Vorticity and Vortex Dynamics 2011 Part 12 ppt

546 10 Vortical Structures in Transitional and Turbulent Shear Flows (1) y x z (2) (3) (4) (5) (6) (a) (b) Fig. 10.23. A global view of the vortical structures from transitional to a turbulent boundary layer. (a) Sequence of transition. (b) Fully developed turbulence corresponding high-speed and low-speed streaks. The averaged spanwise wave- length of the streaks in the sub layer is typically 100 viscous lengths (up to at least Re θ ≈ 6000 based on momentum thickness θ). The streamwise struc- tures are broken from time to time under influence of vortex interaction with surrounding. They will also be reformed and strengthened around the high- shear region through the instability mechanism as stated before. Unlike the early stage of transition, all the aforementioned structures are now coexisting with the background of small random eddies produced by vortex breakdown or burst. From the wall region to the outer edge of the boundary layer, there ap- pear hairpin structures inclined to the wall at an angle of approximately 45 ◦ (Fig. 10.23b). At lower Reynolds number, the vortices are less elongated and more like horseshoe shaped. Low-Reynolds number simulations indicate that they most often occur asymmetrically or even singly (sometimes named hooks), with only occasional instances of counter-rotating pairs. At moder- ate or relative higher Reynolds numbers, the vortices are elongated and more hairpin-shaped. There is a controversy on whether the hairpin vortices could remain up to the fully developed turbulent region downstream and how large an area they can occupy in the outer region. Head and Bandyopahyay (1981) reported that a turbulent boundary layer is filled with hairpin vortices in their smoke tunnel experiment; but it is not so from many other results. Thus, it might be helpful to discuss the Reynolds-number effect on the hairpin structures. The viscous dissipation of a vortex pair of separation λ z depends on the viscous cancelation of the vorticity with opposite sign, which happens due to the vorticity diffusion from both legs at a rate proportional to νω/λ z . 10.3 Vortical Structures in Wall-Bounded Shear Layers 547 The lifetime of the hairpin votices, t life , should be proportional to ω, λ z and inversely proportional to the diffusion rate, and thus t life canbescaledto λ 2 z /ν. On the other hand, the lift-up velocity of the hairpin vortices depends on the induced velocity caused by the mutual induction of the vortex pair and is proportional to Γ/λ z , where the circulation Γ ∼ ωλ 2 z . Furthermore, ω depends on the wall shear, ω ∼ ∂U/∂y| w ∼ u τ /δ and so the time t p required for the hairpin vortices to lift up and penetrate the whole boundary layer of thickness δ canbescaledtoδ 2 /u τ λ z . This gives t p /t life ∼ δ 2 ν/u τ λ 3 z . Then, since λ z is known to be scaled to the viscous length ν/u τ , the ratio t p /t life is actually the square of the Reynolds number, (δu τ /ν) 2 . This argument can at least qualitatively explain why one observes larger number of horseshoe-shaped structures and hairpins in transitional or relatively low-Reynolds number flows than that of hairpin-shaped structures at high Reynolds numbers. For the latter the time required for the hairpin vortices to lift up and penetrate to the outer edge of the boundary layer will be much longer than their life time and most of them would be dissipated before penetrating through the whole layer. The outer edge of the turbulent boundary layer consists of three dimen- sional bulges, the turbulent/nonturbulent interface with the same scale of the boundary-layer thickness δ. Deep irrotational valleys occur at the edges of the bulges, through which free-stream fluid is entrained into the turbulent re- gion (Robinson 1991b). Inside the bulges are slow over-turning motions with a length scale of δ. They have relatively long life times compared with the quasistreamwise vortices that form, evolve, and dissipate rapidly in the near- wall region. These large-scale structures at the outer edge are also related to the induced velocities of groups of hairpin heads. The inner–outer region interaction is one of the major controversial issues in turbulent boundary layer theories. It is now almost a common understand- ing that the outer-region structures have a definite effect on the near-wall production process (Praturi and Brodkey 1978; Nakagawa and Nezu 1981) but not play a governing role (Falco 1983). The large over-turning motions are