، من . التحديات عند تطبيقها للحصول علي الحل العددي للسريان المضطربسريان في تقديم أسلوب منطقي لتحديد مدي تعقيد أو بساطة النموذج المطلوب إستخدامه لحساب ال، ناحية

المضطرب في إحدي التطبيقات المرتبطة بمجال التأثير المتبادل بين الموائع والمنشآت، ومن الناحية الأخري في توضيح آيف تتغير دقة الحل العددي للسريان المضطرب مع زيادة درجة التعقيد للنموذج المستخدم لحساب

ويقوم هذا البحث أيضا بتقديم شرح مختصر . لمنشأاتالسريان المضطرب في مجال التأثير المتبادل بين الموائع واللنماذج المعروفة بإسم نماذج إجهادات رينولدز والتي من المحتمل إستخدامها في الأبحاث القادمة في مجال التأثير

هذا وقد تم أيضا في هذا البحث إستعراض ومناقشة العديد من الموضوعات . المتبادل بين الموائع والمنشآت مثل النماذج المستخدمة بالقرب من الأسطح بشئ من التفصيل ة بالمحاآاة العددية للسريان المضطربالمرتبط

.الصلبة أو الجدران وآذلك الشروط الحدودية المناسبة لتلك الأسطح This paper reviews the recent developments of turbulence modelling and numerical simulations employing in fluid-structure interaction (FSI) applications. A profound investigation of the performance of several turbulence treatment variants applied to complex FSI problems is carried out. This work is motivated by the desire to combine the advantageous numerical methods for FSI with the superior predictive performance of modern turbulence models which is mathematically complex and numerically challenging. The main objectives of the present paper are twofold. On one hand, it is required to provide a rational way for deciding how complex a turbulence model is required for a given FSI problem. On the other hand, to show how accuracy changes with complexity. This review also includes a short description of Reynolds stress models which may be used for unsteady flow in such environments in the coming investigations. Further topics related to the numerical aspects of simulating the turbulent flow, such as the near-wall modelling of turbulence and the appropriate boundary conditions are also discussed in some details.

1. INTRODUCTION 1.1 Fluid-Structure Interaction Problems

The large majority of practically relevant FSI applications are materially affected by turbulent transport. Turbulence has a decisive influence on drag, species transport, vorticity distribution and separation, which in return can affect the qualitative behaviour of the structure. The nature of fluid-structure interaction problems requires detailed measurements that are costly and time-consuming. However, some types of measurements are almost impossible to execute in the

laboratory. Therefore, computers are sometimes more informative in predicting the characteristics of turbulent flow and accurately simulate the flow field rather than measuring it experimentally. Consequently, the coupling of unsteady fluid flow and structure motion is an important field of computational mechanics. Moreover, the problems treated in such area have prominence in aerospace engineering, industrial applications, and many others in daily life.

In order to calculate the fluid-structure interaction in an accurate manner, it is required an accurate computation of unsteady flow and concurrent updating

A. Balabel and D. Dinkler

2 Emirates Journal for Engineering Research, Vol. 11, No.2, 2006

of fluid mesh. Accordingly, a deformation mesh method is developed to treat the FSI with flexible interface. This is achieved by considering the domain of fluid as a virtual solid structure which is deformed with the moving real structure.

Most flows encountered in FSI applications are three-dimensional, highly unsteady, and contain a great deal of vortices. These important properties are considered as an essential characteristic features of turbulent flow, e.g. flows around and after bluff and slender bodies induce vortex and vibration, wind effects on super tall buildings, exited oscillation of high constructions and nonlinear aero-elastic flutter phenomena.

Many efficient numerical methods are already existed, either for solving the governing equations of the fluid or structure dynamics. However, for the solution of FSI problems, the numerical method applied has to be modified accordingly. On fluid-structure interaction problem, the fluid boundaries are the surface of the moving structures. Through the technique of a virtual deformable solid, for the fluid domain, the fluid mesh is deformed simultaneously with the structure. Accordingly, the numerical methods for such problems engage two essential problems. On one hand, it is difficult to maintain the conservation properties of the fluid-structure system. On the other hand, the free-boundary character of the interface could reduce the efficiency of iterative solution methods, as it induces a mutual dependence between the fluid and structure solution and their domains of definition.

By adding a set of turbulence-transport equations to the fluid-structure equations, a new set of difficulties is encountered, especially, for modern nonlinear turbulence models. These difficulties can include stiffness caused by the presence of an additional time scale, singular behaviour near solid boundaries, and non-analytical behaviour at the sharp turbulent interface. According to the prescribed problems of turbulent flow, most of numerical investigations in FSI applications are carried out at low or moderate Reynolds numbers. The high Reynolds number FSI applications are numerically challenging and, consequently, are considered to be partially unresolved issue in FSI problems.

1.2 Turbulent Flow

Turbulent flow is of central importance to many engineering applications, for example the aerospace industry, process engineering, internal combustion engines and environmental engineering. Despite great efforts by research engineers and scientists turbulent flow is not well-understood and remains difficult to predict. Consequently, new concepts and hypothesis are continually being introduced. Emphasis is placed on understanding the physics of turbulent flow and gaining an appreciation of the current concepts and methods.

Currently, the development of computers and Computational Fluid Dynamics (CFD) has made the numerical simulation of complex fluid flow, combustion, aero-acoustics and heat transfer problems possible. Turbulent flow in three-dimensional, complex geometries, unsteady or steady, can be dealt with.

The large majority of practically relevant flows are materially affected by turbulent transport. While in the major aero-or hydrodynamic engineering applications, turbulence has a decisive influence on heat transfer, species transport, drag, vorticity distribution, swirl and separation. Turbulence can also influence even the qualitative behaviour of the flow which contains significant region of separation and recirculation.

In general, the importance of turbulence poses serious challenges to the ability of computational approaches, based on the solution of Reynolds-Averaged transport equations, to give quantitatively reliable predictions for any but relatively simple thin-shear flows. Indeed, it is often concluded that the turbulence modelling and simulations is the pacing item for the rate of progress of CFD in fluids engineering. That is reflected by the enormous amount of research on turbulence predictions over the past three decades. Before proceeding, it is useful to introduce a general classification scheme for methods of predicting turbulent flows.

According to Bardina et al. [ 1] there are six categories, each of which can be divided in sub-categories. Some of these categories are either limited to simple types of flows or are rarely used except for homogenous turbulence, therefore they are not considered further in our review. However, the other categories include the most commonly used methods in predicting turbulence characteristics. All these methods, described below, require the solution of some form of the conservation equations for mass, momentum and energy. The major difficulty is that, the turbulent flow has a much wide range of length and time scales. Therefore, the equations for turbulent flows are usually much more difficult and expensive to solve. In general, these methods are: 1. Direct numerical simulations (DNS) 2. Large eddy simulations (LES) 3. Reynolds-averaged Navier-Stokes equations

(RANS) Nevertheless, no pretence has been made that any

of these numerical methods can be applied to all turbulent flows: such as ´universal´ method may not exist. Each method has its adv-/disadvantages, limitations and applicable regimes. These important related topics are discussed in more details in the following sections. We focus at first on the modern efforts that more directly address the physics of turbulence without introducing Reynolds-closure approximations, namely DNS and LES. Then, we discuss the approximations of Reynolds-averaged models for use in general engineering applications. It

Turbulence Models for Fluid-Structure Interaction Applications

Emirates Journal for Engineering Research, Vol. 11, No.2, 2006 3

is worth mentioning that most of the applications considered in this paper are related only to the area of fluid-structure interaction applications.

The direct numerical simulation DNS of turbulent flow has been applied successfully to simple geometries such as boundary layer, plane channels, backward facing step and square ducts. To attain Reynolds number of about 10000, the DNS involves huge calculations in terms of memory space and computer time. However, the results of the DNS are important to understand the near-wall structure of turbulence and are also useful in the analysis of a priori evaluations of turbulence models, which will be explored in this review.

Up to today, most, if not all, simulations are carried out with traditional RANS (Reynolds-Averaged Navier-Stokes). In RANS, we split the flow variables into one time-averaged (mean) part and one turbulent part. The latter is modelled with a turbulence model such as k-ε or Reynolds Stress Model.

For many flows it is not appropriate to use RANS, since the turbulent part can be very large and of the same order as the mean. Examples are unsteady flow in general and wake flows or flows with large separation. For this type of flows, it is more appropriate to use Large Eddy Simulation (LES). In order to extend LES to high Reynolds number flows new methods have recently been developed, that are called DES (Detached Eddy Simulation), URANS (Unsteady RANS) or Hybrid LES-RANS. They are all unsteady methods and a mixture of LES and RANS [ 2].

In LES, DES, URANS and Hybrid LES-RANS the large-scale part of the turbulence is solved for by the discretized equations whereas small-scale turbulence is modelled. The definition of large-scale differs in the different methods. Furthermore, the limit between large-scale and small-scale is often not well defined. Since turbulence is three-dimensional and unsteady, it means that in all numerical methods the simulations must always be carried out as 3D and unsteady.

Over the past decade, efforts have focused mainly on the construction of two-equation eddy-viscosity models and Reynolds-stress-transport closure. The conventional eddy-viscosity models are based on the set of linear Boussinesq stress-strain relations. This approach is attractive from a computational point of view, especially in terms of numerical robustness, and has a large popularity with CFD practitioners. However, this approach is known to be affected by major weaknesses which can be considered as the source of substantial errors where the gradient of the normal stresses contribute significantly to the momentum balance, or what is called complex strain. Although some of ad-hoc corrections have been made, usually to the length-scale equation, and/or the formulation of alternative equations for different length-scale parameters, none of these corrections can address the fundamental limitations arising from unrealistic constitutive relations.

