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Accepted for publication in J. Fluid Mech. 1

Roughness eﬀects on the Reynolds stress

budgets in near-wall turbulence

Junlin Yuan1†, Ugo Piomelli1

1Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON

K7L3N6, Canada

(Received ?; revised ?; accepted ?. - To be entered by editorial oﬃce)

The physics of the roughness sublayer are studied by direct numerical simulations of

an open-channel ﬂow with sandgrain roughness. A double-averaging approach is used to

separate the spatial variations of the time-averaged quantities and the turbulent ﬂuctua-

tions. The spatial inhomogeneity of velocity and Reynolds stresses results in an additional

production term for the turbulent kinetic energy, the “wake production”; it is the ex-

cess wake kinetic energy, generated from the work of mean ﬂow against the form drag,

that is not directly dissipated into heat, but instead converted into turbulence. The wake

production promotes wall-normal turbulent ﬂuctuations, and increases the pressure work

which ultimately leads to more homogeneous turbulence in the roughness sublayer, and

to the increase of Reynolds shear stress and the drag on the rough wall. In the fully rough

regime, roughness directly aﬀects the generation of the wall-normal ﬂuctuations, while,

in the transitionally rough regime, the region aﬀected by roughness is separated from the

region of intense generation of these ﬂuctuations. The budget of the wake kinetic energy

and the connection between the wake and the turbulence suggest strong interactions

between the roughness sublayer and the outer layer that are insensitive to the variation

of the outer-layer conditions. Furthermore, the present results may have implications for

the relationship between the roughness geometry and the ﬂow dynamics in the region

directly aﬀected by roughness.

Key words: Roughness, turbulence, direct numerical simulation

1. Introduction

Roughness plays an important role in many ﬁelds of study. A substantial amount of

work has been carried out to understand the dynamics of turbulent ﬂows over rough

walls, both for engineering and atmospheric applications. Roughness eﬀects on the ﬂow

are summarised by Raupach et al. (1991) and Jim´enez (2004). Roughness increases the

drag on the wall due to the pressure drag, resulted from the wake created downstream of

a roughness element. Many studies have shown that, far away from the wall, roughness

does not aﬀect the turbulent statistics, but it sets the velocity scale by increasing the

friction coeﬃcient.

Inside the roughness sublayer (deﬁned as the region where roughness causes spatial

variations of time-averaged turbulent statistics), the ﬂow dynamics are signiﬁcantly al-

tered as the turbulent structures become more three-dimensional. In order to capture the

interaction between the turbulence and the spatial variation of the time-averaged velocity

ﬁeld (wake), the space-time double-averaging approach can be used (Raupach & Shaw

†Email address for correspondence: junlin.yuan@queensu.ca

2J. Yuan and U. Piomelli

1982). Earlier studies of the ﬂow over canopies (Finnigan 2000; Raupach et al. 1991)

found that the wake leads to an additional source term in the budget of the turbulent

kinetic energy (TKE), the “wake production”.

Roughness and sediments are diﬀerent from canopies in the vertical distribution of the

density of the roughness elements. Recent studies on ﬂow over gravel beds (Mignot et al.

2009; Dey & Das 2012) found that the eﬀect of the wake on TKE is negligible compared

to the production due to the shear of the mean velocity (shear production).

Direct and large-eddy simulations have studied the roughness eﬀects on wall-bounded

ﬂows, e.g., Kanda et al. (2004), Coceal et al. (2006), Orlandi & Leonardi (2008), Leonardi

& Castro (2010), and Lee et al. (2011). Their main focus has been on the ﬂow above the

roughness and the parameterization of the drag. The layer below the roughness crest,

howeve, is also important; understanding this layer would provide insights to the rough-

ness eﬀects in the regions above. It was observed by Coceal et al. (2006) that the wake

depends on the geometry of the roughness, and that, although smaller than the Reynolds

stresses, the dispersive stresses (resulting from the wake component) are signiﬁcant in the

lower part of the roughness sublayer. However, the eﬀects of roughness on the budgets

of speciﬁc Reynolds normal stresses are not well studied. It is important to understand

the roughness eﬀects on the wall-normal and spanwise turbulent ﬂuctuations, since these

two components are more direct indicators (than TKE) of inner-outer-layer interactions,

near-wall structure instability, and drag generation. To this end, we use direct numerical

simulations of the turbulent ﬂow over sandgrain roughness in the transitionally and fully

rough regimes, to study the roughness eﬀects on the budgets of the Reynolds stresses,

and to investigate the energy transfer between the wake and the turbulence.

