Document [original]

J. Fluid Mech. (2013), vol. 723, pp. 587–603. c

Cambridge University Press 2013 587

doi:10.1017/jfm.2013.143

Simulations of turbulent channels with

prescribed velocity profiles

Florian Tuerke1,2,‡and Javier Jiménez2,3,†

1Institut f¨

ur Str¨

omungsmechanik und Akustik, Technische Universit¨

at Berlin, M¨

uller-Breslau-Strasse 8,

10623 Berlin, Germany

2School of Aeronautics, Universidad Polit´

ecnica de Madrid, 28040 Madrid, Spain

3Centre for Turbulence Research, Stanford University, Stanford, CA 94305, USA

(Received 28 June 2012; revised 25 February 2013; accepted 5 March 2013;

first published online 16 April 2013)

Direct numerical simulations of turbulent channels with artificially prescribed velocity

profiles are discussed, using both natural and purposely incorrect profiles. It is found

that turbulence develops correctly when natural profiles are prescribed, but that even

slightly incorrect ones modify the Reynolds stresses substantially. That is used to study

the dynamics of the energy-containing velocity fluctuations. The stronger (weaker)

structures generated by locally stronger (weaker) mean shears have essentially correct

isotropy coefficients but they are out of energy equilibrium, with the energy imbalance

compensated by turbulent diffusion. The velocity scale in smooth profiles changes

with the distance to the wall, and is best described by a friction velocity derived

from the local total tangential stress. The behaviour across sharper shear jumps is

more consistent with non-equilibrium eddies that relax over wall-normal distances

of the order of the distance to the wall, suggesting that the energy equilibrium in

the logarithmic layer is not local to a given height, but applies to extended layers

homogenized by wall-normal fluxes. Examples of that non-local character are the

large-scale inactive fluctuations near the wall, whose velocities do not scale with the

local shear stress, but with that of their active ‘cores’ farther away from the wall.

Key words: turbulent boundary layers, turbulent flows

1. Introduction

This paper discusses direct numerical simulations of turbulent channels in which

the instantaneous wall-parallel-averaged velocity is artificially prescribed. The rest of

the flow, including all the fluctuations, is computed in the standard manner. For

brevity, and since that procedure may be considered the inverse of Reynolds-averaged

Navier–Stokes simulations in which the fluctuations are modelled while the mean

profile is computed, we will refer to it as ‘inverse RANS’, or IRANS. Our original

motivation, later abandoned, was to shorten the inflow length of direct simulations

of turbulent boundary layers (Simens et al. 2009) by keeping the mean velocity

profile artificially fixed to some empirical approximation over an upstream fraction

† Email address for correspondence: [email protected]

‡ Present address: Laboratorio de Fluidodin´

amica, Facultad de Ingenier´

ıa, Universidad de

Buenos Aires, Paseo Col´

on 850, C1063ACV Buenos Aires, Argentina.

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588 F. Tuerke and J. Jiménez

of the simulation box. The idea was to give the turbulent fluctuations some fetch in

a ‘correct’ environment before the flow was fully released, but the experiment failed.

The Reynolds stresses became substantially different from their ‘natural’ values, and,

although they returned to their correct levels upon release of the mean profile, the

inflow was not shortened.

Irrespective of its merits as a method for generating inflow conditions, the reasons

why the Reynolds stresses grew when the profile was kept fixed remained interesting

and unclear, and are the subject of this paper. The two possibilities examined by us are

that the growth of the fluctuations was due to fixing the mean profile, instead of letting

it evolve according to its own equation of motion, or that the fluctuations are sensitive

to relatively small deviations of the imposed velocity from its ‘correct’ shape, and that

the approximation used in our tests (Nagib, Chauhan & Monkewitz 2006) was slightly

incorrect.

Both results would be interesting, and provide opportunities to explore the dynamics

of turbulence. A central problem of wall-bounded turbulence is how the relative

intensities of structures of different sizes, associated with different wall distances,

adjust themselves so that the mean momentum transfer along the wall-normal direction

remains in balance. We will examine two models. Consider first local equilibrium. It

is not difficult to construct feedback models in which locally weak structures with

insufficient Reynolds stresses result in a local acceleration of the mean velocity, which

in turn leads to the enhancement of the velocity gradient and to the strengthening

of the local fluctuations. It is conceivable that fixing the mean profile disrupts that

mechanism.

For example, consider a open channel between a no-slip wall at y=0, where the

tangential stress is τw=1, and a free-slip surface at y=1. The wall-parallel-averaged

momentum equation can be written as,

∂tS=∂yyτ, (1.1)

where Sis the mean shear and τis the total tangential stress. Equilibrium requires

that τ=1−y, but the stability of small perturbations about that state depends on the

relation between τand S. In the simplest assumption that τ=τ(S), the equation for

stress fluctuations, e

τ, is

∂te

τ=(dτ/dS) ∂yye

τ, (1.2)

and stability requires that (dτ/dS)⩾0 everywhere. The experiments in this paper

can be understood as a way of exploring the relation between the total stress and

the velocity profile by perturbing the latter. In general, they involve modifying the

equations of motion, but, as stressed in previous works, that is one of the strengths of

numerical simulations (Jim´

enez & Pinelli 1999).