weak, though they have influence on bursting and thus on small-scale transition. Although the outer layer also contains energetic structures, recent numerical experiments (Jim´enez and Pinelli 1999) have confirmed that the essential inner-layer dynamics (y + < 60) can operate autonomously. One of the interesting issues relevant to the inner–outer region interaction is whether the large over-turning motion has important influence on the for- mation of streamwise vortices. It was suggested (Brown and Thomas 1977; Cantwell et al. 1978) that the successive passing of the large over-turning motions would cause waviness of near-wall streamlines. The G¨ortler instabil- ity on a concave boundary layer might have influence on the formation or growing of the streamwise vortices. This suggestion is similar to the “G¨ortler- Witting mechanism,” which conjectured that large amplitude T–S waves will locally induce concave curvature in the streamlines and hence a G¨ortler instability (Lesson and Koh 1985). But, by a computation on a wavy wall, 548 10 Vortical Structures in Transitional and Turbulent Shear Flows Saric and Benmalek (1991) showed that the wall section with convex curva- ture had an extraordinary stabilizing effect on the G¨ortler vortex so that the net result of the whole wavy wall (or the large amplitude T–S waves) was sta- bilizing. However, the flow waviness caused by the large over-turning motion is not sinusoidal (or the convex and concave portions of the curvature are not symmetrical), so the net effect of the overturning motion is still to be clarified in the future. 10.3.5 Streamwise Vortices and By-Pass Transition Streamwise vortices are seen in all high Reynolds number shear flows, includ- ing free shear layers (mixing layer, wake, and jet, etc.) and wall-bounded shear layers (boundary layer, wall jet, wall wake, etc.). In the former, the inflectional instability leads to spanwise vortices first. A streamwise vortex is a product of secondary instability of the existing spanwise structures. In the latter, the streamwise vortex starts immediately after the nonlinear process starts in the wall region, so one never sees an observable spanwise vortex. However, the background mechanism of streamwise vortices formation is in common, both due to sufficiently strong shear field and three-dimensional disturbances. The processes described so far are not the only mechanism to form stream- wise vortices. Corotating streamwise vortices can be formed in the boundary layer on a sweepback wing due to the crossflow instability. Counter-rotating streamwise vortices can also be formed due to centrifugal instability, such as the Dean vortices in curved channels (Dean 1928), the G¨ortler vortices near a concave surface (G¨ortler 1940; Drazin and Reid 1981), the Taylor vortices between concentric cylinders with the inner one rotating, or the streamwise vortices in the outer region of the wall jet on a convex wall, etc. Thus, stream- wise vortices are a popular flow phenomenon in turbulent shear layers. It has been shown in Sect. 10.3.2 that the streamwise vortices play a domi- nant role in the self-sustaining mechanism of boundary-layer turbulence. Actu- ally, the momentum transported by the streamwise vortices not only generates the streaks but also account for the increase of skin friction in the turbulent boundary layer (Orlandi and Jim´enez 1994). The dominant roles of stream- wise vortices near the wall in turbulence production and drag generation is now widely accepted (e.g., Kim et al. 1987). In engineering applications, the influences of streamwise vortices in mass transfer (e.g., mixing), momentum transfer (e.g., Reynolds shear stress and skin friction), and energy transfer (e.g., heat transfer) are also significant. Besides, as will be discussed below, streamwise vortices is a key mechanism in the by-pass transition to turbulence. All of these explain why we have to pay enough attention to the specific nature related to streamwise vortices. Figure 10.2 has shown that traveling vortices may be detected as and expressed by waves. This is however not the case for a steady streamwise vor- tex. Correspondingly, the mechanism of disturbance growth related to stream- wise vortices cannot be expressed by the growth of normal modes either. The current understanding of the streak development is the nonmodal growth 10.3 Vortical Structures in Wall-Bounded Shear Layers 549 (transient growth) introduced in Sect. 9.1.2 and discussed in Sect. 9.2.4 in the context of