## Abstract

The flow of a viscous incompressible fluid in a plane channel is simulated numerically with the use of a computational code for the numerical integration of the Navier–Stokes equations, based on a mixed spectral-finite difference technique. A turbulent-flow database representing the turbulent statistically steady state of the velocity field through 10 viscous time units is assembled at friction Reynolds number *Re*_{τ}=180 and the coherent structures of turbulence are extracted from the fluctuating portion of the velocity field using the proper orthogonal decomposition (POD) technique. The temporal evolution of a number of the most energetic POD modes is represented, showing the existence of dominant flow structures elongated in the streamwise direction whose shape is altered owing to the interaction with quasi-streamwise, bean-shaped turbulent-flow modes. This process of interaction is responsible for the gradual disruption of the streamwise modes of the flow.

## 1. Introduction

The properties of turbulence in wall-bounded flows have been investigated by several authors, both experimentally and numerically, with the use of a variety of techniques and methods. A synthetic picture of the subject can be drawn as follows (a considerable amount of results have been reviewed by Robinson 1991 and Panton 2001).

The velocity field in the inner region of a boundary layer is organized into alternating *streaks* of high- and low-speed fluid (Kline *et al*. 1967), persistent, quiescent most of the time and randomly distributed in space. The most relevant part of the turbulent production process occurs in the buffer layer during outward *ejections* of low-speed fluid and *sweeps* of high-speed fluid towards the wall (Corino & Brodkey 1969). The near-wall turbulence production process appears as an intermittent cyclic sequence of turbulent events. The so-called *bursting* phenomenon can be identified in different ways: (i) lift-up, oscillation and breakup of low-speed streaks, (ii) shear-layer interface between sweeps and ejections, and (iii) ejection generating from a low-speed streak. In the outer region, three-dimensional bulges with dimension of the order of the boundary layer thickness form in the turbulent/non-turbulent interface. Irrotational valleys also form at the edges of the bulges, through which free-stream fluid is entrained towards the turbulent region. Weakly irrotational eddies are observed beneath the bulges and fluid at relatively high speed impacts the upstream sides of the large-scale motions, forming shear layers (Cantwell 1981).

Vortex dynamics is also relevant. One of the first contributions to the issue of the presence of vortices in the boundary layer is due to Theodorsen (1952) who introduced the *hairpin* vortex. Robinson (1991) confirmed the existence of non-symmetric *arch* vortices and *quasi-streamwise* vortices on the basis of the evaluation of direct numerical simulation (DNS) results. The composition of a quasi-streamwise vortex with an arch vortex may result in a hairpin vortex, complete or, most frequently one-sided, but this conclusion may strongly depend on the particular technique used for vortex detection. Studies involving the dynamics of hairpin vortices in the boundary layer have been performed experimentally by Perry & Chong (1982), Acarlar & Smith (1987*a*,*b*), Smith *et al*. (1991), Haidari & Smith (1994) and numerically by Singer & Joslin (1994). Based on this, a picture of vortex generation and reciprocal interaction in the boundary layer emerges in which processes of interaction of existing vortices with wall-layer fluid involve viscous–inviscid interaction, generation of new vorticity, redistribution of existing vorticity, vortex stretching near the wall and vortex relaxation in the outer region. Individual vortices advected in a shear flow evolve—nonlinearly and mainly inviscidly—into, in most cases, non-symmetric hairpin-shaped structures, beginning from the portion of the vortex characterized by the highest curvature. During their development, spanwise vorticity is transformed into streamwise vorticity with deformation and birth of subsidiary vortices (Acarlar & Smith 1987*b*; Haidari & Smith 1994; Singer & Joslin 1994). The most relevant vortex-interaction processes occurring in the boundary layer are: (i) spanwise vortex compression and stretching in regions of increasing shear, (ii) spanwise vortex expansion and relaxation in regions of decreasing shear, and (iii) vortex coalescence resulting in larger vortices. The process of evolution of a hairpin vortex involves the development of vortex legs in regions of increasing shear with intensification of vorticity in the legs themselves. The leg of a vortex, considered in isolation, may appear as a quasi-streamwise vortex near the wall. The head of a vortex, instead, rises through the shear flow, entering regions of decreasing shear. As a consequence, the vorticity in the vortex head diminishes (Head & Bandyopadhyay 1981; Perry & Chong 1982). The processes involving multiple vortex dynamics are more complex. An attempt at a description of such phenomena has been made by Smith *et al*. (1991) according to which the coalescence of small vortices into larger structures is described in terms of intertwining, amalgamation and reinforcement occurring when upward-migrating vortices approach one another.

