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Characteristics of self-sustaining mechanisms
To summarize the previous sections, we state that where Reynolds stress −uv is positive, kinetic energy is produced. Since the key structure involved in such production are the streamwise vortices, which appear and disappear, we may understand that there is a cycle governing the generation of such vortices. Theoccurrence of this cycle guarantees the self-sustainment of turbulence. The cycle can be envisioned as follows: many different structures constitute the skeleton of this cycle and they are linked together by some mechanisms. The main effort to provide is to establish the cause-and-effect relationship between these structures and the mechanisms behind their generations. Many studies are concerned with identifying the structures, based on experimental or DNS data. A few others focus on the mechanisms leading from one structure to another (actually there are very few mechanisms, the well known is the lift-up, by which, the streamwise vortices generate the streaks). A self-sustaining cycle has several characteristics. The first might be the time scale of the repetition of events. The time interval between the coherent structures should be the same, or at least comparable between the cycles. Another important characteristic is the spatial features of the coherent structures. It is not clear that the generated streaks have the same amplitude (or strength) and the same spanwise width as the previous streaks. Same question holds for the streamwise vortices, especially if they are or not strong enough to restart the cycle.
Another characteristic concerns the nature of the process. Some of them include an instability, whereas in others, a vortex can generate another vortex. Hence, in discussing self-sustaining mechanisms, we will use two main classes (for a complete review, see Panton (1997) (76) and Panton (2001)(77)). The “parent offspring” class is defined by having a flow structure that develops in time to replicate itself without involving an instability. The second class has, during part of the cycle, a velocity profile that is unstable to small disturbances.
Self-replicating mechanisms
Let us first summarize generally accepted ideas ((77)). First, there are the streaks. It is almost universally agreed that the streamwise vortices near the wall sweep low speed fluid into the low-speed streaks (lift-up mechanism). On the other side of this vortex, high-speed streaks are generated. This produces a characteristic streak velocity profile (u) with spanwise variation. Together with the vertical motion v of the streamwise vortices, Reynolds stress are produced. Many people view the streamwise vortices as the legs of hairpin vortices (see hairpin vortices are produced given the initial situation as a fully developed turbulent state. Here we focus on how a hairpin vortex is generated, or springs from another hairpin vortex.
Brooke & Hanratty (1993)(15) performed computation of a channel flow and examined vorticity patterns in (y, z) planes. Viewing (y, z) planes enables one to see streamwise vortices, even if the vortex axis is not exactly in the (y, z) plane. They found that a new vortex is born at the downstream end of the parent on the down-wash side. This end is lifted from the wall and they refer to it as the detachment point. They remarked that the vorticity in the child vortex is of the opposite sign to that of the parent and that the regeneration is not influenced by outer flow events. Thus, they envisioned a regeneration process that is entirely within the inner region.
An interesting contribution is given by Heist et. al. (2000)(43). They identified a new process that forms about 30 percent of the streamwise vortices. This was accomplished by examining the changes, with time, of the turbulent field obtained from a direct numerical simulation of turbulent flow in a channel. Streamwise vortices create a shear layer by pumping low momentum fluid from the wall. One or more small spanwise vortices are formed at the top of this layer. They grow in size and rotate in the direction of flow. The main focus of Heist’s paper is not on how they form but on what happens after they form. Figure 1.6 is a plan view of their DNS results. Vortex A is followed as a parent that produces vortex B. Vortex B grows in the spanwise direction and then elongates in the streamwise direction and intensifies. Previous investigators have suggested that spanwise vortices could have a direct role in the formation of streamwise vortices. A number of investigators have argued that spanwise vortices play a key role in sustaining turbulence. Different proposals have been made to explain how this occurs. A common assumption is that spanwise vortices evolve into hairpin vortices, but the details of this process are not given (see e.g. (77)).
A common picture (among many others) about the hairpin vortex generation is presented in (76). A streamwise vortex collects fluid from near the wall and creates a low-speed streak. Next, the low-speed streak region forms an obstacle for the faster moving stream. This event takes place as long as the stream1.7 wise vortices exist and pump more fluid into the streak area. Hence, this lifted area produces a streamwise shear, due to the impingement of the faster moving stream on it. This lifted region produces a stronger U(y) shear layer which rolls up, much like a Kelvin-Helmholtz cat’s eye, into a vortex arch or head. The vortex lines in the head extend down. As the head is convected downstream, the legs are stretched and the swirling thereby intensified. Thus, the hairpin formation process consists of streak lift-up, shear layer intensification, and hairpin re-formation.
Instability-based mechanisms
The class of self-sustaining processes involving instability mechanisms is now discussed. During part of the cycle, a velocity profile is unstable to infinitesimal perturbations and a linear theory is developed. The velocity profile, which must exist for sufficient time for the instability to develop, can be the base flow. A perturbation is added to it and the question is, does this disturbance grow or not. More specifically, what kinds of perturbations grow and how fast do they grow.
In order to form a complete cycle, the instability should ultimately lead back to produce the initial velocity profile. A disturbance and a velocity profile are the only common elements of a sta16 bility analysis. The next choice concerns the equations that are used to follow the development of the disturbance. Typical laminar flow stability analysis uses linearized equations under the assumption that they govern the initial development of infinitesimal disturbances. Furthermore, linear equations have many mathematical properties. They allow an analysis by normal modes, i.e., the eigenfunction and eigenvalues of the system. The fastest exponentially growing normal mode determines the characteristics of the most-unstable infinitesimal disturbance.
