**2. Turbulence**

Turbulence has been a long standing challenge for human mind. Five centuries after the first studies of Leonardo da Vinci, understanding turbulence continues to attract a great deal of attention. This may be due to its fascinating complexity and ubiquitous presence in a variety of flows in nature and engineering. The first turbulence references by Leonardo da Vinci are based on visual observations. In 1883, Osborne Reynolds introduced the concept of

Turbulent Boundary Layer Models: Theory and Applications 207

equations, similar to other simpler deterministic equations, can often behave chaotically under certain conditions. Due to the randomness in turbulent flows, it is hopeless to track instantaneous behaviour. Instead, the goal is to measure this behaviour in the temporal or

Most researchers in the turbulence field accept that instantaneous flow variables satisfy the Navier-Stokes equations as an axiom and use it as the basis for the development of models for numerical simulation. Assuming that details of motion at the level small and intermediate scales, which tend to exhibit high randomness levels and peculiar characteristics such as isotropy, are not required in most applications of interest in engineering and geophysics, the establishment of two approaches, which have the potential for being applied to problems of engineering interest, can be defined. The first approach is based on the use of filters for the flow variables of interest, Large Eddy Simulation (LES). The second one relies on the use of statistical averages on the same variables, Reynolds-averaged Navier-Stokes equations. Although the former is formally superior to the latter, its use implies paying a computational price which is too high for applications of practical interest. LES requires less computational effort than direct numerical simulation (DNS), but more effort than those methods that solve the Reynoldsaveraged Navier-Stokes equations (RANS). These equations, derived by Osborne Reynolds in 1985, describe the dynamics of the "mean flow" in terms of a time average, and later defined as average in the probability space "ensemble average". The Reynolds stresses produced by advection terms, which are second order correlations in statistical terms, are determined by exact transport equations for the Reynolds stresses derived from the Navier-Stokes equations. However, third-order correlations appear in such expressions and four-order correlations will appear in the exact transport equations for the third-order correlations. This is called the problem of closure of the statistical treatment. The approach of neglecting correlations of higher order has proved to be unsuccessful because the turbulent flows are not completely random. Experimental investigations have made it possible to identify, through the use of conditional sampling techniques, "coherent structures" such as shear layers imbedded in turbulent flows, and that the degree of coherence is scale dependent. In the solution of complicated sets of nonlinear partial differential equations, the interaction between physics and numerical approach is very strong, and the use of second approach in question makes it possible to have a better understanding of that interaction and, as a consequence, to control it. Four main approaches have been followed to find ways to close the Reynolds equations by introducing hypotheses based on physical insight and observational evidence: 1 transport; 2- mean velocity field; 3- turbulent field, and 4- invariant models. The resulting model equations contain a number of empirical constants which, in general, increase with their complexity. These models have the base on important concepts and hypotheses as the eddy viscosity concept by Boussinesq, in 1877, Prandtl's mixing length concept, in 1925, Kolmogorov's isotropic dissipation assumption, in 1941, and Rotta's energy

redistribution hypothesis, in 1951 (Monin & Yaglom, 1971; Rodi, 1984).

an incompressible and viscous fluid, with sediment in suspension, are written:

The fundamental equations of the Fluid Mechanics applied to a three-dimensional flow of

**3. Governing equations** 

spatial mean.

averages, which became the base of great theoretical-experimental studies. In 20th century, Taylor by the thirties presented the first statistical theory for isotropic turbulence, Kolmogorov by the year 1941 formulated theoretical developments for local turbulence, Batchelor by the year 1953 distinguish himself for theoretical and experimental studies about free turbulence of waves and jets. Then, much more other studies were presented, mainly about wall turbulence, boundary layer and air models. Several resumes can be found in Monin and Yaglom (1971), Tennekes and Lumley (1972), Launder and Spalding (1972), Hinze (1975), Schiestel (1993), Nezu and Nakagawa (1993), Rodi (1980, 1993), Mohammadi and Pironneau (1994), Lumley (1996), Chen *et al.* (1996) and Lesieur (1997), among others.

