1. Modeling of Turbulence

3. Traditional one point closures are based on an implicit hypothesis of single scale description. This hypothesis can be incorrect in turbulent flows with high departures from equilibrium. This is the case when the turbulent flow is subjected to a force varying strongly in space or in time or when energy is fed at particular wavenumbers or frequencies in the spectrum. In these types of flow, multiscale models have been developed in order to account for some spectral information in a simplified way.

4. One point models require the introduction of hypotheses, often numerous, linking unknown correlations to known quantities. The modeler is guided for this by his physical intuition of the acting turbulent mechanisms, dimensional analysis, fundamental theoretical concepts, comparing experimental results, etc. These hypotheses involve numerical constants that are determined against key experiments that are relatively simple and well known experimentally (grid turbulence, wall boundary layer turbulence, pure strain, homogenous shear, etc.). A turbulence model can then be viewed as a quantified summary of our present knowledge on turbulence that is synthesized as an equation set. The validity of these schemes is subsequently validated or invalidated after confrontation between numerical predictions and experimental results for various turbulent flows. It is thus through its consequences that a turbulence model finds its ltimate justification. The domain of application of the model will be, of course, more or less limited depending on the generality level of the hypotheses that have been retained.

5. Properties of Turbulent Flows Irregular signal in space and time

6. Rotational flow The presence of countless swirling eddies is highly rotational. The turbulent motion thus presents strong fluctuations in the curl of velocity. The non-linearities control the interactions between these eddies of differing size.

7. High diffusivity A turbulent field diffuses any transportable quantity, such as temperature or a dye, but also momentum, rapidly. In reality, turbulent diffusion is due to convective terms at the fluctuating level.

8. A three-dimensional phenomenon Fluctuating turbulent motions are always three- dimensional and unsteady. Let us mention here and now that a so-called “two-dimensional” turbulence also exists and can be found in very special situations. Its mechanisms are very different from those prevailing in three-dimensional turbulence.

9. In order for a turbulent flow to be two- dimensional, it is necessary that an external constraint force it to remain two-dimensional, otherwise instabilities will develop and bring it back to three-dimensionality. Two-dimensional turbulent flows are encountered in very particular circumstances: atmospheric turbulence, turbulent flows submitted to high rotation rates, etc.

10. Unpredictable character of trajectories The detailed properties of a turbulent flow present an extreme sensitivity to initial and boundary conditions. This behavior is apparent if we consider a tiny deviation in the initial conditions, we then observe that the two flows become rapidly very different from each other if we look at its instantaneous detailed description. This unpredictable character of the detailed fluid particle trajectories on sufficiently long time intervals corresponds to a loss of the memory of initial conditions. This is the unpredictability phenomenon. It explains, for example, the difficulties encountered in long term weather prediction.

11. Coexistence of eddies of very different scales There exists a whole cascade of eddies of smaller and smaller scales, created by non-linear processes due to inertial terms in the equations of motion.

12. Dissipation Turbulence cannot be sustained by itself, it needs an energy supply. This source of energy can have various origins, the most usual is shear or strain of the mean flow, but the origin can also come from external forces such as Archimedean forces. If turbulence is deprived of any generation process, it decays progressively. Turbulence is dissipative. The mechanism of viscous dissipation of turbulence is related to the existence of strong gradients of the instantaneous velocity field. The instantaneous strain rates indeed become very high inside the smallest eddies and the degradation of the turbulent kinetic energy into heat is thus very strong. We shall consider here only the macroscopic approach which supposes that the molecular scales are largely smaller than the scales of the smallest turbulent eddies, which justifies the method of continuous media and the Navier-Stokes equations in particular.

13. Traditional point of view on turbulence Turbulence is an eddying motion which, at high Reynolds numbers usually reached in practical flows, shows a wide spectrum spreading over a significant range of eddy scales and a corresponding spectrum in frequency. The turbulent motion, always rotational, can be perceived as a muddle of swirling eddies whose rotational curl vectors are directed randomly in space and strongly unsteady.

14. The largest eddies, which are associated with the low frequency range of the energy spectrum, are determined by the boundary conditions of the flow. Their length scale is comparable to the order of magnitude of the whole domain occupied by the flow itself. The smallest eddies, associated with high frequencies in the spectrum, are determined by viscous forces. The energy spectrum width, i.e. the scale difference between the largest and the smallest eddies, increases with the Reynolds number.