A second model, not necessarily inconsistent with the first one, is that the

momentum balance is a property of the correct mean velocity profile as a whole.

The classical theory, as represented by the logarithmic-layer argument of Townsend

(1976), is that the turbulent energy production, which is proportional to the mean

shear and to the tangential Reynolds stress, balances the local dissipation, which

is proportional to the cube of the fluctuation intensities. That would explain why

changing the mean profile, and therefore the production, modifies the intensities, but

it would also suggest that the latter are determined by the local mean shear, with

an interesting lack of interaction among structures at different wall distances. The

present experiments would then be useful in differentiating between local and global

contributions to the equilibrium of energy and momentum, by allowing us to consider

Turbulent channels with prescribed profiles 589

profiles that differ from the canonical one in different ways at different distances from

the wall. In fact, one of the results of this paper will be that energy equilibrium of

the Reynolds-stress-bearing scales has to be understood in terms of production and

dissipation integrated over ‘attached’ layers extending to the wall, in agreement with

the models of Townsend (1961), Perry & Abell (1975) or Perry, Henbest & Chong

(1986), rather than holding at individual wall distances. It will also turn out that the

near-wall ‘inactive’ motions, which carry energy but no tangential stress, derive their

velocity scales from their detached ‘active’ parts, also in agreement with Townsend

(1961) model.

Effects similar to those discussed here had been observed by P. Moin (private

communication) and Jarrin (2007), associated in both cases with attempts to shorten

the initial transient in large-eddy simulations of channels, but they were not analysed

in detail. A distant precursor can also be found in the ‘constrained Euler’ simulations

of isotropic turbulence with prescribed spectra, used to study the energy transfer and

the behaviour of large-eddy simulations by Shtilman & Chasnov (1992), Jim´

enez

(1993), She & Jackson (1993) and Zhou (1993), among others.

The organization of the paper is as follows. The numerical experiments are

described in § 2, followed in § 3by the presentation and discussion of the results, and

by conclusions in § 4. Additional details can be found in earlier reports by Jim´

enez

(2010) and Tuerke (2011).

2. The numerical experiments

The simulations are performed in doubly periodic computational channels with a

half-height h, and streamwise and spanwise periodicities 2πhand πh, as in Moser,

Kim & Mansour (1999). It was shown by del ´

Alamo et al. (2004) that even smaller

boxes have little effect on the spectral energy distribution of the resolved scales, and

that scales longer or wider than the box, which are represented in the simulations

by wavevectors with zero streamwise or spanwise wavenumbers, carry the correct

Reynolds stresses. Flores & Jim´

enez (2010) showed later that the mean and fluctuating

velocities of channel turbulence remain normal up to a wall distance approximately

one-third of the spanwise periodicity, which would include the whole present channels.

We will concern ourselves with the behaviour of the flow above the viscous layer,

and, as a safeguard against possible effects of the box size, mostly centre on the

region below y/h≈0.6. We use u,vand wfor the streamwise, wall-normal and

spanwise components of the velocity vector u, and x,yand zfor the respective spatial

coordinates. Capital and primed symbols respectively represent mean and root-mean-

squared quantities with respect to the average h i, defined over the two homogeneous

directions and time. Instantaneous wall-parallel averages, which are equivalent to the

(0,0)modes of the two-dimensional (x,z)Fourier expansions of the corresponding

quantities, are denoted, for example, by bu00. The ‘+’ superscript refers to quantities

normalized with the friction velocity uτand with the kinematic viscosity ν. Density is

assumed unity and dropped from the equations. Most of the simulations in this paper

have friction Reynolds numbers h+≈950. A second set of simulations was run at

h+≈550, with similar results. They are not emphasized, because of their limited scale

separation. but they are occasionally used to differentiate between inner and outer

length scalings.

Note that even if the Reynolds number is marginal from the point of view of scale

separation, the range between y+≈100 and y/h≈0.5(y+≈500)can be considered

as relatively free from both viscous and outer effects. We will loosely refer to that

590 F. Tuerke and J. Jiménez

range as ‘logarithmic’, in the spectral sense that the wavelength of the peak of the

premultiplied velocity spectrum grows linearly with y(see, for example, figure 1 in

Jim´

enez 2012). A more conservative definition of the logarithmic layer, in terms of

the mean velocity profile, only extends to y/h≈0.15, especially in boundary layers

(Nagib et al. 2006), but that outer limit is mostly due to the relatively strong ‘wake’

component of the mean velocity profile in external flows, which is a consequence of

the rotational–irrotational intermittency along the edge of the boundary layer (Jim´

enez

& Hoyas 2008). The wake is much weaker in internal flows, and it was shown

by Mizuno & Jim´

enez (2011) that the logarithmic fit of the mean profile can be

extended in channels to y/h≈0.5 by a simple modification of the classical law.

Moreover, neither the logarithmic mean profile nor the linear spectral scaling will be

particularly important in most of the following. We will basically only require that

viscous effects be negligible (y+100), which is easily satisfied for most of the

h+=950 simulations.

