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doi: 10.1098/rspa.2006.1785, 593-612463 2007 Proc. R. Soc. A

Giancarlo Alfonsi and Leonardo Primavera wall region of plane channel flowThe structure of turbulent boundary layers in the

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The structure of turbulent boundary layersin the wall region of plane channel flow

BY GIANCARLO ALFONSI1,* AND LEONARDO PRIMAVERA

2

1Dipartimento di Difesa del Suolo, Universita della Calabria, Via P. Bucci 42b,87036 Rende (Cosenza), Italy

2Dipartimento di Fisica, Universita della Calabria, Via P. Bucci 33b,87036 Rende (Cosenza), Italy

The flow of a viscous incompressible fluid in a plane channel is simulated numericallywith the use of a computational code for the numerical integration of the Navier–Stokesequations, based on a mixed spectral-finite difference technique. A turbulent-flowdatabase representing the turbulent statistically steady state of the velocity field through10 viscous time units is assembled at friction Reynolds number RetZ180 and thecoherent structures of turbulence are extracted from the fluctuating portion of thevelocity field using the proper orthogonal decomposition (POD) technique. The temporalevolution of a number of the most energetic POD modes is represented, showing theexistence of dominant flow structures elongated in the streamwise direction whose shapeis altered owing to the interaction with quasi-streamwise, bean-shaped turbulent-flowmodes. This process of interaction is responsible for the gradual disruption of thestreamwise modes of the flow.

Keywords: Navier–Stokes equations; wall-bounded turbulence;proper orthogonal decomposition

*A

RecAcc

1. Introduction

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

The velocity field in the inner region of a boundary layer is organized intoalternating streaks of high- and low-speed fluid (Kline et al. 1967), persistent,quiescent most of the time and randomly distributed in space. The most relevantpart of the turbulent production process occurs in the buffer layer duringoutward ejections of low-speed fluid and sweeps of high-speed fluid towards thewall (Corino & Brodkey 1969). The near-wall turbulence production processappears as an intermittent cyclic sequence of turbulent events. The so-calledbursting phenomenon can be identified in different ways: (i) lift-up, oscillationand breakup of low-speed streaks, (ii) shear-layer interface between sweeps and

Proc. R. Soc. A (2007) 463, 593–612

doi:10.1098/rspa.2006.1785

Published online 31 October 2006

uthor for correspondence ([email protected]).

eived 17 September 2006epted 4 October 2006 593 This journal is q 2006 The Royal Society


G. Alfonsi and L. Primavera594


ejections, and (iii) ejection generating from a low-speed streak. In the outerregion, three-dimensional bulges with dimension of the order of the boundarylayer thickness form in the turbulent/non-turbulent interface. Irrotationalvalleys also form at the edges of the bulges, through which free-stream fluid isentrained towards the turbulent region. Weakly irrotational eddies are observedbeneath the bulges and fluid at relatively high speed impacts the upstream sidesof the large-scale motions, forming shear layers (Cantwell 1981).