15. Momentum and heat transfer are mainly due to large-scale motions which contribute to statistical correlations

16. The large eddies interact with the mean flow (because their characteristic scales have the same order of magnitude), they extract kinetic energy from the mean flow and supply this energy to the large-scale agitations.

17. Turbulent structures can be considered as swirling vortex elements that are stretching each other. This vortex stretching is one of the most important aspects of turbulent motion. It produces the transfer of energy on smaller and smaller scales until the viscous forces become active and dissipate this energy, this is the energy cascade.

18. The amount of energy coming from the mean flow and injected into the turbulent motion is determined by the large scale motions, it is only this amount of energy which will be able to cascade to smaller scales and then to be dissipated. Thus, the dissipation rate of kinetic energy is determined by the large-scale motions, although the dissipation is a viscous process that occurs mainly at the level of small eddies. The fluid viscosity does not determine the dissipation rate itself but only the length of the scale at which it happens. The higher the Reynolds number, the smaller the dissipative eddies.

19. Owing to their interaction with the mean flow, the large eddies are strongly dependent on the boundary conditions of the problem. The mean flow often presents preferred directions that are then imposed on the large- scale turbulent motions. These big eddies may consequently be highly anisotropic. During the cascade process, this directional dependency is however weakened. When the Reynolds number is sufficiently high, then the range of big eddies and the range of small dissipative eddies become clearly separated in the spectrum and this directional dependency is almost completely lost. This is the tendency of fine scale turbulence to local isotropy.

20. The origins of turbulence The occurrence of turbulence may have various causes. In turbulent shear flows, it is generally due to an increase in the flow Reynolds number, which represents the ratio of inertial forces to viscous forces. However, the occurrence of turbulence is also influenced by external forces (Archimedean forces, Coriolis forces, etc.).

21. The various approaches to turbulence A general explanatory theory of turbulence phenomena does not exist, but there are numerous partial and incomplete theories. Among these theories, some of them are very rudimentary and very limited but remain nevertheless useful for an industrial approach; some others are more advanced and require more extensive mathematical developments. The approaches to turbulence are thus numerous and various; turbulence is a subject in continuous evolution which is incessantly enriched by new materials. Sometimes, even theories that are apparently extraneous to the studied domain can bring an unexpected perspective and cast new light in spite of their different formalism.

22. Direct approach Most of the approaches to turbulence assume that the detailed instantaneous motion of the fluid is described by the Navier-Stokes equations. The fluid is then considered a continuum in comparison with the molecular scale. According to this point of view, the equations of turbulence are completely known, and some research works have been directed towards the study of “turbulent solutions” of the Navier- Stokes equations.

23. The study of every turbulent flow could thus be done, in principle, by solving the Navier- Stokes equations directly. In view of predicting a turbulent flow numerically, the question thus arises: why not solve the Navier-Stokes equations? This point of view would consist of making a direct calculation of the turbulent motion for one or several realizations with random initial and boundary conditions and then carrying out a statistical treatment of these solutions. It can be shown however that the number of discretization points necessary to represent the smallest scales of turbulence reaches colossal values.

25. The order of magnitude of these scales has been divided by four in order to be able to describe (at least very roughly) a sine curve on a full period.

27. This direct approach requires powerful computational means, it can be performed at the present time only for turbulent flows in relatively simple geometries and for low Reynolds numbers. Direct solution of the Navier-Stokes equations in turbulent regimes thus remains dedicated to fundamental research studies on turbulence

28. LES The method consists of calculating the large-scale eddying motions of the turbulent flow, solving the filtered Navier-Stokes equations on a coarse grid and modeling the turbulent motions whose scale is smaller than the size of the mesh step (subgrid scale models). It is a large eddy simulation (the large eddies are precisely the range of eddies that would be the most difficult to model). This technique is still costly as regards the amount of numerical calculations (three-dimensional and unsteady) without being always prohibitive. The approach could be particularly useful for flows in which the big eddies play an important role.