The detailed accurate computation of large scale turbulent flows has become increasingly important and considerable effort has been devoted to the development of models for the simulation of complex turbulent flows in several applications over the last decades. The description of turbulence flows is based on the assumption that instantaneous flow variables satisfy the Navier-Stokes equations, which contain a full description of turbulence, given that they describe the motion of every Newtonian incompressible fluid based on conservation principles without further assumptions. Analysing the applicability of continuum concepts to the description of turbulence, Moulden *et al*. (1978) conclude that if the Newtonian constitutive relation is valid, then it is plausible to accept that turbulent flows instantaneously satisfy the same dynamical equations as laminar flows. For laminar flows, analytical or numerical solutions can be directly compared to experimental results in some cases. Moser (2006) declared that despite the increasing range of turbulence spatial scales as the Reynolds number increases, in turbulence, the continuum assumption and the Navier-Stokes equations are an increasingly good approximation.

The aforementioned assumption seems to be well supported as DNS "Direct Numerical Simulation", in which all scales of the motion are simulated using solely the Navier-Stokes equations. It is the most natural approach to the numerical simulation of turbulent flows but, since by Kolmogorov's theory, small scales exist down to O. (Re-3/4), in order to capture them on a mesh, a meshsize 3 4 *h* Re and consequently (in 3D) 9 4 *N* Re mesh points are necessary. Thus, it only could be applied for simple and low-Reynolds number turbulent flows (Kaneda & Ishihara, 2006; McComb, 2011). Even if DNS were feasible for hydraulic practical interest, it is not possible to define, with the precision required by the smallest scales of the motion, proper initial and boundary conditions. This fact is of significant importance due to non-linear character of the advection terms, which results in the production and maintenance of instabilities which in turn excite small scales in the motion. The presence of non-linear terms also precludes the existence, in the most general case, of unique solutions for a given set of initial and boundary conditions. Thus, as a large Reynolds number turbulent flow is inherently unstable, even small boundary perturbations may excite the already existing small scales, with possible corresponding perturbation amplifications. The lack of solution uniqueness and the infeasibility of defining precise initial and boundary conditions combine themselves in a way that the resultant flow appears random in character. Indeed, the uncontrollable nature of the boundary conditions (in terms of wall roughness size and distribution, wall vibration, etc.) forces the analyst to characterize them as "random forcings" which, consequently, produce random responses (Aldama, 1990). The Navier-Stokes equations can then exhibit great sensitivity to initial and boundary conditions leading to unpredictable chaotic behaviour. Although the fundamental laws behind the Navier-Stokes equations are purely deterministic, these 206 Advanced Fluid Dynamics

averages, which became the base of great theoretical-experimental studies. In 20th century, Taylor by the thirties presented the first statistical theory for isotropic turbulence, Kolmogorov by the year 1941 formulated theoretical developments for local turbulence, Batchelor by the year 1953 distinguish himself for theoretical and experimental studies about free turbulence of waves and jets. Then, much more other studies were presented, mainly about wall turbulence, boundary layer and air models. Several resumes can be found in Monin and Yaglom (1971), Tennekes and Lumley (1972), Launder and Spalding (1972), Hinze (1975), Schiestel (1993), Nezu and Nakagawa (1993), Rodi (1980, 1993), Mohammadi and Pironneau (1994), Lumley (1996), Chen *et al.* (1996) and Lesieur (1997),

The detailed accurate computation of large scale turbulent flows has become increasingly important and considerable effort has been devoted to the development of models for the simulation of complex turbulent flows in several applications over the last decades. The description of turbulence flows is based on the assumption that instantaneous flow variables satisfy the Navier-Stokes equations, which contain a full description of turbulence, given that they describe the motion of every Newtonian incompressible fluid based on conservation principles without further assumptions. Analysing the applicability of continuum concepts to the description of turbulence, Moulden *et al*. (1978) conclude that if the Newtonian constitutive relation is valid, then it is plausible to accept that turbulent flows instantaneously satisfy the same dynamical equations as laminar flows. For laminar flows, analytical or numerical solutions can be directly compared to experimental results in some cases. Moser (2006) declared that despite the increasing range of turbulence spatial scales as the Reynolds number increases, in turbulence, the continuum assumption and the