In spite of the remarkable amount of scientific work accomplished, there are still no definite conclusions on the character of the phenomena occurring in the near-wall region of wall-bounded turbulent flows. This fact leads to the impossibility of building a satisfactory model for the prediction of the turbulent velocity field based on the behaviour of the turbulent-flow structures.

In the numerical field, a valuable approach to the calculation of turbulent flows is the DNS, in which the objective of computing all turbulent scales is pursued and the Navier–Stokes equations are numerically integrated with no modifications of any kind. The critical aspect of this method is the numerical accuracy of the calculations that has to be sufficiently high as to resolve the essential turbulent scales. DNS results for the case of the plane channel have been reported by Kim *et al*. (1987), Lyons *et al*. (1991), Rutledge & Sleicher (1993) and Moser *et al*. (1999). Modern techniques for the numerical integration of the Navier–Stokes equations (advanced numerical methods and high-performance computing) have the ability of remarkably increasing the amount of data gathered during a research of computational nature, bringing to the condition of managing large amounts of data. A typical turbulent-flow database includes all three components of the fluid velocity in all points of a three-dimensional domain, evaluated for an adequate number of time-steps of the turbulent statistically steady state. Mathematically founded methods for the identification of vortical structures in a turbulent-flow database have been introduced by: (i) Perry & Chong (1987), based on the complex eigenvalues of the velocity-gradient tensor, (ii) Hunt *et al*. (1988), based on the second invariant of the velocity-gradient tensor, (iii) Zhou *et al*. (1999), based on the imaginary part of the complex eigenvalue of the velocity-gradient tensor, and (iv) Jeong & Hussain (1995), based on the analysis of the Hessian of the pressure.

In turn, it is recognized that not all turbulent scales contribute to the same degree in determining the physical properties of a turbulent flow. Methods can be applied to extract from a turbulent-flow database only the relevant information for the physical understanding of a turbulent phenomenon, i.e. to separate the effects of appropriately defined modes of the flow from the background flow or, finally, to extract the coherent structures of turbulence, irrespective of the definition of the coherent structure adopted. A powerful technique for the eduction of the coherent structures of turbulent flows is the proper orthogonal decomposition (POD). The POD has been first introduced in turbulence research by Lumley (1971) and is extensively presented in Sirovich (1987) and Berkooz *et al*. (1993). The method has been used (among others): (i) in Rayleigh-Bénard turbulent convection problems by Park & Sirovich (1990), Sirovich & Park (1990), Deane & Sirovich (1991) and Sirovich & Deane (1991)), (ii) in studies of free shear flows by Sirovich *et al*. (1990*a*,*b*) and Kirby *et al*. (1990), (iii) in studies of separated flows by Manhart (1998) and Alfonsi *et al*. (2003), and (iv) in the analysis of wall-bounded turbulent flows. In the field of wall-bounded flows, an early application of the method can be found in Bakewell & Lumley (1967). Aubry *et al*. (1988) used the POD in studying and modelling the turbulent boundary-layer problem starting from the experimental eigenfunctions of pipe-flow data. Moin & Moser (1989), Sirovich *et al*. (1990*a*,*b*) and Ball *et al*. (1991) applied the method of the POD to the case of the turbulent channel flow. Webber *et al*. (1997) used the method for the analysis of a numerical database obtained using the minimal channel flow domain of Jiménez & Moin (1991).