Vortex dynamics is also relevant. One of the first contributions to the issue ofthe presence of vortices in the boundary layer is due to Theodorsen (1952) whointroduced the hairpin vortex. Robinson (1991) confirmed the existence of non-symmetric arch vortices and quasi-streamwise vortices on the basis of theevaluation of direct numerical simulation (DNS) results. The composition of aquasi-streamwise vortex with an arch vortex may result in a hairpin vortex,complete or, most frequently one-sided, but this conclusion may strongly dependon the particular technique used for vortex detection. Studies involving thedynamics of hairpin vortices in the boundary layer have been performedexperimentally by Perry & Chong (1982), Acarlar & Smith (1987a,b), Smithet al. (1991), Haidari & Smith (1994) and numerically by Singer & Joslin (1994).Based on this, a picture of vortex generation and reciprocal interaction in theboundary layer emerges in which processes of interaction of existing vortices withwall-layer fluid involve viscous–inviscid interaction, generation of new vorticity,redistribution of existing vorticity, vortex stretching near the wall and vortexrelaxation in the outer region. Individual vortices advected in a shear flowevolve—nonlinearly and mainly inviscidly—into, in most cases, non-symmetrichairpin-shaped structures, beginning from the portion of the vortex characterizedby the highest curvature. During their development, spanwise vorticity istransformed into streamwise vorticity with deformation and birth of subsidiaryvortices (Acarlar & Smith 1987b; Haidari & Smith 1994; Singer & Joslin 1994).The most relevant vortex-interaction processes occurring in the boundary layerare: (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 ahairpin vortex involves the development of vortex legs in regions of increasingshear 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 ofdecreasing shear. As a consequence, the vorticity in the vortex head diminishes(Head & Bandyopadhyay 1981; Perry & Chong 1982). The processes involvingmultiple vortex dynamics are more complex. An attempt at a description of suchphenomena has been made by Smith et al. (1991) according to which thecoalescence of small vortices into larger structures is described in terms ofintertwining, amalgamation and reinforcement occurring when upward-migrating vortices approach one another.

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

Proc. R. Soc. A (2007)


595The structure of turbulent boundary layers


In the numerical field, a valuable approach to the calculation of turbulentflows is the DNS, in which the objective of computing all turbulent scales ispursued and the Navier–Stokes equations are numerically integrated with nomodifications of any kind. The critical aspect of this method is the numericalaccuracy of the calculations that has to be sufficiently high as to resolve theessential turbulent scales. DNS results for the case of the plane channel havebeen reported by Kim et al. (1987), Lyons et al. (1991), Rutledge & Sleicher(1993) and Moser et al. (1999). Modern techniques for the numerical integrationof the Navier–Stokes equations (advanced numerical methods and high-performance computing) have the ability of remarkably increasing the amountof data gathered during a research of computational nature, bringing to thecondition of managing large amounts of data. A typical turbulent-flow databaseincludes all three components of the fluid velocity in all points of a three-dimensional domain, evaluated for an adequate number of time-steps of theturbulent statistically steady state. Mathematically founded methods for theidentification of vortical structures in a turbulent-flow database have beenintroduced by: (i) Perry & Chong (1987), based on the complex eigenvalues ofthe velocity-gradient tensor, (ii) Hunt et al. (1988), based on the second invariantof the velocity-gradient tensor, (iii) Zhou et al. (1999), based on the imaginarypart 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 samedegree in determining the physical properties of a turbulent flow. Methods can beapplied to extract from a turbulent-flow database only the relevant informationfor the physical understanding of a turbulent phenomenon, i.e. to separate theeffects of appropriately defined modes of the flow from the background flow or,finally, to extract the coherent structures of turbulence, irrespective of thedefinition of the coherent structure adopted. A powerful technique for theeduction of the coherent structures of turbulent flows is the proper orthogonaldecomposition (POD). The POD has been first introduced in turbulence researchby Lumley (1971) and is extensively presented in Sirovich (1987) and Berkoozet al. (1993). The method has been used (among others): (i) in Rayleigh-Benardturbulent convection problems by Park & Sirovich (1990), Sirovich & Park(1990), Deane & Sirovich (1991) and Sirovich & Deane (1991)), (ii) in studies offree shear flows by Sirovich et al. (1990a,b) and Kirby et al. (1990), (iii) in studiesof separated flows by Manhart (1998) and Alfonsi et al. (2003), and (iv) in theanalysis of wall-bounded turbulent flows. In the field of wall-bounded flows, anearly application of the method can be found in Bakewell & Lumley (1967).Aubry et al. (1988) used the POD in studying and modelling the turbulentboundary-layer problem starting from the experimental eigenfunctions of pipe-flow data. Moin & Moser (1989), Sirovich et al. (1990a,b) and Ball et al. (1991)applied the method of the POD to the case of the turbulent channel flow. Webberet al. (1997) used the method for the analysis of a numerical database obtainedusing the minimal channel flow domain of Jimenez & Moin (1991).