shear-layer instability, which has been shown to have potential im- portance for studies of by-pass transition (e.g., Gustavsson 1991; Butler and Farrel 1992). A pair of counter rotating streamwise vortices in a boundary layer will cause wall-normal velocity disturbance that accumulates (or grows) alge- braically along the streamwise direction x (Fig. 10.24). Even if the stream- wise vortices decay along x, the normal velocity disturbance could still grow as an integrated effect. The closely related phenomenon is the occurrence of low-speed streaks and the surrounding high shear layers. Actually we have already come across similar phenomenon in the discussion of self-sustaining mechanism in boundary layers (Fig. 10.18). The later breakdown of low speed streaks occurs through a secondary instability, which is developed on the local shear layer between high- and low-speed streaks when a critical Reynolds number based on their size is sufficiently large (Sect. 9.1.2 and Sect. 9.2.4). If this mechanism overwhelms the normal-mode transition, there occurs by-pass transition. Let us discuss in a little more detail. The T–S waves in a boundary layer on a smooth plate will start when the Reynolds number reaches certain crit- ical value. The disturbances with frequencies within the unstable region will grow exponentially in the linear regime. If, by any mechanism, there occurs a pair of relatively weak streamwise vortices, then their induced velocity dis- turbances cannot compete with those induced by the T–S waves (the normal mode) because the former grows algebraically. However, if the flow is stable to normal-mode disturbances or there are sufficiently strong initial streamwise vortices for the transient growth to be overwhelming, transition to turbulent flow will take place without passing through the stage of exponential grow of T–S waves. This is called by-pass transition. The transition of the Couette flow and circular-pipe flow are good examples where the velocity profiles are linearly stable to normal modes. Subcritical transition in an ordinary boundary layer is another example where the T–S y z x v(x ) v(z ) Fig. 10.24. Transient growth and counter rotating vortices 550 10 Vortical Structures in Transitional and Turbulent Shear Flows wave is linearly stable due to the low Reynolds number. For all these cases, the transition scenario can occur only if there is a mechanism other than passing through the exponential growth of T–S waves. Besides, if the initial disturbance amplitude exceeds a threshold level, by- pass transition will take place (Darbyshire and Mullin 1995; Draad et al. 1998), such as in a boundary layer on a rough surface or a boundary layer under a surrounding of high turbulence intensity (e.g., a turbine blade). This result is independent of whether the shear flow is unstable to exponential growth of wave-like disturbances. As discussed above, a boundary layer subjected to a free-stream turbulence of moderate levels would develop unsteady streamwise oriented streaky structures with high and low streamwise velocity. This phe- nomenon was observed even as early as Klebanoff et al. (1962) who observed a by-pass of linear stage whenever the initial amplitude of the perturbation was large, and also discovered the existence of streamwise vortices in the flow field near the surface by measuring two velocity components. Subcritical tran- sitions have recently been investigated in more detail for a variety of flows, for examples, in circular pipes (e.g., Morkovin and Reshotko 1990; Morkovin 1993; Reshotko 1994), in plane Poiseuille flows and in boundary layer flows (e.g., Nishioka and Asai 1985; Kachanov 1994; Asai and Nishioka 1995, 1997; Asai et al. 1996; Bowles 2000). 10.4 Some Theoretical Aspects in Studying Coherent Structures Having seen the significant role of coherent structures in the development of the two example flows, their physical understanding, prediction, and control have become a very active area in turbulence studies. However, a turbulent flow is full of vortical structures of various scales, which can all cause the stretching or tilting of local vorticity. It is not an easy job to calculate all these influences unless a direct numerical simulation is performed, which up to now is still limited to relatively low Reynolds number flows. Thus, the traditional way in turbulence studies is the statistical method. The famous Kolmogorov (1941, 1962) theory and the recent development of the universal scaling law of cascading (She and