The alternative route thus pursued extensively over the past decade has been second-momentum closure which is mathematically complex and numerically challenging and often computationally expensive. Accordingly, it has some limitations in the context of industrial CFD. This has thus motivated efforts to construct models which combine the simplicity of the eddy-viscosity formulations and the superior fundamental strength of second-moment closure. These efforts have given rise to the group of nonlinear eddy-viscosity models which are discussed later in this review.

2. PREDICTION METHODS FOR TURBULENT FLOWS

2.1 Direct Numerical Simulation

Direct numerical simulation (DNS) is the most accurate and simplest approach to turbulence simulation. This approach involves the numerical solution of the Navier-Stokes equations that govern fluid flow without modelling or approximations with its accuracy is only bounded by the accuracy of numerical scheme adopted, boundary conditions and spatial and temporal discretizations. The main problem is that, the large number of grid points and the small size of time steps required to capture the so small time and space scale of turbulent motion make the advanced turbulent computations outside the realm of possibility for present computers. Massive computer resources are required for 3D computations and increase markedly with Reynolds number. According to Kolmogrove's scaling law, the number of grid points required for 1-D turbulent computations is proportional to the ratio of the size of the largest eddies to the viscous scale, e.g. for 3D turbulent calculations with Re≈5000, a nearly 107 grid points are required with estimated calculation time of ≈46 days as indicated in Le et al. [ 3]. In general, the number of grid points required NG and the number of arithmetic operations NOP must be performed are estimated as:

411

49

Re,Re ∝∝ OPG NN Due to the overwhelming volume of literature on

this topic [ 4], in the following review we restrict ourselves only to the FSI applications.

In relatively earlier attempt Karniadakis and Triantafyllou [ 5] have studied the transition to turbulence and 3D dynamics in the wake of bluff objects using DNS at Reynolds number up to ≈500. Further, DNS is applied to study flow-induced cable vibration at Reynolds numbers 100, 200 and 300 [ 6]. By comparing the flow-induced vibration case with the forced vibration case, relatively similar responses are observed at Re=100, while the discrepancies become larger by increasing the Reynolds number. Through a 2D direct numerical simulation, Nair and Sengupta [ 7] have investigated the flow past elliptic cylinders for two Reynolds numbers Re=3000 & Re=10000 at



different angles of attack and thickness-to-chord ratios. The Navier-Stokes equations are solved in stream-function-vorticity formulation using a finite difference method with third- and fifth-order upwind scheme for the convection term. Upwind schemes are preferred for reasons of high accuracy of the results and numerical stability at high Reynolds number.

According to Williamson [ 8], and based on the previous investigations of Braza et al. [ 9], and Sengupta [ 10, 11], they have dictated that the 2D Navier-Stokes equations can be solved to understand the formation and shedding of vortices in the Reynolds number range between 1000 and 200000. This idea has been used in Nair and Sengupta [ 12] for studying the flow field around circular and elliptic cylinders at zero angle of attack at high Reynolds number of 10000 and extended in their investigation. They found that the fifth order upwind method will be far more effective in controlling errors at high wave numbers. The resulting truncation error is indistinguishable from the machine round-off error compared with other central and upwind schemes. The comparison with experimental visualization showed the ability of their method to predict flow details and overall features of the location and size of recirculation zones. However, no quantitative comparisons for the computed results were given.

Evangelinos, et al. [ 13] have used spectral direct numerical simulation to simulate turbulent flow past rigid and flexible cylinders subjected to vortex-induced vibrations (VIV) at Reynolds number Re=1000. Depending on the numerical confirmation by Evangelinos and Karniadakis [ 14], the flow is considered to be turbulent at that Reynolds number. Both short and long cylinders corresponding to length-to-diameter ratio of 4π and 387, respectively, are investigated. The coupled Navier-Stokes/structure dynamics equations are discretized in space using a new spectral method that employs a hybrid grid and Fourier complex modes. They concluded that their DNS simulations are prohibitive expensive to be used in engineering design of VIV. Moreover, higher Reynolds number simulations could not be performed as they were also prohibitive expensive.

Accordingly, the cylinder flow at high Reynolds number has subjected to a large volume of DNS investigations as it has become a canonical test case for DNS. For low Reynolds numbers regimes, Re ≤ 500, a good understanding of the physics has been obtained [ 15]. However, at relatively higher (but still subcritical) Reynolds number, 500 ≤ Re ≤ 1000, considerably less is known due to the numerical problems encountered. A series of careful DNS studies in the subcritical Reynolds number regime has significantly improved the understanding of flow transition from 2D to 3D states [ 16].

Therefore, most of the recent investigations in this area are carried out experimentally and have focused on the shear layer instability and its frequency scaling

with respect to the Reynolds number [ 17]. However, these investigations could not frame an answer for many important questions related to the extent of vortex dynamics phenomena and their influence on the resulting unsteady fluid forces.

Ma et al.[ 18] have performed a DNS of the cylinder flow at Re=3900, which is considered as the highest Reynolds number DNS achieved. They employed a Fourier expansion and a spectral element discretization on unstructured mesh in the spanwise direction and in the streamwise-crossflow planes, respectively. The mean velocity profiles and the power spectra are in good agreement with the experimental data in the near wake as well as far downstream. In the vicinity of the cylinder surface, there is also a good agreement with experimental measurements. However, discrepancies in the base pressure coefficient and the recirculation bubble length have been observed by comparing with large eddy simulations[ 19]. Franke and Frank[20] have referred that to the different averaging time in computing statistics. In DNS, the statistics are accumulated over 600 convective time units (D/Uo), while in LES the statistics are accumulated over about 35 convective time units. It was also observed that as the averaging time increases, the value of the base pressure and recirculation bubble length start from those in LES and gradually change to those in DNS.

The three-dimensionality effect on the accuracy of the numerical simulations has been also investigated through a number of 2D and 3D direct numerical simulations at low Reynolds numbers. It is concluded that the over prediction of fluid force in 2D simulations is primarily an intrinsic 3D effect. This fact is exemplified in the 2D-DNS performed by Mittal and Kumar[ 21] at Reynolds numbers ranging from Re=103 to Re=104.

More recently, a 3D-DNS has been pursued by Dong and Karniadakis [ 22] for the turbulent flow past a stationary cylinder as well as a rigid cylinder undergoing forced harmonic oscillations in the cross-flow direction at high Reynolds number, Re=104. This Reynolds number is about one order of magnitude higher than that considered in previous DNS of VIV in Evangelinos and Karniadakis [ 14] and is about 2.5 times the highest Reynolds number in previous DNS for fixed cylinder flow performed by Ma et al. [ 18]. In order to tackle the computational cost associated with this high Reynolds number, an efficient multilevel-type parallel algorithm is employed. In this simulation, the Navier-Stokes equations are solved in a coordinate system attached to the cylinder, which avoids the difficulty of a moving mesh in the oscillating-cylinder case. The time integration is calculated via the stiffly-stable pressure correction-type scheme with third-order accuracy in time. A Fourier spectral expansion in the homogeneous direction is employed for the spatial discretizations, while a spectral element approach in the streamwise and cross-flow directions is adopted. Different grid resolutions are tested by varying the



number of grid points in the spanwise-direction and the order of spectral elements in the x-y planes. The computed parameters, such as drag coefficient, base pressure coefficient, Strouhal number are in quite good agreement with the experimental data for all resolutions. However, the lift coefficient demonstrates a higher sensitivity to the grid resolution. With high-resolution mesh, the values of the lift coefficient fall within the range of experimentally measured values. Unfortunately, the CPU time required for each grid resolution ranges from (2 to about 25) ×104 hrs as the resolution increases.

Although of the development of massively-parallel machines over the last few years has reduced the DNS execution times, but storage is still a problem, both during the computation and for later archiving ´fields´ of raw data at selected time steps. On the other side, and from the point of view of a numerical treatment of the spatial directions, there are two primary concerns in this regard. Firstly, it is required to achieve accurate representations of derivative, especially at the highest wave numbers. Secondly, is required to avoid a phenomenon known as ´aliasing´, where waves are produced with wave number greater than the maximum wave number due to nonlinear interactions among the resolved wave numbers [ 23]. Spectral method-Fourier series in the spatial directions can be used to insure accurate computation of derivatives, rather than finite-difference methods that usually underestimate derivatives of a given function, leading to inaccuracies in the smallest (dissipating) scales. However, the application of spectral method with unstructured grids, which now well established in modern algorithms, is a difficult task. Therefore, there has been a general swing towards finite-difference methods with a higher order of accuracy to use with arbitrarily non-uniform grids.

2.2 Large Eddy Simulations

The underlying assumption of large eddy simulations (LES) is based on that the large scale motions in turbulent flow are more energetic than the small scale ones, therefore, it carry most of the Reynolds stress and must be computed. The small scale turbulence is weaker, contributing less to the Reynolds stresses and is more nearly isotropic having nearly universal characteristics, consequently, it is thus amenable to modelling.

Large eddy simulations is considered as a promising computational approach, in which, the large scale turbulent motion is resolved from the first principles by numerically solving a filtered set of equations governing this large scale, while turbulence modelling is employed to approximate the effects of subgrid scale turbulence. LES is three-dimensional, time dependent and expensive but much less costly than a DNS of the same flow, and it is preferred when the DNS is not feasible; e.g. for high Reynolds number flows or when the geometry is too complex such as

flow over an airplane, or inside a combustor. Recently, Rodi [ 24] presents a review on LES for flows with large regions of separation.

In LES approach, a velocity field that contains only the large scale components is resolved. That can be obtained by filtering the Navier-Stokes equations by means of a specified filter system, associated with it a length scale ∆ (or width of the filter) that defines the limit between the large and small scale eddies. The small scaled, called subgrid scales, are dynamically modelled by means of subgrid scale models. It is obviously that, when ∆→ 0, LES approaches DNS.