The study of the behaviour of the coherent structures of turbulence in theboundary layer of wall-bounded flows offer the possibility of clarifying thephysical mechanisms through which turbulent energy of a mechanical nature isdissipated into heat. The understanding of these mechanisms brings newperspectives on two important objectives of modern fluid technology, the control





of turbulence and the development of new predictive models for the numericalcalculation of high-Reynolds-number turbulent flows. Relevant implications ofturbulence control are the reduction of skin friction in wall-bounded flows, thedelay of separation in wake flows, the enhancement of mixing in free shear flowsand the controlled sediment transport in multiphase flows. The development ofpredictive models is related to both devise new subgrid scale (SGS) models in thelarge eddy simulation (LES) approach to turbulence modelling and thederivation from the system of the governing equations, of low-dimensionalmodels for the turbulent phenomena.

In this work, the issue of the coherent structures of turbulence in the wall regionof turbulent channel flow is addressed. The turbulent structures are educed withthe technique of the POD from the fluctuating portion of a numerical databasethat has been built using a three-dimensional time-dependent computational codefor the numerical integration of the Navier–Stokes equations at friction Reynoldsnumber RetZ180, following the DNS approach.

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

2. Methods

(a ) Numerical simulations

The numerical simulations are performed using a computational code based on amixed spectral-finite difference technique. The system of the unsteadyNavier–Stokes equations for incompressible fluids is considered (conservativeform, i, jZ1, 2, 3)

vuivt

Cv

vxjðuiujÞZK

vp

vxiC

1

Ret

v2uivxjvxj

; ð2:1aÞ

vuivxi

Z 0; ð2:1bÞ

where ui(u, v, w) are the velocity components in the Cartesian coordinate systemxi(x, y, z). Equations (2.1) are non-dimensionalized by the channel half-height h,the friction velocity utZ

ffiffiffiffiffiffiffiffiffiffitw=r

p, ru2

t for pressure and h/ut for time, beingRetZ(uth/n) the friction Reynolds number, r the fluid density and n the fluidkinematic viscosity. The fields are admitted to be periodic in the streamwise (x)and spanwise (z) directions and equations (2.1) are Fourier transformedaccordingly. The nonlinear terms of the momentum equation are evaluatedpseudospectrally by antitransforming the velocities back to physical space toperform the products (FFTs are used) and the 2/3s dealiasing procedure isapplied 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 tothe walls), incorporating a grid-stretching law of hyperbolic tangent type for abetter spatial resolution near the walls. For time advancement, a third-orderRunge–Kutta algorithm is implemented and time marching is accomplished with



Table 2. Predicted versus computed mean-flow variables.

Ret Reb Rec Ub/u t Uc/u t Uc/Ub Cf

predicted values180 2800 3240 15.56 18.02 1.16 8.44!10K3

computed values (present work)178.74 2786 3238 15.48 17.99 1.16 8.23!10K3

Table 1. Characteristic parameters of the numerical simulations.

computing domain computational grid grid spacing

Lx Ly Lz LCx LCy LCz Nx Ny Nz DxC DyCcentre DyCwall DzC

2ph 2h ph 1131 360 565 96 129 64 11.8 4.4 0.87 8.8



the fractional-step method introduced by Kim et al. (1987). No-slip boundaryconditions at the walls and cyclic conditions in the streamwise and spanwisedirections are enforced. In table 1, the characteristic parameters of thesimulations are reported (in tables 1 and 2, local wall units are xCi Zxi=dt,tCZtu t/dt, u

CZU/u t, where U is a streamwise velocity averaged on a x–z planeand time and dtZn/u t is the viscous length). The Kolmogorov spatialmicroscale, estimated using the average rate of dissipation per unit mass acrossthe width of the channel, results hCz1.8. The stretched grid along y allows thepresence of eight grid points within the viscous sublayer (yC%7), so that theusually followed requisites for the numerical accuracy of DNS calculations(Grotzbach 1983) are satisfied, in particular: (i) to select a normal-to-the-wallgrid width distribution able to resolve the steep gradients of the velocity fieldnear 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 D�y resultssmaller than the relevant turbulent elements (i.e. D�y%ph), and (iii) to haveDt%th (see subsequent paragraph).