The numerical code integrates the incompressible Navier–Stokes equations for the

wall-normal vorticity and for the Laplacian of the wall-normal velocity, as in Kim,

Moin & Moser (1987), with the mass flux kept constant. The spatial discretization

is dealiased Fourier in the homogeneous wall-parallel directions, and Chebychev in

the wall-normal one, using 768 ×385 ×768 collocation points along x,yand z.

The collocation resolution is 1+

x=7.8, 1+

z=3.9 and 1+

y=0.03–7.7, and time

advancement is third-order semi-implicit Runge–Kutta (Spalart, Moser & Rogers

1991). The simulations were run for 10 eddy-turnover times, h/uτ, and statistics were

accumulated over the last 6h/uτ. The initial condition for all the simulations was the

same statistically converged field of a standard simulation at the same h+.

In all but the reference numerical experiments, or in the case F92 discussed

below, the evolution equations for the (0,0)Fourier modes of the three velocity

components are substituted by fixed prescribed mean profiles. Their evolution equation,

e.g. ∂tbu00 =b

Nx,00, where b

N00 is the wall-parallel-averaged right-hand side of the

Navier–Stokes equation, is therefore substituted by

∂tbu00 =∂yb

τx,00 −c

∂xp00 +fx=b

Nx,00 +fx,(2.1)

where τx=ν∂yu−uvis the total streamwise shear stress. Requiring

∂tbu00 =0,(2.2)

defines a body force per unit mass,

fx=bfx,00 = −b

Nx,00,(2.3)

that depends only on yand on time. Since the net effect of (2.3) is to enforce (2.2),

and therefore to keep bu00 identical to its initial condition, (2.1) is implemented in the

code by prescribing bu00 at each time step, using (2.3) to compute fx. The spanwise

component is similarly set to b

w00(y,t)=0, while b

v00(y,t)vanishes from continuity.

Note that the only direct effect of bu00 on the stress b

Nx,00 is through the viscous

term ν∂yybu00, which is negligible above the buffer layer. Its direct contribution to

the Reynolds stress, hbu00vi=hbu00b

v00i, vanishes identically. The force in (2.3) is only

defined up to a constant, which can be absorbed into the mean pressure gradient, c

∂xp00,

which is adjusted every time step to maintain a constant mass flux. From now on, we

Turbulent channels with prescribed profiles 591

Symbol βyb/h h+Ubh/ν Prescribed

C90 –4– 0 N/A 934 18518 Profile

C91 —— +0.5 N/A 950 20012 Profile

C92 – – – −1 N/A 950 17654 Profile

F92 - ◦- - −1 N/A 950 17654 Force

B9L –H–−1 0.25 931 18518 Profile

TABLE 1. Parameters of the IRANS profiles. The mixing parameter βis defined in (2.5),

and ybis the centre of the blending layer.

will assume that

Z2h

fxdy=0.(2.4)

In the simplest IRANS case C90, the streamwise mean velocity is fixed to the long-

term average, Un(y), of a standard simulation in a similar computational box, which is

assumed to be ‘correct’, and that will be denoted from now on as ‘natural’. It agrees

with the larger channel at the same Reynolds number by del ´

Alamo et al. (2004).

The IRANS experiments, C91 and C92, summarized in the upper part of table 1, use

a mixed mean profile

bu00 =U(y;β) =(1−β)Un(y)+βUC(y), (2.5)

where UCis the profile obtained by integrating ∂yUC=u2

τ(1−Y)/νt, where Y=y/h,

and the total viscosity,

νt

ν=1

2+1

2(1+κ02h+2

92Y−Y223−4Y+2Y221−exp −y+

A2)1/2

,(2.6)

depends on κ0and A(Cess 1958). The best fit to the natural profile at h+=950

is κ0=0.438 and A=27.76, but the IRANS mean profiles are constructed using

(2.5)–(2.6) with the ‘unnatural’ values A=51.5 and κ0=0.5, which tend to increase

the shear near the wall if β > 0, and near the centreline if β < 0. The friction velocity

and the molecular viscosity are adjusted to fix h+, and to keep the same mass flux as

the natural simulations.

A second set of profiles was constructed from C90 and C92, blending β=0 near

the wall to β= −1 away from it, using a cubic spline in |y−yb|⩽0.075h, continuous

and with continuous first derivatives at its two end points (see figure 1). The profiles

were later made symmetric with respect to the channel centreline. Three blending

locations were used to represent the buffer, logarithmic, and outer layers, but only the

‘logarithmic’ blend B9L is discussed here (yb=0.25h, see table 1). The viscosity and

mass flux of these simulations are kept the same as in the natural one, which requires

rescaling uτand h+slightly.

The mean velocity and shear profiles for the IRANS simulations are shown in

figure 1. The wall-scaled profiles in figure 1(a) differ appreciably from each other,

mostly because uτis changed by (2.5), but figure 1(b), where the velocities are

normalized with the natural profile, shows that the difference is never more than 5%.

Note the closer agreement in figure 1(a) of C90 and B9L, which share the same wall

shear; the slight mismatch in figure 1(b) between the inner parts of C90 and B9L

is due to the rescaling of the friction velocity mentioned at the end of the previous

592 F. Tuerke and J. Jiménez

(a)(b)(c)

101102103

0.95

1.00

1.05

0 0.5 1.0 0

0.6

1.0

1.4

0.5 1.0

FIGURE 1. Prescribed profiles for the IRANS simulations. (a) Mean velocity in wall units.