The system of equations used in LES, consists of the continuity equation and the filtered Navier-Stokes equations for incompressible flow, which reads:

0=∂∂

i

i

xu (1)

j

iji

ij

jii

xu

xp

xuu

tu

∂

∂+∇+

∂∂

−=∂

∂+

∂∂ τ2

Re1 (2)

Where, iu is the filtered velocity introduced by Leonard [ 25] and defined by the general spatial filtering operation as:

``)(`),( dxxuxxGu ii ∫= (3)

Where `),( xxG is a localized function known as the filter kernel. Different filter kernels have been applied in LES include a top-hat-filter, cut-off-filter and a Gauss-filter. The width of the filter needs not have anything to do with grid size, h, other than the obvious condition that ∆ > h. A commonly used homogeneous, symmetric, non-negative filter is the box filter [ 26]. In the above context, τij is called the subgrid scale Reynolds stress (SGS) and its modelling should be based on the local velocity field or, perhaps, on the past history of the local fluid. That can be accomplished by solving a set of partial differential equations to obtain parameters needed to determine the SGS Reynolds stress.

Tejada and Jansen [ 26] presented some important results concerning the development of new spatial test filters required in the dynamic model of small-scale motion in LES. Through performing simulations of decaying isotropic turbulence using tetrahedral and hexahedral meshes, they found that, despite the different filters used, results are invariant as long as the filter width are consistently computed on a fixed topology. They referred the difference in previous simulation results by using different-shapes test filters to the incorrect computing of test filter width with a proper definition. However, the results can vary as the mesh changed from hexahedral to tetrahedral one regardless of whether or not the test filter width is consistently computed.

The overwhelming majority of the previous investigations in complex flows using LES have been carried out with structured grid finite difference



methods. These methods suffer from two major problems. On one hand, it is rather difficult to represent most of arbitrarily complex geometry. On the other hand, the extension to higher Reynolds number needs an impractical number of grid points. Currently, unstructured-grid methods are applied in most of CFD codes offering a release from both of those constraints.

Through the ability of unstructured-grid, a great reduction in computational effort is obtained by adapting it locally to fine scale structures in one flow region while remaining coarse grid in others. The extension of LES to finite element method on unstructured grids has recently been done in Jansen [ 27] to study the flow over NACA 4412 airfoil. The associated problem due to the application of the unstructured grid, discussed in that research, is that the definition of points location that approximate the filtering operator is not simply determined through a recursive formula as in the case of a structured grid. Consequently, four filtering operators are developed and discussed in details.

Sun and Dalton [ 28] have adopted the vorticity/ stream function formulation along with two different subgrid scale models at different values of Keulegan-Carpenter number (KC) defined as KC=UT/D, where D is cylinder diameter and T is oscillation period. They have proved that 2D finite-difference large eddy simulation method is inadequate to describe the oscillating flow over a circular cylinder. However, for KC≤2, the calculated lift coefficient compared favourably with experimental data.

Following, a stabilized finite element formulation to solve LES equations is developed by Jansen [ 29]. The numerical method along with the subgrid scale model adopted in such simulation is firstly validated against two canonical flows; the decay of isotropic turbulence and the turbulent channel flow, and then secondly it is applied to a complex flow around NASA 4412 airfoil. The simulation results did not yield a complete agreement between the LES and the experimental measurements.

Because the turbulent von Karman vortex street past circular cylinders involves most of the characteristics features of technical applications, more attention has been given for cylinder flow as it becomes a canonical test case for LES. Mittal and Moin [ 30] and Kravchenko and Moin [ 31] are among the first to undertake LES studies for cylinder flow at Re=3900. Although of the good agreement with the experimental data for the mean velocity and Reynolds stresses profiles far downstream, however, several issues exist regarding the power spectra and the mean velocity in the recirculation region. On the other hand, no clear inertial range is obtained in either of these two simulations. However, the inertial range is accurately captured in Kravchenko and Moin [ 31], who employed a high-order scheme based on B-spline and zonal grid. The mean velocity profiles from all three simulations agree with one another away from the cylinder surface,

however, they all differ from experimental data in the vicinity of the cylinder.

Consequently, a variety of aspects that affect the quality of LES has further been examined. Breuer [ 32] has employed five different numerical schemes and two subgrid-scale models in order to investigate the numerical and modelling aspects of LES for steady flow past a circular cylinder (Re=3900). The main objective was the investigation of numerical effect of the discretization of the nonlinear convective fluxes on LES solutions. The near-wall model is neglected, (owing to the low Re assumed), and moreover, computations without any subgrid scale models are performed to prove the performance of SGS models. The results showed a better agreement with the experimental data when the central difference schemes are used rather than dissipative methods, concluding that, the central difference advection scheme being recommended. In addition, the low-order upwind schemes showed that they could not predict accurately many important physical quantities.

Following, Breuer [ 33] has performed the same case but at high Reynolds number (Re=140,000) to evaluate the applicability of LES for practically relevant high-Re-flows. The influence of grid resolution and SGS models are also investigated. Using a paralyzed finite-volume Navier-Stokes solver, a series of grids applying Smagorinsky model and the dynamic SGS models is used in the computations. The results of integral parameters and mean velocity profiles showed fairly agreement with experimental data especially in the near wake. However, a large deviation is observed in the far wake due to its coarse resolution. Nevertheless, two aspects of LES were explained in that work; the increased importance of SGS models for high-Re-flows and the questionable results of grid refinement which did not automatically lead to an improved agreement with experimental measurements.

Considering this high-Re cylinder flow, Jordan [ 34] studied the shear layer instability at (Re=8000) using an upwind finite difference scheme and dynamic SGS model. Several parameters have been captured correctly by the LES solution.

More recently, Wang, et al. [ 35] have investigated the viability and accuracy of LES in supercritical regime (Re=5×105, to 2×106) with a simple wall stress model providing boundary conditions to the LES. The mean pressure distribution is predicted reasonably well at low Reynolds number, while the solution becomes less accurate at the higher end of Reynolds number. In general, the LES solutions are shown to be considerably more accurate than RANS simulations by comparing it with the experimental measurements concerning the delayed boundary layer separation and the reduced drag coefficient after the drag crisis.

Other effects of some numerical aspects concerning LES are studied further in more details. Breuer [ 33] has studied the effect of high order compact and explicit upwind difference schemes. He concluded that it is



important to resolve the inertial range in LES as the upwind scheme coupled with dynamic subgrid-scale model. Also, Frank and Frank [ 20] have investigated the effect of the averaging time on the accuracy of statistical quantities in LES. They observed that the accumulation time has an important effect on the value of the base pressure coefficient and the recirculation bubble length.

The study of square cylinders using LES has received much more attention in the last decade. Although the square cylinder has the physical advantages that upstream separation is fixed by sharp corner, but however, this case is considered as a standard test case for LES solutions [ 36]. The obtained results, in such simulation, appeared to be sensitive to numerical details.

Murakami and Iizuka [ 37] have studied the flow over a square cylinder conducting 3D-LES. The results of different SGS models, namely: the conventional standard Smagorinsky model, the dynamic Smagorinsky model and the Lagrangian dynamic Smagorinsky model, are compared with experimental measurements. The latter model is found to be useful more than the others. It overcomes most of the drawbacks associated with the other models concerning the unstable computation applied to flow past bluff bodies.

Recently, LES has considered dealing with the problem of vortex-induced vibration (VIV) on a circular cylinder. In forced-oscillation study, Tutar and Holdo [ 38] have used LES to evaluate the transverse flow around an oscillating circular cylinder normal to uniform flow. The numerical simulations were carried out using LES method in 2D and 3D with a developed near-wall approach without using a "law of the wall" for finite element code. They did extensive calculations at moderate Reynolds number (Re=2.4×104) and with a relatively small oscillation. Their results showed that 3D-LES gives more realistic flow field predictions and can further remove overconservatism in the prediction of hydrodynamic force coefficient.

Dalton et al. [ 39] have discussed the effect of a small control cylinder on the transverse force acted upon a large primary cylinder when the control cylinder takes different locations in the emanated shear layer. A 2D-LES calculation was used in to include the effects of wake turbulence at (Re=100, 1000, 3000) in addition to flow-visualization studies. The calculated results showed that the LES model predicts essentially an elimination of the transverse force on the primary cylinder for an appropriate placement of the control cylinder.

A combination of 3D-LES and the finite volume method with an unstructured mesh has been presented by Li Chen et al. [ 40] . The overall solution procedure is second-order accurate in both space and time. Their investigation is focused on the prediction of the turbulent wake dynamics behind two side-by-side

circular cylinders at Re=750. The numerical simulation confirmed the experimental observation of the formation of two symmetric wake streets behind the cylinders and the presence of gap flow deflection in case of large and intermediate cylinder spacing,.

For the self-exited case, AlJamal and Dalton [ 41] have performed a 2D-LES calculation on VIV of a circular cylinder at Re=8000 and with a range of damping ratios and natural frequencies. In spite of the shortcoming of a 2D representation, the results showed the expected vibratory response of the cylinder in specified range of vortex shedding. They did not find details of shedding pattern in their calculated results. However, these results are seen to represent the VIV behaviour of a cylinder reasonably well for the parameter values used in their calculations.

In general, LES has shown much promise, but the technique is much too costly at present to be considered as an engineering tool. The LES requires a hug amount of computer capacity and is not applicable to most engineering problems. LES is 3D, and hence a very computationally demanding approach, which is probably realistic away from a solid boundary, but however, close to a boundary, has all the limitations of a simple eddy-viscosity model. Therefore, LES is conceptually very close to Reynolds-averaged Navier-Stokes equations (RANS), but as a method it is more closely related to DNS. In RANS the turbulence model damps out the smallest scale of motion irrespective of the grid resolution. However, through the use of dense grid in LES, the turbulence model includes less damping as the filter width defines the damping amount. Although RANS behaves very well in boundary layer flow and needs considerably less grid points than LES, however, RANS is not able to describe properly flows over non-streamlined bodies. Moreover, for many flows it is not appropriate to use RANS, since the fluctuation quantities can be very large and of the same order as the mean quantities.