The initial transient of the flow in the channel is simulated by inserting aninitial velocity profile that varies with time. The turbulent statistically steadystate is reached (the behaviour of the total shear stress across the section of thechannel is monitored for this scope) and simulated for a time tZ10h/u t (t

CZ1800).Twenty thousand time-steps are calculated with a temporal resolution ofDtZ5!10K4h/u t (DtCZ0.09). The estimated Kolmogorov time-scale resultsthz1.89!10K2h/u t.

With regard to the numerical methods used, it is to be noted that the presentmixed spectral-finite difference technique is less expensive in the amount ofcomputations with respect to fully spectral codes and that the grid-stretching lawin 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 differentparallel computing systems (Passoni et al. 1999, 2001).



0

5

10

15

20

1 10 100

u+

y+

Figure 1. Mean velocity profile normalized by the friction velocity in wall coordinates. Presentwork: solid line. Data from Moser et al. (1999): plus. Law of the wall uCZyC anduCZ2:5 ln yCC5:5: dotted line.



In table 2, the predicted and computed values of a number ofmean-flowvariablesare reported (Ub is the bulk velocity; RebZUbh/n is the related Reynolds number;Uc is the mean centreline velocity; RecZUch/n is the related Reynolds number).The predicted values of Uc/Ub and of the skin friction coefficient Cf are obtainedfrom the experimental correlations suggested by Dean (1978),

Uc

Ub

� �Z 1:28ð2 RebÞK0:0116; ð2:2Þ

Cf Z 0:073ð2 RebÞK0:25; ð2:3Þ

while the computed skin friction coefficient [CfZð2tw=rU 2b Þ; twZmðvU=vyÞwall] is

calculated using the value of the shear stress at the wall actually obtained in thecomputations (a three-point finite difference routine is used). In figures 1–3, meanvelocity profile, turbulence intensities and Reynolds shear stress (in wall units) ofthe present work are compared with the results of Moser et al. (1999) at RetZ180,respectively. The agreement between the present results (obtained with a mixedspectral-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 ui(xj, t)

on a finite domain D, one wants to find the highest mean-square correlatedstructure to the elements of the ensemble on average. This corresponds to finda deterministic vector function 4i(xj) that maximizes the normalized innerproduct of the candidate structure with the field. A necessary condition for thisproblem is that 4i(xj) is an eigenfunction, solution of the eigenvalue problem and



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10 20 30 40 50 60 70 80

u′v′

___

y+

Figure 3. Reynolds shear stress ðKu 0v 0 Þ normalized by the friction velocity in wall coordinates.Present work: solid line. Data from Moser et al. (1999): plus.

0

0.5

1.0

1.5

2.0

2.5

3.0

10 20 30 40 50 60 70 80

u′rm

s′ v′

rms′

w′ rm

s

y+

Figure 2. Rms values of the velocity fluctuations normalized by the friction velocity in wallcoordinates. Present work: solid line, ðu 0

rmsÞ; dashed line, ðv 0rmsÞ; dotted line, ðw 0rmsÞ. Data from

Moser et al. (1999): plus, ðu 0rmsÞ; times, ðv 0rmsÞ; asterisk, ðw 0

rmsÞ.