(b) Velocity and (c) shear, normalized with the unconstrained profiles. Symbols as in table 1.

–3

0 0.5 1.0 0.5 1.0

(a)(b)

FIGURE 2. (a) Average body force hfxi, obtained from (2.3). The solid dots are obtained

from the overall momentum balance in (3.3), as a test for the statistical equilibrium of the

simulation. (b) Root-mean-squared body force fluctuations. Symbols for C90–C92 are as in

table 1.

paragraph, but is difficult to appreciate in either figures 1(a) or 1(c) because their

vertical scales are respectively 100 and 10 wider than in figure 1(b). Figure 1(c) shows

that the shear ∂yUvaries more than the velocity, ∼20 %, although the changes are

still an order of magnitude weaker than the natural vorticity fluctuations, which can be

estimated from the energy dissipation to be |ω0| ≈ (κy+)1/2∂yU, where κis the K´

arm´

constant, and are therefore at least 4–6 stronger than ∂yUabove the buffer layer.

Figure 2(a) presents the time-averaged IRANS force, Fx= hfxi, obtained from (2.3).

Comparing figures 1(c) and 2(a) suggests that Fxis more closely related to the

difference between the natural and prescribed mean shears, which intersect one another

near Y=0.1, than to the mean velocities, which intersect near Y=0.5. That was

confirmed by testing a case whose shear intersects the natural one at a different

location from the ones above. The sign of the force tracks approximately the excess

in shear with respect to the natural profile. We will briefly return to this point when

discussing figure 4.

3. Results

The first interesting result is that fixing the mean velocity profile to its natural

value, as in case C90, does not change the intensity of the fluctuations, showing that a

Turbulent channels with prescribed profiles 593

0 0.5 1.0

0 0.5 1.0 0 0.5 1.0

(a)(b)

(c)(d)

0.6

0.8

1.0

1.2

0 0.5 1.0

0.5

1.0

1.5

FIGURE 3. (a) Streamwise, (b) wall-normal, and (c) spanwise velocity, and (d) pressure

fluctuation intensities, for the simulations in table 1. Symbols as in the table.

feedback cycle between the mean profile and the fluctuations, such as the one in the

example (1.1), is not required for the stability of the flow, and that any effect of the

fluctuations on the mean profile can be replaced by a hard prescription. Note that this

is not a trivial result. Figure 2(a) shows that the mean IRANS force vanishes in C90,

but its instantaneous value does not. Figure 2(b) shows that the root-mean-squared

temporal fluctuations of the force, f0

x, are of the same order as the driving pressure

gradient u2

τ/hover most of the channel, and substantially higher near the wall.

On the other hand, the velocity fluctuation intensities are very sensitive to minor

variations of the mean profile. Figure 3shows intensity profiles for the IRANS

simulations. The ‘flat’ case C91, which has a stronger shear near the wall than the

natural one, has stronger intensities in the buffer layer and slightly weaker ones above

it. The ‘round’ profile C92 which has a stronger shear in the central part of the

channel and a milder one near the wall, shows the opposite trend, and the same is true

for the pressure fluctuations in figure 3(d).

Figure 3also includes the blended profile B9L, which will be discussed later,

and case F92, which was run by adding to the right-hand side of (2.1) a constant

volumetric force equal to the mean Fxcomputed for C92, instead of fixing the mean

velocity. That simulation is also included among the mean profiles of figure 1(a) and

there, as well as in figure 3, the results are close enough to C92 for the lines to

overlap and to be hard to distinguish visually. The prescribed-force simulation has

also been added to figure 1(b,c), where the differences are more visible, suggesting

a slightly stronger wake component in F92 than in C92. However, note again the

594 F. Tuerke and J. Jiménez

0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00

(a)(b)

FIGURE 4. (a) Tangential Reynolds stress. (b) Total stress. Symbols as in table 1.

different vertical scales of figure 1(a–c). The similarity between fixing the force and

the velocity is consistent with the previous observation that fixing the natural IRANS

profile, which involves a force with zero average and large temporal fluctuations, gives

similar results to the unconstrained simulations, in which the forcing is kept fixed to

the same (zero) average.

Those results suggest that the forcing acts mostly through the effect of the mean

profile on the fluctuations, rather than by injecting energy directly into them. That this

is the case can be shown from the equation for the fluctuating kinetic energy,

∂th|u|2i/2− hu·Ni=hu·fi.(3.1)

The right-hand side is the energy injected by the IRANS force directly into the

fluctuations. It vanishes identically because fis spatially uniform over wall-parallel

planes. It only has a (0,0)Fourier mode, and is instantaneously orthogonal under

spatial averaging to all but the (0,0)Fourier component of the velocity, so that

hu·fi=hbu00 ·fi.(3.2)

The average in the right-hand side of this equation involves only time, but, in each

of the two types of simulations mentioned above, constant-velocity or constant-force,

one of the two factors is constant in time, and can be taken out of the average, while

the other one has zero mean. The direct effect of the force on the fluctuating energy

therefore vanishes.