29. Statistical method (statistical equations, open system, closure problem) The statistical method, using mean value operators, leads to the closure problem of the system equations. This system of equations is closed by introducing modeling hypotheses. Generally, two approaches can be distinguished in the statistical method. They are the one point closures based on a single point moment hierarchy and the two point closures or spectral models based on a two point moment hierarchy or their Fourier transforms which are spectral tensors. The first method has been largely used for the calculation of non- homogenous turbulent shear flows encountered in real situations, the second method is usually limited to homogenous isotropic or anisotropic turbulence.

30. A turbulent flow solution is always an unsteady complicated solution of the equations of motion, presenting irregular fluctuations in space and time. Faced with the disorderly aspect of turbulent evolutions and this apparent complexity of the phenomenon, the natural and most useful standpoint was to introduce statistical methods. This seemingly randomness of the turbulent evolutions originates in the irregular features in the initial and boundary conditions which are not precisely determined in detail and in which a very small deviation causes a complete change in the detailed structure of the whole flow. The statistical method is not justified by the absence of causality but rather by the ignorance of overabundant causes that are hardly accessible.

31. The decomposition of an instantaneous characteristic quantity of the turbulent flow into a macroscopic part corresponding to the mean average value and a fluctuating part corresponding to an apparently random part, allows us to develop the statistical treatment of the equations of motion. This treatment, applied to the Navier-Stokes equations which describe the detailed instantaneous motion of the fluid, introduces new unknown terms coming from the convective derivative and which are interpreted as turbulent stresses.

32. Several averaging operators can be used: time averaging, space averaging, stochastic averaging (probability theory) defined from a probability density function, statistical averaging or ensemble averaging (mean value on an ensemble of realizations). Ensemble averaging is defined as the statistical limit of the arithmetic mean taken over several experimental realizations made in the same general flow conditions.

33. Reynolds rules

34. Periodic unsteady flows In the case of periodic unsteady turbulent flows, we also define phase averaging, which is an ensemble average considered for a fixed value of the phase of the periodic motion. This definition leads to a three-term decomposition:

39. The Navier-Stokes equations can also be used to derive new general equations governing the various statistical averages of higher order (which are statistical moments in one or several points and at one or several instants of time). The evolution equations of second order moments (Reynolds stresses) involve triple velocity products (or triple correlations) which come from the non-linear terms. The process can be repeated indefinitely in principle, in order to get a statistical hierarchy of moment equations.

40. However, each new equation introduces an even greater number of unknowns; we have an open hierarchy. In this case also, for achieving the closure of the system, additional hypotheses involving higher order moments, will be necessary.

41. It is also possible to develop spectral methods using the same point of view. It is thus necessary to consider homogenous anisotropic turbulence (HAT). The open character of the system is revealed on the spectral tensors which are Fourier transforms of two point correlations. Spectral closure methods more refined than one point moment closures, must be amenable to dealing with a turbulent field in which the energy spectrum varies strongly, but the closure hypotheses are all the more delicate to introduce because they require a more detailed knowledge of the acting mechanisms.

42. The probability density function contains in particular the complete statistical information in one or several points on the turbulent quantity under consideration. The statistical treatment developed on the basis of probability theory introduces a probability density function (or PDF) P defined by:

43. The statistical moments of order k:

44. Joint PDF

45. To summarize, in the usual statistical approaches of turbulence, it is possible to distinguish very schematically two main groups: One of these two groups makes use of two or more point statistics but useful results are limited to homogenous turbulence and in most cases isotropic, the other group relates to one point statistics but can be applied to inhomogenous turbulent shear flows. One point statistical models, in spite of their shortcomings, are particularly useful to the engineer because they allow the numerical prediction of non- homogenous complex flows in various geometries at a moderate calculation cost. The choice of a particular method of approach depends mainly on the type of problem under consideration and the answers that are requested (global mean information or detailed description).

48. Homogenous and isotropic turbulence (HIT) Turbulent flows encountered in industrial practice are generally nonhomogenous and non-isotropic. The local isotropy theory leads us to assume that the fine-grained structures in a non-isotropic turbulence field tends towards isotropy, so this fact shows the usefulness of studying homogenous isotropic turbulence (HIT). Because of its simplified features, homogenous and isotropic turbulence is relatively more easily tractable by theoretical means and experimental comparisons can be made (decay of grid turbulence). Most experimental studies are performed in a wind tunnel in a mean uniform current with a velocity U which is large compared to the turbulent fluctuations.