The study of the behaviour of the coherent structures of turbulence in the boundary layer of wall-bounded flows offer the possibility of clarifying the physical mechanisms through which turbulent energy of a mechanical nature is dissipated into heat. The understanding of these mechanisms brings new perspectives on two important objectives of modern fluid technology, the *control of turbulence* and the development of new *predictive models* for the numerical calculation of high-Reynolds-number turbulent flows. Relevant implications of turbulence control are the reduction of skin friction in wall-bounded flows, the delay of separation in wake flows, the enhancement of mixing in free shear flows and the controlled sediment transport in multiphase flows. The development of predictive models is related to both devise new subgrid scale (SGS) models in the large eddy simulation (LES) approach to turbulence modelling and the derivation from the system of the governing equations, of low-dimensional models for the turbulent phenomena.

In this work, the issue of the coherent structures of turbulence in the wall region of turbulent channel flow is addressed. The turbulent structures are educed with the technique of the POD from the fluctuating portion of a numerical database that has been built using a three-dimensional time-dependent computational code for the numerical integration of the Navier–Stokes equations at friction Reynolds number *Re*_{τ}=180, following the DNS approach.

The present work is organized as follows. In §2*a*, the numerical techniques implemented in the computational code and the criteria followed for the build-up of the database are described. In §2*b*, the technique of the POD is reviewed. In §3, the results of the decomposition are presented and the flow dynamics analysed. Section 4 contains the concluding remarks.

## 2. Methods

### (a) Numerical simulations

The numerical simulations are performed using a computational code based on a mixed spectral-finite difference technique. The system of the unsteady Navier–Stokes equations for incompressible fluids is considered (conservative form, *i*, *j*=1, 2, 3)(2.1a)(2.1b)where *u*_{i}(*u*, *v*, *w*) are the velocity components in the Cartesian coordinate system *x*_{i}(*x*, *y*, *z*). Equations (2.1) are non-dimensionalized by the channel half-height *h*, the friction velocity , for pressure and *h*/*u*_{τ} for time, being *Re*_{τ}=(*u*_{τ}*h*/*ν*) the friction Reynolds number, *ρ* the fluid density and *ν* the fluid kinematic viscosity. The fields are admitted to be periodic in the streamwise (*x*) and spanwise (*z*) directions and equations (2.1) are Fourier transformed accordingly. The nonlinear terms of the momentum equation are evaluated pseudospectrally by antitransforming the velocities back to physical space to perform the products (FFTs are used) and the 2/3*s* dealiasing procedure is applied to avoid errors in transforming the results back to Fourier space. Second-order finite differences are used in the *y*-grid points (the direction orthogonal to the walls), incorporating a grid-stretching law of hyperbolic tangent type for a better spatial resolution near the walls. For time advancement, a third-order Runge–Kutta algorithm is implemented and time marching is accomplished with the fractional-step method introduced by Kim *et al*. (1987). No-slip boundary conditions at the walls and cyclic conditions in the streamwise and spanwise directions are enforced. In table 1, the characteristic parameters of the simulations are reported (in tables 1 and 2, local wall units are , *t*^{+}=*tu*_{τ}/*δ*_{τ}, *u*^{+}=*U*/*u*_{τ}, where *U* is a streamwise velocity averaged on a *x*–*z* plane and time and *δ*_{τ}=*ν*/*u*_{τ} is the viscous length). The Kolmogorov spatial microscale, estimated using the average rate of dissipation per unit mass across the width of the channel, results *η*^{+}≈1.8. The stretched grid along *y* allows the presence of eight grid points within the viscous sublayer (*y*^{+}≤7), so that the usually followed requisites for the numerical accuracy of DNS calculations (Grötzbach 1983) are satisfied, in particular: (i) to select a normal-to-the-wall grid width distribution able to resolve the steep gradients of the velocity field near the wall (i.e. to have at least three grid points in the viscous sublayer), (ii) to select a normal-to-the-wall grid such as the mean grid width results smaller than the relevant turbulent elements (i.e. ), and (iii) to have Δ*t*≤*τ*_{η} (see subsequent paragraph).