The aforementioned assumption seems to be well supported as DNS "Direct Numerical Simulation", in which all scales of the motion are simulated using solely the Navier-Stokes equations. It is the most natural approach to the numerical simulation of turbulent flows but, since by Kolmogorov's theory, small scales exist down to O. (Re-3/4), in order to capture them on a mesh, a meshsize 3 4 *h* Re and consequently (in 3D) 9 4 *N* Re mesh points are necessary. Thus, it only could be applied for simple and low-Reynolds number turbulent flows (Kaneda & Ishihara, 2006; McComb, 2011). Even if DNS were feasible for hydraulic practical interest, it is not possible to define, with the precision required by the smallest scales of the motion, proper initial and boundary conditions. This fact is of significant importance due to non-linear character of the advection terms, which results in the production and maintenance of instabilities which in turn excite small scales in the motion. The presence of non-linear terms also precludes the existence, in the most general case, of unique solutions for a given set of initial and boundary conditions. Thus, as a large Reynolds number turbulent flow is inherently unstable, even small boundary perturbations may excite the already existing small scales, with possible corresponding perturbation amplifications. The lack of solution uniqueness and the infeasibility of defining precise initial and boundary conditions combine themselves in a way that the resultant flow appears random in character. Indeed, the uncontrollable nature of the boundary conditions (in terms of wall roughness size and distribution, wall vibration, etc.) forces the analyst to characterize them as "random forcings" which, consequently, produce random responses (Aldama, 1990). The Navier-Stokes equations can then exhibit great sensitivity to initial and boundary conditions leading to unpredictable chaotic behaviour. Although the fundamental laws behind the Navier-Stokes equations are purely deterministic, these

Navier-Stokes equations are an increasingly good approximation.

among others.

equations, similar to other simpler deterministic equations, can often behave chaotically under certain conditions. Due to the randomness in turbulent flows, it is hopeless to track instantaneous behaviour. Instead, the goal is to measure this behaviour in the temporal or spatial mean.

Most researchers in the turbulence field accept that instantaneous flow variables satisfy the Navier-Stokes equations as an axiom and use it as the basis for the development of models for numerical simulation. Assuming that details of motion at the level small and intermediate scales, which tend to exhibit high randomness levels and peculiar characteristics such as isotropy, are not required in most applications of interest in engineering and geophysics, the establishment of two approaches, which have the potential for being applied to problems of engineering interest, can be defined. The first approach is based on the use of filters for the flow variables of interest, Large Eddy Simulation (LES). The second one relies on the use of statistical averages on the same variables, Reynolds-averaged Navier-Stokes equations. Although the former is formally superior to the latter, its use implies paying a computational price which is too high for applications of practical interest. LES requires less computational effort than direct numerical simulation (DNS), but more effort than those methods that solve the Reynoldsaveraged Navier-Stokes equations (RANS). These equations, derived by Osborne Reynolds in 1985, describe the dynamics of the "mean flow" in terms of a time average, and later defined as average in the probability space "ensemble average". The Reynolds stresses produced by advection terms, which are second order correlations in statistical terms, are determined by exact transport equations for the Reynolds stresses derived from the Navier-Stokes equations. However, third-order correlations appear in such expressions and four-order correlations will appear in the exact transport equations for the third-order correlations. This is called the problem of closure of the statistical treatment. The approach of neglecting correlations of higher order has proved to be unsuccessful because the turbulent flows are not completely random. Experimental investigations have made it possible to identify, through the use of conditional sampling techniques, "coherent structures" such as shear layers imbedded in turbulent flows, and that the degree of coherence is scale dependent. In the solution of complicated sets of nonlinear partial differential equations, the interaction between physics and numerical approach is very strong, and the use of second approach in question makes it possible to have a better understanding of that interaction and, as a consequence, to control it. Four main approaches have been followed to find ways to close the Reynolds equations by introducing hypotheses based on physical insight and observational evidence: 1 transport; 2- mean velocity field; 3- turbulent field, and 4- invariant models. The resulting model equations contain a number of empirical constants which, in general, increase with their complexity. These models have the base on important concepts and hypotheses as the eddy viscosity concept by Boussinesq, in 1877, Prandtl's mixing length concept, in 1925, Kolmogorov's isotropic dissipation assumption, in 1941, and Rotta's energy redistribution hypothesis, in 1951 (Monin & Yaglom, 1971; Rodi, 1984).
