WALL-BOUNDED AND FREE-SURFACE TURBULENCE
AND ITS COMPUTATION
(July - December 2004)
~ Abstracts ~
Decaying turbulence in an
active-grid-generated flow and comparisons with large-eddy
simulation
Hyung-Suk Kang, Johns Hopkins University
CFD for complex industrial
flows: Strategies for rurbulence modelling
Brian Launder, The University of Manchester
The lecture begins by arguing that the usual CFD approach to the computation of industrial flows, namely two-equation eddy-viscosity models matched to log-law ‘wall functions’ close to solid boundaries, is an inadequate basis in most applications. Alternatives are therefore discussed: non-linear eddy viscosity models and second-moment closures. It is argued that the former approach is only worthwhile if one includes up to cubic products of strain and vorticity in the constitutive equation while the second is generally beneficial only if one adopts special practices for handling the effects of the wall on the pressure-containing correlations. Particular attention is given to the Two-Component Limit approach to modelling these terms adopted at UMIST. Two new strategies for wall functions are presented, one an analytical approach, the other a numerical treatment. Both are shown to be far more generally applicable than the log-law approach that they supersede. Numerous applications are shown of three-dimensional flows.
Turbulence modelling of
buoyancy-affected flows
Brian Launder, The University of Manchester
The earth’s gravitational field has very important and wide-ranging effects on turbulent flows. The most important sub-classification is between flows where the effects of buoyancy are felt principally on the mean flow and those where the direct effect is on the turbulent fluctuations (which in turn has an impact on the mean motion). The former flows may often be represented with quite rudimentary turbulence models while the latter generally require second moment closure and, in some cases third-moment closure or an unsteady RANS approach. The lecture provides a survey of alternative models and methodologies and the types of flow to which they may be successfully applied. Applications range from vertical flows through heated pipes and annuli to the mixing of horizontal streams of different densities.
Large eddy simulations on
unstructured grids
Maria Vittoria Salvetti, University of Pisa, Italy
In the perspective of the application of large-eddy simulation (LES) to problems of industrial or engineering interest, the use of unstructured grids becomes particularly attractive, because of their friendliness when applied to complex realistic geometries. Although less numerous than those using structured grids, the examples of LES on unstructured grids have increased in the last decade.
The aim of the present seminar is to outline the main difficulties related to LES on unstructured grids and to briefly review the different choices adopted in the literature.
Then, the implementation of the LES approach in a code which may be considered as a prototype of industrial numerics is presented. The numerical solver [1] is based on a mixed finite-volume/finiteelement formulation, applicable to unstructured grids. The Roe flux difference splitting is the basic scheme for the approximation of the convective fluxes and the MUSCL reconstruction method is employed to obtain second-order accuracy in space. Either implicit or explicit schemes are available to advance the equations in time.
The choices adopted for numerical viscosity, SGS modeling and time-advancing are described. In particular, a MUSCL reconstruction technique is presented [2] for which the resulting numerical viscosity is proportional to sixth-order spatial derivatives and, thus, more concentrated on the highest resolved frequencies than in classical stabilization methods.
The capabilities of the presented method and of the adopted choices are illustrated by examples of application to flows around bluff bodies and lifting surfaces at high angle of attack.
References
[1] C. Farhat, B. Koobus, and H. Tran. Simulation of vortex shedding dominated flows past rigid and flexible structures. In Computational Methods for Fluid-Structure Interaction, pages 1–30. Tapir, 1999.
[2] S. Camarri, M. V. Salvetti, B. Koobus, and A. Dervieux. A low-diffusion MUSCL scheme for LES on unstructured grids. Computers and Fluids, 33(9):1101–1129, 2004.
Hybrid RANS/LES simulations on
unstructured grids
Maria Vittoria Salvetti, University of Pisa, Italy
The motivations leading to hybrid RANS/LES simulations and the different approaches are briefly reviewed. Then, a hybrid RANS/LES approach based on the idea of Limited Numerical Scales (LNS) [1], is presented.
The basic idea of LNS consists in a blending between an eddy-viscosity RANS closure model and an LES one, depending on the local grid resolution. In the proposed approach, the main ingredients are the standard k – ε model for compressible flows for the RANS part and the subgrid scale Smagorinsky model for LES. Moreover, the blending function is such that the lowest value of eddy viscosity is used among those given by RANS and LES closure respectively.
The hybrid RANS/LES approach is implemented in a numerical solver [2] for the simulation of compressible flows, in which the discretization is carried out on unstructured grids through P1 Galerkin finite elements for the viscous terms and a finite volume approach for the convective ones. The Roe flux difference splitting is the basic scheme for the approximation of the convective fluxes and the MUSCL reconstruction method is employed to obtain second-order accuracy in space. A low diffusion version stabilized with sixth-order spatial derivatives is used [3]. Either implicit or explicit schemes are available to advance the equations in time.
The LNS model is applied to the simulation of the flow around a square cylinder at a Reynolds number of 20000. The results are compared with those obtained with RANS and LES respectively and with experimental data. On fine grids, as those typically used in LES, the LNS approach recovers the LES accuracy. Conversely, on coarse grids, it gives better results than both LES and RANS simulations and, in particular, an excessive damping of the flow unsteadiness is avoided by the use of the blended LNS eddy-viscosity. Finally, it is shown that, at least for the considered test-case, a completely RANS treatment of the near wall region is obtained in LNS, while the Smagorinsky eddy viscosity is used in the wake.
References
[1] P. Batten. LNS - An approach towards embedded LES. In AIAA Paper 2002-0427. American Institute of Aeronautics and Astronautics, 2002.
[2] C. Farhat, B. Koobus, and H. Tran. Simulation of vortex shedding dominated flows past rigid and flexible structures. In Computational Methods for Fluid-Structure Interaction, pages 1–30. Tapir, 1999.
[3] S. Camarri, M. V. Salvetti, B. Koobus, and A. Dervieux. A low-diffusion MUSCL scheme for LES on unstructured grids. Computers and Fluids, 33(9):1101–1129, 2004.
Flow control
Mohamed Gad-el-Hak, Virginia Commonwealth University
The ability to actively or passively manipulate a flow field to effect a desired change is of immense technological importance. In its broadest sense, the art of flow control probably has its roots in prehistoric times when streamlined spears, sickle-shaped boomerangs, and fin-stabilized arrows evolved empirically by archaic Homo sapiens. As defined by Flatt (1961), the term boundary layer control includes any mechanism or process through which the boundary layer of a fluid flow is caused to behave differently than it normally would were the flow developing naturally along a smooth straight surface. In this presentation, methods of control to achieve transition delay, separation postponement, lift enhancement, drag reduction, turbulence augmentation, or noise suppression are considered. Emphasis is placed on external boundary-layer flows although applicability of some of the methods reviewed for internal flows will be mentioned. An attempt is made to present a unified view of the means by which different methods of control achieve a variety of end results. Performance penalties associated with a particular method such as cost, complexity, or trade-off will be elaborated. The presentation will emphasize flow control technology developed specifically for the aviation industry.
Turbulence: The Taming of the
Shrew
Mohamed Gad-el-Hak, Virginia Commonwealth University
Considering the extreme complexity of the turbulence problem in general and the unattainability of first-principles analytical solutions in particular, it is not surprising that controlling a turbulent flow remains a challenging task, mired in empiricism and unfulfilled promises and aspirations. Brute force suppression, or taming, of turbulence via active control strategies is always possible, but the penalty for doing so often exceeds any potential savings. The artifice is to achieve a desired effect with minimum energy expenditure. Spurred by the recent developments in chaos control, microfabrication and neural networks, efficient reactive control of turbulent flows, where the control input is optimally adjusted based on feedforward or feedback measurements, is now in the realm of the possible for future practical devices. But regardless of how the problem is approached, combating turbulence is always as arduous as the taming of the shrew. The former task will be emphasized during the oral presentation, but for this abstract we reflect on a short verse from the latter.
From William Shakespeare's The Taming of the Shrew.
Curtis (Petruchio's servant, in charge of his country
house): Is she so hot a shrew as she's reported?
Grumio (Petruchio's personal lackey): She was, good Curtis,
before this frost. But thou know'st winter tames man, woman,
and beast; for it hath tamed my old master, and my new
mistress, and myself, fellow Curtis.
Flow physics in microdevices
Mohamed Gad-el-Hak, Virginia Commonwealth University
Interest in microelectromechanical systems (MEMS) has experienced explosive growth during the past few years. Such small devices typically have characteristic size ranging from 1 mm down to 1 micron, and may include sensors, actuators, motors, pumps, turbines, gears, ducts and valves. Microdevices often involve mass, momentum and energy transport. Modeling gas and liquid flows through MEMS may necessitate including slip, rarefaction, compressibility, intermolecular forces and other unconventional effects. In this presentation, I shall provide a methodical approach to flow modeling for a broad variety of microdevices. The continuum-based Navier-Stokes equations-with either the traditional no-slip or slip-flow boundary conditions-work only for a limited range of Knudsen numbers above which alternative models must be sought. These include molecular dynamics (MD), Boltzmann equation, Direct Simulation Monte Carlo (DSMC), and other deterministic/probabilistic molecular models. The present talk will broadly survey available methodologies to model and compute transport phenomena within microdevices.
Reynolds number effects in
wall-bounded turbulent flows
Mohamed Gad-el-Hak, Virginia Commonwealth University
Most studies on turbulent wall-bounded flows are being carried out at rather low Reynolds numbers. Since many practical flows have very high Reynolds number, the question is how relevant are the low-Reynolds-number data to field situations? Except for the mean flow review of Coles (1962), the Reynolds number effects on higher-order turbulence statistical quantities and on coherent structures have not been adequately researched; and there are some unresolved aspects of the effects on even the mean flow at very high Reynolds number. The key points of the present survey of published data are:
- The widely accepted asymptotic state of the wake component is present only in the range 5,000 < Reθ < 15,000. At higher values, it drops although at a much slower rate.
- The Reynolds number dependence of the post-transition relaxation length is not well understood. (3) The Clauser's shape parameter is Re dependent at Req < 1000.
- Unlike the mean flow, the statistical turbulence quantities do not scale accurately with the wall-layer variables over the entire inner layer. Such scaling applies over only a very small portion of the inner layer adjacent to the wall.
- While the variously defined (small) length scales differ greatly from each other at low Reynolds number, they all asymptote to the mixing length at Reθ > 10,000.
- The outer layer structure changes continuously with Reynolds number, and very little is known about the structure of very high-Reynolds-number turbulent boundary layers.