Figure 4(a) shows that the tangential Reynolds stresses, −huvi, of the IRANS

simulations differ significantly from one another, and figure 4(b) shows that those

differences are not compensated by the viscous stress. In natural simulations, the

gradient of the total stress hτxi = u2

τ(1−Y)balances the mean pressure gradient

(Townsend 1976, p. 133), and it follows from (2.1) that the deviation of the observed

hτxifrom that linear law is due to the averaged IRANS force. Averaging (2.1)

over time, integrating over yto eliminate the mean pressure gradient, and using

the normalizing condition (2.4), one obtains

hhfxi/u2

τ= −1−∂Yhτ+

xi,(3.3)

which can be used as a check on the average force computed directly from (2.3). The

results are included as heavy dots in figure 2(a). Note that, after integrating (3.3),

figure 4(b) can be interpreted as the total force exerted over a layer extending from

Turbulent channels with prescribed profiles 595

0.2

0.4

0 0.5 1.0

0.5

1.0

1.5

0.6

0.8

1.0

0 0.5 1.0

(a)

(c)

(b)

(d)

FIGURE 5. (a) Stress correlation coefficient, −huvi/u0v0. (b) Streamwise, (c) wall-normal,

and (d) spanwise velocity fluctuation intensities, normalized with the local friction velocity

(3.4). Symbols as in table 1.

the wall to a given distance, and therefore as the contribution of the forcing to the

momentum flux τx. Comparing it with figure 1shows even more clearly than figure 2

that the IRANS force follows the mean shear, rather than the velocity.

3.1. Active and inactive motions

Figure 5(a) reveals that, even with the differences just discussed, the correlation

coefficient of the fluctuations, cuv= −huvi/u0v0, is very similar for all the cases,

including the natural one, especially above the buffer layer. Since cuvcan be

interpreted as a measure of the efficiency of the turbulent eddies in transporting

momentum, this suggests that the structure of the Reynolds-stress-bearing eddies in

the IRANS channels, or at least the way that they transfer momentum, is not very

different from those in natural ones. It also suggests that the fluctuation profiles of

the different simulations should collapse better when normalized with a ‘local’ friction

velocity defined as

u∗(Y)= [hτxi/(1−Y)]1/2,(3.4)

which takes into account the stresses introduced by the IRANS force, instead of with a

single uτ. That is confirmed by figure 5(b–d), which includes flat and rounded IRANS

profiles, as well as the blended case B9L, whose stresses have an almost discontinuous

jump near y=0.25h. Some non-equilibrium effects visible in figure 5within and

above the blending layer of B9L will be analysed in more detail in the next section,

596 F. Tuerke and J. Jiménez

but, even in that extreme case, the collapse is much better than the classical scaling in

figure 3.

It is interesting that the collapse with the local stress degrades near the wall for the

wall-parallel intensities and for cuv. That behaviour is consistent with the influence of

the ‘inactive’ motions proposed by Townsend (1961), who noted that the intensities

of the wall-parallel velocity components do not need to decay as they approach

the wall, except within the viscous layer, but that vand the tangential Reynolds

stress are blocked by the impermeability condition. Therefore, flow structures with

wall-parallel dimensions O(λ) would be inactive from the point of view of momentum

transfer below some y∼O(λ), carrying energy but no tangential stress. Townsend

(1961) also proposed that structures that are inactive near the wall are maintained by

Reynolds stresses further up, so that inactive motions are the ‘roots’ of detached active

structures. Spectra of an attached (u)and a detached (−uv) variable are represented in

figure 6(a,b) as context for the discussion below. The attached spectrum Euu extends

all the way to the wall, even for the longest scales of the large computational boxes,

Lx=8πh, in figure 6(a), but Euvis restricted to wavelengths λ∼y, leaving an inactive

‘wedge’ of long structures near the wall (Perry & Abell 1975; Perry et al. 1986). A

recent compilation of similar data over a wide range of Reynolds numbers can be

found in Jim´

enez & Hoyas (2008).

The classical theory (Townsend 1976, pp. 135–139) for the scaling of the

velocity fluctuations with uτis that they have to carry the tangential stresses,

τ≈u2

τ≈ −huvi ∼ u02. That argument would not seem to apply to inactive structures,

but is restored if those structures are the roots of detached active ones, and inherit

their scaling from them. The progressive accumulation near the wall of increasingly

long and wide inactive scales as the Reynolds number increases, each of them with

intensities proportional to u2

τ, is behind the well-known observations of logarithmic

intensity profiles of attached velocity fluctuations, (Townsend 1961; Jim´

enez & Hoyas

2008; Hultmark et al. 2012) of the logarithmic growth of the near-wall maximum of

u02+(deGraaff & Eaton 2000; Metzger & Klewicki 2001; Hoyas & Jim´

enez 2006), and

of the k−1spectrum of the streamwise velocity (Perry & Abell 1975; Perry et al. 1986;

Perry & Li 1990). Recent reviews are Smits, MacKeon & Marusic (2011) and Jim´

enez

(2012).