Leveque 1994; She 1997, 1998) belong to the statistical method. They both revealed the multiscale structures in turbulence and contributed firmly to the physical background of cascading. Recently, the latter theory has made progresses in combining the knowledge of their universal scaling law with those of coherent structures in shear flows (Gong et al. 2004). However, there is still a long way to go before it can help turbulence modeling to solve the problem of turbulence development in a flow field. So, the most convenient statistical method to date is still based on the Reynolds decomposition. As has been pointed out in the context of Fig. 10.2 and Sect. 10.3.5, turbu- lent disturbances related to steady components of streamwise vortices cannot 10.4 Some Theoretical Aspects in Studying Coherent Structures 551 be expressed by the temporal fluctuations of the velocity field. This brings us to a further discussion on the limitation of the Reynolds decomposition. A combination of triple decomposition and vortex dynamics has shed light on building up statistical vortex dynamics and may be a more powerful way out in turbulence studies. But more detailed studies on the vortical structures in turbulence require DNS or deterministic theories. Many achievements have been made on the relevance of vortex dynamics to turbulence. Theoderson (1952) was the first to predict theoretically the generation of hairpin-shaped structures in a boundary layer as early as 1952. Since then, abundant experimental and computational results have been ob- tained in the past half century, which have prepared a condition for applying vortex dynamics to predict the coherent structures or explain their evolu- tion (e.g., Saffman and Baker 1979; Leonard 1985; Hunt 1987; Ashurst and Meiburg 1988; Virk and Hussain 1993; Hunt and Vassilicos 2000; Lesieur et al. 2000; Schoppa and Hussain 2002; Lesieur et al. 2003). As mentioned in Sect. 1.2, these efforts have naturally in turn enriched the content of vorticity and vortex dynamics (e.g., Melander and Hussain 1993a and 1994, Pradeep and Hussain 2000; Hussain 2002). We expect that the present section can offer readers some brief concepts related to the basic theories that are important in handling coherent structures. 10.4.1 On the Reynolds Decomposition The Reynolds decomposition has been the most popularly applied statisti- cal method and has contributed tremendously to turbulence studies. While extended to triple decomposition of the velocity field, it has shown its poten- tial also in studies of coherent structures. In the triple decomposition method, one expresses any instantaneous quan- tity ϕ as ϕ = Φ + ϕ c + ϕ r , (10.1) where Φ is its time-mean, and ϕ c and ϕ r are its coherent and random compo- nents, respectively. Neglecting the correlation between the coherent and random motions, the coherent energy equation can be written as (Hussain 1983) 12 U j ∂ ∂x j  1 2 u ci u ci  = − ∂ ∂x j  u cj p c + 1 2 u ci u ci u cj  − u ci u cj ∂U i ∂x j 345 + u ri u rj  ∂u ci ∂x j − ∂ ∂x j u ci u ri u rj − c , (10.2) where u i = U i + u ci + u ri . 552 10 Vortical Structures in Transitional and Turbulent Shear Flows The viscous diffusion term and the energy production due to normal stresses have been neglected in (10.2) due to their little contribution to the coherent energy balance. The left-hand side of the equation is the advection of coherent energy by the mean. The terms on the right-hand side are: (1) the diffusion of the coherent energy by coherent velocity and pressure fluctuations; (2) the coher- ent production by the mean shear; (3) the intermodal energy transfer that expresses the rate of energy transfer from coherent motions to random ones; (4) the diffusion of the coherent energy by random velocity fluctuations; and (5) the viscous dissipation of coherent energy that is usually negligible. Equation (10.2) shows very clearly the energy transfer between mean, coherent, and random motions and is helpful in understanding, prediction, and control of coherent structures (see Sect. 10.5.3). However, due to the prob- lem revealed by Fig. 10.2 and discussed in Sect. 10.3.5, one should be able to imagine that the existence of streamwise vortices would also cause problem on both the traditional Reynolds decomposition and the triple decomposition of the velocity field as discussed later. The most representative product from the Reynolds decomposition is the Reynolds shear stress − u  v  that is a particular correlation function in turbu- lence