In order to extend LES to high Reynolds number flows new methods have recently been developed. These methods are known as DES (Detached Eddy Simulation), URANS (Unsteady RANS) or Hybrid LES-RANS. They are all based on a mix of LES and RANS, where they apply RANS to calculate wall-bounded region entering a mixing section and uses a LES procedure to calculate the mixing-dominated regions. That makes the RANS modelling an efficient and robust tool in computing complex flows at very high Reynolds numbers. Therefore, in the next section a short review of RANS modelling is presented.

2.3 RANS Modelling

As seen previously that DNS and LES are the most realistic approaches for turbulence simulations and have attracted considerable attention during the past decade. However, they have not in general lived up to initial expectations. This state of affairs leaves methods that are rely on RANS equations as the most



promising alternative for practical engineering computations. Recently, RANS modelling is used to predict most of complex flows encountered in engineering applications in conjunction with near- wall and non-isotropic turbulence closure. Although a complete agreement with experiments is not achieved, these calculations succeeded in resolving complex features of both the mean flow and turbulence field, which simple isotropic turbulence models had failed to describe even in a gross qualitative sense.

So, in this section we seek to summarize the experience gained from recent applications of advanced RANS methods, which resolve the near-wall flow and account for the anisotropy of the Reynolds stresses. We should to clarify that our goal herein is not to provide a complete review of advanced turbulence models [ 42], instead we will give a short review on the classification of the turbulence models then we will focus on a few selected models that have been successfully applied to simulate the fluid-structure interaction problems. For incompressible flows, the RANS equations read in tensor notation, where repeated indices imply summation, as follows:

0=∂∂

i

i

xU (4)

)(1jiij

jij

ij

i uuxx

pxUU

tU

−∂∂

+∂∂

−=∂∂

+∂

∂ τρ

(5)

The molecular stress is defined as:

)(21,2

i

j

j

iijijij x

UxUSS

∂

∂+

∂∂

== ντ

Upper case denotes ensemble-mean quantities and lower case fluctuating or turbulence quantities; p is pressure, ρ is density, ν is the kinematic viscosity and Sij is the strain rate tensor. The additional quantities known as the Reynolds stresses must be modelled in order to close the system.

3. CLASSIFICATION OF TURBULENCE MODELS

Turbulence models to close the RANS equations can be divided into two categories according to whether or not the Boussinesq assumption is applied, proposing a hypothesis to model the turbulent stresses, analogous to the molecular one, and based on the assumption that the apparent turbulent shearing stresses might be related to the rate of mean strain through an apparent scalar turbulent or "eddy" viscosity. For the general Reynolds stress the Boussinesq assumption gives:

δνδν kSkx

UxUuu ijtij

i

j

j

itji 3

2232)( −=−

∂

∂+

∂∂

=− (6)

With δij is the Kroneker delta and k= jiuu5.0 is

the turbulent kinetic energy of fluctuations. The scalar quantity νt is known as the turbulent viscosity or eddy viscosity. Accordingly, the first category known as

turbulent or eddy viscosity models (EVM). However, models which affect closure to the Reynolds equations without this assumption will be referred to as Reynolds stress or stress-equation models (RSM). More details will be followed.

3.1 Linear and Nonlinear Eddy Viscosity Models

Eddy viscosity models are still the ´standard´ approach for industrial CFD applications for turbulent flow as the others require far more computational resources than are available for routine work. The common classification of EVM's can be considered according to the number of supplementary partial differential equations which must be solved in order to supply the modelling parameters. This number ranges from zero for the simplest algebraic models to twelve for the most complex of the Reynolds stress models. Reference is also sometimes made to the ´order´ of the closure. According to this terminology, a first-order closure evaluates the Reynolds stresses through functions of the mean velocity and geometry alone. A second-order closure employs a solution to a modelled form of a transport partial differential equation for one or more of the characteristics of turbulence.

Two different groups belong to EVM are characterized; the linear and nonlinear eddy viscosity models. The LEVM's are based on the Boussinesq-hypothesis described previously, while NLEVM's are based on nonlinear constitutive relations in which extra terms (e.g. the quadratic and cubic terms) are adopted to capture the flow features.

The recent novel attempts to improve the EVM's are focusing on near-wall modelling and nonlinear constitutive relation used in the associated models. The near-wall modelling is motivated by the disagreement of EVM's models near-wall with the data deduced from several experiments. The modified low-Reynolds-number models (LRN) suggest ways of improving EVM's. The basic idea is based on the implementation of the near-wall viscous effects by damping the turbulent viscosity toward a wall with introducing a damping function. That can be seen in Jones and Launder [ 43], who devised a LRN version of the two-equation eddy viscosity k-ε model. Many recent versions of such models can be found in Rodi and Mansour [ 44].

The NLEVM is another important topic in the recent modelling of turbulent flow. This approach is based on the nonlinear extension of the linear stress-strain relation, Eq. (6). Nonlinear constitutive equations have been proposed to overcome the limitations of linear eddy-viscosity models in describing complex turbulent flows. NLEVM's are thus made to mimic the physics of turbulence by means of mathematical artefacts and calibration.

LEVM's perform reasonable well in attached boundary layer flows as only one component of the Reynolds stress tensor is of significant importance. However, it is already known that LEVM's fail in a



number of flow situations where the normal stresses are anisotropic, e.g. wall bounded flow. The LEVM's do not produce meaningful difference between the normal stresses. For example, in shear flows where only S12 is nonzero, then Eq. (6) leads to isotropic turbulence as:

332211 uuuuuu == (7) However, the values of normal stresses are very

different from one another in actual flow cases. Thus, LEVM's lack the capability of predicting anisotropic turbulence in many industrially important flows. As a result, LEVM's are unable to account for irrotational strains as well as the curvature effects, related either to curved boundary or streamline curvature [ 45].

Most NLEVM's are quadratic, while those models

[46-48] are cubic and that of Gatski and Speziale [ 49] is quartic. These differences in order are of considerable significance. In particular, the cubic fragments play an essential role in capturing the strong effect of curvature on the Reynolds stresses, while the quadratic terms are responsible for the ability on nonlinear models to capture anisotropy.

The comparison of many quadratic stress-strain relations with experimental data showed that none achieves much greater width of applicability. Craft et al. [ 46] have proposed a cubic relation between the strain and vorticity tensor and the stress tensor, which does much better than conventional eddy-viscosity schemes in capturing the streamline curvature effects over a range of flows. This model has been further employed by Saghafian, et al. [ 50], with adjustment of the coefficients of the 'cubic' terms, to predict forces in steady and oscillatory flows past a circular cylinder for a wide range of Reynolds numbers, involving turbulent separation.

Although the NLEVM's overcome some deficiencies of the LEVM's by adding extra terms which takes into account of flow features like curvature effect, anisotropy and swirl, however, the prediction of the normal stress anisotropy is quantitatively smaller compared to the DNS results. This kind of formulation of the constitutive relation is ad hoc basis and mathematical. Hence, it has fewer roots in physics.

3.2 Reynolds Stress Models (RSM's)

In Reynolds stress models (sometimes called stress-equation models) the Boussinesq hypothesis described previously is not used. These models have been used to date largely as tools or subjects in turbulence research rather than to solve engineering problems.

In Reynolds stress model, exact transport equations can be derived for the unknown Reynolds stresses

jiuu . Naturally, these equations contain terms

which must be modelled during its solution with the usual system of conservation equations. Therefore, the

solution of the system of these equations requires numerous modelling assumptions and, consequently, this approach is computationally expensive.

An Algebraic Reynolds stress model (ARSM) was firstly introduced by Rodi [ 51] and further extended by Launder et al. [ 52]. In ARSM scheme, the transport of Reynolds stresses is approximated in terms of that of turbulence energy k to reduce the differential equation for

jiuu to a set of algebraic ones. ARSM has been

found to be numerically and computationally cumbersome since there is no diffusion or damping present in the equations. This usually means convergence problems in the numerical algorithm. In many applications the computational effort has been found to be excessively large and the benefits of using ARSM instead of the full Reynolds stress transport form are then successively lost. Therefore, a more simple and straightforward way to calculate the Reynolds stress anisotropy is needed. It should be noted that a sort of NLEVM is sometimes called algebraic Reynolds stress models (ARSM's) which is considered as a more extended modelling concept. These models are mathematically very close to EVM's and link some recent nonlinear eddy viscosity approaches. There are similarities between NLEVM's and ARSM's in the constitutive relations. The difference is only in the formulations; the NLEVM class is derived on ad hoc basis while the ARSM's is systematically derived from Reynolds stress model (RSM). Therefore, the ARSM class inherits the features of the RSM's.

A model, where the Reynolds stresses are explicitly related to the mean flow field, is called an explicit algebraic Reynolds stress model (EARSM). It is much more numerically robust and has been found to have almost a negligible effect on the computational effort as compared to EVM's. The EARSM class has some similarities with the NLEVM class in the constitutive relation. The former class has its root in RSM while the later class is purely mathematical.

RSM's are often referred to as `second-order closure` or `second-moment closure` models. They include terms that are expected to correct some of Boussinesq approximation's shortcomings, such as the automatically response to effects of streamline curvature, system rotation, stratification, and the inequality of the normal stresses.

In general, numerical stability of RSM's is often problematic and, with seven transport equations and often a relatively small time step to ensure numerical stability, it can also be very computationally demanding in FSI applications, especially for a large 3D case. Therefore, some attempts have been recently carried out in order to develop new models to be used. In the model development, attention is especially paid to the model sensitivity to pressure gradients, model behaviour at the turbulent/laminar edges, and to calibration of the model coefficients for appropriate flow phenomena.