2. Problem formulation

2.1. Governing equations and numerical techniques

The incompressible ﬂow of a Newtonian ﬂuid is governed by the equations of conservation

of mass and momentum,

∂ui

∂xi

= 0,(2.1)

∂ui

∂t +∂uiuj

∂xj

=−∂P

∂xi

+ν∂2ui

∂xj2+Fi,(2.2)

where uiand Pare the velocities and pressure.

x1,x2and x3(or x,yand z) are, respectively, the streamwise, wall-normal and spanwise

directions, and ui(or u,vand w) are the velocity components in those directions; P=p/ρ

is the modiﬁed pressure, ρthe density and νthe kinematic viscosity. We performed

direct numerical simulations of (2.1) and (2.2) using a staggered grid and second-order,

central diﬀerences for all terms, a second-order semi-implicit time advancement, and MPI

(Message Passing Interface) parallelization (Keating et al. 2004). An immersed-boundary

method (IBM) based on the volume-of-ﬂuid approach (Scotti 2006; Yuan & Piomelli

2014) is applied: a body force, Fiis used to impose the no-slip boundary condition on

the rough surfaces; it is non-zero in the interface cell only. Details on the model can be

found in Yuan & Piomelli (2014).

2.2. Averaging approach

Following Mignot et al. (2009), a double-averaging (DA) approach is applied to decom-

pose a ﬂow quantity, θ, into space-time average, h(·)i(bar and brackets denote temporal

Roughness eﬀects on near-wall turbulence 3

and spatial averages, respectively), the spatial disturbance of the temporal average, f

(·),

and the turbulent ﬂuctuation (denoted by the prime),

θ(x, y, z, t) = hθi(y) + e

θ(x, y, z) + θ′(x, y, z , t).(2.3)

Two spatial averaging approaches are used (Nikora et al. 2007). The intrinsic spatial

average, h·i, is carried out in the (x, z )-plane, in the ﬂuid domain only, while the superﬁcial

average, h·is, is carried out over the whole (x, z)-plane.

Applying the intrinsic average to the time-averaged u-momentum equation for the

open-channel ﬂow, one obtains

−∂P

∂x +ν∇2u−∂u v

∂y −∂u′v′

∂y = 0.(2.4)

Note that the IBM force Fidoes not appear in (2.4) since it is zero inside the ﬂuid

domain. Equation (2.4) can be written as

−∂hPis

∂x +ν∂2huis

∂y2−∂hu′v′is

∂y −∂heuevis

∂y −*∂e

P

∂x +s

+νh∇2euis= 0,(2.5)

where the last two terms on the left hand side are, respectively, the pressure drag, fp, and

the viscous drags, fν(Raupach & Shaw 1982); they arise due to the non-commutativity

between the operators ∂ /∂xiand h·i below the roughness crest. The sum of these two

terms is the total drag. It can be shown that F1, averaged in the interface cell in the

(x, z)-plane, equals the total drag, −h∂e

P /∂xis+νh∇2euis.

2.3. Simulation parameters

An open-channel ﬂow of domain size 6h×1h×3hin x,y, and zis simulated; this size

is suﬃcient to accommodate the largest turbulent structures on a smooth-wall channel

ﬂow. No-slip and symmetric boundary conditions are applied to the bottom wall and

to the top boundary, respectively. Periodic conditions are used in both the x- and z-

directions. A constant pressure gradient is applied to drive the ﬂow. A virtual sandgrain

model (Scotti 2006) is used to impose roughness at the bottom boundary: the bottom

domain is separated into tiles of size around 2ks×2ks; in each tile a roughness element is

planted, which constitutes a randomly rotated ellipsoid with semiaxes equaling ks, 1.4ks,

and 2ks. Note that, for this virtual sandgrain surface, ksis known a priori since the

model is calibrated, in a speciﬁc range of roughness Reynolds numbers, to give the same

roughness function as the Nikuradse sandgrain with a height ks; however, the shape of

the elements is not exactly the same as those of Nikuradse. A fraction of the surface

is visualized in Figure 1(a) as iso-surface of the ﬂuid volume fraction equaling 0.5. The

Reynolds number based on the half-channel height (h) and the friction velocity (uτ) is

Reτ= 1000. Two roughness heights are used to yield a transitionally rough and a fully

rough ﬂow: a low-roughness case (R1, ks/h = 0.02, k+

s= 22), and a high-roughness one

(R2, ks/h = 0.07, k+

s= 72). The roughness crests (kc) for R1 and R2 are 3% and 9% of

h, respectively. A smooth-wall case (case SM) is also included for comparison.