Immersed boundary method for flow
and heat transfer inside/over a complex geometry
Haecheon Choi, Seoul National University
A new immersed-boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source/sink as well as a momentum forcing. Also, a heat source/sink is introduced to simulate the temperature fields over or inside complex geometries. The present method is based on a finite-volume approach on a staggered mesh together with a fractional-step method. Both momentum forcing and mass source/sink are applied on the body surface or inside the body to satisfy the no-slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary. Heat source/sink is applied to satisfy the thermal conditions such as the iso-thermal and iso-heat-flux conditions. Several examples of applying the immersed boundary method will be presented: e.g. large eddy simulation of turbulent flow inside a ribbed channel, large eddy simulation of turbulent flow over a sphere, RANS of flow past a triangluar cylinder, etc.
Active and passive control of
turbulence
Haecheon Choi, Seoul National University
In this talk, the results from both numerical and experimental studies on active and passive controls for drag reduction are presented. We consider the form drag reduction for flow over a bluff body. The bluff bodies considered in our study are the circular cylinder, sphere and two-dimensional model vehicle at a wide Reynolds number range. The control methods investigated are the suboptimal control, single frequency forcing, distributed forcing, and control with a passive device. It is shown that some of the control methods suggested significantly reduce the form drag for various Reynolds numbers.
Large eddy simulation of enclosed rotor-stator flow
Chao-An Lin, National Tsing Hua University
Large Eddy simulation in
support of RANS modelling
Michael Leschziner, Imperial College London
LES is regarded by some as a prediction method that will eventually replace RANS modelling in a practical environment. However, those engaged in major studies of both strategies are well aware that both have serious (though different) sets of limitations, which constrain their general use and which require careful consideration and discrimination. While LES obviously returns a ‘full’ temporal and spatial representation of turbulence, and with it the turbulence dynamics, the numerical resolution needed to achieve adequate accuracy is formidable in sheared near-wall flows at elevated Reynolds numbers, for which the required near-wall grid density does not differ drastically from that in DNS. Separation from gently curved surfaces is a case in point, for in this, the evolution of the boundary layer towards separation needs to be carefully resolved. Experience with LES applied to marginally stalling aerofoils, for example, suggests that grids of the order 107 nodes are needed for a spanwise domain that is even a small fraction of the chord. The development of more economical hybrid RANS/LES schemes is being pursued vigorously, worldwide, but there are a number of important uncertainties and problems that need to be resolved. The lecture will address these, among other topics. RANS will therefore remain the mainstay of CFD for many years to come, especially for flows dominated by near-wall shear layers.
Whatever its future in practical CFD, LES has an important role to play in the development and validation of RANS models, at a cost significantly lower than DNS. The important constraint to respect in its exploitation to this end is that the filtered-out motion and the subgrid-scale stresses do not affect the accuracy of the simulation, to the extent that the second and third moments are returned with sufficient accuracy within the primary regions of interest. If this is done, LES can provide extremely useful results for model validation, including budgets for the Reynolds-stress components. In fact, such data are potentially even more useful than experimental measurements, because the conditions at the boundaries of the simulation domain can be controlled more carefully than in experiments. Thus, if statistical spanwise homogeneity (i.e. perfect statistical two-dimensionality) is desired, periodic conditions can be applied to a sufficiently wide spanwise slab. Streamwise periodicity – for example, in the case of a nominally infinite streamwise sequence of obstacles – can similarly be prescribed virtually perfectly.
The lecture will deal with some of the issues noted above. LES studies will be reviewed, in which near-wall structure and resolution are important issues. In most cases, stress budgets have been extracted, in addition to other quantities, and these have been used (and continue to be used) as references for RANS modelling studies. The lecture will also deal with the effectiveness of approximate near-wall treatments for high-Re flows, aiming to reduce substantially the resource requirements of wall-resolving simulations.
Second-moment and related
turbulence closure models: potential, achievements and
prospects for complex flows
Kemal Hanjalic, Delft University of Technology
Despite almost three decades of development and indisputable progress, turbulence models are still regarded as the major weakness of CFD packages. A variety of models with different levels of sophistication and potential are available nowadays, but most industrial CFD codes still use the rudimentary two-equation eddy-viscosity k-ε or k-ω models with linear stress-strain relationship. While recognizing the limitation of the simple linear models, recent interest of industry in model improvements has been confined mainly to the revival of the non-linear eddy-viscosity and explicit algebraic models. Three equation models, notably with elliptic relaxation, or with equation for one of the turbulent stress invariants, are also gaining in popularity. However, differential second-moment (Reynolds-stress) models, long expected to become industrial standard, are still viewed as a development target rather than as a proven and mature technique.
Eddy-viscosity models - irrespective of their level - have serious deficiencies and limitations, as will be illustrated by a brief overview of some notable failures. The rationale for employing more advanced models for computation of complex flows (primarily at the second-moment closure level) will be discussed, focusing on predicting effects of stress anisotropy, streamline curvature, rotation, compression and transition. Several current approaches to improving the second-moment closures will be discussed, which include variable-coefficients quasi-linear and non-linear pressure-strain models, elliptic relaxation, and viscous and near-wall modifications.
The superiority of these models will be illustrated by a series of computational examples with a new second-moment closure, which includes non-equilibrium attached, separating and reattaching flows, impingement, secondary motion, cyclic compression, swirl and system rotation. The modeling of molecular effects both near and away from a solid wall and associated laminar-to-turbulent and reverse transition will also be discussed. Examples include by-pass and separation-induced transition and revival of laminarized turbulence in oscillating flows.
Closure models for turbulent
flows driven by thermal buoyancy and other body forces
Kemal Hanjalic, Delft University of Technology
Turbulent flows driven or influenced by thermal buoyancy are frequently encountered in many technological applications, such as building structures, space heating and cooling, smoke and fire spreading, nuclear reactor containment, radioactive waste containers, electronics equipment, solar collectors, crystal growth. Environmental flows in the atmosphere and water accumulations are also dominated by buoyancy force. These flows are, however, featured by phenomena that pose special challenge to conventional one-point closure models. Inherent unsteadiness, energy non-equilibrium, counter-gradient diffusion, strong pressure fluctuations, lack of universal scaling, all believed to be associated with distinct large-scale coherent eddy structures, are hardly tractable by Reynolds-type averaging. Buoyancy produces a unidirectional stratification and, depending on the orientation of the temperature-gradient vector imposed by the boundary conditions with respect to the gravitation vector, a variety of regimes may coexist. Second-moment closures, though inadequate for providing information on eddy structure, offer better prospects for capturing at least some of the phenomena than eddy-viscosity models, though they require a large number of equations to be solved.
We review the rationale and some specific modeling issues related to buoyant flows within the realm of one-point closures. The inadequacy of isotropic eddy-diffusivity models is first discussed, followed by the rationale of the second-moment modeling and its term-by-term scrutiny based on direct numerical simulations (DNS). Obvious deficiencies are identified and some new proposals for model improvement are presented, illustrated by a priori validation in several generic cases, including flows with heating from sides and from below. Algebraic models based on a rational truncation of the differential second-moment closure are proposed as the minimum closure level for complex flows. Despite deficiency in this assumption, it shown that a quasi-linear algebraic model reproduces well the mean flow properties, turbulence second moments and wall heat transfer in a variety of wall-bounded buoyant flows. The concept is extended to forced and buoyancy-driven flows of conductive fluids subjected to magnetic field.
Transient RANS for “ultra hard”
thermal convection and environmental flows at extreme Rayleigh
numbers
Kemal Hanjalic, Delft University of Technology
Much controversy surrounds the physics of turbulent thermal convection, especially over heated horizontal surfaces at very high Rayleigh numbers. The interest is motivated not only by scientific curiosity, but also by the importance in understanding thermal convection in atmosphere, oceans, earth mantle, and in many technological applications. Major turbulence source is the surface boundary layer, which controls the plume release and heat transfer, but because of extremely small thickness, neither measurements nor numerical simulations (DNS, LES) can at present resolve its structure if Ra is beyond 1010. On the other hand, the well-organised large-scale convective structures act as the major carrier of momentum and heat across the outer domain and their resolution in time and space is essential for predicting any property of interest.
We show that a time-dependent Reynolds-Averaged Navier-Stokes (T-RANS) approach can reproduce not only the long-term averaged parameters and turbulence statistics over a much higher Ra range than possible by LES, but also to capture the coherent roll/cell structure (VLES) and their reorganization due to a change in Ra number, boundary configuration (plane-versus wavy walls), or the action of another (Lorentz) body force. The large scale deterministic motion is fully resolved in time and space, whereas the unresolved stochastic motion is modelled by a 'subscale' model for which a conventional 3-equations (k-ε-θ) algebraic stress/flux closure was used. The validation against DNS data for low Ra, and experiments for moderate Ra for Rayleigh-Bénard and penetrative convection show excellent agreement. The method was then used to simulate very high Ra numbers (up to 1017), revealing a consolidation and dramatic thinning of the wall thermal boundary layer with an increase in the Ra number, with finger-like plumes between planform structures becoming also thinner, more distant, but much more vigorous. Application of the method to a real-life-scale environmental problems is demonstrated in examples of air movement and smoke spreading in heated furnished and occupied residential space, in summer air conditioned room and in diurnal air movement and pollutant spreading over a town situated in a mountainous valley of complex orography, topped by an inversion layer.
Hybrid RANS/LES approaches for
high Re number turbulent flows
Kemal Hanjalic, Delft University of Technology
Formidable resolution challenge in LES of wall-bounded turbulent flows for high Reynolds numbers and complex geometry has triggered significant research in exploring alternative computational approaches, directed primarily towards merging LES and RANS strategies. The research pursued by various groups over the past few years has disclosed however, that the problem is more difficult than originally envisaged: two methods differ in essence and a simple merging poses serious problem in reconciling different physics. The problem is equally acute in the zonal approach with a conventional RANS in the near-wall region and the conventional LES with a grid-size-based subgrid scale model in the outer flow, as it is in the “seamless” approach using the same model throughout the flow with a continuous sub(grid)scale model modification. Although common RANS models seem to reproduce the velocity response to external perturbations fed from LES across the RANS/LES interface surprisingly close to the genuine LES fluctuations, some dynamically important small scale structure in the buffer layer are missing, leading to failure in reproducing even the standard log-law in equilibrium wall flows. This physical inconsistency seems intractable to simple merging because the same form of equations are solved irrespective of their interpretation – filtered or averaged, and the modifications of the subs(grid)scale model generally failed to cure the problem. Some extra forcing has been recently proposed to compensate for RANS deficiency and thus to reconcile the physical inconsistency between the two approaches. These issues will be discussed and illustrated by some tests conducted by the author’s and other groups, using different approaches.