49. We say that turbulence is homogenous and isotropic if its statistical properties are invariant under translation, rotation and reflection of the coordinate system.

50. Kolmogorov hypotheses and the local isotropy theory Let us, for example, consider the case of decay of turbulence behind a grid. During the evolution of the turbulence field, the big eddies will generate smaller and smaller eddies by non-linear interactions, thus transferring the energy to the smallest eddies. At the same time, the fluid viscosity will act and the viscous dissipation becomes higher and higher for the small eddies. Thus, the inertial forces are effective in transferring the energy on a longer and longer range of wavenumbers as long as the viscous dissipation does not come into play for damping them at large values of wavenumbers. The equilibrium between these two opposite trends is controlled by the turbulence Reynolds number: the higher the turbulence Reynolds number, i.e. the smaller the molecular viscosity, the stronger the efficiency of inertial forces to transfer energy to higher and higher wavenumbers will be.

51. When turbulence is created, only the smaller wavenumbers are excited. The wavenumbers which are involved correspond to the size of the whole mechanical system that create turbulence. The effect of inertial forces being to transfer energy to higher wavenumbers until it reaches the sink due to viscosity, there is thus an entire spectral zone which obtains energy by transfer only and not directly from the mean flow. We can then suppose that the smaller eddies will be independent from the external conditions which have produced the initial big eddies.

52. The very big eddies contain relatively low amounts of energy and moreover their characteristic frequency, of order u.Κ, becomes very small when approaching the origin of the spectrum. This is why we are led to assume the permanent character of very big eddies.

53. Vortex stretching One of the most essential mechanisms responsible for the energy cascade is the vortex stretching phenomenon. Let us consider the equation for the swirl vector, obtained by taking the curl of the velocity vector in the Navier-Stokes equations.

54. Or equivalently:

56. Corresponding to a pure strain applied in an inviscid fluid

58. This mechanism is effective for extracting energy from the mean flow. In this case, the most efficient vortices for extracting energy from the mean flow are those who have their principal axes almost aligned with the principal axes of the mean strain field This same mechanism also acts on smaller scale vortices, transferring energy to higher wavenumbers.

59. In the cascade process, the small eddies tend to get free from the anisotropy characteristics of the bigger ones and then the microturbulence approaches isotropy. Considering the effect of vortex stretching, an initial stretching in the z-direction enhances motions in the x- and y-directions while reducing the scale. After repeating several times the isotropy is approached. This property of tendency to isotropy of the microturbulence will be used in one point closures, in particular for representing the dissipation term in the Reynolds stress transport equations.

62. The velocity differences appearing in the structure functions when the two points are sufficiently close to each other are determined by the small eddies, and the big eddies have no influence on these differences. Previous discussions allow us to assume that, when the turbulent Reynolds number is sufficiently high, there exists a domain in the turbulent fluid, which is small compared to the size of the big eddies and considered during a time interval which is short compared to the characteristic period of turbulence evolution, in which the turbulence field is locally homogenous and isotropic. The concept of local isotropy does not require homogenity and isotropy in the whole flow, but only inside a small domain with characteristic size L during a time interval T.

63. Kolmogorov first hypothesis

64. Kolmogorov second hypothesis

65. Spectral point of view

68. When the Reynolds number is sufficiently high, there may exist a region of considerable extent, located inside and at the beginning of the equilibrium range and in which the dissipation is negligible. Inside this zone, the energy transfer by inertial forces is the dominant process and the effect of viscosity vanishes; this is the inertial subrange

70. One point closures One point theories are based on the development of a hierarchy of one point moment equations and they are mainly useful for studying non-homogenous turbulent shear flows, i.e. those usually encountered in industrial practice. These models are commonly called “RANS” (“Reynolds averaged Navier-Stokes”). The term “URANS” refers to the case of unsteady flows (“Unsteady RANS”). A calculation based on URANS makes use of a statistical model and predicts mean values (statistical means based on ensemble averaging) that are dependent on time. It thus allows us to reveal organized vortices or evolving structures which are not turbulent big eddies because they are preserved by ensemble averaging. This must not be confused with the LES method which allows us to simulate the large-scale part of turbulent fluctuations. The two approaches are thus conceptually different.