In natural flows, it is difficult to distinguish between active and inactive motions

from their intensities because, as we have just seen, they share the same velocity scale,

but the fluctuations in IRANS simulations scale differently near and far from the wall,

and active motions, whose scale is determined locally, can be easily separated from

inactive ones, whose velocity scales are determined by their remote active cores. That

is seen in figure 6(c), which is an enlargement of the near-wall region of figure 5(a),

with added cases at different Reynolds numbers. All the correlation coefficients cuv

have minima below y+≈100, where u0does not scale well with the tangential stress,

partly because viscosity carries some of the total stress, but mainly because of the

inactive component just discussed. The minima of the two IRANS cases, both of

which correspond to ‘rounded’ profiles with stronger outer fluctuations, are deeper than

in the natural cases, and it is tempting to interpret that observation as evidence that

the inactive components of those cases scale with their stronger outer u∗instead of

with the local ones. That the effect is confined to the inactive wedge of wavelengths

is confirmed by figure 6(d), which presents the difference between the streamwise

spectra Euu(λx,y)of the IRANS simulation C92 and the natural one C90. Both are

normalized with their local u∗, which works well in the active core that includes most

Turbulent channels with prescribed profiles 597

103

102

103

102

102103104102103104

103

102

102103

0.3

0.4

0100 200 300

0.5

1.0

1.5

0.5

1.0

1.5

50 100 50 100

(a)(b)

(c)(d)

(e)(f)

FIGURE 6. (a) Normalized premultiplied spectrum, kxEuu/u02of natural channels, against

the streamwise wavelength. Wall scaling. Contours are 0.08(0.04)0.2. ——, h+=934 (del

Alamo et al. 2004); – – –, h+=2003 (Hoyas & Jim´

enez 2006). (b) Same as (a) but

for −kxEuv/u0v0. Contours are 0.05(0.02)0.11. (c) Stress correlation coefficient, −huvi/u0v0.

——, Natural channels; – – –, IRANS with β= −1. Lines with circles are h+=550 and

those without are h+≈950. –O– natural channel at h+=2003, included for comparison.

(d) Premultiplied spectrum of the excess streamwise velocity, in local friction scaling,

kx(E∗

uu,C92 −E∗

uu,C90). Contours are −0.05(0.05)0.5. The thicker contour is 1Euu =0, and the

dashed one is negative. The dashed diagonal in (a,b,d) is λx=9y. (e) Mean-squared intensity

of the large-scale streamwise-velocity fluctuations for wavelengths λx⩾2hand λz⩾h, in

local friction scaling. Symbols as in table 1. (f) As in (e) but normalized with the local

friction velocity measured at y/h=0.4.

598 F. Tuerke and J. Jiménez

of the energy, but fails in the large-scale inactive wedge, where the IRANS normalized

spectrum exceeds the natural one.

That this is due to the scaling of the inactive motions with the stress of the outer

fluctuations is tested in figure 6(e,f), which presents the behaviour of the large-scale

velocity fluctuations near the wall. Those streamwise fluctuations have been filtered

to include only wavelengths longer and wider than λx=2hand λz=h, and are only

plotted below y+=100. The figures therefore only include contributions from the

inactive wedge. In figure 6(e) they are scaled with their local u∗, and the collapse is

poor. Inspection of figure 6(b) suggests that the active cores of structures with those

wavelengths are located at y/h≈0.4–0.5, whose stresses would provide their velocity

scales. Figure 6(f) includes the same data as figure 6(e), but scaled with u∗(0.4h). The

scaling is much better, and the same is true when u∗is taken at y=0.5h. Note that

the almost discontinuous stress jump in the blended case B9L, which lies between the

intensities in the figure and their assumed scaling velocities, constitutes a particularly

stringent test for the non-local scaling of the inactive motions.

Bradshaw (1967) used a similar technique to test the influence of the intensity of the

outer velocity fluctuations on the near-wall inactive motions, modifying the outer shear

of his boundary layer by means of a pressure gradient.

3.2. Energy balance

The plateau of cuvover y/h≈0.1–0.6 is usually associated with the equilibrium

logarithmic layer. We saw in the previous section that the part of the classical

theory linking the velocity fluctuations with the tangential Reynolds stress holds in

the IRANS simulations. On the other hand, the argument stating that the mean profile

adjusts itself so that the production of turbulent energy, Π= −huvi∂yU, balances the

dissipation, does not hold. Figure 7(a–c) shows the production, the dissipation, and

the ratio Π/. In the natural case, production exceeds dissipation in the buffer layer,

and both are roughly in equilibrium across the logarithmic layer. Above y≈0.5h, the

production decreases as both huviand the mean shear approach zero at the centreline,

while the dissipation does not vanish. The resulting energy deficit is compensated by

the gradient of a turbulent diffusion flux, ∂yhvKi, where Kis the kinetic energy, whose

ultimate source is the production excess near the wall (figure 7d). Other contributions,

such as the pressure and viscous diffusion terms, are negligible except very close to

the wall (Mansour, Kim & Moin 1988; Hoyas & Jim´

enez 2008; Tuerke 2011).