Analysis of the spectral
Variational Multiscal Method
Pierre Sagaut, LMM - University of Paris VI (Pierre et
Marie Curie)/(CNRS)
The search for self-adaptive subgrid turbulence models yields the definition a a large number of methods and models. But a comon feature shared by all recent approaches is that they rely on a two-scale/two-level decomposition of the resolved velocity field, properties of the subgrid motion being "extrapolated" thanks to some self-similarity assumptions. A very recent appraoch, coined as the Variational Multiscale Methods, was proposed by T.J.R. Hughes and his co-workers in 2001. It is based on an orthogonal (in some sense) decomposition of the resolved velocity field and the subgrid model is built using the smallest resolved scales only. This model is local in terms of wave number, rendering it "self-adaptive" or "dynamic". The lecture will deal with the sensitivity of the model to its components in the case of fully developed turbulence in the limit of infinite Reynolds numbers
Towards large Eddy simulation of
real wall-bounded flows
Christer Fureby, The Swedish Defence Research Agency (FOI)
Large Eddy Simulation (LES) of complex wall bounded flows becomes prohibitively expensive at high Reynolds (Re) numbers if one attempts to resolve the small but dynamically important vortical structures, especially in the near wall region. The LES wall boundary condition problem is to account for the effects of the near-wall turbulence between the wall and the first node and its transfer of momentum to the wall. In LES of high-Re-number flows we may either try to resolve the near-wall dynamics or we may try to model it. The first approach calls for very fine grids in the near-wall region, taking advantage also of local grid refinement and unstructured grids. For flows in engineering applications (high Re and complex, and possibly moving, geometries) this approach is expensive but technically feasible. The second approach calls for more sophisticated modeling of the complex flow in the near-wall region, but is currently the only afford-able method. When using wall-models the dynamics of the energy-containing eddies in the wall layer are determined from a separate wall-model calculation, that provides the outer flow LES with a set of approximate boundary conditions. Consequently, high Re-number flows can be studied using LES if more sophisticated models are used that better takes into acount the subgrid flow physics, in particular close to the wall. In this lecture this problem will be discussed in detail, and a set of applications ranging from fully developed turbulent channel flows to the flow past a fully appended ship in model scale will be presented to illustrate both the problem itself, and possible solutions. Accordingly, a range of models will be described that have the potential of allowing such LES calculations. This range of models will cover the range from extended wall models (typically based on the log law) to subgrid simulation models, typically based on multi-scale concepts, in which equations for the subgrid scale flow physics are solved on embedded grids.
Dynamic models in Large Eddy
Simulation of turbulent flows
Charles Meneveau, Johns Hopkins University
We review the dynamic model for LES and comment on the required modifications when high Reynolds number boundary layer flows are considered. Specifically, the scale-dependent dynamic model (Porte-Agel et al. 2000) is described. For applications to flows in complex geometries, the usual method of averaging over regions of statistical homogeneity is not applicable. We discuss possible generalizations of the scale-dependent model in the context of the Lagrangian model, where averages are accumulated over pathlines of the flow rather than directions of statistical homogeneity. With a particularly simple, although as yet incomplete, version of this model, we study turbulent boundary layer flow over surfaces with varying roughness scales.
Dynamics and statistics of
velocity gradients in the inertial range of turbulence, and
implications for LES
Charles Meneveau, Johns Hopkins University
One important facet of turbulence relates to typical local deformation rates of the flow as described by the velocity gradient tensor. Its properties determine, for instance, if the deformation produces vorticity stretching along a particular direction (possibly yielding tubular vortex structures), or to produce structures that are flattened out in one direction while expanding in the other two (possibly yielding pancake-like objects). On the basis of three-dimensional measurements in turbulent duct flow using holographic PIV techniques (Tao, Katz & Meneveau, J. Fluid Mech. 2002), we consider the dynamics of the velocity gradient tensor filtered at inertial-range scales. In addition to self-interactions and the filtered pressure Hessian, the evolution of the filtered velocity gradient tensor is determined by the subgrid-scale stress tensor. As in so-called Restricted Euler dynamics, the evolution equations can be simplified by considering two invariants R and Q. The effects of the subgrid-scale stress tensor on them can be quantified unambiguously by evaluating conditional averages that appear in the evolution equation for the joint PDF of the invariants. The HPIV measurements show that the SGS stresses have significant effects, e.g. along the so-called Vieillefosse tail they oppose the formation of a finite-time singularity that occurs in Restricted Euler dynamics. Motivated by practical modeling needs in the context of large eddy simulations, we examine the behavior of various closures for the subgrid stresses. Analysis of the Smagorinsky, nonlinear, and mixed models show that all reproduce the real SGS stress effect along the Vieillefosse tail, but that they fail in several other regions. An attempt is made to optimize the mixed model by letting the two model coefficients be functions of the two invariants R and Q. (work performed with F. Vanderbos and J. Katz, funded by ONR).
Two scales asymptotics for
turbulence modeling
Olivier Pironneau, University of Paris VI (Pierre et Marie
Curie)
We shall show how multiple scale expansions can be justify Reynolds Average models (RANS) like the k-epsilon model. We shall also show how homogenization of rough surfaces can lead to an effective mean surface . Finally we shall discuss new tentatives in the combination of LES and RANS.
DNS of turbulent flow over dimpled surfaces
Zhengyi Wang, National University of Singapore
Flows over dimple surfaces have attracted increasing attention in recent years. The flow structures induced by dimpled walls have been recognized to have potential industrial applications. For examples, vortex shedding from dimples on golf balls has helped to reduce and stabilize flow separation behind the balls, leading to improved flying characteristics. Vortex pairs generated from dimple edges and central portion may also enhance surface heat transfer without increasing the total drag. Recently, Russian researches suggested that surface dimples may alter near-wall turbulence with positive effect on drag. In our study, direct numerical simulation (DNS) is carried out to study flow structures over dimpled walls. These simulations offer rich information on the flow structures over dimpled surfaces, which will help us to gain deeper insight into dimple-based control mechanisms.
Continuous mode transition
Paul A. Durbin, Stanford University
The process by which free-stream, vortical disturbances induce transition to turbulence in an underlying boundary layer, without the intervention of instability waves, is an instance of `bypass transition'. That terminology is all-encompassing; it is defined by what does not occur: Tollmein-Schlichting waves are bypassed. Laboratory experiments and DNS have shed considerable light on what does occur.
Some aspects can be understood in terms of the continuous spectrum of the Orr-Sommerfeld and Squire equations. Instead of discrete mode (Tollmein-Schlichting wave) precursors, transition by free-stream turbulence can be studied starting from continuous mode precursors. An examination of mode shapes leads to a theory of how free-stream disturbances penetrate the boundary layer and, ultimately, provoke transition. The basic idea is that low-frequency modes penetrate the boundary layer, while high frequencies are expelled --- a result referred to as shear sheltering. Low frequency penetration can be characterized by a `coupling coefficient'. While this is a route into the boundary layer, transition subsequently involves an interaction between low and high frequency modes, to produce break down near the top of the boundary layer.
Continuous mode transition is illustrated by numerical simulations of mode interaction. Either one or two modes are prescribed at the inlet to the computational domain. One low frequency mode will generate perturbation jets in the boundary layer. Transition does not occur. One low and one high frequency mode suffice to induce transition.
These studies of mode interactions provide a fundamental perspective on the transition mechanism seen in full simulations with turbulent inflow. DNS of transition induced by grid turbulence and by swept wakes will also be presented.
Periodic motions in turbulence
Shigeo Kida, Kyoto University
Turbulence is intrinsically chaotic and any fluid motions never repeat. It is this chaotic nature that makes it difficult to analyze turbulence dynamics such as generation and sustenance mechanisms, mixing and diffusion, and so on. A well-defined coherent motion, if any, would be useful to analyze the spatiotemporal structure of turbulence and to educe its general properties. It has long been known for low-dimensional dynamical systems that there exist infinitely many periodic orbits close to any chaotic orbits. This motivates us to search for periodic motions in fluid turbulence, which is regarded as a high-dimensional dynamical system. Since any periodic motions in turbulence are unstable, they never manifest themselves naturally. Therefore, they can only be captured by iterations such as the Newton-Rapson method. Here, we present periodic motions recently obtained in the Couette turbulence and in isotropic turbulence. In order to save the necessary memory to describe the periodic motions we use a minimal box (J. Jim´enez and P. Moin 1991, J. Fluid Mech. 225, 213-240) for the Couette turbulence and the high-symmetric flows (S. Kida 1985, J. Phys. Soc. Jpn. 54, 2132-2136) for the isotropic turbulence. Among those periodic motions, the ones which exhibit the same statistical properties as the turbulence are of our chief concern because they may be used as stardards to analyze the turbulence dynamics. Such periodic motions are discussed by comparing with turbulent motions.
Elementary vortices in turbulence
Shigeo Kida, Kyoto University
Turbulence is composed of many vortical motions of different sizes and shapes. The knowledge of the physical characteristics of individual vortices is helpful for understanding of turbulence dynamics. It is generally recognized that tubular swirling regions of high vorticity exist in various kinds of turbulence. They are nicknamed worms in isotropic turbulence, streamwise vortices in wall turbulence, rolls and ribs in free-shared turbulence. Although the shape of the cross-section and the length of such tubular vortices vary from turbulence to turbulence, the size of the cross-section and the swirl velocity obey, in average, the Kolmogorov scaling law. They are regarded as the smallest vortical structure in turbulence, so that they are called the elementary vortex. The elementary vortex plays significant roles in turbulence dynamics, such as generation and sustenance of turbulence, diffusion and mixing, and so on. Here, we review a series of studies on the elementary vortex performed these years by our research group. An eduction method of the elementary vortex from a turbulent field is introduced, and a vortex is identified as a low-pressure region and the physical characteristics are discussed. The dynamical roles of the elementary vortex are examined in deformation and stretching of fluid lines. A future direction in the study of the elementary vortex is viewed.