The initial transient of the flow in the channel is simulated by inserting an initial velocity profile that varies with time. The turbulent statistically steady state is reached (the behaviour of the total shear stress across the section of the channel is monitored for this scope) and simulated for a time *t*=10*h*/*u*_{τ} (*t*^{+}=1800). Twenty thousand time-steps are calculated with a temporal resolution of Δ*t*=5×10^{−4}*h*/*u*_{τ} (Δ*t*^{+}=0.09). The estimated Kolmogorov time-scale results *τ*_{η}≈1.89×10^{−2}*h*/*u*_{τ}.

With regard to the numerical methods used, it is to be noted that the present mixed spectral-finite difference technique is less expensive in the amount of computations with respect to fully spectral codes and that the grid-stretching law in the *y*-direction allows a great flexibility in the grid-refinement operations (Alfonsi *et al*. 1998; Passoni *et al*. 2002). In order to save computational time, parallel versions of the code have been developed and implemented on different parallel computing systems (Passoni *et al*. 1999, 2001).

In table 2, the predicted and computed values of a number of mean-flow variables are reported (*U*_{b} is the bulk velocity; *Re*_{b}=*U*_{b}*h*/*ν* is the related Reynolds number; *U*_{c} is the mean centreline velocity; *Re*_{c}=*U*_{c}*h*/*ν* is the related Reynolds number). The predicted values of *U*_{c}/*U*_{b} and of the skin friction coefficient *C*_{f} are obtained from the experimental correlations suggested by Dean (1978),(2.2)(2.3)while the computed skin friction coefficient [; ] is calculated using the value of the shear stress at the wall actually obtained in the computations (a three-point finite difference routine is used). In figures 1–3, mean velocity profile, turbulence intensities and Reynolds shear stress (in wall units) of the present work are compared with the results of Moser *et al*. (1999) at *Re*_{τ}=180, respectively. The agreement between the present results (obtained with a mixed spectral-finite difference computational code) and the results of Moser *et al*. (1999) (obtained with a fully spectral code) is rather satisfactory.

### (b) Proper orthogonal decomposition

An overview of the method is given.

By considering an ensemble of temporal realizations of a velocity field *u*_{i}(*x*_{j}, *t*) on a finite domain *D*, one wants to find the highest mean-square correlated structure to the elements of the ensemble on average. This corresponds to find a deterministic vector function *φ*_{i}(*x*_{j}) that maximizes the normalized inner product of the candidate structure with the field. A necessary condition for this problem is that *φ*_{i}(*x*_{j}) is an eigenfunction, solution of the eigenvalue problem and Fredholm integral equation of the first kind(2.4)where is the two-point velocity correlation tensor. To each eigenfunction is associated a real-positive eigenvalue *λ*^{(n)} and every member of the ensemble can be reconstructed by means of the modal decomposition,(2.5)where the random coefficients, are(2.6)The contribution of each mode to the kinetic energy content of the flow is given by(2.7)*E* being the kinetic energy in the domain *D*.

The POD is optimal for modelling or reconstructing a signal in the sense that, for a given number of modes, the projection on the subspace used contains the most kinetic energy possible on average (Ball *et al*. 1991; Berkooz *et al*. 1993).

In the present work, the POD is used for the analysis of the fluctuating portion of the velocity field. The two homogeneous directions are handled in Fourier space (POD and Fourier decomposition coincide along homogeneous directions), so that the optimal representation of the velocity field in the statistical sense outlined above, is sought in the direction *y*, normal to the walls.