3.3 RANS Modelling and FSI Applications

I- Flow around bluff bodies Recently, many physical phenomena in FSI applications can be well captured in robust and accurate manner as a result of the great advances achieved in turbulence modelling, mesh generation and linear system resolution. Flow around moving and/or oscillating bodies, which can be rigid or flexible with a prescribed law of deformation, is one of the most interdisciplinary fields of interest in such area.

The aerodynamic characteristics of elongated rectangular cylinders (bluff bodies) were investigated in detail, since it is a common configuration which can be found in many structures, such as tall buildings and bridges. The separated shear layer generating at the leading edge of the bluff bodies plays an important role in the production of aerodynamic forces. The behaviour of the shear layer separated from the windward corner and vortices shedding into the weak are dependent on the cross-section ratio B/D, where B is the chord length along the direction of flow and D is the depth of the section. Problems with large cross-section ratio pose a tremendous challenge to the development of numerical simulations due to the evidence of discontinuities in Strouhal number.

Many numerical procedures are applied to predict flow characteristics around rectangular cylinders with different cross-section ratios. Despite of the meaning-ful results obtained from the numerical methods employing either direct simulations or subgrid scale models, the RANS model has been used in most applications. In the former case, both methods are examples of 3D analysis and their results are meaningful only when a sufficient spatial resolution is employed; therefore, these methods are memory intensive and require a lot of processing time. Contrarily, RANS model enables 2D computations to obtain a physically, smooth, and periodic vortex shedding even in flows with high Reynolds number.

Based on this hypothesis, Frank and Rodi[ 53] showed that, for a square cylinder in a smooth flow, the Strouhal number and mean drag coefficient are reproduced well by using the Reynolds-stress equation model, while the conventional standard k-ε model failed to predict vortex shedding. This was explained by Murakami et al.[ 54], where the separation is weaken-ed as excessive amount of turbulent kinetic energy is generated near the frontal corner by effects of isotropic eddy viscosity. Improved predictions of aerodynamic characteristics and turbulent statistics are obtained by Kato and Launder [ 55], who proposed a modified k-ε model in which production of turbulent kinetic energy is expressed in terms of a vorticity tensor and a velocity strain tensor. Later, using the same model combined with generalized log-law boundary conditions, Kato [ 56] has obtained satisfactory results concerning the prediction of the unsteady wind force and vortex-induced vibrations of a square cylinder.

Figure 1. Streak lines for flow over a rectangular cross-section cylinder [ 57]

However, with regards to that, most applications of

RANS model have been confined to small cross-section ratios. Shimada and Ishihara [ 57] have used the two-layer k-ε model with a modified k-production term to predict the aerodynamic characteristics of rectangular cross-section cylinders having infinite spanwise length with various B/D ratios (ranging from B/D= 0.6 to 8), as seen in Fig.1.

The numerical model adopted is described as follows. The RANS equations are expressed in the following form:

)32()])([( kP

xxU

xU

xDtDU

ii

j

j

it

j

i +∂∂

−∂

∂+

∂∂

+∂∂

= νν (8)

Where the eddy viscosity coefficient εν µ /2kCt = ,

and the kinetic energy and its dissipation rate are obtained by the following transport equations:

εσν

ν −−∂∂

+∂∂

= kjk

t

j

Pxk

xDtDk )])([( (9)

kCPC

xxDtD

kj

t

j

εεεσν

νεεε

ε

)()])([( 21 −−∂∂

+∂∂

= (10)

All coefficients in the equations are identical to those identified from temporal means of experiment-ally measured values and used in the conventional standard k-ε model, described as given in Table 1.

In the standard k-ε model, the production term is given by:

22

)(21

i

j

j

ik x

UxUkCP

∂

∂+

∂∂

=εµ

(11)

Table 1. Coefficients of the standard k-ε turbulence model

Cµ Cε1 Cε2 σk σε 0.09 1.44 1.92 1.0 1.3



By using the above formula in the numerical procedure of Shimada and Ishihara [ 57] it is noticed that excessive amounts of turbulent kinetic energy are produced near the leading edges of the cross-section. Therefore, the production term is replaced by the modified one based on the assumption of flow irrotationality proposed by Kato and Launder [ 55];

)(21)(

212

i

j

j

i

i

j

j

ik x

UxU

xU

xUkCP

∂

∂−

∂∂

∂

∂+

∂∂

=εµ

(12)

Although the previous approach is 2D, it was in a good agreement with results of experiments and 3D analysis in the considered range of B/D for drag coefficients and mean surface pressure distribution even in the range of high Reynolds number. However, the prediction of fluctuations in pressure and forces was underestimated in some cases as a result of the inability of the RANS model to asses the stochastic components (defined as: the stochastic component of parameter is evaluated as its variance). Consequently, it is concluded that the RANS model can only account for the periodic components, which is directly solved from the simulation.

Consequently, research studies aimed at extending the applicability of EVM's have been highly demand by industry. In fact, the continuous research efforts have been significantly extending the performance of EVM's. The recent novel attempts to improve the EVM's are focusing on wall detecting parameters and nonlinear constitutive relations used in the models.

Although the ideas of NLEVM themselves emerged back in the 70's by the work of Pope [ 58], until recently, the models of this type were not widely explored. Many attempts at developing and using such schemes have been recently made [59-61].

NLEVM's are applied for the first time in FSI to the circular cylinder case [ 50]. The nonlinear turbulence model [ 46] is applied to predict forces in steady and oscillatory flows past a circular cylinder for a wide range of Reynolds number, involving laminar and turbulent separation. Initially, four turbulence models were applied to steady ambient flow around a cylinder: the linear Launder-Sharma (LS) model [ 62] (which is known as the industry-standard low-Reynolds number model); the nonlinear Speziale model [ 59] with quadratic terms; the nonlinear model [ 46] with cubic term, and Lien, Chen and Leschziner (LCL) model [ 47], also with cubic term. The linear and nonlinear k-ε models are described below [ 50].

linear model The linear stress-strain relation is defined as

follows:

ijtijji Skuu νδ 232

−= (13)

The eddy viscosity νt is defined as:

εν µµ ~

2kfCt = (14)

The coefficient Cµ and fµ is defined according to various models used (LS, CLS, LCL or Speziale model). The turbulent kinetic energy is given by k and ε~ is the homogenous part of the dissipation rate ε, where D+= εε ~ and D is defined as 25.0 )/(2 ixk ∂∂ν . The transport equations for k and ε~ are:

εσν

ν −−∂∂

+∂∂

=∂∂

+∂∂

kjk

t

jjj P

xk

xxkU

tk )])([( (15)

Ek

fCPfCxxx

Ut k

j

t

jjj +−−

∂∂

+∂∂

=∂∂

+∂∂ εεε

σν

νεεεε

ε

~)~()]

~)([(

~~2211

(16) Where E is defined as 22 )/(2 kiit xxU ∂∂∂νν and the production rate of the turbulent kinetic energy is given by:

ijijtj

ijik SS

xUuuP ν2)( =

∂∂

= (17)

This can be written by using Boussinesq assumption as:

j

i

i

j

j

itk x

Ux

UxU

P∂∂

∂

∂+

∂∂

= )(ν (18)

In high Reynolds number k-ε models, fµ=1, D=E=0, εε ~= .

nonlinear models For nonlinear models, (in quadratic and cubic eddy-viscosity models), the stress-strain relation can be written in the following form [ 46]:

klklijklklij

ijlmklmklikljkljklikkliljkjlik

ijklkljkikikjkjkik

ijklkljkikijijji

SkCSSSkC

SSSkCSSSkC

kCSSkC

SSSSkCSkfCkuu

ΩΩ+

+ΩΩ−ΩΩ+ΩΩ+Ω−Ω

+ΩΩ−ΩΩ+Ω+Ω

+−+−=

3

4

73

4

6

3

4

53

4

4

2

3

32

3

2

2

3

1

2

~~

)32(~)(~

)31(~)(~

)31(~~2

32

εε

δεε

δεε

δεε

δ µµ

Where the main strain and vorticity tensor are defined by:

)(21),(

21

i

j

j

iij

i

j

j

iij x

UxU

xU

xUS

∂

∂−

∂∂

=Ω∂

∂+

∂∂

=

The values of the coefficients used in the above formula are varying according to the turbulence model considered [ 50]. It should be noticed that, the above nonlinear components of Reynolds stresses are included in the RANS momentum equations as source terms.

The quadratic C1~C3 terms produce discrepancies between the normal stresses. These quadratic NLEVM's thus successfully reproduced turbulence driven secondary flows, however, they did not have sensitive to streamline curvature (including swirl). Therefore, cubic terms C4~C7 are introduced in order to capture the streamline curvature effect.

(19)



The most appropriate turbulence model was that of Craft et al. [ 46] with adjustment of its coefficient of the cubic term to fit the experimental data. In general, the forces predicted are in reasonable agreement with experimental data up to Re=2×105. The inability to predict attached separation bubbles is however a marked weakness. At super critical Reynolds number, the agreement of the predicted results with experiments or LES is dependent on Keulegan-Carpenter number. It should be pointed out that the model [ 46] was, afterwards, modified in order to correctly mimic near-wall turbulence [ 63].

II- Vortex-induced vibrations Vortex shedding behind bluff bodies arises in many fields of engineering, such as heat exchanger tubes, marine cables, flexible risers in petroleum production and other applications where vortex-induced vibrations are important. The case of an elastically mounted cylinder vibrating as a result of fluid forcing is one of the most basic and revealing cases in the general subject of vortex-induced vibration in fluid-structure interaction applications. Consequently, determination of the unsteady forces on the cylinder is of central importance to the dynamics of such interaction.

Many experimental measurements have been carried out for measuring the unsteady force exerted on a cylinder undergoing forced vibration. However, there have appeared no direct lift-force measurements in the literature for an elastically mounted arrangement

[ 64]. Therefore, the calculation of flows where vortex shedding occurs presents one of the most challenging tasks in CFD modelling. On one side, modelled equations must be solved accurately in both, space and time. On the other side, turbulence models employed must correctly capture the complex physics of such flows.