Mechanism of Turbulence transition in
wall bounded shear flows
Hua-Shu Dou, National University of Singapore
Linear stability theory represents the state-of-the-art of the research for flow stability problems, it obtains agreement with experiments for some problems such as Rayleigh-Benard and Taylor-Couette flows, partly agreement with experiment for 2D boundary layer problem. However, the subcritical condition of turbulent transition predicted with it differs largely from the experiments for wall bounded parallel flows. Other theories including energy method, weak nonlinear theory, and second instability theory, can not obtain satisfactory agreement with experiments either for this issue. The author proposed a new theory based on the energy gradient concepts. In this theory, the whole flow field is treated as an energy field. For given initial disturbance, the stability of the flow is dominated by an energy gradient parameter, K, which is defined as the ratio of the energy gradient in the transverse direction to that in the streamwise direction. The distribution of K in the flow field and the property of disturbance determine the disturbance amplification or decay in the flow. It is suggested that the flow instability should first occur at the position of Kmax which may be the most dangerous position. For the experiments determined subcritical condition, it is found that the transition takes place at a critical value of K (Kc=380) for wall bounded parallel flows (plane and pipe Poiseuille flows and plane Couette flow). The most dangerous position is at y/h=0.5774 for plane Poiseuille flow, r/R=0.5774 for pipe Poiseuille flow, and near the wall for plane Couette flow (about y/h=1). These observations have been confirmed by some experimental results in the literature.
Experimental detection of the new
phenomenon of turbulent thermal diffusion
Tov Elperin, Ben-Gurion University of the Negev
A new phenomenon of turbulent thermal diffusion, which was predicted theoretically by Elperin et al. (1996, 1997), has been detected experimentally in oscillating grids turbulence with an imposed mean temperature gradient in air flow. This effect implies an additional mean flux of particles in the direction opposite to the mean temperature gradient and results in formation of large-scale inhomogeneities in the spatial distribution of particles. We used Particle Image Velocimetry to determine the turbulent velocity field and an Image Processing Technique to determine the spatial distribution of particles. Analysis of the intensity of laser light Mie scattering by particles showed that they are accumulated in the vicinity of the mean fluid temperature minimum due to turbulent thermal diffusion.
Instabilities near the
attachment line of swept wings
Joern Sesterhenn, Field of activity fluid mechanics, TUM
In this talk I first review the stability characteristics of the flow at the leading edge of a swept wing. There, one finds the attachment line instability, centrifugal instabilities and a crossflow instability. Next I report on the numerical investigation of the swept leading edge flow for a compressible fluid for several Reynolds numbers and nose radii under subsonic and supersonic conditions.
Deep-water surface-wave breaking
and upper-ocean dynamics
W. Kendall Melville, University of California, San Diego
Surface waves play an important role in coupling the atmosphere and the ocean at scales from millimeters (capillary waves) to megameters, with swell generated in the Southern Ocean propagating across the Pacific to support the surfing communities of Southern California. Wave breaking limits the height of surface waves and in so doing transfers momentum from waves to currents; makes energy available for mixing the surface layers of the ocean; enhances heat and gas transfer, and generates sound as entrained air breaks up into bubbles. As a two-phase, unsteady, free-surface transitional flow, breaking poses profound difficulties for experimentalists and theoreticians alike. In order to make progress in understanding breaking, we must use some of the classical tools of fluid mechanics; combining plausible physical and scaling arguments along with careful laboratory experiments. However, since breaking is also a stochastic process, whose statistics cannot be reliably simulated in the laboratory, we must also go into the field to measure the incidence of breaking and test statistical models. In this talk, I will present an overview of current work on all these aspects of deep-water wave breaking, including recent progress in including a simple model of breaking in a direct numerical simulation of the marine boundary layer.
The initial generation of
langmuir circulations-and waves and currents
W. Kendall Melville, University of California, San Diego
In 1938, Irving Langmuir, a Nobel laureate in chemistry, published what is perhaps the first modern observational paper on the coherent fluid dynamical structures that would subsequently take his name. He showed that the surface signatures of Langmuir circulations (LCs), lines of foam roughly aligned with the wind, were regions of convergence between counter rotating vortices. Since that time it has become clear that LCs play an important role in the marine boundary layer, transporting heat, mass (gas), chemical species, momentum and phytoplankton. However, it was not until 1976 that Craik and Leibovich (CL) presented a rational theory for LCs that depended on the interaction between the Stokes drift of the surface wave field and the mean vorticity of the wind-driven surface currents. It was posed as a stability theory with the LCs growing as unstable perturbations. The CL theory came in several flavours, and it was the CLII theory which assumed that at the scales that applied in the ocean, say O(10 – 100) m, the currents associated with the LCs were very weak, so that there was no feedback from the LCs to the waves. However, increasingly sophisticated field measurements and numerical solutions were unable to confirm the predictions of the stability theory. In retrospect, early laboratory experiments by Faller and co-workers appear to have held the key to observing the initial generation of LCs; but it is only in recent years that modern quantitative imaging techniques have permitted a more complete investigation of small-scale LCs. In this talk I will present and discuss recent measurements of the initial generation of Langmuir circulations and show that they are in agreement with the most recent theory which takes into account the feedback between LCs and the wave field. I will discuss this work in the context of the initial generation of surface waves and currents.
Intermittent mixing by breaking
of wind waves and Stokes drift effects on the turbulent upper
layer in the sea: Implications for oil transport and
dispersion
Vladimir Maderich, Institute of Mathematical Machine and
System Problems, Ukraine
Oil spilled at sea often entrained by breaking waves in stormy conditions and forms clouds of oil droplets which are dispersed by subsurface turbulence, wind and wave driven shear currents. Aims of this seminar is • to consider role of turbulence injected by breaking waves, wind stress and Stokes forcing in mixing of near-surface layer; • to estimate importance of these mechanisms in oil dispersion in the near-surface layer. We simulated joint action of wind stress, Stokes drift and wave breaking on the near-surface turbulent layer with use of time-dependent 1-D model with two-equation turbulence closure. The model equations are derived by horizontal averaging of Langmuir circulation model Mc Williams et al., 1997). An injection of turbulence by penetrating breakers in wave-breaking layer was parameterized by source terms in the turbulent kinetic energy and dissipation rate equations. The Monte-Carlo simulations of intermittent mixing support assumption that observed quasi-lognormal distribution of dissipation rate is associated with breaking of waves in many scales. The based on Kolmogorov (1941) approach model of the oil droplet breakup was proposed to reproduce observed log normal distribution of oil droplet sizes. A new 3-D Lagrangian oil spill model [1] that describes main transport and weathering processes is briefly described. The results of simulations of turbulence and droplet concentration in the wave enhanced layer for stormy conditions by linked model of sea dynamics and model of oil spill are presented.
References 1. Brovchenko I., Kuschan A. , Maderich V., Shliakhtun M. , Koshebutsky V. , Zheleznyak M. Model for oil spill simulation in the Black Sea. Proc. 3rd Int. Conf. Oil Spills, Oil Pollution and Remediation, 16-18 Sept. 2003, Bogazici Univ., Istanbul, p. 101-112.
3-D non-hydrostatic free-surface
models and their applications
Vladimir Maderich, Institute of Mathematical Machine and
System Problems, Ukraine
The 3-D hydrostatic free-surface primitive equations are used in most numerical models of estuaries, coastal seas and ocean circulation. However, non-hydrostatic effects remain important in a wide spectrum of stratified flows even when the horizontal scale of the process is much more than vertical scale. In the lecture the non-hydrostatic models are reviewed and a new three-dimensional numerical model [1] for simulation of the unsteady free-surface density stratified turbulent flows is presented. Unlike other non-hydrostatic models, 2-D depth-integrated momentum and continuity equations are integrated explicitly, whereas 3-D equations are solved implicitly with time-splitting technique for internal and external modes. The pressure and velocity fields are decomposed into hydrostatic and non-hydrostatic counterparts and components are found sequentially. The general vertical coordinate and orthogonal curvilinear horizontal coordinates are used in this model. The model includes DNS, LES and RANS versions. It can be considered as non-hydrostatic extension of most ocean free-surface models with mode splitting and terrain–following coordinates. The geophysically motivated applications are presented and the comparison with laboratory experiments is given. These applications include surface waves’ transformations over the bar, nonlinear degeneration of basin-scale internal waves in a closed basin, Kelvin-Helmholtz instability, exchange flows in the sea strait and near-bed currents generated by ships’ propulsion.
References 1. Kanarska Y., Maderich V. (2003) A non-hydrostatic numerical model for calculating of free-surface stratified flows. Ocean Dynamics, 51, N3, 176-185.
On incompressible Navier-Stokes
dynamics - a new approach for analysis and computation
Jian-Guo Liu, University of Maryland
The pressure term has always created problems for understanding the Navier-Stokes equations of incompressible flow. It enforces the incompressibility constraint like a Lagrange multiplier, and updating it dynamically is a problem due to the lack of a useful evolution equation or boundary conditions. Existing methods able to handle these difficulties are sophisticated and lack the robustness and flexibility that would be useful to address more complex problems.
We will show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, for three-dimensional flow in a general domain with no-slip boundary conditions, the computation of incompressible Navier-Stokes dynamics is basically reduced to solving a heat equation and a Poisson equation at each time step. The computation of the momentum equation is completely decoupled from the computation of the kinematical pressure Poisson equation used to enforce incompressibility. This class of methods is very flexible and can be used with all kinds of spatial discretization methods, apparently without taking special care to satisfy inf-sup conditions.
The FIK identity and its
implication for turbulent skin-friction reduction
Nobuhide Kasagi, Tokyo University
The identity equation derived by Fukagata, Iwamoto and Kasagi (Phys. Fluids, Vol. 14, pp. L73-L76, 2002) leads to a general strategy for accomplishing turbulent skin friction drag reduction. This is demonstrated by referring to several typical examples of recently studied control schemes including local blowing/suction and polymer additives. Based on the FIK identity and numerical experiment of direct numerical simulation, the performance of active feedback control of wall turbulence, which has been paid much attention over the decade but only been studied at low Reynolds numbers, is estimated at higher Reynolds numbers. A possible control scheme for enhancing heat transfer while keeping skin friction reasonably low is also discussed.
Some geometric and
function-analytic properties of the set of steady solutions to
the Navier-Stokes equations past an obstacle
Giovanni P. Galdi, University of Pittsburgh
Consider a body B (compact set of R^3) moving with constant velocity V in an incompressible Navier-Stokes fluid subject to an external force f and filling the whole space. No restrictions are imposed on f and V, other than f belonging to a suitable Lebesgue class L and V being nonzero.Let S denote the set of corresponding steady solutions to the Navier-Stokes equation with velocity field having a finite Dirichlet integral. In this talk we show that for any generic force f in L and for any nonzero V, S is constituted by a finite (odd) number of solutions. Moreover, we show that when f=0, S is homeomorphic to a compact set of R^N, where N depends only on V, B and on the kinematic viscosity of the fluid.