Table 1 tabulates the parameters and spatial resolutions. Uniform grids are used in x

and z, while the grid is reﬁned near the wall in ydirection. For the rough cases, ∆y+<1

for y6kc, resulting in more than 31 and 51 grid points in ybelow the mean roughness

height and the crest, respectively. For the smooth case, ∆y+

min = 0.3, with 4 points below

y+= 1. The total number of grid points varies between 70 and 130 millions. The grid

sizes in xand zdirections are less than or equal to 12 and 6 wall units, respectively,

suﬃcient for DNS resolution.

4J. Yuan and U. Piomelli

Figure 1. (a) Visualization of 1/8 of the surface R2; wall-normal proﬁles of (b) DA

streamwise velocity ( huiand huis) and (c) roughness geometry function.

Case k+

sks/h kc/h k/h Nx×Nzni×nk∆x+×∆z+

SM 0 0 0 0 – 512 ×512 12 ×6

R1 22 0.02 0.03 0.013 7 ×7 1024 ×512 6 ×6

R2 72 0.07 0.09 0.040 20 ×20 1024 ×512 6 ×6

Table 1. Parameters for all cases. kc: roughness crest; k: mean roughness height. njranges

from 128 to 256. Nxi: grid points per sand-grain element.

The shape resolution of a roughness element is quantiﬁed by Nxi, the number of grid

points (in xi) used to resolve the shape of each sandgrain element. A grid-convergence

study (not shown) was carried out, and it was found that the present resolution is suﬃ-

cient to obtain the Reynolds stresses and their budgets. For all cases, data are collected

from a total simulation time of T= 50h/uτafter the transient. A convergence study

showed that decreasing the simulation time to 0.5Tleads to errors of less than 1% for

third- or lower order turbulent statistics.

3. Results

3.1. Turbulent statistics

The DA streamwise velocity is shown in Figure 1(b). Below the roughness crest, the

velocity varies almost linearly, consistent with Model III of the velocity distribution pro-

posed by Nikora et al. (2004) applied to rough-wall ﬂows with relatively low submergence

(h/ks). The superﬁcial average (huis= Φhui) is lower than the intrinsic one, since the

roughness geometry function, Φ(y), deﬁned as the fraction of space occupied by ﬂuid

in the horizontal plane at y, is less than one (Figure 1(c)). The distribution of Φ(y) is

not aﬀected by the variation of the spatial resolution. The zero-plane displacement (d),

representing the eﬀective elevation of the boundary layer due to roughness, is obtained

as the location of the centroid of the wall-normal averaged drag-force proﬁle (Yuan &

Piomelli 2014); d≈0.8ksfor both rough cases.

Figure 2(a) shows the DA velocity proﬁles in wall units. The roughness functions,

∆U+, obtained as the oﬀsets of hui+

sproﬁles in the logarithmic region from the universal

logarithmic law, are 3.5 and 7.8 for cases R1 and R2, respectively, yielding k+

s= 22 and

72 based on the Colebrook correlation (Colebrook 1939); case R2 is in the fully rough

regime, while case R1 is transitionally rough.

The Reynolds stresses in cases R1 and R2 are compared in Figure 2(b) with smooth-

Roughness eﬀects on near-wall turbulence 5

Figure 2. (a) Streamwise velocity and (b) Reynolds stresses for cases SM ( ), R1 (◦), and

R2 (△). + smooth-wall experiment (Schultz & Flack 2013). kR: top of the roughness sublayer.

Figure 3. Form-induced stresses for cases R1 (empty symbols) and R2 (ﬁlled symbols): (a)

heu2i+

s, (b) ▽hev2i+

s,△ h ew2i+

s, and ⋄ heuevi+

s.

wall experimental data from channel-ﬂow studies with the same Reynolds number (Schultz

& Flack 2013); agreement is obtained in the outer layer, supporting the wall similarity

hypothesis. Case R1 preserves a high near-wall peak of the streamwise normal Reynolds

stress, while the peak is signiﬁcantly damped for case R2, due to the destruction of the

buﬀer layer. The top of the roughness sublayer is denoted by kR; its location will be

discussed below. Inside the roughness sublayer, all normal Reynolds stresses in wall units

are damped, compared to case SM. The magnitudes of the other two shear stresses are

negligible compared to hu′v′i, due to the randomness of the rough surface.