Fredholm integral equation of the first kindðDRij xl ; x

0k

� �4j x 0k� �

dx 0k Z

ðD

uiðxl ; tÞuj x 0k ; t� ��

4j x 0k� �

dx 0k Z l4iðxlÞ; ð2:4Þ

where RijZhuiðxl ; tÞujðx 0k; tÞi is the two-point velocity correlation tensor. To each

eigenfunction 4ðnÞi ðxjÞ is associated a real-positive eigenvalue l(n) and every

member of the ensemble can be reconstructed by means of the modaldecomposition,

uiðxj ; tÞZXn

anðtÞ4ðnÞi ðxjÞ; ð2:5Þ





where the random coefficients, are

anðtÞZðDuiðxk ; tÞ4

ðnÞi ðxkÞdxk : ð2:6Þ

The contribution of each mode to the kinetic energy content of the flow isgiven by

E Z

ðDhuiðxj ; tÞuiðxj ; tÞidxj Z

Xn

lðnÞ: ð2:7Þ

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 usedcontains the most kinetic energy possible on average (Ball et al. 1991; Berkoozet al. 1993).

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

More in particular, Fourier decomposition is performed along x and z and foreach Fourier mode m, n the following problem is solved:ðC1

K1um;ni ðy; tÞum;n

j ðy 0; tÞD E

4m;n;qj ðy 0Þdy 0 Z lm;n;q4

m;n;qi ðyÞ: ð2:8Þ

For each pair m and n, the POD gives 3Ny complex eigenfunctions, being thegeneric eigenfunction represented by the index q. The velocity field isreconstructed as follows:

uiðx; y; z; tÞZX3Ny

qZ1

XCðNx=2Þ

mZKðNx=2Þ

XCðNz=2Þ

nZKðNz=2Þam;nq ðtÞ4m;n;q

i ðyÞeimxeinz ; ð2:9Þ

where

am;nq ðtÞZ

ðC1

K1um;ni ðy; tÞ4m;n;q

i ðyÞdy: ð2:10Þ

3. Results

(a ) Decomposition

As a result of the decomposition of the velocity fluctuations, 3Ny(387) PODmodes and the correspondent eigenvalues are determined for each wavenumberindex pair. Table 3 reports the individual fraction of the turbulent kinetic energyand the cumulative energies of the velocity fluctuations of the first 16 moreenergetic eigenfunctions of the decomposition (m is the wavenumber along x; n isthe wavenumber along z; and q is the generic POD mode) that alone account for



Table 3. Energy content of the first 16 POD eigenfunctions.

index mode (m, n, q) energy fraction energy sum

1 (0, 1, 1) 0.03220 0.032202 (0, 2, 1) 0.02173 0.053933 (0, 2, 2) 0.01535 0.069294 (1, 1, 1) 0.01508 0.084375 (1, 2, 1) 0.01454 0.098916 (0, 3, 1) 0.01197 0.110897 (1, 3, 1) 0.01196 0.122868 (1, 2, 2) 0.01160 0.134469 (0, 4, 1) 0.01053 0.1449910 (1, 4, 1) 0.00972 0.1547211 (0, 4, 2) 0.00945 0.1641712 (0, 5, 1) 0.00933 0.1735013 (1, 2, 3) 0.00863 0.1821414 (0, 3, 2) 0.00806 0.1902015 (1, 4, 2) 0.00776 0.1979616 (0, 3, 3) 0.00765 0.20561

x

y

z

Figure 4. Surfaces of constant streamwise fluctuation reconstructed from the first 7260 most energeticPOD modes (light surfaces are positive, dark surfaces are negative streamwise velocity).



about 20% of the total energy. About 6.9% of the energy resides in the first threestreamwise independent modes (mZ0), while the first mode that exhibits astreamwise dependence (ms0) is the fourth.