Some simple immersed boundary
techniques for simulating complex flows with rigid boundary
Ming-Chih Lai, National Chiao Tung University
In this talk, we shall review some immersed boundary (IB) techniques for the simulation of flow interacting with solid boundary. The IB formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. A new boundary force calculation on the Lagrangian marker is introduced to ensure the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a sparse linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Moreover, if the boundary configuration is stationary, then the above matrix is time-independent and is inverted once. Thus, the extra computational effort is small. Four different test problems are simulated using the present technique (decaying vortex, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results.
Evaluation of different
non-linear constitutive relations of Reynolds stresses using
direct numerical simulation data
Varangrat Juntasaro, Kasetsart University, Thailand
The present paper aims to improve the accuracy of the non-linear eddy-viscosity turbulence models via the constitutive relation of the Reynolds stresses using the direct numerical simulation data. The direct numerical simulation data is used directly in the constitutive relation to ensure that the error in the predicted results is actually comes from the constitutive relation of the Reynolds stresses not from the modeled transport equations of the turbulent kinetic energy and its dissipation rate. The comparative study of different constitutive relations is made. The new damping function in terms of turbulent Reynolds number and the model constants based on the chosen non-linear constitutive relation is proposed to fit the direct numerical simulation data. The new constitutive relation of the Reynolds stresses proves to have higher accuracy in predicting the Reynolds stresses and the mean velocity profile than the previous constitutive relations. It is therefore the suitable relation to be used in the non-linear eddy-viscosity turbulence models to improve the accuracy of the models in predicting the flow field.
Some pathological phenomena of finite difference
approximations for parabolic blow-up problems
Hisashi Okamoto, Kyoto University
Non-local features of stably
stratified turbulent boundary layers
Sergej S. Zilitinkevich, University of Helsinki and Uppsala
University
Pat 1 - Boundary-layer depth
Traditional and recently proposed formulations for the depth of the stable planetary boundary layer (SBL), including the bulk-Richardson-number method, diagnostic equations for the equilibrium SBL height and relaxation-type prognostic equations, are discussed from the point of view of their physical grounds and relevance to experimental and LES data. An advanced diagnostic formulation is derived accounting for the free-flow stability and baroclinicity (the mechanisms overlooked in prior models). Its extension to non-steady regimes provides a prognostic equation recommended for use in practical applications.
Part 2 - Resistance laws
The planetary boundary layer (PBL) resistance and heat-transfer laws express the surface fluxes of momentum and heat through the PBL governing parameters. Since the late sixties, the dimensionless coefficients (A, B and C) in these laws were considered as single-valued functions of an internal stability parameter (controlled by the surface buoyancy flux). Numerous studies revealed very wide spread of experimental data on the A, B and C functions. It is not surprising that the above laws, although included in all modern textbooks on boundary-layer meteorology, are not used in practical applications. In the proposed advanced theory the resistance and heat-transfer laws are revised accounting for non-local effects caused by the free-flow stability, baroclinicity and the rise of capping inversion. The coefficients A, B and C become functions of the four independent dimensionless arguments (including the surface buoyancy flux, the free-flow Brunt Vaisala frequency, the baroclinic shear, and the ratio of the actual to the equilibrium PBL depths). Moreover the coefficient C (in the heat transfer law) is redefined to accounting for the effect of capping inversion. The advanced laws are validated through large-eddy simulation (LES) of different types of PBLs: truly neutral, conventionally neutral, nocturnal and long-lived. This new development explains why prior formulations performed so poor and promotes advanced resistance and heat transfer laws as a practical tool for use in environmental modelling applications.
The influence of large convective
eddies on the surface layer turbulence
Sergej S. Zilitinkevich, University of Helsinki and Uppsala
University
Large-scale coherent structures in the shear-free convective boundary layer (CBL) consist of narrow strong plumes and wider but weaker downdraughts. Near the surface they cause local convective winds blowing towards the plume axes in internal boundary layers. The typical life-times of these structures are much larger that the eddy-overturning time scale. The critical features of the flow in the internal boundary layer (IBL) addressed here are the interactions between the eddy motions driven by convection in the bulk of the IBL and small-scale turbulence in the thin near-surface shear stress layer scaled with the local Monin-Obukhov length (based on the local friction velocity inherent to the IBL flow, often referred to as the “minimum friction velocity”). Matching the coherent structure velocity and turbulence between these two layers leads to the dependence of the minimum friction velocity on the surface buoyancy flux, the CBL depth and the surface roughness length. The proposed model accounts for the role of the surface roughness in two different flow-roughness interaction regimes: (i) where the heights of the roughness elements are less than the thickness of the surface shear stress layer, and (ii) where the drag of the high roughness elements directly affects the velocity profile in the IBL, so that no logarithmic profile develops near the surface. In the present paper, a consistent theoretical model for these two regimes is developed and comprehensively validated against data from measurements in different sites over the sea and the land and also through large-eddy simulation of convective boundary layers over a range of surfaces from very smooth to very rough. Excellent correspondence between model results, field observations and large-eddy simulations is achieved over a very wide range of the surface roughness lengths and boundary-layer heights. The obtained resistance and heat/mass transfer coefficients are recommended for practical use in weather-prediction, climate and other environmental models.
Control theory approach to
aerodynamic shape optimization
Siva Nadarajah, McGill University
Summary
Engineers continually strive to improve their designs, both to increase their operational effectiveness and their market appeal. While some qualities such as aesthetics are hard to measure, the factors contributing to operational performance and cost are generally amenable to quantitative analysis. In the design of a complex engineering system, relatively small design changes can sometimes lead to significant benefits. For example, small changes in wing section shapes can lead to large reductions in shock strength in transonic flow. Changes of this type are unlikely to be discovered by trial and error methods, and it is in this situation that optimization methods can play a particularly important role. The potential benefit of using optimization theory has only been realized recently with the advent of faster methods to obtain the gradient. The control theory approach to shape optimization has revolutionized the concept of utilizing computational fluid dynamics as a design tool. The ability to obtain gradients cheaply has allowed researchers to attempt new problems in aircraft design. This course introduces you the current state-of-the-art methods for shape optimization. You will be presented with various numerical optimization methods and the control theory approach to shape optimization with applications for both steady and unsteady flows. You will learn how to formulate and derive the adjoint equations for simple problems using both the continuous and discrete adjoint approach.
Key Topics
1. Traditional Methods to Compute Sensitivity. 2. Numerical Optimization Methods. 3. Control Theory Approach to Aerodynamic Shape Optimization. 4. Effect of Design Variables in Optimization. 5. Optimal Shape Design for Unsteady Flows
Course Outline
1. Aerodynamic Design Methods
a. Inverse Surface Methods
b. Inverse Field Methods
c. Gradient Based
d. Non Gradient Based
2. Review of Numerical Optimization Methods
e. Steepest Descent
f. Conjugate Gradient
g. Newton’s Method
h. Quasi-Newton’s Method
3. Design using the Euler Equations
i. Formulation of the Continuous Adjoint Equation
j. Formulation of the Discrete Adjoint Equation
k. Comparison between the Continuous and Discrete Adjoint Method
l. Adjoint Boundary Conditions
m. Cost Functions: Inverse Design and Drag Minimization
4. Design Variables
n. Mesh Points
o. Hicks-Henne Functions
p. B-spline control points
q. Alternative Design Variables
5. Example Problems
r. Brachistochrone Problem
s. 2D Euler equation (Internal Flow)
6. Optimal Shape Design for Unsteady Flows
t. Time Accurate Methods.
u. Frequency Domain Methods.
Analysis of multi-plane PIV
measurements in a turbulent boundary layer: large scale
structures, and implications for control
Ivan Marusic, University of Minnesota
In this lecture results will be reported from an experimental study of zero-pressure-gradient turbulent boundary layers using single-plane and simultaneous dual-plane stereoscopic PIV (Particle Image Velocimetry). Measurements were made on multiple planes in the boundary layer, including cross-stream planes inclined at 45 and 135 degrees to the streamwise direction, together with streamwise-wall-normal and streamwise-spanwise planes. The results show clear evidence of large-scale organization with long streamwise low-momentum zones consistent with the scenario of spatially coherent packets of hairpin vortices in the logarithmic region of the flow. Statistical correlation analysis across the boundary layer indicates the occurrence of a distinct two-regime behavior, in which streamwise-velocity-fluctuation correlation contours either appear to be coupled to the buffer region, or decoupled from it. The demarcation between these two regimes is found to scale well with outer variables. The results are consistent with a coherent structure that becomes increasingly uncoupled (or decorrelated) from the wall as it grows beyond the logarithmic region, providing additional support for a wall-wake description of turbulent boundary layers. The measurements also show evidence of significant spanwise organization, that manifests as a persistent spanwise stripiness in the streamwise velocity component of the PIV vector fields. The implications of the above for skin-friction drag reduction strategies will be discussed.
Long-lived coherent structure in a
transitional boundary layer
Cun Biao Lee, Peking University
The dynamics in transitional boundary layers involve complex interactions between many different flow structures. Even the simple case of flat plate boundary layer flow involves various dynamic processes that occur during transition from laminar (regular) to turbulent (irregular) flow. However, it is not easy to identify unambiguously the flow structures, so the dynamic processes are still not fully understood. This paper describes flat plate boundary layer flow experiments that reveal these processes using two-dimensional hot film velocity measurements and flow visualization. The results present direct evidence of the existence of solitons-like coherent structures which are a kinds of dominant flow structures in dynamic processes in both the early and later stages of a boundary layer transition as well as in turbulent boundary layer. In boundary layer flows, solitons-like coherent structures were considered as the building blocks to produce other coherent structures.
Dynamics of transitional boundary
layer : measurement and visualization
Cun Biao Lee, Peking University
A physical model is presented to describe the dynamic processes that occur in a transitional boundary layer flow. The solitons-like coherent structures ( called SCS), the closed vortex, secondary closed vortex, streamwise vortices and the chain of ring vortices are postuated to be the basic flow structures of the transitional boundary layer as well as the turbulent bounbdary layer. It is argued that the central features of transitional and developed turbulent boundary layer flows can be explained in terms of how the series of vortices interact each other, and with the SCS. The physical process that leads to the regeneration of the closed vortex along the border of the SCS is described, as well as the process of evolution of the vortices from low frequency to high frequency. The model is supported by important developments in a number of experiments illustrated in both transitional and developed turbulent boundary layer[1-4].