The standard k-ε model did not yield satisfactory results in such applications; therefore some attempts to modify it have been developed [ 55]. The RNG k-ε (Renormalization Group Theory) has been probably the most used eddy-viscosity model for this type of flow due to the acceptable results [ 65]. Notably, the accuracy of eddy-viscosity turbulence models for vortex shedding flows from simple cylinders depends mainly on their ability to control the production of turbulent kinetic energy in the stagnation region. Some of these models were in that sense ad-hoc tuned and consequently not performing well in some other flow situations. Other models have been also used for solving this problem such as RSM in conjunction with wall functions [ 53], or large eddy simulation [ 36].

Since the CFD users seek less complicated and more robust solutions, RSM's have not been widely used for industrial applications. Recently, Basara and Jakirlic [ 66] proposed a new hybrid turbulence modelling strategy for industrial CFD. In this model, the Reynolds stress and mean rate of strain tensors are coupled via Boussinesq formula as in the standard

two-equation k-ε model. The model has been computationally validated in a number of well-proven fluid flow benchmarks. However, such modelling strategy must be further checked whether it can be beneficial for FSI applications with increasing of robustness and decreasing the computation time when working on different grid orientations. On the other side, the accuracy of this model in predicting a wide range of flows must be compared with that obtained from the full Reynolds-stress modelling concept.

The Vortex-induced vibration is generally associated with the so-called `lock-in` phenomenon where the motion of the structure is believed to dominate the shedding process, thus synchronizing the shedding frequency. Lock-in is characterized by a shifting of the vortex shedding frequency to the system natural frequency and it can also refer to the coalescence of the shedding, the cylinder oscillation and natural frequency. Therefore, the case of an elastically mounted cylinder vibrating as a result of fluid forcing is one of the most basic and revealing cases in the general subject of vortex-induced bluff body fluid-structure interactions.

Guilmineau and Queutey [ 64] have studied the VIV of an elastically mounted rigid circular cylinder with low mass damping. The incompressible 2D-RANS equations are used to predict the vortex shedding around the cylinder by solving it using finite volumes on structured grids. They adopted the low-Reynolds number turbulence model [ 67], called shear-stress transport (SST) k-ε model. The Menter Shear Stress Transport (SST) model is a blend of k-ε and k-ω models, where one equation for the turbulent kinetic energy k and a second equation for the specific turbulent dissipation rate ω are solved without implementation of wall functions. Near the boundary the k-ω model is used, and the k-ε model is used away from the wall and in shear layers. Free stream values of k and ω may be specified. Each equation is solved individually and an iterative method has been used to reduce the factorization error. In practice, the factorization error dominates when the time step is too large and the scheme is not amenable to parallel processing, thus it limits the potential for fast convergence. However, the model is robust, and may be more accurate in adverse pressure gradients than some of the other two equation models.

Three different initial conditions are used: from rest, increasing velocity and decreasing velocity resulting in a different response of the cylinder. All simulations predicted correctly the amplitude of vibration in the lower branch, while the upper branch did not match the experiments. Guilmineau and Queutey [ 64] considered that these results are encouraging as no previous simulations have predicted such high amplitude of vibration.

Recently, Leroyer and Visonneau [ 68] proposed a numerical simulation of a self-propelled fish-like body by using the flow solver ISIS. This flow solver solves



the incompressible unsteady Reynolds-Averaged Navier-Stokes equations (URANS) based on the finite volume method with several turbulence closures. The one-equation turbulence model [ 69] along with different other closures based on k-ω formulation have been used for turbulence modelling in their study. Newton's law is coupled with RANS equations to calculate the temporal evolution of the kinematic characteristics of the flexible body.

Two different coupling methods for FSI are introduced, in the first one, called weakly coupling, hydrodynamic forces and moments provided by RANS equations are calculated only at the end of every time step, and used to obtain the new position of the body. This approach showed unstable coupling, where instabilities appear as soon as the body density is close to the fluid density. In the second coupling method, called nonlinear coupling, the calculation process of forces and moments acting on the body is iterated until nonlinear RANS equations and Newton's laws residual are reduced by an appropriate factor. The latter procedure was definitely more stable for the test cases involving solid bodies; however, divergent oscillations are encountered for flexible bodies. Therefore, a specific treatment of the Newton's laws equations is applied to achieve a stable coupling.

4. NUMERICAL CONSIDERATIONS For convection-dominated flows at high Reynolds numbers smaller scales become important, and any numerical scheme should avoid introducing spurious dispersion at high wave number. In numerical computations, the sub-grid scale kinetic energy is smaller than the sub-grid scale diffusion and thus a second-order accurate scheme appears to be sufficient. However, for large Reynolds number flows, it is important to describe the mean advection problem accurately. For these practical situations, pseudo-spectral or higher-order schemes are necessary. These higher-order methods are finding more and more applications in direct and large eddy simulations [ 70, 71]. In such simulations, the upwind schemes will provide a 2D direct numerical simulation.

The numerical simulations of turbulent flows, in particular, pose some numerical difficulties that must be optimized in order to obtain reliable, robust, and stable numerical solutions. The associated difficulties can include stiffness caused by the presence of initial and boundary conditions and the singular behaviour near solid boundaries. This section will focus on some of these numerical problems.

4.1 Boundary Conditions

In any Navier-Stokes calculation, DNS, LES or RANS, a set of 3D stiff differential equations is solved numerically according to prescribed initial and boundary conditions. Since initial and boundary conditions are usually specified in some extent of

presumption, it is important to minimize their influence in the model calculations.

The initial conditions are determined based on observations. Such high-resolution observations are generally not available. Alternatives are to use results of a previous simulation to give initial conditions at the upstream boundary; Le et al.[ 3], or use suitably re-scaled results of the outgoing flow at the down stream boundary [ 72]. Some DNS and LES formulations have shown that the turbulent statistics appeared to depend strongly on initial conditions chosen. Some resulting truncation errors are due to the effect of ignoring the energetic scales in the initial condition. The growth in energy in those scales due to mean shear is not represented, and the net result is an under prediction of the Reynolds stresses [ 73].

The second primary difficulty is the boundary conditions, especially at open boundaries. Because of the elliptic nature of the problem, the flow at such boundaries depends on the unknown flow outside the computational domain. This problem is circumvented in DNS and LES by imposing periodic boundary conditions for directions in which the flow is statistically homogenous (e.g., the streamwise and spanwise directions in channel flow). The most previous simulations have been homogenous or periodic in at least one spatial direction. However, flows that grow in the streamwise direction in a nearly self-similar manner, such as equilibrium boundary layer flow, can be reduced to approximate homogeneity by a coordinate transformation [ 74].

In nonperiodic flows, the downstream boundary conditions are usually taken as zero streamwise gradients for all variables. This can be applied under the assumption of the slowly changing of the statistics of real flow in the streamwise direction, inducing negligible upstream influence. At the solid boundaries, the no-slip boundary condition is applied, which pose no special problems. These comments regarding the initial and boundary conditions are hold for RANS, DNS and LES. Moreover, for LES there is an additional major difficult near the solid boundary where all eddies are small, and it must be resolved. The resolving of the smallest eddies by LES make the grid spacing and the time step used gradually reach that needed for DNS as the surface is approached. This is a serious limitation on Reynolds number for LES and it will be discussed in the next section.

4.2 Turbulence Models near Solid Boundary

Treatment of turbulence models near solid walls is complex. Near-wall turbulent flow is not like that in free stream; instead have a boundary layer consisting of laminar sublayer, transition region and log-law region. Proper treatment of boundary conditions is important for a simulation of flow field around bluff bodies, as the flow is involved with separation and reattachment as it passes through the bodies. Hence is required to treat this region of the flow differently.



Previous work on the k-ε model used the generalized log-law near a solid boundary or employing a sufficient number of grid points. As near-wall flow is universal, it can be introduced a separate wall model to describe the flow in the layer of cells close to the wall. The velocity in this cell is given by usual log-law:

Byuuu +== ++ log1

* κ (20)

Where *u is the shear velocity or friction velocity defined by means of the wall shear stress wτ and the fluid density ρτ /*

wu = . The parameter +y is defined

as ν/* yuy =+ and κ is the von Kàrmàn constant usually taken 0.41. The constant B can be evaluated from velocity measurements. For reason of economy, the log-law is preferred in most of the numerical simulations. However, the validity of the generalized log-low is questionable for special flows (e.g. around a rectangular cross-section) and the effect of Reynolds number on such flows is not precisely reflected. Moreover, a wide scatter in values of the constants of the logarithmic velocity profile was found in the literature for both the von Kàrmàn constant k and the additive constant B.

In the study [ 57] concerning rectangular cylinders, the low Reynolds-number one-equation model is employed. This model is referred to as the two-layer model [ 75, 76]. In this model, the k-equation is solved by assigning k=0 on the solid boundary. The turbulent dissipation rate ε near the wall is determined, instead of solving ε equation, by the turbulent kinetic energy using a length scale lε as follows:

ε

εl

k 5.1

= (21)

The eddy viscosity νt is calculated in that region using the turbulent kinetic energy and a length scale lµ:

µµν

lkCt

5.1

= (22)

The length scales lε and lµ are defined as follows:

)/Re25exp(1,

Re/3.51 +−=

+=

AAyClyCl

y

l

y

l

µµε

(23)

Where the constants are given as Cl=κCµ(-0.75),

Aµ=50.5, A+=25. The turbulent Reynolds number Rey is defined as Rey=k0.5 y/ν. The two layer model [ 77] is successfully predicting the variation of drag coefficient of circular cylinder.

5. VALIDATION AND COMPARISON OF TURBULENCE MODELS

For the validation of the recently developed turbulence models in different flow geometries and operating conditions [78-80]. These investigations showed the most important features and advantages of RSM, EARSM, and LES in comparison with the standard k-ε model

and the ARSM models in or over a fixed geometry including separation and curved flows.