Figure 4 shows a representation of the flow field in the plane channel in termsof isosurfaces of fluctuating streamwise velocity reconstructed from the first 7260most energetic modes of the decomposition, which correspond to about 95% ofthe total energy content. A number of flow structures are visible, mainly



x

y

z

Figure 5. Surfaces of constant streamwise fluctuation reconstructed from the first most energeticeigenfunction of the decomposition (the light surface is positive, the dark surface is negativestreamwise velocity).



elongated in the streamwise direction (light surfaces are positive, dark surfacesare negative streamwise velocity). The visualization shown in figure 4 actuallyinhibits any possible interpretation of the flow phenomena in terms of dominantstructures. In order to filter out small-scale effects, a lower number of modes isselected for the visual flow representations presented in the subsequent sections.

In the subsequent sections, the first five most energetic POD modesare considered. This number of modes has been chosen because the first three arex-independent, while the second two are x-dependent. The interaction betweenthese two types of modes is actually the relevant phenomenon to be studied withregard 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 adecreasing possibility of detecting the energetically dominant flow modes.

(b ) Eigenfunctions’ analysis

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

The first eigenfunction accounts for 3.22% of the total energy content of theflow (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 ispositive, the dark surface is negative streamwise velocity) reconstructed from thefirst eigenfunction. The visualization represents the average flow structure in theinterval 0%tC%1800 owing to the first POD mode and shows two x-independentstructures elongated in the streamwise direction Cx. These structures are highlysimilar to the so-called roll modes identified by Webber et al. (1997) in their workof POD in the minimal channel flow domain of Jimenez & Moin (1991).



x

y

z

Figure 6. Surfaces of constant streamwise fluctuation reconstructed from the second most energeticeigenfunction of the decomposition (light surfaces are positive, dark surfaces are negativestreamwise velocity).



Figures 6 and 7 show surfaces of constant streamwise velocity reconstructedfrom 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 inthese cases, the visualizations show flow structures elongated in the streamwisedirection similar to those of figure 5, with appropriate repetitions according tothe values of m and n.

Figure 8 shows surfaces of constant streamwise velocity reconstructed fromthe fourth most energetic eigenfunction of the decomposition (1, 1, 1) thataccounts for 1.51% of the total energy of the flow. This is the first streamwise-dependent eigenfunction. The visualization shows couples of positive andnegative bean-shaped flow structures aligned in the streamwise direction,where one of the structures of each couple is more displaced towards the centreof the channel (Ky-direction). Figure 9 shows surfaces of constant streamwisevelocity reconstructed from the fifth most energetic eigenfunction of thedecomposition (1, 2, 1) that accounts for 1.45% of the total energy of the flow.This is the second streamwise-dependent eigenfunction. The visualization showsbean-shaped, quasi-streamwise flow structures aligned in the streamwisedirection similar to those shown in figure 8, with appropriate repetitionsaccording to the value of n. The regularity of the positions of these modes in thecomputational domain is impressive. The bean-shaped structures are slightlytilted outward (figure 10) and this result is in agreement with the results ofWebber et al. (1997) who found that the so-called propagating modes tilted awayfrom the wall by an angle of about 308, and also with the results reported byRobinson (1991), in which the existence of quasi-streamwise, outward-tiltedvortical structures is mentioned.



x

y

z

Figure 7. Surfaces of constant streamwise fluctuation reconstructed from the third most energeticeigenfunction of the decomposition (light surfaces are positive, dark surfaces are negativestreamwise velocity).

x

y

z

Figure 8. Surfaces of constant streamwise fluctuation reconstructed from the fourth most energeticeigenfunction of the decomposition (light surfaces are positive, dark surfaces are negativestreamwise velocity).





x

y

z

Figure 9. Surfaces of constant streamwise fluctuation reconstructed from the fifth most energeticeigenfunction of the decomposition (light surfaces are positive, dark surfaces are negativestreamwise velocity).

xz

y

Figure 10. Surfaces of constant streamwise fluctuation reconstructed from the fifth most energeticeigenfunction of the decomposition. Side view (light surfaces are positive, dark surfaces arenegative streamwise velocity).