C. B. Lee 1998 Phys letts A 247, 397-402
C. B. Lee 2000 Phys Rev. E 63, 3659-70
C. B. Lee 2001 Experiments in Fluids, 30, 354-57
C. B. Lee & Shiyi Chen, 2004 submitted.
An augmented approach for stokes
equations with discontinuous viscosity and singular forces
Zhilin Li, North Carolina State University
For Stokes equations with a discontinuous viscosity across an arbitrary interface, the pressure is known to be discontinuous and the velocity is known to be non-smooth. These discontinuities are coupled together which makes it difficult to develop algorithms to obtain an accurate numerical solution. In this paper, an second order accurate augmented approach that decouples the jump conditions through two intermediate variables has been developed. The GMRES iterative method is used to solve the augmented variables which are only defined on the interface. The augmented approach also rescales the problem in such a way that fast Poisson solvers can be used at each iteration. The GMRES iterations seems to be independent of mesh sizes. Numerical examples against exact solutions show the new method has average second order accuracy in the infinity norm. An example of a moving interface problem is also presented.
Modeling for bypass transition
in boundary layers on a flat plate
Ekachai Juntasaro, Suranaree University of Technology,
Thailand
Bypass transition is one of the transitional flows that is practically encountered in the aerodynamic and turbomachinery applications. Better performance of transition and turbulence modeling is essentially needed for a significant improvement in the efficiency of engineering systems. However, transition modeling is substantially developed on a practical point of view of turbulence models which are ad hoc in a regime of fully turbulent flows. This drawback motivates many research groups to include the physics of transition into turbulence models to capture the transitional phenomena properly. This includes a variety of physical aspects, for instance, intermittency factor, turbulent spots, criteria of transition onset, Taylor microscale, time scale and pressure diffusion. Mostly, transition and turbulence modeling is based on the model and modifications are proposed to take into account those physical effects on turbulence models when the flow changes from laminar to fully turbulent state. Some research groups account for those physical effects via the transport equation which is derived from the experimental correlation and rather complicated for implementing into the CFD software. The others remain working on improving the conventional model by modifying the relevant parameters with appropriate functions to capture the transition-to-turbulence change. The present study is motivated by the latter research groups in which case the intermittency factor is included into the model to precisely capture the transition onset and to control the growth rate of transition to turbulence within the natural length found in experiments.
Turbulent transport of passive
scalar in waves
Adrian W.K. Law, Nanyang Technological University
The dispersion of passive scalar in an Eulerian wave field is of fundamental interest to many physical phenomena, including the transport of pollutants in the coastal environment. The situation is complex due to the interplay between the organized oscillatory flow field and the diffusion movement. Basic analysis can be performed by taking the incremental change of the trajectory of the passive scalar as a superposition of the wave-induced orbital displacement and the stochastic displacement due to the random Brownian fluctuation. Perturbation can then be performed afterward on their relative importance.
The talk reviews the current understandings on the potential interactions. The first part addresses the turbulent diffusion under a regular wave field. The presence of the Brownian fluctuations reduces the Stokes drift as pointed out by Jansons and Lythe (1998), while the oscillating flow field in turn modifies the diffusive variance. The vertical mode profile of Stokes drift also gives rise to a longitudinal dispersion of the passive scalar similar to the river flows (Law, 2000). The second part summarises some current issues in a random wave field, particular on the transport of surface pollutants. The contribution of the randomness of the wave motion to the diffusion was originally suggested by Herterich and Hasselmann (1982) and reinforced by recent literatures such as Weichman and Glazman (2000). However, experimental observations so far, including those by the author, had not been able to offer a satisfactory confirmation. The difficulties will be highlighted in the presentation.
References
Herterich, K. and Hasselmann, K., 1982. “The horizontal
diffusion of tracers by surface waves.” J. Phys. Oceanogr.,
12, 704-711.
Jansons, K. and Lythe., G.D., 1998. “Stochastic Stokes’
drift.” Physical Review Letters, 81, 3136-3139.
Law, A.W.K., 2000. “Taylor dispersion of contaminants due to
surface waves.” J. Hyd. Res., IAHR, 38(1), 41-48.
Weichman, P.B., and Glazman, R.E., 2000. “Passive scalar
transport by traveling wave fields.” J. Fluid Mech., 420,
147-200.
Candidates for vortex structures
in the inertial range of turbulence
Anthony Leonard, California Institute of Technology
The evolution of initially weak structures of vorticity as they evolve in an incompressible turbulent flow is investigated. Such objects are candidates for important structures in the inertial range and in the dissipation range of scales. As these structures are strained by the flow, fine-scales of vorticity are produced along the direction of maximum compression with a consequent flow of energy to the high wavenumbers. It is shown that, under certain circumstances, the self-energy spectrum of such a structure may be time-averaged, producing a fractional power law. The exponent of the power law depends on the ratio of the first two Lyapunov exponents of the strain tensor.
Curvature and torsion of
material lines in chaotic flows
Anthony Leonard, California Institute of Technology
The deformation of a material line as it evolves in a chaotic flow is considered. Of particular interest are the curvature and torsion as a function of arclength along the curve. These quantities are sufficient to define the intrinsic geometry of the line. Regions of high curvature, once they start to develop, are essentially permanent, and in fact have a universal structure. Regions of high torsion, on the other hand, are transitory and correspond to a near singularity in the coordinate system of the Frenet frame rather than an exotic shape of the curve.
Compatible differential
constraints to an infinite chain of transport equations for
cumulants
V.N. Grebenev, Russian Academy of Science
In many cases, semiempirical closed models of turbulence, which include differential equations for the cumulants (moments) of hydrodynamical quantities, are among the basic methods of describing turbulence. The closure procedures for the cumulants (moments) often implicitly assume that the closed system of differential equations allow the existence of invariant sets (manifolds). As it is pointed out by Chorin, closure relations are, as a rule, derived using empirical hypotheses and implying certain assumptions, which are often poorly justified.
To the best of our knowledge no regular mathematical methods exist for resolution of the closure problem. The method based on studying the systems of differential equations by means of the transformation groups keeping the system invariant makes it possible to suggest an alternative approach to the study of the above mentioned problem (Ibragimov, Khabirov, Oberlack, Torrissi and et al.). The main problems of this analysis are as follows: search for the transformation group of the symmetries of the system, construction of an optimal system of subalgebras of the Lie algebra corresponding to the transformation group of the system; and classification of the submodels. A closed submodel is obtained by representing a solution through the invariants of the subalgebra. As a rule, only for subalgebras of small dimensions we can calculate all necessary invariants, and then to realize a reduction. For subalgebras of higher dimensions, the point invariants are insufficient for construction of the submodels and in this case other invariants are necessary to use. Moreover, for such subalgebras no classification results of invariant submodels are available at present.
Physically valid algebraic closure relationships which are used in certain turbulence models, as a rule, correspond to invariants of infinite-dimensional Lie subalgebras. In particular, it was established (Grebenev and Iluyshin) that in the problem of the development of a shearless mixing layer the classical algebraic-invariant Hanjalic–Launder model for an un-stratified flow and Zeman–Lumley algebraic model for a stratified flow coincide with the equation of invariant manifold generated by the original model. It means that the model under consideration admits an infinite-dimensional Lie subalgebra. Therefore the method based on the group analysis of differential equations cannot be used for a full classification of invariant submodels.
We would like to propose an approach based on using the so-called direct method in symmetry analysis of differential equations. We suggest to study the relevant overdetermined system which is constructed by adding to the system of differential equations under consideration an additional equation (algebraic closure relationship). Here the problem of reduction of the overdetermined system obtained to an involution should be solved. There are general algorithms (Cartan and et. al.) which can be apply to solve this problem and the classification of the differential equations with such constraints may be obtained.
The aim of the present article is to apply the above mentioned approach for formulating integrability properties of an infinite chain of transport equations for the cumulants which appear in modeling the dynamics of a momentumless turbulent planar wake. We expose the conditions of compatibility of the original infinite system of partial differential equations for cumulants with the gradient-type algebraic relation for the so-called triple correlations (the differential constraint) The compatibility conditions obtained make it possible to realize a reduction of the original infinite chain of transport equations for the cumulants and to present an algorithm for calculating cumulants of arbitrary order. This algorithm is based on a recursion relation. An illustrative example of applying the compatibility conditions obtained for examining a third-order closure model of turbulence is given.
Experiments and theory on
near-wall turbulence control
Kenny Breuer, Brown University
Part I: Adaptive feedforward control of turbulence
We present recent results on the use of adaptive feedforward control techniques for the control of fully turbulent shear flows. Arrays of Sensors and Actuators are used in conjunction with a fully adaptive control system to drive the flow towards a state of lower intensity turbulence. We will present theoretical results on the performance of the adaptive control system as well as experimental results, including issues for the practical implementation of control systems in a real turbulent flow.
Part II: Turbulence control using Lorentz forces
We present recent experimental results on the nature of a turbulent shear flow subjected to oscillatory forcing using Lorentz Force actuators - actuators that induce motion in the near wall of the turbulent boundary layer through the interaction between a stationary magnetic field and a time-dependant electric field. The performance of the actuators and the effect of the control on the turbulent flow (including drag reduction and the inhibition of coherent structures) are presented. Other issues concerning the fabrication and performance of the actuators are presented along with rather pessimistic predictions for the ability of such actuators to efficiently control high Reynolds number flows.
A curious phenomenon in a model
problem, suggestive of the hydrodynamic inertial range and
smallest scale of motion
John Heywood, University of British Columbia
The Kolmogorov theory of turbulence, based on a scaling argument, predicts a smallest significant length scale, beyond which a flow is very smooth. Stated spectrally, there is a largest significant wave number, beyond which the energy decays very rapidly with respect to further increases in the wave number. However, this scaling argument has never made contact with the rigorous mathematical theory of the Navier-Stokes equations. If it were understood in the context of rigorous theory it would have many important consequences, a particular corollary being the global in time regularity of solutions of the Navier-Stokes equations.