More in particular, Fourier decomposition is performed along *x* and *z* and for each Fourier mode *m*, *n* the following problem is solved:(2.8)For each pair *m* and *n*, the POD gives 3*N*_{y} complex eigenfunctions, being the generic eigenfunction represented by the index *q*. The velocity field is reconstructed as follows:(2.9)where(2.10)

## 3. Results

### (a) Decomposition

As a result of the decomposition of the velocity fluctuations, 3*N*_{y}(387) POD modes and the correspondent eigenvalues are determined for each wavenumber index pair. Table 3 reports the individual fraction of the turbulent kinetic energy and the cumulative energies of the velocity fluctuations of the first 16 more energetic eigenfunctions of the decomposition (*m* is the wavenumber along *x*; *n* is the wavenumber along *z*; and *q* is the generic POD mode) that alone account for about 20% of the total energy. About 6.9% of the energy resides in the first three streamwise independent modes (*m*=0), while the first mode that exhibits a streamwise dependence (*m*≠0) is the fourth.

Figure 4 shows a representation of the flow field in the plane channel in terms of isosurfaces of fluctuating streamwise velocity reconstructed from the first 7260 most energetic modes of the decomposition, which correspond to about 95% of the total energy content. A number of flow structures are visible, mainly elongated in the streamwise direction (light surfaces are positive, dark surfaces are negative streamwise velocity). The visualization shown in figure 4 actually inhibits any possible interpretation of the flow phenomena in terms of dominant structures. In order to filter out small-scale effects, a lower number of modes is selected for the visual flow representations presented in the subsequent sections.

In the subsequent sections, the first five most energetic POD modes are considered. This number of modes has been chosen because the first three are *x*-independent, while the second two are *x*-dependent. The interaction between these two types of modes is actually the relevant phenomenon to be studied with regard to the structure of the boundary layer (Webber *et al*. 1997; among others). A larger number of eigenfunctions leads to a more complex flow field with a decreasing possibility of detecting the energetically dominant flow modes.

### (b) Eigenfunctions' analysis

In this section, the first five most energetic eigenfunctions of the decomposition are singularly analysed.

The first eigenfunction accounts for 3.22% of the total energy content of the flow (table 3) and is characterized by the triplet (0, 1, 1) of the indices *m*, *n*, *q*. Figure 5 shows surfaces of constant streamwise velocity (the light surface is positive, the dark surface is negative streamwise velocity) reconstructed from the first eigenfunction. The visualization represents the average flow structure in the interval 0≤*t*^{+}≤1800 owing to the first POD mode and shows two *x*-independent structures elongated in the streamwise direction +*x*. These structures are highly similar to the so-called *roll modes* identified by Webber *et al*. (1997) in their work of POD in the minimal channel flow domain of Jiménez & Moin (1991).

Figures 6 and 7 show surfaces of constant streamwise velocity reconstructed from the second (0, 2, 1) and the third (0, 2, 2) most energetic eigenfunctions, that account for 2.17 and 1.53% of the total energy, respectively (table 3). Also in these cases, the visualizations show flow structures elongated in the streamwise direction similar to those of figure 5, with appropriate repetitions according to the values of *m* and *n*.

Figure 8 shows surfaces of constant streamwise velocity reconstructed from the fourth most energetic eigenfunction of the decomposition (1, 1, 1) that accounts for 1.51% of the total energy of the flow. This is the first streamwise-dependent eigenfunction. The visualization shows couples of positive and negative bean-shaped flow structures aligned in the streamwise direction, where one of the structures of each couple is more displaced towards the centre of the channel (−*y*-direction). Figure 9 shows surfaces of constant streamwise velocity reconstructed from the fifth most energetic eigenfunction of the decomposition (1, 2, 1) that accounts for 1.45% of the total energy of the flow. This is the second streamwise-dependent eigenfunction. The visualization shows bean-shaped, quasi-streamwise flow structures aligned in the streamwise direction similar to those shown in figure 8, with appropriate repetitions according to the value of *n*. The regularity of the positions of these modes in the computational domain is impressive. The bean-shaped structures are slightly tilted outward (figure 10) and this result is in agreement with the results of Webber *et al*. (1997) who found that the so-called *propagating modes* tilted away from the wall by an angle of about 30°, and also with the results reported by Robinson (1991), in which the existence of quasi-streamwise, outward-tilted vortical structures is mentioned.