(c ) Behaviour with time

The behaviour with time of each of the first five most energetic POD modeshas been singularly analysed. The analysis has shown a fundamental differencebetween 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:

(i) the first three streamwise-independent modes propagate with time:— for relatively long time-intervals along the positive streamwise direction;



x

y

(a)

z

(b)

x

y

z

Figure 11. Surfaces of constant streamwise fluctuation reconstructed from the first three mostenergetic, streamwise-independent, POD modes at: (a) tCZ50.4; (b) tCZ198 (light surfaces arepositive, dark surfaces are negative streamwise velocity).



— for relatively short time-intervals (50.4%tC%324 over 1800 in oursimulations) along the negative spanwise direction Kz (reference frameshown in the figures);

(ii) the second two streamwise-dependent modes propagate with timealways following rigorously the positive streamwise direction Cx.

In order to better illustrate this phenomenon, additional visualizations arepresented.

Figure 11a,b shows the flow field—identified in terms of surfaces of constantstreamwise fluctuations—produced by the sum of the first three most energeticx-independent POD modes in a time frame in which they propagate following theKz-direction. At tCZ50.4 (figure 11a), two streamwise-independentmain structures aligned in the x-direction are visible. The position ofthe positive (light) structure is at the Cz limit of the computationaldomain, while at tCZ198 (figure 11b) the dark (negative) structure has reachedthe Kz limit of the computational domain. At tCZ324, the negative structure isre-entering into the computational domain along the periodic z-direction (notshown).

Figure 12a,b shows the flow field produced by the sum of the second twomost energetic x-dependent POD modes at different instants. At tCZ25.2(figure 12a), a number of bean-shaped quasi-streamwise flow structures arevisible. In the centre of the computational domain, a negative (dark) structurehas almost reached the Cx limit of the computational domain, followed by apositive (light) structure. At tCZ61.2 (figure 12b), the aforementionedpositive structure has almost reached the Cx limit of the computationaldomain and is followed by a negative turbulent structure along the periodicx-direction. The propagation of these two structures (like that of the otherstructures visible in the figures) rigorously develops along the positivestreamwise direction.



x

y

(a)

z

(b)

x

y

z

Figure 12. Surfaces of constant streamwise fluctuation reconstructed from the second two mostenergetic, streamwise-dependent, POD modes at: (a) tCZ25.2; (b) tCZ61.2 (light surfaces arepositive, dark surfaces are negative streamwise velocity).



(d ) Flow dynamics

Some of the time-steps of the numerical simulation are followed with referenceto the evolution in time of the flow field produced by the sum of the first fiveabove-considered POD modes, in a time frame in which the streamwise-independent modes propagate along the Kz-direction.

Figure 13a,d shows the flow structures through the sequence of instantstCZ54, 64.8, 72 and 82.8, respectively (the streamwise direction is from left toright). Two dominant structures elongated in the streamwise direction arevisible. With reference to the positive (light) structure, it appears that the basicstreamwise-elongated structure—formed by the first three x-independentmodes—is deformed owing to the interaction with the flow structure formedby the second two bean-shaped, quasi-streamwise modes.

In figure 13a, a first group of bean-shaped modes is coming into thecomputational domain while a second group is going out. The phenomenonevolves as shown in figure 13b,c, where the second group of bean-shaped modescompletes the exit from the computational domain (along the Cx-direction) andthe first group of quasi-streamwise modes reaches the centre of thecomputational domain, causing a remarkable alteration of the shape of thestreamwise-independent flow structure. At tCZ82.8 (figure 13d), the group ofquasi-streamwise modes continues the movement along the streamwise direction,causing the propagation in that direction of the perturbation of the flow structurealigned in the streamwise direction.