Here, we consider a certain infinite system of ordinary differential equations, regarding it as a highly simplified model of how energy might be passed up the spectrum in the Navier-Stokes equations, into the smaller scales of motion. Numerical experiments with this system of equations reveal a very striking "inertial range" and "smallest scale" phenomenon. One observes the apparent determination of a "largest significant mode number" by an abrupt change over just a very few mode numbers in the character of the energy decay with respect to mode number. This translates in the spectral analogy to a "smallest significant length scale". We formulate corresponding mathematical definitions and prove much of what is observed in these experiments, especially concerning the decay of steady solutions with respect to mode number. While our results for nonstationary solutions are not as complete as for steady solutions, their proofs seem more relevant to Navier-Stokes theory. We conclude by describing and conjecturing about the results of some further experiments with related equations, in which the coefficients are varied or the viscosity is set equal to zero. Experimentally, for the systems of equations that we have dealt with so far, the "inertial range / smallest scale phenomenon" has been found to persist, quite dramatically. We intend to demonstrate this during the talk with some computations using a laptop. Our objective is to begin a rigorous investigation of smallest scale phenomena in simple model problems, hoping for insights that might generalize to the rigorous mathematical theory of the Navier-Stokes equations.
Impact of wall function modelling
on meshing, simulation, optimization and control with fluid
flows
Bijan Mohammadi, University of Montpellier and Institut
Universitaire de France
We would like to show how to develop generic wall function approaches for turbulent flows. The algorithm for wall function development can be seen as domain decomposition by space and time dimensions reduction. However, compatibility issues must be observed as in traditional domain decomposition. Unfortunately, this has often been lost in traditional wall function development where the low-complexity model used near the wall is sometime incompatible with the model in the flow and is obtained from a previous study with a different low-Reynolds continuous model. This usually leads to poor numerical results and brings the impression that wall function development is a dead-end. We show several examples of development starting from different global models. In particular, we are interested by flows over rough surfaces with large temperature gradients for high speed flows. These flows are impossible to compute without a reduced complexity modelling of near-wall regions because the multi-scale nature of the phenomenon involved.
In addition to the impact of wall function modelling on the simulation and the fact that resulting discrete systems happen to be easier to solve due to a more favourable conditioning and also the fact that much less mesh generation ability is required we would like to show some extra positive effects of this approach on shape optimization and flow control issues. In particular, we show how to use wall functions as a low-complexity model in sensitivity evaluations with incomplete sensitivities.
The different ingredients will be illustrated on various academic and industrial configurations.
Active flow control of a NACA 0015
airfoil using a planar micro ZNMF jet-in-crossflow near the
leading edge
Julio Soria, Monash University
This lecture presents the results of an investigation of the effect of using a wall-normal micro zero-net-mass-flow jet-in-crossflow (ZNMF-JICF) located at the leading edge of a NACA 0015 airfoil as an active flow control device. The study involved parametric investigations conducted using a two-dimensional airfoil in the LTRAC horizontal 500 mm water tunnel at a Reynolds number of 3.08 x 104. In this component of the investigation, the lift was measured as a function of angle of attack for a range of micro ZNMF-JICF frequencies and amplitudes. The largest lift increases were found when a non-dimensional frequency of 1.3 and an oscillatory momentum blowing coefficient of 0.14% were employed. Under these forcing conditions the stall angle of the airfoil was delayed from an angle of attack of 10o to an angle of attack of 18o, resulting in a maximum lift coefficient increase of up to 50% above the uncontrolled lift coefficient. For these conditions, a more detailed study of the underlying flow structure was undertaken using Planar laser induced fluoroscence (PLIF) flow visualisations at a reduced Reynolds number of 1.54 x 104. This revealed that the lift increments were the result of the generation of a train of large-scale, span-wise vortices over the upper surface of the airfoil. The study was further extented by employing multigrid digital cross-correlation particle image velocimetry (MCCDPIV) to quantify the flow structure observed in the PLIF flow visualization. The results of the PLIF flow visualisation and MCCDPIV experiments will be presented and discussed. Diffulties associated with undertaking MCCDPIV in these type of flows will also be discussed.
Simulation for vorticity
generation and evolution in oblique shock wave interaction
with multifluid interface
Weijun Tang, Institute of Applied Physics and Computational
Mathematics, China
The results of direct numerical simulations of planar shock-accelerated multifluid interfaces in two dimensions are presented and compared with shock tube experiments of Haas [1] and Sturtevant [3]. Different parameters in equations of state can be used instead of which used with the same parameters in [2,4]. This difference for equations of state is emphasized for multifluid instead of stratified density fluid [2,4]. A mixed type algorithm for volume-fraction model is added to the Euler equations. High resolution Godunov scheme piecewise parabolic method with multifluid is used, which describes the interface and vorticity evolution more sharper. Heavy-to-light ("slow/fast" or s/f) and light-to-heavy ("fast/slow" or f/s) gas interfaces are examined and early-time impulsive vorticity deposition and the evolution of coherent vortex structures are checked and quantified. The effect of baroclinic vorticity generation from secondary shock-interface interaction, which can describe the evolution of vortex layer on interface, is quantified.
[1] J.F. Haas, Private communication, 1988.
[2] J.F. Hawley, N.J. Zabusky, Vortex paradigm for shock
accelerated den-sity stratified interfaces. Phys. Rev. Lett.
63, 1989, 1241-1244.
[3] B. Sturtevant, Rayleigh-Taylor instability in
compreessible fluids, in Shock Tubes and Waves, edited by H.
Gronig, VCH, Berlin, 1987, 89.
[4] X. Yang, LL. Chern, N.J. Zabusky, R. Samtaney, J.F.
Hawley, Vor- ticity generation and evolution in
shock-accelerated density-stratified interfaces. Phys. Fluids.
A 4(7), 1992, 1531-1540.
Optimal shape design for
fluids: theoretical and practical aspects
Olivier Pironneau, University of Paris VI (Pierre et Marie
Curie)
Optmal Shape design for the Navier-Stokes equations is required either for optimization problems such as drag reduction or for identification problems such as inverse design. This talk will
- Give some examples of applications from industry
- Recall the theoretical results for existence and differentiability
- Present the latest algorithms, including topological optimization
- Show some realisations
Spectral element/spectral vanishing
viscosity method for large eddy simulation of turbulent flows
Chuanju Xu, Xiamen University
A spectral vanishing viscosity (SVV) method for the spectral element computation of incompressible turbulent flow is investigated. This method is based on adding a SVV stabilization term in the original formulation of the Navier-Stokes equations. The SVV stabilization term, introduced initially in [JCP, 2004, 196(2), p680], can be interpreted as a SVV-filter [ICOSAHOM04] and provides an alternative approach to the SGS term. A number of numerical example, such as the turbulent wake behind a cylinder at Reynolds number Re=3900, are presented to show the accuracy and efficiency of the proposed method.
Mean momentum balance:
implications for wall-turbulence control
J. C. Klewicki, University of Utah
The common depiction of turbulent wall-flow structure is largely derived from the observed properties of the mean velocity profile and the relative magnitudes of the mean viscous stress and Reynolds shear stress. Together these can be used to motivate the familiar picture of wall-flows as being composed of a viscous sublayer, buffer layer, logarithmic (overlap) layer and wake layer. Perspectives that view the boundary layer as a dynamical “machine” are derived from this structural framework, and thus provide a basis for implementing strategies intending to modify mean dynamics. For example, independently of Reynolds number this picture naturally focuses attention to the thin viscous wall-layer (buffer layer and below). With increasing Reynolds number, a particular challenge associated with this focus is the diminishingly small fraction of the flow occupied by the wall layer.
Recent data analyses and complementary theoretical developments[1] are employed to ascertain flow structure as directly specified by the mean dynamical equation. Available high quality data reveal a four-layer description that is a considerable departure from the four layer structure (discussed above) traditionally and nearly universally ascribed to turbulent wall flows. Each of the four new layers is well-characterized relative to the contributions required to balance the governing equation, and thus the mean dynamics of these four layers are unambiguously defined. The inner normalized physical extent of three of the layers exhibit significant Reynolds number dependence. The scaling properties of these layer thicknesses are determined. Particular significance is attached to the viscous/Reynolds stress gradient balance layer. For example, contrary to the common view, the existence of this layer reveals that viscous effects remain as dynamically significant as turbulent inertia out a wall-normal position beyond the peak in the Reynolds stress. Multiscale analyses substantiate the four layer structure in developed turbulent channel flow. In particular, the analysis verifies the existence of a third layer, with its own characteristic scaling, between the traditional inner and outer layers.
The implications of these results are discussed relative to the problem of wall-flow control. Attention is paid to the utility of the present framework for both implementing and assessing the effectiveness of candidate flow control strategies. In this context, a new model of the boundary layer, whose dynamical attributes are in accord with the properties of the mean momentum balance, is presented and discussed.[2] A brief discussion is provided that identifies connections between the present findings to previous efforts that have utilized the Lamb vector in assessing the effectiveness of strategies intending to modify wall-flow dynamics.[3]
[1] Wei, T., Fife, P., Klewicki, J. and McMurtry, P. 2005 “Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows,” J. Fluid Mech., 522, 303.
[2] Klewicki, J., McMurtry, P., Fife, P. and Wei, T. 2004 “A physical model of the turbulent boundary layer consonant with the structure of the mean momentum balance,” 15th Australasian Fluid Mechanics Conference., University of Sydney, Sydney Australia, 13-17 December.
[3] Crawford, C., Marmanis, H and Karniadakis, G. 1998 “The Lamb vector and its divergence in turbulent drag reduction,” in, Proc. International Symposium on Seawater Drag Reduction, Newport, RI, July 22-24.
Turbulent boundary layers and
their control: quantitative flow visualization results
Michele Onorato, Politecnico di Torino
The strong development of PIV in the last few years has opened a new field for the experimental study of turbulent flows. The main importance of PIV with respect to classical point measurement techniques in the study of wall turbulence is the ability of producing instantaneous maps of the velocity field, giving information about organized motions and their relation to the skin friction grow and turbulence production. It should be fair to say that the bulk of our recent understanding of the structure of turbulent wall flows has been achieved by studies based on the analysis of data obtained by direct numerical simulations. While the capabilities of current computers restrict such simulations to relatively low Reynolds numbers and relatively simple geometry, the results have been extremely illuminating because they provide full instantaneous fields of velocity, vorticity, rate-of-strain and pressure with spatial and temporal resolutions. One of the goal of using PIV is to extend such knowledge to higher Reynolds numbers of greater interest for industrial and environmental applications and to more complex flows as, e.g., turbulent boundary layers manipulated by external forces in order to control turbulence, skin friction, wall heat exchange and aeroacoustic emission. In this presentation results obtained by applying PIV to canonical turbulent boundary layers and to wall flows under the action of external forcing will be shown and analysed. In particular results obtained at the Modesto Panetti Aerodynamic Laboratory of the “Politecnico di Torino” will be discussed, with reference, where it is possible, to similar results produced in other laboratories. In addition to PIV results for the basic canonical zero-pressure-gradient turbulent boundary layer, data will be here shown and discussed for two manipulated boundary layer flows. In the first the control mechanism consists in a forced transversal oscillation of the wall, in the second the near wall flow is manipulated by large scale longitudinal vortices. The reasons for the choice of these external forcing mechanisms are that both are examples of large scale manipulations and both are significant of forcing by transversal motions superimposed to the mean flow. All details about measurement procedures will be omitted during the presentation.