72. Classification of one point closures 1- Local definitions for L and V – L: mixing length given by an algebraic function – V: velocity scale given by the mean strain rate of the flow field

75. 4) Several transport equations for L – L: multiple scales obtained from transport equations – V: transport equations

76. Methods for Obtaining Reynolds Stresses

80. Different types of models

81. Moreover, the gradient hypothesis may fail in asymmetric turbulent flows which exhibit a zone with “negative production” of turbulent energy (in these flows, the turbulent shear stress does not vanish exactly at the point where the maximum mean velocity occurs). In spite of their theoretical limitations, these simple closure schemes still remain useful for the industrial approach of the engineer. J. Boussinesq, 1877 was the first to represent the effect of turbulence through a turbulence viscosity coefficient named eddy viscosity (Boussinesq model). Initially, the eddy viscosity was assumed to be a constant. Afterwards, several authors introduced variations in the eddy viscosity according to the flow parameters.

82. b) One-equation models In the case of models using a single transport equation, it is still necessary to specify analytically a characteristic length scale L which will be dependent on the geometry of the specific flow under consideration. This group of models accounts in some ways for a memory effect, but it cannot be easily applied to complex geometries because of the difficulty to choose correctly the length scale L throughout the entire flow. Because of this, they do not reach a much better degree in universality.

83. c) Two- and three-equation models In these models, the characteristic length scale L can be determined by a transport equation, and thus these models allow an improved degree of generality and wider universality. These models have been largely used (the k- model being the most popular) in various flow configurations and they offer an interesting compromise between universality and accuracy for global predetermination of turbulent shear flows.

84. The k-ε model, has been largely used in practical applications, in this model the turbulent field is described using the two parameters that are the turbulence kinetic energy k and its dissipation rate ε. These two parameters are obtained in the flow field by solving their modeled partial differential transport equations (written here in two dimensions):

86. d) Turbulent Reynolds stress transport models These models allow us to account more precisely for the anisotropy of the Reynolds stress tensor. They are potentially more general and universal than the previous models but more difficult to develop because the closure hypotheses are more numerous and represent more detailed mechanisms and numerical constants to be determined are also more numerous. The underlying philosophy in the introduction of more and more transport equations remains the search for a more universal model based on a finer and more realistic description of turbulence interaction phenomena.

87. e) Algebraic Reynolds stress models These can be considered as a simplified expression of the previous Reynolds stress transport models, which can be more easily solved numerically. They are composed of a set of algebraic equations for the individual components of the Reynolds stress tensor coupled with a two- equation transport model for k and for example

88. f) Multiscale models In all the previous approaches there is an underlying hypothesis of spectral equilibrium. Indeed, in these models which are single-scale models, all the turbulent eddies are characterized from a single length scale L. The multiple scale models allow us to get free from any spectral equilibrium hypothesis. Consequently, the number of equations to solve is greater. These models are more recent and their development is still in progress

89. Transport of a passive scalar (temperature)

90. Considering that the turbulent fluxes of a passive scalar are highly influenced by the stresses and the characteristic scales of the dynamic field, it is impossible to get a good model for the turbulent fluxes of a scalar before having settled a good model for the Reynolds stresses themselves. As for the dynamic turbulence field, various levels of closure have been developed for scalar turbulence up to high levels.

91. The development turbulence models Turbulence transport modeling based on evolution equations of one point statistical moments require the introduction of closure hypotheses, often numerous, that link the unknown correlations to the known quantities. To do that, the modeler is guided by his physical intuition of the acting mechanisms, dimensional analysis, tensorial properties, equation compatibility, etc. The main techniques for the development of one point closures, generally combine the formalist approach which emphasizes the mathematical and functional aspects, the phenomenological approach which mainly relies on physical intuition for modeling turbulent interaction processes and sometimes also the integral approach which follows from the two point closures.

92. One of the preliminary options, prior to any turbulence model development is the specification of known quantities chosen for describing the turbulence field and also consequently the closure level which will determine the number of equations in the system. Second order closures correspond to a level of description which offers a good compromise between potentialities for representing turbulent phenomena of practical interest and the possibility of using efficient numerical treatment. Lower order closures remain limited in their field of application, while higher order closures are heavy to carry on and generally difficult to develop. In every process of closure of the moment equations, there is however the underlying idea that higher order moments are relatively less influential than the first moments, a hypothesis that justifies the truncation of the hierarchy of equations.