In the case of fluid-structure interaction applications, Bunge[ 81] has used RANS equations for predicting flows around an inextensible, massless sail (flexible membrane airfoil) at different angle of attacks, as shown in Fig. 2. His numerical Results are presented for fully turbulent conditions employing closure models of different degree of complexity in comparison with experimental and analytical results. The numerical model was able to capture the experimental results at low angle of attack for all turbulence models investigated. However, severe deviations and differences between turbulence models have been observed in case of massive separation, as it can be seen in Fig.3. Consequently, he concluded that higher-order closure models are required.

Recently, Bunge et al.[ 82] have investigated numerically the flow-induced oscillatory behaviour of a rectangular body in the range of (Re= 2 ~ 25 ×105). They selected a mass-spring-dashpot system with a model of length to height ratio L/H=2, see Fig.4.

A finite-volume based Navier-Stokes CFD-code is used for computing the flow field, while a finite-difference-based algorithm is employed to solve the differential equation for the body vibration.

The turbulence models used in his previous study

[ 81] are also used to describe the effect of turbulence modelling on the quality of the simulation. These are the Wilcox k-ω model [ 83], the SA-model [ 69], LLR k-ω model [ 84] and SALSA-model (Strain Adaptive, Linear SA model [ 85]). The last two models are a modified version of the first two models with variable coefficients and are especially tailored for non-equilibrium flows. The results produced by the one-equation models (SA & SALSA models) are not as periodic as desired and do not exhibit two distinct frequencies. The two-equation models exhibited sufficient unsteadiness. The LLR k-ω model exhibits the highest frequency and smoothest curve; therefore, it is preferred in the computations. Some of the obtained results [ 82] are presented in Figures 5 and 6.

6. CONCLUSIONS The state-of-the-art of utilizing turbulence models in fluid-structure interaction applications is reviewed. The fundamental characteristics and specific predictive qualities of different turbulence models are discussed. Four types of numerical methods for analyzing turbulent flow (DNS, LES, RANS and RSM models) are presented. New models and techniques are noted. Revision of turbulence models for adopting the bluff body aerodynamics is also described. Some numerical results, illustrated in previous investigations, for some FSI applications with different turbulence models are illustrated. The following aspects are summarized through this review covering recent research works of turbulence modelling in FSI applications:



Figure 2. Configuration of the two-dimensional sail, structural model

and computational grid [ 82].

Figure 3. Lift coefficient vs. angle of attack for different excess

length and different turbulence models [ 82].

Figure 4. Mass-spring-dashpot system [ 82].

Figure 5. Lift coefficient for different turbulence models [ 82].

Figure 6. Maximum amplitude of motion vs. flow velocity [ 82].

DNS of turbulent flows has been applied

successfully only to simple geometries at critical Reynolds numbers. To attain high Reynolds number, DNS involves huge calculations in terms of memory space and computing time. However, the results of DNS are important to understand the near-wall structure of turbulence and in the analysis of a priori evaluations of developed turbulence models.

Turbulence simulations employing LES have shown much promise, but the technique is much too costly at present to be considered as an engineering tool. For high Reynolds number turbulent flow, new methods have been developed which are all unsteady methods and based on the combination of LES and RANS.

Although of the superior predictive performance of the Reynolds stress models, however they are mathematically complex and numerically challenging. They still must utilize approximations and assumptions in modelling terms which presently can not be measured. These RSM's are perhaps still in their infancy and it may be some time yet before they have been tested and refined to the point that they become commonplace in engineering calculations.

Overall, it may be concluded that the NLEVM's are the most promising scheme in the eddy viscosity models of turbulence. They overcome some deficiencies of the linear EVM's by adding extra terms which take into account flow features like curvature effect, anisotropy, and swirl. The improvements are significant for some cases. The prediction of the normal stress anisotropy is qualitatively correct, but quantitatively smaller compared to the DNS results. This kind of formulation of the constitutive relation is ad hoc basis and mathematical. Hence, it has fewer roots in physics. However, the continuous research efforts can significantly extend the performance of NLEVM's in the future to be considered as an engineering tool.



ACKNOWLEDGMENT This work has been carried out during my stay in Institute for Structural Analysis, TU-Braunschweig, Germany. It has been supported by DFG "Post-Doc" scholarship in the group of Fluid-Structure Interaction. We are gratefully acknowledged DFG.

REFERENCES 1. Bardina, J., Ferziger, J.H. and Reynolds, W.C. (1980)

Improved subgrid models for large eddy simulations. AIAA paper 80-1357.

2. Camarri, S. and Salvetti, M.V. (2005) Hybrid RANS/LES simulations of a bluff-body flow. Wind and structures, 8, No.6, 407-426.

3. Le, H., Moin, P. and Kim, J. (1997) Direct numerical simulation of turbulent flow over a backward-facing step. Journal of fluid mechanics, 330, 349-374.

4. Moin, P. and Mahesh, K. (1997) Direct numerical simulation- A tool in turbulence research. Annu. Review of fluid mechanics, 30, 539-578.

5. Karniadakis, G.E. and Triantafyllou, G.S. (1992) Three-dimensional Dynamics and transition to turbulence in the wake of bluff objects. Journal of fluid mechanics, 138, 1-30.

6. Newman, D. and Karniadakis, G.E. (1996) Simulations of flow over a flexible cable: a comparison of forced and flow induced vibration. Journal of fluids and structures, 10, 439-453.

7. Nair, M.T. and Sengupta, T.K. (1997) Unsteady flow past elliptic cylinders. Journal of fluids and structures, 11, 555-595.

8. Willimson, C.H.K. (1996) Vortex dynamics in the cylinder wake. Annu. Review of fluid mechanics, 28, 477-539.

9. Baraza, M., Chassaing, P. and Ha Minh, H. (1986) Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. Journal of fluid mechanics, 165, 79-130.

10. Sengupta, T.K. and Sengupta, R.S. (1992) Dynamics of near wake of circular cylinder at high Reynolds number. IUTAM symposium, Göttingen, Germany.

11. Sengupta, T.K. and Sengupta, R.S. (1994) Flow past an impulsively started circular cylinder at high Reynolds number``, Computational mechanics, 14, 298-310.

12. Nair, M.T. and Sengupta, T.K. (1996) Onset of asymmetry: flow past circular and elliptic cylinders. International journal of numerical methods in fluids, 23, 1327-1345.

13. Evangelinos, C., Lucor, D. and Karniadakis, G.E. (2000) DNS-Derived force distribution on flexible cylinders subjected to vortex-induced vibration. Journal of fluids and structures, 14, 429-440.

14. Evangelinos, C. and Karniadakis, G.E. (1999) Dynamics and flow structures in the turbulent wake of rigid and flexible cylinders subjected to vortex-induced vibrations. Journal of fluid mechanics, 400, 91-124.

15. Williamson, C.H.K. and Govardhan, R. (2004) Vortex-induced vibrations. Annual Review of fluid mechanics, Annu. Review of fluid mechanics, 36, 413-455.

16. Handerson, R.D. (1997) Nonlinear dynamics and pattern formation in turbulent wake transition. Journal of fluid mechanics, 352, 65-112.

17. Prasad, A. and Williamson, C.H.K. (1997) The instability of shear layer separating from a bluff body. Journal of fluid mechanics, 333, 375-402.

18. Ma, X., Karamanos, G.S. and Karniadakis, G.E. (2000) Dynamics and low dimensionality of a turbulent near wake. Journal of fluid mechanics, 410, 29-65.

19. Kravchenko, A.G. and Moin, P. (2000) Numerical studies of flow over a circular cylinder at ReD=3900. Physics of fluids, 12, 403-417.

20. Franke, J. and Frank, W. (2002) Large eddy simulation of the flow past a circular cylinder at ReD=3900. Journal of wind engineering and industrial Aerodynamics, 90, 1191-1206.

21. Mittal, S. and Kumar, V. (2001) Flow-Induced vibrations of a light circular cylinder at Reynolds number 103 to 104. Journal of sound and vibrations, 245, 923-946.

22. Dong, G.E. and Karniadakis, G.E. (2005) DNS of flow past a stationary and oscillating cylinder at Re=10000. Journal of fluids and structures, 20, 519-531.

23. Fertziger, J.H. (1976) Large eddy numerical simulations of turbulent flows. AIAA paper 76-347, SanDiego, CA.

24. Rodi, W. (1997) Comparison of LES and RANS calculations of the flow around bluff bodies. Journal of wind engineering and industrial Aerodynamics, 69-71, 55-75.

25. Leonard, A. (1974) Energy cascade in large eddy simulation of turbulent fluid flows. Adv. Geophys. 18A, 237-248.

26. Tejada, A.E. and Jansen, K.E. (2000) Spatial test filters for dynamic model large-eddy simulation with finite elements. Commun. Numer. Meth., 1-15.

27. Jansen, K.E. (1994) Unstructured grid large eddy simulation of flow over an airfoil. Ann. Research Brief, Center of Turbulent Research, 161-173.

28. Sun, X. and Dalton, C. (1996) Application of the LES method to the oscillating flow past a circular cylinder. Journal of fluids and structures, 10, 8, 851-872.

29. Jansen, K.E. (1999) A stabilizing finite element method for computing turbulence. Comp. Meth. Appl. Mech. Engng., 174, 299-317.

30. Mittal, R. and Moin, P. (1997) Suitability of upwind-biased finite-difference schemes for large eddy simulations of turbulent flows. AIAA Journal, 35, 1415-1417.

31. Kravchenko, A.G. and Moin, P. (1998) B-spline method and zonal grids for numerical simulation of turbulent flows. Report TF-73, Dept. of Mech. Engng., Stanford Uni., USA.

32. Breuer, M. (1998) Numerical and modelling influences on large eddy simulations for the flow past a circular cylinder. Int. J. of Heat and Fluid Flow, 19, 512-521.