### (c) Behaviour with time

The behaviour with time of each of the first five most energetic POD modes has been singularly analysed. The analysis has shown a fundamental difference between the first three streamwise-independent modes (0, 1, 1), (0, 2, 1), (0, 2, 2) and the second two quasi-streamwise modes (1, 1, 1), (1, 2, 1), as follows:

the first three streamwise-independent modes propagate with time:

for relatively long time-intervals along the positive streamwise direction;

for relatively short time-intervals (50.4≤

*t*^{+}≤324 over 1800 in our simulations) along the negative spanwise direction −*z*(reference frame shown in the figures);the second two streamwise-dependent modes propagate with time always following rigorously the positive streamwise direction +

*x*.

In order to better illustrate this phenomenon, additional visualizations are presented.

Figure 11*a*,*b* shows the flow field—identified in terms of surfaces of constant streamwise fluctuations—produced by the sum of the first three most energetic *x*-independent POD modes in a time frame in which they propagate following the −*z*-direction. At *t*^{+}=50.4 (figure 11*a*), two streamwise-independent main structures aligned in the *x*-direction are visible. The position of the positive (light) structure is at the +*z* limit of the computational domain, while at *t*^{+}=198 (figure 11*b*) the dark (negative) structure has reached the −*z* limit of the computational domain. At *t*^{+}=324, the negative structure is re-entering into the computational domain along the periodic *z*-direction (not shown).

Figure 12*a*,*b* shows the flow field produced by the sum of the second two most energetic *x*-dependent POD modes at different instants. At *t*^{+}=25.2 (figure 12*a*), a number of bean-shaped quasi-streamwise flow structures are visible. In the centre of the computational domain, a negative (dark) structure has almost reached the +*x* limit of the computational domain, followed by a positive (light) structure. At *t*^{+}=61.2 (figure 12*b*), the aforementioned positive structure has almost reached the +*x* limit of the computational domain and is followed by a negative turbulent structure along the periodic *x*-direction. The propagation of these two structures (like that of the other structures visible in the figures) rigorously develops along the positive streamwise direction.

### (d) Flow dynamics

Some of the time-steps of the numerical simulation are followed with reference to the evolution in time of the flow field produced by the sum of the first five above-considered POD modes, in a time frame in which the streamwise-independent modes propagate along the −*z*-direction.

Figure 13*a*,*d* shows the flow structures through the sequence of instants *t*^{+}=54, 64.8, 72 and 82.8, respectively (the streamwise direction is from left to right). Two dominant structures elongated in the streamwise direction are visible. With reference to the positive (light) structure, it appears that the basic streamwise-elongated structure—formed by the first three *x*-independent modes—is deformed owing to the interaction with the flow structure formed by the second two bean-shaped, quasi-streamwise modes.

In figure 13*a*, a first group of bean-shaped modes is coming into the computational domain while a second group is going out. The phenomenon evolves as shown in figure 13*b*,*c*, where the second group of bean-shaped modes completes the exit from the computational domain (along the +*x*-direction) and the first group of quasi-streamwise modes reaches the centre of the computational domain, causing a remarkable alteration of the shape of the streamwise-independent flow structure. At *t*^{+}=82.8 (figure 13*d*), the group of quasi-streamwise modes continues the movement along the streamwise direction, causing the propagation in that direction of the perturbation of the flow structure aligned in the streamwise direction.