Figure 7(a–c) shows that the same general balance applies to the IRANS simulations

C91 and C92, although the details are different. The production and the dissipation

follow trends similar to the intensities; both are stronger where the shear is higher

than normal, and lower where it is weaker, but those changes do not cancel each

other everywhere. The ratio Π/ in figure 7(c) deviates more from equilibrium in

the IRANS simulations than in the natural one, specially below y≈0.3hand in

the blended case. The turbulent diffusion fluxes change to compensate the imbalance

(figure 7d), to the point that their gradients occasionally switch sign with respect to

their natural values. The overall flux, hvKi, remains positive in all our simulations.

The importance of the advective energy fluxes argues against local equilibrium

models, which was one of the questions posed in the introduction, and supports a

more distributed view of the energy balance. For example, the ratio Ladv= hvKi/ is

a ‘diffusion length’ representative of the thickness of the layer whose dissipation could

be fully accounted for by the advective energy flux, and figure 7(e) shows that Ladvis

of the order of or larger than yacross the equilibrium layer. The implication is that

the energy of the turbulent structures does not dissipate locally, but is advected over

Turbulent channels with prescribed profiles 599

0.2

0.4

0.2

0.4

0.5 1.0 0.5 1.0

0.5

1.0

1.5

0 0.25 0.50 0 0.25 0.50

–0.5

0.5

0.25 0.50 0.75 1.00

0.5 1.0

(a)(b)

(c)(d)

(e)(f)

FIGURE 7. (a) Compensated turbulent energy production. (b) Compensated dissipation. (c)

Ratio of the turbulent kinetic energy production to the dissipation. (d) Turbulent transport

contribution to the energy budget. (e) Turbulent diffusion energy length. Symbols in (a–e) as

in table 1. (f) Transition in the blended case B9L, from the natural profile C90 below the

blend, to C92 above it, in terms of the fractional coefficient defined in (3.5). –◦–, kinetic

energy production; – – –, streamwise fluctuations, u02; ——, transverse fluctuations, v02+w02;

–O–, dissipation. The dashed horizontal lines are φ=0, C90 and φ=1, C92.

layers of the order of their distance to the wall before equilibrating. That advection is

what appears as a turbulent diffusion flux. It was shown by Flores & Jim´

enez (2010)

that individual eddies, marked in their case by a particularly strong tangential stress,

move towards and away from the wall with wall-normal advection velocities of the

order of uτ, which agrees with the observed values of v0, and, since the plane-averaged

wall-normal velocity b

v00 vanishes at all times, the flux hvKiis a weighted average of

the energy being advected upwards and downwards. The positive fluxes in figure 7(e)

600 F. Tuerke and J. Jiménez

imply that the eddies moving away from the wall are stronger on average than those

moving towards it, which is consistent with the shape of the intensity profiles. In

natural channels such processes are difficult to disentangle because all eddies share

a common velocity scale, but IRANS simulations provide a way of manipulating the

fluxes by imposing different velocity scales at different wall distances.

The blended IRANS simulation B9L was undertaken for that purpose. As described

in § 2, it is constructed from the natural profile C90 near the wall and the rounded one

C92 above the blend, with a sharp increase in the shear across the blending layer. Its

energy budget near the wall is essentially normal, presumably because the time scales

in the vicinity of the buffer layer are fast enough to equilibrate the flow. Above the

‘discontinuity’, both the production and the dissipation increase, but it is interesting

that, while the production in figure 7(a) adapts almost immediately to that of C92,

the change of the dissipation in figure 7(b) is more gradual. In particular, note that

the dissipation profile does not show any trace of the dip in the energy production

within the blending layer. As a result of those different adaptation rates, the ratio Π/

increases sharply above the blend (figure 7c), and the diffusion fluxes in figure 7(d)

change accordingly.

Figure 7(f) compares the different relaxation rates by means of the ratio

φξ=ξ+

B9L−ξ+

C90

ξ+

C92 −ξ+

C90

,(3.5)

which is plotted for different (ξ) quantities, and which changes from φ=0 for C90 to

φ=1 for C92. It is difficult not to interpret that plot as an evolution from left to right.

The generation and transfer of turbulent energy in parallel shear flows starts with the

tangential Reynolds stress, which feeds energy into the streamwise velocity component,

from where it is redistributed by pressure to the transverse velocities, and finally to

viscous dissipation across the inertial energy cascade (Pope 2000, pp. 315–317). Each

of those steps takes a fraction of an eddy-turnover time (Kolmogorov 1941; Kerr

1990), which is O(y/uτ)in wall turbulence.

The simplest interpretation of figure 7(f) is that, as the eddies move across the blend

into the region of stronger shear, their energy production rate and streamwise velocity

fluctuations, u02, increase almost immediately. After a lag, which can be estimated

from the figure as 1y≈y/3, the transverse components, v02+w02, begin to receive

energy and grow, and it is only after a further lag of the same magnitude that the

dissipation begins to adjust. Note that the velocities, which start to receive energy

before the dissipation has been fully established, overshoot their asymptotic values,

and only decay towards it after the viscous dissipation has had time to act. That is

especially clear in u02, while the transverse velocities never have time to start decaying

in this particular simulation. Note that the spatial offsets just mentioned are consistent

with temporal lags of the order of an eddy turnover with an advection velocity of

the order of uτ, as discussed above. Note also that an alternative interpretation

emphasizing the advection of fluid particles instead of coherent eddies would result

in essentially the same observations. In that sense, the two interpretations could be

considered equivalent, although the long relaxation lengths in figure 7(f) probably

require that the energy of the particles stays ‘unmixed’ for some time.