The phenomenon described in figure 13a–d is very similar to the so-calledwave-like disturbance shown by Webber et al. (1997) in the minimal channel flowdomain (in their case a large number of modes has been used for thevisualizations). In their work, the wave-like disturbance seems to be morelocalized onto the x–y plane, while in our work it is of a fully three-dimensionalnature. The reason for this may be that the minimal channel-flow domain ofWebber et al. (1997) is not sufficient to give a complete representation of the flowphenomena. According to our results, this kind of wave-like disturbance is clearly



x

y

(a)

z

(b)

x

y

z

x

y

(c)

z

(d )

x

y

z

Figure 13. Surfaces of constant streamwise fluctuation reconstructed from the first five mostenergetic POD modes at: (a) tCZ54; (b) tCZ64.8; (c) tCZ72; (d) tCZ82.8 (light surfaces arepositive, dark surfaces are negative streamwise velocity).



owing to the process of interaction between the dominant streamwise-elongatedflow structures propagating along the negative spanwise direction, and the bean-shaped straightforwardly propagating POD modes.

The following group of visualizations (figure 14a–d) shows another sequence ofevents through times tCZ136.8, 165.6, 172.8 and 190.8, respectively. Also, in thiscase, two dominant structures elongated in the streamwise direction are visible.With reference to the negative (dark) structure, figure 14a shows a surfaceextending away from the streamwise-elongated structure. This phenomenon is verysimilar to the so-called extended surface shown by Webber et al. (1997) in theminimal channel flow domain (they used a large number of modes for thevisualizations). In their case, the extended surface seems to be more localized ontothe x–y plane, while in our case it mainly lies onto the x–z plane. Again, thesedifferences may be due to the different sizes of the computational domains.According to our results, the phenomenon of the extended surface is an episode ofthe process of interactionbetween themain streamwise-elongated structure and thebean-shaped, quasi-streamwise modes. In particular, it represents the onset of the



x

y

(a)

(b)

z

x

y

z

x

y

(c)

(d )

z

x

y

z

Figure 14. Surfaces of constant streamwise fluctuation reconstructed from the first five mostenergetic POD modes at: (a) tCZ136.8; (b) tCZ165.8; (c) tCZ172.8; (d) tCZ190.8 (light surfacesare positive, dark surfaces are negative streamwise velocity).



occurrence of the break-up of the main flow structure. The physical reason for thedisruption of the main flow structure is again the result of the interaction betweentheKz propagating, x-independent modes and the straightforwardly propagating,x-dependent turbulent structures. Figure 14b shows the detachment of theextended 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 14c,d,the process of disruption of the original structure becomes more evident.

There are practical implications in turbulence control related to a deeperunderstanding of the above phenomena. Sirovich & Karlsson (1997) performed alaboratory experiment in which randomized arrays of appropriately designedprotrusions on the bottom of a channel—devised on the basis of the results of theSirovich’s group—resulted in a measured drag reduction of about 10% withrespect to the smooth-wall case. This result actually represents the firstsuccessful laboratory-tested application of a turbulence-control technique forskin-friction reduction, based on the analysis of the coherent structures ofturbulence (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 PODmodes in turbulent flows, may result in more advanced technologies for passiveturbulence control.

4. Concluding remarks

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

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

(i) elongated, streamwise-independent turbulent-flow structures, aligned inthe x-direction and

(ii) bean-shaped, streamwise-dependent modes, slightly tilted outward.

The analysis of the behaviourwith time of the first fivemost energetic PODmodeshas revealed the way in which the different structures propagate.More in particular:

(i) the first three streamwise-independent modes propagate for relatively longtime frames in the streamwise direction and for relatively short time framesalong the negative spanwise direction and

(ii) the second two streamwise-dependent modes propagate following thestreamwise direction.

The analysis of the evolution in time of the flow field produced by the sum of thefirst five most energetic POD modes has revealed the existence of dominantstructures elongated in the streamwise direction. The shape of these structures isaltered owing to the process of interaction with the bean-shaped, quasi-streamwiseturbulent-flow modes. This process of interaction is responsible for the gradualbreak-up of the x-elongated structures.

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