In some cases as pCf, this approach fails. However, the linearized Navier– Stokes equations are non-normal and their eigenfunction are not orthogonal. This allows certain disturbances that are composed of several normal modes to grow to large amplitudes. Factors of 1000 or more have been found. At large times the disturbance approaches the behavior of the last decaying eigenfunction. This transient growth is algebraic and for that reason this phenomenon is also called
algebraic growth (as we see later on, Schoppa & Hussain used this transient growth to show that normal modes growth is not sufficient to trigger nonlinearity). Trefethen & Trefethen (1993) (100) make an analogy with vectors. Consider two almost oppositely directed basis vectors that have large amplitudes. These are normal mode components. Their resultant, the perturbation, has only a
small magnitude. Then, one basis vector magnitude decreases with time while the other is unchanged. This causes the resultant perturbation to increase in amplitude. Ultimately, the behavior of the resultant perturbation follows the dominant eigenvector and decreases (stable case).
Transient growth was ignored for many years. Recently, it was realized that it could lead to profiles that, because of their large amplitude, are subject to secondary instability and or nonlinear effects. Transient growth has a history of development and a nice introductory review is in (100).Hence, in the instability mechanism class there are many questions. They concerns the nature of the perturbations (finite or infinitesimal), the equations to be used and the approach (normal mode or transient growth). Note finally that what happens when nonlinear effects become important is another (difficult) issue.
Reynolds stress −U0V1
The evolution equation for the disturbance kinetic energy is the Reynolds-Orr equation: dE dt = P − D, where E(t) = 1 2 R V (u02 + v02 + w02) dV and the two terms on the right-hand side represent the exchange of energy with the base flow U = Uby i.e. the production P = − R V u0v0@yU dv and energy dissipation D due to viscous effects. The velocity components (u0, v0,w0) are the perturbations to the basic flow U, given by (Eq. 2.9-Eq. 2.10). After integrating over y, we have P = − R Dx,z U0V1 ds, where is a positive constant. One reason this equation is interesting is that the occurrence of the two events (U0 negative, V1 positive) or (U0 positive, V1 negative) produce positive Reynolds stress −U0V1 and hence are very significant events in the production of turbulence (in the literature, they are called Q2 and Q4 events, as seen in §1.4.2) . The linear deformation of the base flow by a vertical velocity is the well known lift-up mechanism (or “effect”) (see e.g.(88)). The generated streamwise velocity by this mechanism is called a streak. In our model, this mechanism is represented by the linear term −a2UbV1 in the equation (2.11) and U0 represents the streak. It is clear that when the lift-up feeds energy into the system (i.e., V1 generates U0), we have positive Reynolds stress, −U0V1 > 0. The origin of this lift-up are the streamwise vortices, which justifies their dominant role in turbulence production. The central issue addressed in the chapter IV concerns the generation of the streamwise vortices.
Table of contents :
1 Introduction
1.1 Transient lifetimes
1.1.1 Introduction
1.1.2 Numerical and experimental studies
1.2 Turbulent spots
1.2.1 Introduction
1.2.2 Characteristics of turbulent spots
1.3 Spreading mechanisms
1.4 Self-sustainment of wall-turbulence
1.4.1 Definition of coherent structures
1.4.2 Kinetic energy production
1.5 Characteristics of self-sustaining mechanisms
1.6 Self-replicating mechanisms
1.7 Instability-based mechanisms
1.7.1 Inflectional profiles
1.7.2 Minimal flow unit
1.7.3 Streaky velocity profiles
1.7.4 Conclusions on instabilities
1.8 Conclusions of the chapter
2 Modeling plane Couette flow
2.1 Introduction
2.2 No-slip model
2.2.1 Reynolds stress −U0V1
2.3 Free-slip model
2.3.1 Derivation
2.3.2 Formal comparison of the models
2.3.3 Main results for the free-slip models
2.4 Conclusions
3 Numerical results
3.1 Introduction
3.2 Numerical implementation
3.2.1 Numerical scheme
3.2.2 The algorithm and numerical validations
3.3 Results
3.3.1 Global sub-criticality
3.3.2 Extensivity of the sustained turbulent regime
3.3.3 Transient lifetimes
3.3.4 Mean turbulent flow
3.4 Conclusions
4 On the outskirts of a turbulent spot
4.1 Introduction
4.2 Numerical simulations of turbulent spots
4.3 Generation of large scales from small scales
4.4 Conclusion
5 Generation mechanism for streamwise vortices
5.1 Introduction
5.2 Generation of streamwise vorticity
5.2.1 Previous works
5.2.2 Comparisons
5.3 Generation of streamwise vortices
5.3.1 Equations of the vorticities
5.3.2 Generation of !x and the key-structure
5.3.3 Different roles of both tilting terms: Complementarity
5.3.4 Nonlinear advection of !x by (U0,W0)
5.3.5 From spanwise to streamwise vortices
5.3.6 Statistical tools
5.3.7 Conclusions on the generation of streamwise vortices
5.4 From spanwise to streamwise vortices: an illustrative model
5.4.1 Derivation
5.4.2 Numerical integrations
5.5 Generation mechanism for the spanwise vortices
5.5.1 Study of the spanwise vorticity equation
5.5.2 Generation of the quadrupolar flow (U0,W0)
5.5.3 Generation of the spanwise vortex
5.6 Discussion and conclusion
6 Spreading mechanism
6.1 Introduction
6.2 Growth of a turbulent spot
6.2.1 Coherent structures on the front
6.3 The spreading mechanism
6.3.1 Origin of the dipole motion (U0,W0)
6.3.2 Entrainment of the perturbations: an illustrative model
6.4 Discussion and conclusion
A The symmetries of the model