The form-induced (or dispersive) stresses are shown in Figure 3 for both rough cases;

the wall-normal and spanwise components are smaller than the corresponding compo-

nents of the Reynolds stresses, consistent with previous experimental rough-wall and

canopy studies (Nikora et al. 2001; Mignot et al. 2009); the streamwise form-induced

stress is comparable to the Reynolds stress, a characteristic of relatively low submer-

gence (Manes et al. 2007). The peak of |heuevi+

s|is around 0.15, consistent with previous

DNS study by Coceal et al. (2006). The region with non-negligible heu2

iisis the rough-

ness sublayer, whose thickness, kR, is shown to be around 2ks. The magnitude of the

form-induced stresses relative to the Reynolds stresses increases with k+

s, since, in the

roughness sublayer, hu′2

ii+decreases while heu2

ii+increases with higher k+

s, due to the

lower submergence.

Figure 4 shows the stress balance for case R2; the mean pressure gradient is balanced

6J. Yuan and U. Piomelli

Figure 4. Stress balance of case R2. Total stress from momentum balance, △total

drag from momentum balance, total drag from F1integral, Reynolds shear stress,

◦form-induced shear stress, + viscous stress due to mean shear.

by the sum of other stresses,

τ+=Zh+

y+

(f+

ν+f+

p)dy+− hu′v′i+

s+∂hui+

s

∂y+− heuevi+

s= 1 −y+

h+.(3.1)

The viscous stress is small, consistent with fully rough ﬂow. The form-induced shear

stress is also small compared to the Reynolds shear stress. The total drag, (f+

ν+f+

p),

calculated from momentum balance, agrees with the IBM force, hF1is, which can be used

to calculate the total drag, as shown by Yuan & Piomelli (2014).

3.2. Budgets of Reynolds and dispersive stresses

In the turbulent ﬂow over roughness, the total kinetic energy can be decomposed into

three parts,

1

2huiuii=1

2huiihuii+1

2heuieuii+1

2hu′

iu′

ii,(3.2)

where the terms on the right hand side are, respectively, the mean-ﬂow kinetic energy

(MKE), the wake kinetic energy (WKE) and the TKE. The MKE is converted to small-

scale TKE (and consequently heat) in two ways (Finnigan 2000): ﬁrst, the energy goes

through the whole energy cascade; second, MKE ﬁrst generates WKE of scale ksdue to

the work of the mean ﬂow against the total drag, and the WKE generates TKE of scales

smaller than ksthrough the energy cascade (“short-circuited” cascade (Raupach et al.

1991)). The interactions between WKE and TKE are also two-ways: WKE generates

TKE of scales smaller than ks, while TKE is converted to WKE through the work of

large-scale turbulent structures against the form drag.

The budget of Reynolds stresses and TKE were derived by Raupach & Shaw (1982)

for ﬂow over canopy and by Mignot et al. (2009) for ﬂow over roughness. The budgets of

the normal Reynolds stresses are (no summation over Greek index)

−2hu′

αv′i∂huαis

∂y −2]

u′

αu′

j

∂euα

∂xjs

−∂

∂y h

]

u′

αu′

αevis−∂

∂y hu′

αu′

αv′is

−2u′

α

∂P ′

∂xαs

+ν∂2

∂y2hu′

αu′

αis−ǫαα = 0.(3.3)

Roughness eﬀects on near-wall turbulence 7

Figure 5. TKE budgets for cases SM (line), R1 (empty symbol), and R2 (ﬁlled symbol). All

terms normalized by u4

τ/ν.kc, roughness crest. ⋄Ps,△Pw,◦ǫ,⊲ Tt,⊳ Tν,▽Tp. Note

that the actual bottom-wall location varies.

Summing over the three components yields the budget of the TKE,

−hu′

iv′i∂huiis

∂y −g

u′

iu′

j

∂eui

∂xjs

−∂

∂y hg

u′

iu′

ievis/2

−∂

∂y hu′

iu′

iv′is/2−∂

∂y hP′v′is+ν∂2

∂y2hu′

iu′

i/2is−ǫk= 0,(3.4)

where the terms on the left hand side are, respectively, shear production (Ps), wake

production (Pw), transport due to wake, turbulent transport (Tt), pressure transport

(Tp), viscous transport (Tν), and viscous dissipation (ǫ). ǫis obtained as the residual of

the sum of all other terms. In the normal-stress budget the pressure work, Παα, is also

present.