Least-squares based finite element
models for fluid flows
Junuthula Narasimha Reddy, Texas A & M University
A new computational methodology based on least-squares variational principles and the finite element method is discussed for the numerical solution of the non-stationary Navier-Stokes equations governing viscous incompressible and compressible fluid flows. The use of least-squares principles leads to variationally unconstrained minimization problems, where compatibility conditions between approximation spaces – such as inf-sup conditions – never arise. Furthermore, the resulting linear algebraic problem will always have a symmetric positive definite (SPD) coefficient matrix, allowing the use of robust and fast preconditioned conjugate gradient methods for its solution. In the context of viscous incompressible flows, least-squares based formulations offer substantial improvements over the (traditional) weak form Galerkin finite element models – where the finite element spaces for velocities and pressure must satisfy an inf-sup compatibility condition and one must deal with an un-symmetric and indefinite coefficient matrix. In contrast, least-squares formulations circumvent the inf-sup condition, thus allowing equal-order interpolation of velocities and pressure, and result (after suitable linearization) in linear algebraic systems with a SPD coefficient matrix. We exploit the SPD nature of the coefficient matrix by implementing a parallelized, matrix-free, preconditioned conjugate gradient algorithm for efficient and fast solution of the associated system of equations. In this lecture, the basic theory of least-squares finite element formulations of the Navier-Stokes equations governing viscous incompressible flows will be presented, and their application through some benchmark problems will be discussed.
The research is supported by Computational Mathematics program of the Air Force Office of Scientific Research (AFOSR). Dr. Juan Pontaza is the principal collaborator of the research reported herein.
Computational methods for
distributed parameter estimation with application to
inversion of 3D electromagnetic data
Uri Ascher, University of British Columbia
Inverse problems involving recovery of distributed parameter functions arise in many applications. Many practical instances require data inversion where the forward problem can be written as a system of elliptic partial differential equations. Realistic instances of such problems in 3D can be very computationally intensive and require care in the selection and development of appropriate algorithms. The problem becomes even harder if the model to be recovered is only piecewise continuous.
In this talk I will describe work we have been doing in the context of inverting electromagnetic data in frequency and time domains for geophysical mining applications with the objective of making such computations practically feasible. Our techniques are applicable in a wider context, though.
A second part of the talk will describe our efforts to accommodate discontinuities using modified TV and Huber function regularization. These two variants are shown to be very similar when parameters are chosen well (we show how). The application to diffusive foward models such as Maxwell's equations in low frequencies is perilous, though.
This is joint work with E. Haber.
Mathematical analysis of certain
analytic sub-grid scale models of turbulence
Edriss Titi, Weizmann Institute of Science, Israel and
University of California, Irvine
In this talk I will discuss the mathematical difficulty in proving global regularity for the three-dimensional Navier-Stokes equations. Furthermore, I will show the global regularity for certain analytic three-dimensional sub-grid scale models of turbulence. This will include the Smagorinsky model, Navier--Stokes-alpha model, the Leray-alpha model and the Clark model. All these models are of nonlinear parabolic type and each has a finite dimensional global attractor. In some cases I will provide explicit bounds for the fractal and Hausdorff dimensions of these attractors, in terms of the relevant physical parameters. In addition, I will also prove the global regularity for the "shell model" of turbulence and show that it has a finite dimensional inertial manifold. Hence, this model can be reduced to a finite dimensional ordinary differential system. As a result, one can show that this model has a unique invariant measure associated with its dynamics when it is kicked randomly.
Using Navier-Stokes equations for
digital image denoising and restoration
Xue-Cheng Tai, University of Bergen
Abstract: In this talk, we shall try to summarize the work we have done recently about using partial differential equations for image analysis and processing, we shall concentrate on noise removal and restoration. We shall start by the well-known total variation denoising techniques. From numerical experiments and some analysis, we show that better properties can be kept if we use some modified higher oder partial differential equations. In the end, we show that Navier-Stokes equations is a naturally choice for noise removal and image inpainting. However, the equations we derive here is not coming from fluid mechanics, but from some geometrical considerations for digital images.
Duality techniques in numerical
flow simulation: error estimation, flow control and
stability analysis
Rolf Rannacher, Universität Heidelberg
We present a general approach to a posteriori error control and mesh adaptation for solving flow problems by the finite element method.
Traditionally, a posteriori error estimation in Galerkin finite element methods aims at estimating the error with respect to some natural energy norm in terms of the 'residual' of the computed solution. The problem is then to localize the residual norm in order to make it computable as a sum of cell-wise 'error indicators'. This approach seems rather generic as it is based on the variational formulation of the problem and allows us to exploit inherent coercivity properties. However, in most applications the error in the energy norm does not provide useful bounds on the errors in the quantities of real physical interest. Meshes generated on this basis may not be economical for the purpose of the computation. A more versatile method for 'goal-oriented' a posteriori error estimation is obtained by employing concepts from optimal control theory. Using duality techniques, we derive a posteriori error representations in terms of primal and dual cell-residuals and associated sensitivity factors. The resulting local error indicators assist the construction of economical meshes for evaluating and optimizing the quantities of physical interest. This method is described in the context of flow simulation, i.e., the computation of flow quantities like drag and lift, their optimization by boundary control and the corresponding stability analysis. Also the potential of this approach for the computation of turbulent flow is discussed.
Literature:
[1] R. Becker and R. Rannacher. An optimal control approach
to error estimation and mesh adaptation in finite element
methods, Acta Numerica 2000 (A. Iserles, ed.), pp. 1-102,
Cambridge University Press, 2001.
[2] W. Bangerth and R. Rannacher. Adaptive Finite Element
Methods for Differential Equations, Birkhaeuser:
Basel-Boston-Berlin, 2003.
[3] R. Becker. An optimal-control approach to a posteriori
error estimation for finite element discretizations of the
Navier-Stokes equations. East-West J. Numer. Math. 9,
257-274 (2000).
[4] R. Becker. Mesh adaptation for stationary flow control.
J. Math. Fluid Mech. 3, 317-341 (2001).
[5] R. Becker, V. Heuveline, and R. Rannacher. An optimal
control approach to adaptivity in computational fluid
mechanics, Int. J. Numer. Meth. Fluids. 40, 105-120 (2002).
[6] V. Heuveline and R. Rannacher. Eigenvalue computation in
hydrodynamic stability by adaptive finite elements,
Preprint, SFB 359, University of Heidelberg, December 2004.
Motion of a liqeuid drop on a
solid surface
Huaxiong Huang, York University
In this talk we will describe two methods for computing the motion of a liquid drop on a solid surface. For thin drops with small contact angles, we formulate the problem using lubrication theory. The time-dependent 4th order PDE is solved using the method of lines. For fat drops with large contact angles, we apply a front tracking method based on Perkin's immersed boundary formulation. Numerical results and some premilnary analytical results for the moving contact line will be presented.
Finite element simulation of
flow-structure interaction for a moving tuna
Tony Wen-Hann Sheu, National Taiwan University
A convection-diffusion-reaction finite element model is proposed to simulate the transient Navier-Stokes equations in moving quadratic elements. For stability reason, the streamline upwind Petrov Galerkin (SUPG) finite element model with accurate dispersion nature is developed by preserving the analytic dispersion relation. The resulting dispersion-relation-preserving and SCL-preserving l-SUPG finite element model is applied to solve the fluid flow equations with the linear elastic equations to analyze the flow-structure interaction problems.
Artificial boundary conditions for
the Navier-Stokes equations in unbounded domains
Weizhu Bao, National University of Singapore
Many incompressible viscous fluid flow problems are given in unbounded domains, such as fluid flow around obstacles and fluid flow in a channel. One of the numerical difficulties in these problems is the unboundedness of the physical domain. In this talk, we will show how to design high order nonlocal/local artificial boundary conditions at a given artificial boundary for the Navier-Stokes equations in unbounded domain. Then the original problem is reduced to a problem defined in a bounded computational domain. Well-posedness of the reduced problem is proven. Finite element approximation for the reduced problem and error estimate are presented. Numerical results are reported to demonstrate the efficiency of our high order artificial boundary conditions.
A tentative mathematical
definition of LES and related mathematical issues
Jean Luc Guermond, Texas A&M University
In this talk I will discuss the mathematical foundations of Large Eddy Simulation (LES). I'll try to convince the audience that the mathematical program of LES is not clear enough to yield significant mathematical advances. In particular, it seems to me that a clear unequivocal definition of LES is yet missing. In an attempt to spur discussion on this matter, I'll propose a mathematical definition of Large Eddy Simulation (LES) for three-dimensional turbulent incompressible viscous flows.
I propose to call LES approximation of the Navier--Stokes equations any sequence of finite-dimensional approximations of the velocity and pressure fields that converge (possibly up to subsequences) in the Leray class to a suitable weak solution. I'll show that this definition is constructive by giving examples based on hyperviscosity, the Leray regularization, the so-called Leray-alpha model, the nonlinear Galerkin method, and low-order finite element approximations.
A sequential regularization
formulation for incompressible Navier-Stokes equations
Ping Lin, National University of Singapore
Many mathematical models arising in Science and Engineering lead to computationally challenging problems involving differential equations with constraints. The incompressible fluid flow governed by the Navier-Stokes equations is one example of such problems. The direct discretization of such models is typically fraught with difficulties. We thus need to reformulate the original problem to obtain a better behaved problem before discretization. In this talk, I will use an ODE example to introduce the concept of such equations and various reformulation methods and its correspondence to existing formulations in Navier-Stokes equations. I will then discuss a particular family of such methods, the sequential regularization method (SRM). The penalty method is a special example. Advantages of the SRM over usual methods will be highlighted. Convergence analysis is based on asymptotic expansion. Finite difference (fully explicit without solving any linear or nonlinear system) and Finite element methods (standard finite elements) can be used and numerically analyzed for the reformulated problem. Implementation issues are discussed. A number of flow examples (cavity flow, jet flow and flow passing a cylinder, etc) are computed to demonstrate the method.