studies. For generality, we take the correlation function between veloc- ity components measured at two separate points to discuss the influence of streamwise vortices. In a statistically steady turbulence, it is defined as R ij (x k ; r, τ )=u  i (x k ,t)u  j (x k + r, t + τ), (10.3) where u  i (x k ,t) is the instantaneous value of the ith component of the tem- poral velocity fluctuation at position x k and time instant t; r and τ are the spatial and temporal spacing between the measuring location of u  i and u  j respectively. The over-bar expresses time averaging. For example, −R 12 (x k ;0, 0) just represents the Reynolds shear stress −u  v  at location x k . As is known, the correlation function can usually characterize coherent structures in turbulent flows. However, it has a fundamental defect if stream- wise vortices are involved. Without loss of generality, consider the simulta- neous two-point spatial correlation of spanwise velocity components w with spanwise spacing ∆z in a statistically two-dimensional flow, i.e., i = j = k =3 and τ = 0, that is the most characteristic quantity related to streamwise vortices. Thus, we have: R 33 (z; z, 0) = w  (z,t)w  (z + z,t), (10.4) where the velocity fluctuation w  is a temporal fluctuation. Now, the problem comes because an ideally steady streamwise vortex will generate only a steady induced velocity, but no temporal velocity fluctuations. Even if in real flows the so-called streamwise vortices are not entirely stream- wise and not ideally steady, at least their steady streamwise component will generate no temporal velocity fluctuations. Therefore, the above correlation 10.4 Some Theoretical Aspects in Studying Coherent Structures 553 function cannot reflect the full contribution of turbulence structures, espe- cially, the influence of the steady components of streamwise vortices. Thus, the traditional correlation function has to be reconsidered. A possible way to express the fluctuations caused by streamwise vortices in a statistically steady two-dimensional flow is to replace the temporal fluc- tuations of the velocity components by spatial ones. Namely, instead of (10.4) we set g 33 (z;∆z, 0) = [w(z, t) −w(t)][w(z +∆z,t) −w(t)], (10.5) where g 33 is an instantaneous value of the spatial correlation and denotes the spanwise spatial averaging. In order to obtain a satisfactory statistical quantity, the procedure used to obtain the instantaneous spatial correlation function should be repeated for enough times to form an ensemble average. In statistically steady flows, the ensemble-averaged quantity may be replaced by a time-averaged value and we obtain G 33 (z;∆z, 0) = (w 1 − w av )(w 2 − w av ), (10.6) where we use the following abbreviations for neatness, w 1 = w(z,t), w 2 = w(z +∆z,t)andw av = w(t). By further decomposing w 1 , w 2 , w av into time means and temporal fluc- tuations, the following expression can be obtained G 33 = w  1 w  2 + w 1 · w 2 + w av · w av − w 1 · w av − w 2 · w av + w 2 av − w  1 w  av − w  2 w  av , (10.7) where w 1 and w 2 are the instantaneous values of the spanwise velocity at location 1 and 2, respectively, w  1 and w  2 are the corresponding temporal fluc- tuations and w  av is the time fluctuation of w av . This decomposition contains many additional terms since (10.6) is nonlinear. In a statistically steady two-dimensional flow, w av = 0 at any time so that we have G 33 = w  1 w  2 + w 1 · w 2 = w 1 · w 2 , (10.8) Here w 1 and w 2 are the instantaneous values instead of the temporal velocity fluctuations. We suggest that this G 33 is referred to as the total correlation to distinguish it from the traditional one. If and only if the turbulent flow is ideally two-dimensional with no steady component of w caused by streamwise vortices, can it then recover to the traditional correlation function: G 33 = w  1 w  2 . (10.9) A comparison of the two correlation functions obtained in two extreme cases is shown in Fig. 10.25 (Xu et al. 2000). The results in figure (a) were taken in a wall jet at a sufficient downstream distance of the jet exit, where 554 10 Vortical Structures in Transitional and Turbulent Shear Flows X = 150 mm, y = 0.4 mm b = 5 mm, Uj = 21m s -1 , U ϱ =0 X = 200 mm downstream of vortex generators Conventional correlation Total correlation Conventional correlation Total correlation 0 -1.00 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 (a) (b) -1.75 -1.50 -1.25 0.00 0.25 0.50 0.75 1.00 1.25 20 40 60 80 DZ mm 100 120 140 160 0 20406080 DZ mm 100 G 