93. In spite of difficulties that are fundamental in nature, significant progress has been accomplished in the numerical prediction of turbulent shear flows encountered in industrial practice, aerodynamics and the environment. The closure hypotheses involve “universal” numerical constants. These constants are determined (normally once and for all) by applying the model to key experiments, that are relatively simple experiments, well known experimentally, such as the decay of grid turbulence, wall turbulence, pure homogenous strain and homogenous shear. The domain of application of the model will be, of course, more or less limited depending on the degree of generality of the hypotheses introduced.

94. The role of turbulence models As in every domain in physics, the aim of a turbulence model is the predetermination of the observable behavior of a turbulent flow, using a calculation scheme. The aim of turbulence models is thus practical, the numerical prediction of turbulent flows, and not really to explain the fundamental mechanisms of the phenomenon. Turbulence models can only give an approximate description and they are only applicable to some classes of flows.

95. A model in which the numerical constants need to be adjusted for each particular flow is in its essence almost nothing more than an interpolation method between experimental data. A good turbulence model must allow extrapolation from the empirical information it contains. It is necessary of course to examine the limits beyond which the extrapolation is no longer valid.

96. The qualities that will be appreciated in a turbulence model will be: a large domain of application, the accuracy of the predictions it allows, the simplicity in the implementation and computer time efficiency. However, a more universal model is not necessarily a better choice in a very specific flow case. The choice of the model must be made in each problem considered according to the objectives in view.

97. The choice of a turbulence model among the numerous proposals available in the scientific literature may be often a delicate matter for the final user. This choice is mainly dependent on the expected answers, i.e. on the type of information that we wish to get and also on the variety of flows under consideration, that dictate the level of complexity in modeling and its degree of universality.

98. One of the main advantages in numerical prediction of turbulent flows is the possibility of easily varying the geometrical, dynamic and thermophysical parameters of the problem which is being treated. We can thus study the influence of these parameters, avoiding repeating numerous experiments that are often long and costly.

99. Experiments… A turbulence model is generally built from a conceptual scheme stemming from physical intuition, theoretical data and mathematical properties of the equations (invariance, tensoriality, etc.). The experimental information is embodied globally in the models through physical intuition by the synthetic and comparative knowledge of the data coming from measurements of various flow situations. In fact, experiments are more directly involved at two levels: firstly, for adjusting the closure hypotheses (numerical constants) in the case of fundamental

100. experiments intended to exhibit a particular phenomenon, which is as “pure” as possible, secondly, it is then involved in testing the models that are already built. More complex turbulent flows encountered in practice may often serve as a test for turbulence models and sometimes suggest empirical corrections that allow us to get round some weaknesses they have been revealed in modeling. However, turbulent flows encountered in practice often present several complex features whose effects interfere, making the interpretation of measurements complicated. Thus, it becomes difficult to take advantage directly of the measurements made in such a particular flow if we have in view modeling improvements of some general significance. This practice, in spite of its undeniable interest, thus remains of limited scope.

101. From the viewpoint of the development of new models, it seems necessary, after figuring out and separating out the main difficulties, to devise fundamental experiments performed with the aim to study as far as possible a “pure” phenomenon or a more specific mechanism. This approach, very attractive in its principle, still entails, however, some weaknesses: experimental investigations and quantities to be measured are dependent on the prior choice that has been made on the type of mathematical description of the turbulent field in the model (the mathematical formalism and the level of closure of the turbulent field must be chosen in order to offer a fruitful framework able to account for phenomena, to reveal the true acting parameters and to introduce consistently the closure hypotheses); the second difficulty comes from the fact that the turbulent field does not behave like a “turbulent material” and the various basic acting phenomena simply do not add up. Second order closures seems to be the adequate level of closure to study the various turbulent shear flows encountered in industrial practice and the environment.

102. Turbulent shear flows generally contain swirling large eddies that are strongly dependent on the particular geometry of the flow. From this point of view, each type of turbulent flow always retains some specific characters. The study of the structure of the turbulent field and the interactions between organized eddies is essential to the understanding of acting mechanisms in free turbulent shear flows such as the mixing layer.