4. Conclusions

We have shown that simulations of plane turbulent channels with prescribed mean

velocity profiles can be used to study the physics of turbulent fluctuations above the

Turbulent channels with prescribed profiles 601

buffer layer. In the first place, we have shown that fixing the mean velocity to its

correct value does not change the magnitude of the velocity fluctuations, while even

slightly incorrect profiles have a large effect on the intensities, following the local

mean shear rather than the mean velocity. Fixing the velocity profile is equivalent

to applying a volumetric force, uniform over wall-parallel planes, but with relatively

strong temporal fluctuations. Only the temporally averaged force appears to matter, and

a simulation in which the fixed velocity profile was substituted by an equivalent fixed

force gave similar results. We have interpreted those results as disproving models, such

as the one discussed in the introduction, in which the feedback of the fluctuations

on the mean profile, or more generally of the velocity profile averaged over relatively

large regions, is essential to the stability of the equilibrium of the flow.

Nevertheless, the most interesting results refer to the dynamics of the smaller eddies,

rather than to the mean velocity profile. The equilibrium of the latter requires the

interplay of relatively weak forces, such as the mean viscous force or the Reynolds

stresses, and evolves over correspondingly slow time scales. In most of the present

experiments, the mean profile is simply fixed. The fluctuations have faster time scales,

and respond to stronger local forces, mainly pressure. Consider an eddy, defined in

the sense of Obukhov (1941) as a fluid parcel of characteristic size λand internal

velocity differences uλ, which stays identifiable over a turnover time of order λ/uλ.

It undergoes accelerations of order u∇u∼u2

λ/λ, implying forces per unit mass of

the same magnitude. Assuming inertial scaling, uλ∼(λ)1/3≈uτ(λ/y)1/3, the implied

forces are of order u2

τ(y2λ)−1/3, and are stronger than the Reynolds stress gradient,

O(u2

τ/h), as long as λ/y< (h/y)3. A similar argument shows that the eddy forces are

stronger than the mean viscous force, ν∂yU∼νuτ/y2, if λ/y<y+3. Both are easily

satisfied for all reasonably isotropic scales in the overlap region, for which λ.y.

That is probably the reason why we find that the details of how the mean profile is

modified are irrelevant to the behaviour of the fluctuations, which only feel its effect

when integrated over an eddy turnover.

The experiments in this paper have been designed to explore the internal structure of

eddies, in the sense just defined, by creating profiles that interact with them differently

at different heights, such as with higher shear near or far from the wall, or with sharp

shear changes. That allows us to untangle internal energy-redistribution paths which

are difficult to separate otherwise, without necessarily tracking the eddies individually.

Consistent with the decoupling predicted by the previous arguments, the isotropy

coefficients of the eddies weakened or strengthened by the profile-fixing experiments

are similar to those of natural channels above the buffer layer, even if the intensities

are incorrect. In particular, the correlation coefficient between uand vchanges little,

and the intensity profiles collapse well when normalized with a y-dependent friction

velocity that takes into account the extra stresses introduced by fixing the profile.

The correlation coefficient near the wall is lower than normal in simulations that

have a higher shear in the central part of the channel, and vice versa, reflecting the

effect of the inactive motions carrying energy but no tangential stresses, as originally

hypothesized by Townsend (1961). In particular, we have shown that those inactive

eddies are restricted to a spectral ‘wedge’ of large scales near the wall, whose

intensities are not controlled by the local shear stress, but by the stresses in ‘active’

cores located farther from the wall at the same wall-parallel scales.

The production and dissipation of the kinetic energy are not in local equilibrium,

with the imbalance being compensated by turbulent diffusion, suggesting that it is due

to the different rates at which the various properties adjust to the shear when the fluid

602 F. Tuerke and J. Jiménez

is advected towards or away from the wall. Particularly instructive are simulations in

which the shear increases relatively sharply above a blending layer. The tangential

Reynolds stress is the fastest to respond to the higher shear, followed closely by the

energy of the streamwise fluctuations, then by that of the transverse fluctuations, and

finally by the dissipation, in agreement with the classical understanding of the energy

transfer in shear flows. The offsets between the evolution of the different variables

are of the order of the wall distance, strongly suggesting that the equilibrium between

production and dissipation in the inertial region has to be understood in terms of

the integrated energy over wall-attached layers, rather than locally at a single wall

distance.

If the offsets are interpreted as temporal delays in the evolution of eddies advected

with wall-normal velocities of the order of uτ, as suggested both by the v0profile and

by previous eddy-tracking studies (Flores & Jim´

enez 2010), the implied lags are of

the order of the local eddy-turnover time, in agreement with the classical Kolmogorov

(1941) cascade model.

Acknowledgements

This work was supported in part by the CICYT grant TRA2009-11498, by the

European Research Council grant ERC-2010.AdG-20100224, and by an allocation of

computer time from the Spanish Supercomputing Network. F.T. was supported by the

Erasmus program. We are indebted to A. G. Gungor for many fruitful discussions.

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