33 G 33 120 140 160 Fig. 10.25. The conventional and total correlation. (a) The correlation coefficient in a two-dimensional wall jet. (b) The correlation coefficient in a two-dimensional boundary layer with a spanwise row of symmetrical vortex generators (wavelength = 70 mm). From Xu et al. (2000) the turbulence was almost statistically steady and two-dimensional. The two curves computed by conventional and total correlations are almost identical. It indicates that even if there were streamwise vortices in the flow, they migrated or appeared and disappeared in a random way so that the streamwise vortices did not cause significant steady component of w. Figure (b) shows the opposite extreme where the results were obtained in a boundary layer with a row of symmetrical vortex generators, which were so arranged that all odd-number generators were tilted to one side at a given angle relative to the x-axis and those in even numbers were in the opposite side and symmetrical to the former. 10.4 Some Theoretical Aspects in Studying Coherent Structures 555 The total correlation reaches the level of O(1) while the conventional one is very low in spite of the existence of strong streamwise vortices along with their steady component. In a real turbulent flow region, the steady component of streamwise vor- tices could vary between these two extreme experimental conditions. On the relatively more serious side, for example, Saric (1994) points out that the G¨ortler-vortex motion produces a situation in spatially developing flows where the disturbance is inseparable in three dimensions from the basic-state motion and that it seems as if all interesting phenomena associated with G¨ortler vortices share this three-dimensional inseparability. Actually, they are only inseparable from the time-mean value because the disturbances them- selves involve steady components. On the less serious side, for example, in a turbulent boundary layer, the streamwise vortices have limited lifetime, within which there would be more obvious steady component, but less or even none in long time average (Bernard et al. 1993). This is believed to be the rea- son why this problem did not attract enough attention and people have been confined to the conventional correlation in turbulence studies for so long. Since the Reynolds stresses −ρ u  v  , −ρv  w  , −ρu  w  , and turbulence energy −ρ u  u  , −ρv  v  , −ρw  w  , etc. are all correlation functions, a logical extension of the above argument is that inherent defect may exist in the traditional concept on turbulence quantities based solely on the Reynolds decomposition. Within that framework, all turbulence quantities are expressed only in terms of temporal fluctuations and are supposed to represent all the actions that the turbulence adds to the mean field. The major efforts of the traditional turbulence modeling have been trying to model these quantities. However, once the steady component of streamwise vortices appears, the tra- ditional definition of turbulence energy and turbulent shear stresses will miss an invisible fraction. This is believed to be one of the basic reasons for the difficulties in modeling the wall region where the streamwise vortices are so critical. One might argue that there is nothing wrong with the Reynolds equation. The lost fraction of turbulence contained in the steady components of stream- wise vortices should enter the mean field. This is true. But in doing so the steady components of the streamwise turbulence structures are not expressed as turbulence. Many physical and technical problems would then follow. For example, the entire concept based on the turbulence energy equation has to be reconsidered. How can one count the turbulence production, advection, diffusion, and dissipation if the steady component of the streamwise vortic- ity has to be ruled out from turbulence? Besides, if one tried to absorb the steady component of the streamwise vortices into mean flow, the traditional Reynolds-averaged Navier–Stokes (RANS) solution for the mean field of a nominally two-dimensional turbulent flow would become three-dimensional and hence lose its simplicity. As the above total correlation suggests, one of the ways out could be to ap- ply both temporal decomposition and spatial decomposition in the spanwise [...]