The phenomenon described in figure 13*a*–*d* is very similar to the so-called *wave-like disturbance* shown by Webber *et al*. (1997) in the minimal channel flow domain (in their case a large number of modes has been used for the visualizations). In their work, the wave-like disturbance seems to be more localized onto the *x*–*y* plane, while in our work it is of a fully three-dimensional nature. The reason for this may be that the minimal channel-flow domain of Webber *et al*. (1997) is not sufficient to give a complete representation of the flow phenomena. According to our results, this kind of wave-like disturbance is clearly owing to the process of *interaction between the dominant streamwise-elongated flow structures propagating along the negative spanwise direction, and the bean-shaped straightforwardly propagating POD modes*.

The following group of visualizations (figure 14*a*–*d*) shows another sequence of events through times *t*^{+}=136.8, 165.6, 172.8 and 190.8, respectively. Also, in this case, two dominant structures elongated in the streamwise direction are visible. With reference to the negative (dark) structure, figure 14*a* shows a surface extending away from the streamwise-elongated structure. This phenomenon is very similar to the so-called *extended surface* shown by Webber *et al*. (1997) in the minimal channel flow domain (they used a large number of modes for the visualizations). In their case, the extended surface seems to be more localized onto the *x*–*y* plane, while in our case it mainly lies onto the *x*–*z* plane. Again, these differences may be due to the different sizes of the computational domains. According to our results, the phenomenon of the extended surface is an episode of the process of interaction between the main streamwise-elongated structure and the bean-shaped, quasi-streamwise modes. In particular, it represents the onset of the occurrence of the break-up of the main flow structure. The physical reason for the disruption of the main flow structure is again the result of the interaction between the −*z* propagating, *x*-independent modes and the straightforwardly propagating, *x*-dependent turbulent structures. Figure 14*b* shows the detachment of the extended surface at one end of the negative streamwise-elongated structure and, at the other end, the first manifestation of the structure's break-up. In figure 14*c*,*d*, the process of disruption of the original structure becomes more evident.

There are practical implications in turbulence control related to a deeper understanding of the above phenomena. Sirovich & Karlsson (1997) performed a laboratory experiment in which randomized arrays of appropriately designed protrusions on the bottom of a channel—devised on the basis of the results of the Sirovich's group—resulted in a measured drag reduction of about 10% with respect to the smooth-wall case. This result actually represents the first successful laboratory-tested application of a turbulence-control technique for skin-friction reduction, based on the analysis of the coherent structures of turbulence (the POD modes of the flow in this context). A better understanding of the behaviour, and in particular of the processes of interaction of the POD modes in turbulent flows, may result in more advanced technologies for passive turbulence control.

## 4. Concluding remarks

The flow of a viscous-incompressible fluid in a plane channel has been simulated with the use of a computational code for the numerical integration of the Navier–Stokes equations, following the DNS approach. A numerical database of the velocity field has been built and the coherent structures of turbulence have been educed with the POD technique.

The analysis of the first five most energetic POD modes has shown the existence two kinds of different structures:

elongated, streamwise-independent turbulent-flow structures, aligned in the

*x*-direction andbean-shaped, streamwise-dependent modes, slightly tilted outward.

The analysis of the behaviour with time of the first five most energetic POD modes has revealed the way in which the different structures propagate. More in particular:

the first three streamwise-independent modes propagate for relatively long time frames in the streamwise direction and for relatively short time frames along the negative spanwise direction and

the second two streamwise-dependent modes propagate following the streamwise direction.

The analysis of the evolution in time of the flow field produced by the sum of the first five most energetic POD modes has revealed the existence of dominant structures elongated in the streamwise direction. The shape of these structures is altered owing to the process of interaction with the bean-shaped, quasi-streamwise turbulent-flow modes. This process of interaction is responsible for the gradual break-up of the *x*-elongated structures.

## Footnotes

- Received September 17, 2006.
- Accepted October 4, 2006.

- © 2006 The Royal Society