33. Breuer, M. (2000) A challenging test case for large eddy simulation: high Reynolds number circular cylinder flow. Int. J. of Heat and Fluid Flow, 21, 648-654.

34. Jordan, S.A. (2003) Resolving turbulent wakes. ASME Journal of Fluid Engineering, 125, 823-834.

35. Wang, M., Catalano, P. and Iaccarino, G. (2001) Prediction of high Reynolds number flow over circular cylinder using LES with wall modelling. Centre of turbulence research, Ann. Research Briefs.

36. Rodi, W., Ferziger, J.H., Breuer, M. and Porquie, M. (1996) Status of large eddy simulations: results of a workshop. ASME Journal of fluids Engng., 119, 248-262.



37. Murakami, S. and Iizuka, S. (1999) CFD analysis of turbulent flow past square cylinder using dynamic LES. Journal of fluids and structures, 13, 1097-1112.

38. Tutar, M. and Holdo, A.E. (2000) Large eddy simulation of a smooth circular cylinder oscillating normal to uniform flow. ASME Journal of fluids Engng., 122, 694-702.

39. Dalton, C. , Xu, Y. and Owen, J. C. (2001) The suppression of lift on a circular cylinder due to vortex shedding at moderate Reynolds number. Journal of fluids and structures, 15, 3-4, 617-628.

40. Li Chen, J.Y.Tu. and Yeoh, G.H. (2001) Numerical simulation of turbulent wake flows behind two-side-by-side cylinders. Journal of fluids and structures, 18, 387-403.

41. AlJamal, H. and Dalton, C. (2004) Vortex-induced vibrations using large eddy simulation at a moderate Reynolds number. Journal of fluids and structures, 19, 73-92.

42. Speziale, C.G. (1998) Turbulence modelling for time-dependent RANS and VLES. A review, AIAA journal, 36, (2), 173-184.

43. Jones, W.P. and Launder, B.E. (1972) The prediction of Laminarization with a two-equation models of turbulence. Int. J. Heat Mass Transfer, 15, (2), 301-314.

44. Rodi, W. and Mansour, N. (1993) Low Reynolds number k-ε modelling with the aid of direct simulation data. Journal of Fluid Mechanics, 250, 509-529.

45. Apsley, D., Chen, W., Leschziner, M. and Lien, F. (1997) Non-linear eddy-viscosity modeling of separated flows. Journal of Hydraulic Research, 35, 723-748.

46. Craft, T.J., Launder, B.E. and Suga, K. (1996) Development and application of a cubic eddy-viscosity model of turbulence. Int. J. Heat and Fluid Flow, 17, 108-115.

47. Lien, F.S., Chen, W.L. and Leschziner, M.A. (1996) Low-Reynolds-number eddy strain-viscosity modelling based on non-linear stress-strain/vorticity relations. Proceedings of the third international Symposium of Engineering Turbulence Modelling and Measurements, Crete.

48. Apsley, D. and Leschziner, M. (1997) A new low-Re non-linear two-equation turbulence model for complex flows. Proc. 11th Symposium on Turbulent Shear Flows, Grenoble.

49. Gatski, T.B. and Speziale, C.G. (1993) On explicit algebraic stress models for complex turbulent flow. Journal of Fluid Mechanics, 254, 59-78.

50. Saghafian, M., Stansby, P.K., Saidi, M.S. and Apsley, D.D. (2003) Simulation of turbulent flows around a circular cylinder using nonlinear eddy-viscosity modelling: steady and oscillatory ambient flows. Journal of fluids and structures, 17, 1213-1236.

51. Rodi, W. (1972) The prediction of free turbulent boundary layers by use of a two-equation model of turbulence. Ph.D. Thesis, Univ. of London.

52. Launder, B.E., Reece, G.J. and Rodi, W. (1975) Progress in the development of a Reynolds stress turbulence closure. Journal of Fluid Mechanics, 68, 537-566.

53. Franke, R. and Rodi, W. (1991) Calculation of vortex shedding past a square cylinder with various turbulence models. Eighth symposium on turbulent shear flow, paper 20-1, Univ.-Munich, Germany.

54. Murakami, S., Mochida, A. and Hayashi, Y. (1990) Examining the k-ε model by means of a wind tunnel test and large-eddy simulation of the turbulence structure around a cube. Journal of wind engineering and Industrial Aerodynamics, 35, 87-100.

55. Kato, M. and Launder, B.E. (1993) The modelling of turbulent flow around stationary and vibrating square cylinders. Ninth symposium of turbulent shear flows, paper 10-4, Kyoto, Japan.

56. Kato, M. (1997) 2D turbulent flow analysis with modified k-ε around a stationary square cylinders and vibrating one in the along and a cross wind direction. Journal of structural mechanics and Earthquake engineering, I-41, 577, 217-230.

57. Shimada, K. and Ishihara, T. (2001) Application of a modified k-ε model to the prediction of aerodynamic characteristics of rectangular cross-section cylinders. Journal of fluids and structures, 16, 465-485.

58. Pope, S. (1975) A more general effective viscosity hypothesis. Journal of Fluid Mechanics, 2, 331-340.

59. Speziale, C.G. (1987) On nonlinear k-l and k-ε models of turbulence. Journal of Fluid Mechanics, 178, 459-475.

60. Nisizima, S. and Yoshizawa, A. (1987) Turbulent channel and couette flows using an anisotropic k-ε model. AIAA J., 25-3, 414-420.

61. Myong, H.K. and Kasagi, N. (1990) Prediction of anisotropy of the near-wall turbulence with an anisotropic Low-Reynolds number k-ε turbulence model. ASME J. fluid Eng., 112, 1472-1476.

62. Launder, B.E. and Sharma, B.I. (1972, 1974) Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Letters in Heat and Mass Transfer, 1, 131-138.

63. Craft, T.J., Launder, B.E. and Suga, K. (1997) Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model. Int. J. Heat and Fluid Flow, 18, 15-28.

64. Guilmineau, E. and Queutey, P. (2004) Numerical simulation of vortex-induced vibration of a circular cylinder with low mass-damping in a turbulent flow. Journal of fluids and structures, 19, 449-466.

65. Przulj, V. (1998) Computational modelling of vortex shedding flows. PhD Thesis, City University, London.

66. Basara, B. and Jakiric, S. (2003) A new hybrid turbulence modelling strategy for industrial CFD. International Journal for Numerical Methods in Fluids, 42, 1, 89-116.

67. Menter, F.R. (1993) Zonal two-equation k-ω turbulence models for aerodynamic flows. AIAA 24th fluid dynamic conference, Orlando, FL, USA.

68. Leroyer, A. and Visonneau, M. (2005) Numerical methods for RANS simulations of a self-propelled fish-like body. Journal fluids and structures, 20,7, 975-991.

69. Spalart, P. and Allmaras, S. (1992) A one-equation turbulence model for aerodynamic flows. AIAA 30th Aerospace Sciences Meeting, AIAA paper 92-0439, Reno, NV, USA.

70. Rai, M.M. and Moin, P. (1991) Direct simulation of turbulent flow using finite difference schemes. Journal of computational physics, 96, 15-53.

71. Najjar, F. M. and Vanka, S.P. (1995) Simulation of the unsteady separated flow past a normal flat plate. International journal of numerical methods in fluids, 21, 525-547.



72. Bertolotti, F., Herbert, T. and Spalart, P.R. (1992) Linear and nonlinear stability of the Blasius boundary layer. Journal of fluid mechanics, 242, 441-474.

73. Mahesh, K., Yucheng, H. and Babu, P. (2005) Three problems in the large eddy simulation of complex turbulent flows. Complex effects in LES, Limassol.

74. Spalart, P.R. (1988) Direct numerical simulation of a turbulent boundary layer up to Re=1400. Journal of fluid mechanics, 187, 61-98.

75. Norris, L.H. and Reynolds, W.C. (1975) Turbulent channel flow with moving wavy boundary. Report No. FM-10, Stanford University, Dept. of Mech. Eng., USA.

76. Rodi, W. (1991) Experience with two-layers models combining the k-ε model with a one-equation model near the wall. AIAA paper 91-0216.

77. Shimada, K. and Meng, Y. (1997) Two-dimensional numerical analysis for a circular cylinder in the uniform flow by the k-ε model. Summaries of Technological papers of annual meeting architectural institute of Japan B-1, 327-328.

78. Lakehal, D. and Rodi, W. (1997) Calculation of the Flow Past a Surface-Mounted Cube with Two-Layer. Journal of Wind Engineering and Industrial Aerodynamics, 67, 65-78.

79. Wallin, S. and Johansson, A.V. (2000) An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows. Journal of Fluid Mechanics, 403, 89-132.

80. Hellsten, A. (2004) New Two-Equation Turbulence Model for Aerodynamics Applications. PhD thesis. Report A-21, Department of Mechanical Engineering, Helsinki University of Technology, Finland.

81. Bunge, U., Rung, T. and Thiele, F. (2001) A two-dimensional sail in turbulent flow. Advances in Fluid Mechanics, 30, 245-256.

82. Bunge, U., Gurr, A. and Thiele, F. (2003) Numerical aspects of simulating the flow-induced oscillations of a rectangular bluff body. Journal of fluids and structures, 18, 405-424.

83. Wilcox, D.C. (1994) Simulation of transition with a two-equation turbulence model. AIAA journal, 32, 247-255.

84. Rung, T. and Thiele, F. (1996) Computational modelling of complex boundary-layer flows. Proceedings of Ninth International SYMPOSIUM OF Transport Phenomena in Thermal-Fluid Eng., Singapore, 321-326.

85. Rung, T., Bunge, U., Schatz, M. and Thiele, F. (2002) Restatement of the Spalart-Allmaras eddy-viscosity model in a strain-adaptive formulation. AIAA Journal, 41, 1396-1399.

turbulence models for fluid-structure interaction …

Documents