The budget of WKE, 1/2heuieuii, was derived by Raupach & Shaw (1982). Taking into

account the vertically varying roughness geometry function, Φ(y), the budgets of the

normal dispersive stresses (heu2

αi) read

−2heuαevi∂huαis

∂y + 2 ]

u′

αu′

j

∂euα

∂xjs

−∂

∂y heuαeuαevis−2∂

∂y heuαg

u′

αv′is

−2*euα

∂e

P

∂xα+s

+ 2νheuα∇2euαis= 0 (3.5)

where the six terms on the left hand side are, respectively, shear production of WKE,

wake production (−Pw), transport due to wake, transport due to turbulence, work done

by pressure, and viscous diﬀusion and dissipation at the wake level.

The term Pwappears with opposite signs in the budgets of TKE and WKE; it is

the net transfer from WKE to TKE as the result of their interactions. Previous studies

on ﬂows above canopies found that Pwis usually positive (WKE converts to TKE)

with magnitudes comparable to Ps(Raupach et al. 1991; L´opez & Garc´ıa 2001). In the

experimental study of ﬂow over gravel beds (Mignot et al. 2009), however, the magnitude

of Pwwas found to be less than 5% of Ps.

8J. Yuan and U. Piomelli

Figure 6. Comparison of normal Reynolds stress budgets between cases R1 (empty symbols)

and R2 (ﬁlled symbols): (a) streamwise, (b) wall-normal, and (c) spanwise components. ⋄Ps,

△Pw,◦ǫ, and ▽Π; Ttfor case R1, Ttfor case R2 . All terms normalized by u4

τ/ν.

The TKE budgets normalized by wall units are shown in Figure 5. Case SM has been

validated with the smooth-wall channel-ﬂow DNS results reported by Hoyas & Jim´enez

(2008) (not shown). Above the roughness sublayer (y > kR), the R1 budget terms agree

well with case SM, but in the roughness sublayer, they diﬀer from the smooth case

mainly in that, approaching the wall, both ǫkand Tνbecome zero, indicating a quiet

region without turbulence at the root of the roughness.

Figure 5 also shows that, for both rough cases, the location of the peak of the shear

production is close to the roughness crest, connected to the shear layers formed near

the crest (Mignot et al. 2009; Ikeda & Durbin 2007). For case R2, Pspeaks at 0.73kc

from the actual bottom wall (where Φ = 0), while, for case R1, it peaks at 0.93kc. It was

observed by Mignot et al. (2009) that, for ﬂow over gravel beds, Pspeaks around 0.9kc.Ps

decreases from its peak location to the virtual wall. Dissipation peaks slightly below the

production peak; the transport terms are non-negligible in the roughness sublayer only.

Both the turbulent diﬀusion and the viscous diﬀusion take energy from the region with

high production (near kc) and transport it towards the wall to balance dissipation. These

observations are consistent with the literature (Finnigan 2000; Mignot et al. 2009; Hong

et al. 2011). Furthermore, the pressure transport removes TKE from near the roughness

crest, and transports it to the lower part of the roughness sublayer; for case R2, the

pressure transport is the most important source term in the lowest third of the sublayer,

consistent with LES results in ﬂow within forest canopy (Dwyer et al. 1997), while in the

transitionally rough case R1, the viscous diﬀusion is the most important source term in

the lowest part of the sublayer. Compared to Psand ǫ,Pwis much smaller, consistent

with Mignot et al. (2009). The sign and magnitude of Pware aﬀected by the roughness

height, which will be explained later.

Selected terms of the normal Reynolds stress budgets (in wall units) are shown in

Figure 6. The wall-normal distance is normalized using ks. As the ﬂow approaches the

fully rough regime from R1 to R2, P+

s,uu decreases signiﬁcantly; so do the viscous dissi-

pations of hu′2iand hw′2i, while, below the roughness crest, Π+

vv increases. This indicates

that roughness results in more energy being redistributed to hv′2i, and that stronger

wall-normal ﬂuctuations distort the near-wall quasi-streamwise vortices, homogenizing

the turbulence near the wall, consistent with the previous observations from DNS that

roughness disturbs the near-wall low-speed streaks (Lee et al. 2011).