... apply vortex dynamics to study more detailed coherent structures in turbulence, there are yet two major difficulties: the influence of internal vorticity distribution in a vortex core on the dynamics of the vortex is not well understood; and, the structure and dynamics of a large-scale coherent structure in a turbulent environment are not clear For these purposes vortex core dynamics and polarized vorticity. .. terms on the right-hand side, which are the curl of the coherent and random Lamb vectors and have very clear physical meaning The third term represents the time-averaged effect of the interaction (i.e., stretching and advection) between the coherent vorticity and coherent velocity fluctuations The fourth term is the time-mean effect of the interaction between the random vorticity and velocity fluctuations... dynamics would be helpful, of which the basic theories have been discussed in Sect 8.1.2–8.1.4 (see also Melander and Hussain 1994, and Melander and Hussain 1993a) Here, we only list some results to show their contributions in understanding turbulence Figure 10.27 is a typical result from the core dynamics showing periodical deformation of a coherent vortex core Assume that the initial shape of a vortex. .. From the coherent vorticity equation (10.14), this mechanism can be easily examined Considering Dωxc /Dt, a small normal coherent vorticity component ωyc in a region of strong mean shear ∂U/∂y will lead to a significant value of ωyc (∂U/∂y), and so to a dominant first term to produce streamwise vorticity (see also Williamson 1996) 10.4.3 Vortex Core Dynamics and Polarized Vorticity Dynamics The discussions... vortex core is distorted as (A) The vorticity lines are being uncoiled because the two ends of the vortex segment in the figure are thinner and rotate faster Vorticity surface (a) (b) (c) Vorticity line Streamline (d) (e) Fig 10.27 Schematic of the coupling between swirling and meridional flows From Melander and Hussain (1994) 560 10 Vortical Structures in Transitional and Turbulent Shear Flows than the... and the turbulence surroundings, there are always secondary structures (threads) spun azimuthally around it The vorticity in the threads is mostly azimuthal 10.5 Two Basic Processes in Turbulence 561 (a) right-handed left-handed right-handed left-handed right-handed (b) (c) Fig 10.28 Schematic illustration of coherent-random interaction (a) polarized structures, (b) primary, (c) secondary From Melander... Vortical Structures in Transitional and Turbulent Shear Flows to enhance the lifting vortex and delay its breakdown on a delta wing (Gadel-Hak and Blackwelder 1986; Teng et al 1987) But there is an essential difference between this three-dimensional flow and its two-dimensional counterpart: the three-dimensional vorticity transportation Due to the axial flow, a part of the vorticity can be advected downstream... evolution of the one handed mode (say the left-handed) is coupled with the other (say the right-handed) In developing the polarized vorticity dynamics, the basic analytical tool is the complex helical wave decomposition (HWD) introduced in Sect 2.3.4 One of the major achievements from the polarized vorticity equation is the structure of a coherent vortex column in an environment of random eddies (Fig 10.28)... statistical vorticity dynamics Instead of applying the Reynolds decomposition and triple decomposition to the velocity field only, the statistical vorticity dynamics applies the triple decompositions to both velocity and vorticity field: u(x, t) = U (x, t) + uc (x, t) + ur (x, t), ω(x, t) = Ω(x) + ω c (x, t) + ω r (x, t), (10.10) where u, ω are the instantaneous quantities, U , Ω are time mean quantities, and. .. polarized structures, (b) primary, (c) secondary From Melander and Hussain (1993b) and the threads are highly polarized (Melander and Hussain 1993a and b) It not only gives a clear view on the turbulence cascade, but also enriches the concept of a coherent vortex: in a turbulent flow a coherent vortex should not be only an isolated single vortex Rather, it is always coupled with a group of surrounding . efforts have naturally in turn enriched the content of vorticity and vortex dynamics (e.g., Melander and Hussain 1993a and 1994, Pradeep and Hussain 2000; Hussain 2002). We expect that the present. significant value of ω yc (∂U/∂y), and so to a dominant first term to produce streamwise vorticity (see also Williamson 1996). 10.4.3 Vortex Core Dynamics and Polarized Vorticity Dynamics The discussions. dynamics of the vortex is not well understood; and, the structure and dynamics of a large-scale coherent structure in a turbulent environment are not clear. For these purposes vortex core dynamics and polarized