For case R1, Π+

vv exhibits two peaks; the inner peak is located slightly below the

Roughness eﬀects on near-wall turbulence 9

Figure 7. Comparison of normal form-induced stress budgets between cases R1 (empty sym-

bols) and R2 (ﬁlled symbols): (a) streamwise, (b) wall-normal, and (c) spanwise components.

Line: shear production ( R1, R2), ▽pressure work, △ −Pw, and ◦viscous dissipation

and diﬀusion. All terms normalized by u4

τ/ν.

roughness crest, where the shear production peaks, while the outer peak is outside the

roughness sublayer, at (y−d)+≈30. The outer peak exceeds dissipation, with the excess

energy transported into the region of the inner peak. As the roughness height increases

from R1 to R2, the inner peak of Π+

vv moves upwards and merges into the outer peak.

Meanwhile, excess energy is gained around the roughness crest, and transported both

towards the wall and towards the outer layer. This indicates that, in a transitionally

rough ﬂow, the roughness is somewhat isolated from the most important region for the

generation of wall-normal turbulent ﬂuctuations, while, in a fully rough ﬂow, roughness

directly aﬀects its generation.

The wake production for hu′2iis mostly negative below kcfor R1 and and negligible

for R2; for the other two normal components, it is positive, with non-negligible peak

magnitudes compared to the pressure work (17%Πvv and 23%Πww in case R2). Such

diﬀerence may be due to the fact that hu′2iis mostly associated with turbulent structures

with larger scale than the roughness length scale, converting signiﬁcant amount of TKE

to WKE as they work against the form drag; therefore, more hu′2iis converted to WKE

compared to the amount being converted from WKE, hence the negative Pw,uu . On

the other hand, hv′2iand hw′2ireceive a more signiﬁcant contribution from turbulent

structures smaller than ks; thus hv′2iand hw′2igenerate less WKE than the amount

being converted from WKE through energy cascade. For v′and w′intensities, the wake

production is important, despite a weaker wake contribution to the TKE budget.

Figure 7 shows the dispersive stress budgets. Two signiﬁcant sources are present: the

shear production and the pressure work. The shear production is due to the work of the

mean ﬂow against the form-induced stress; it contributes to heu2ionly. The pressure work

of WKE can be written as

*eui

∂e

P

∂xi+s

=−∂he

Pevis

∂y +huis*∂e

P

∂x +,(3.6)

with the two terms on the right hand side being the pressure transport and the energy

gained from the mean ﬂow working against the form drag; the latter is shown contributing

signiﬁcantly to heu2iand is the only source of hev2iand hew2i. The WKE is partly converted

to TKE through wake production, partly dissipated into heat. As the roughness height

increases from R1 to R2, a larger portion of hev2iand hew2iis converted to Reynolds

10 J. Yuan and U. Piomelli

stresses instead of being directly dissipated, due to the further separation between the

roughness length scale and the viscous length scale.

Conclusions

Direct numerical simulations are carried out in open-channel ﬂows over sand-grain

roughness in the transitionally and the fully rough regimes. A double-averaging (DA)

technique is applied to separate the spatial disturbance of the time-averaged ﬁeld and

the turbulent ﬁeld. Signiﬁcant spatial variations of time-averaged ﬂow quantities are

generated due to the three-dimensionality of the roughness geometry. The form-induced

stresses are small compared to their Reynolds stress counterparts, but the magnitude of

this diﬀerence becomes smaller as the roughness height increases.

The work of the form drag near kcconverts mean-ﬂow energy to wake energy, which is

partly dissipated into heat and partly converted to turbulence through wake production.

The intensiﬁed wall-normal turbulent ﬂuctuations lead to higher pressure work near kc,

and more homogeneous energy redistribution between the Reynolds stresses. In the fully

rough regime, the wake dynamically aﬀects the region of hv′2igeneration, while, in the

transitionally rough regime, its eﬀect is limited below kc.

Despite signiﬁcant transfer from the wake to the turbulent component in the wall-

normal and spanwise directions, for the streamwise component, the conversion is mostly

from turbulence to wake energy, further contributing to a more even TKE redistribution;

this is possibly due to the larger turbulent scales that are associated with this Reynolds

stress component. The wake production of TKE, however, is negligible, because its values

for the three normal Reynolds stresses tend to cancel out, and the magnitudes of total

TKE production and dissipation are greater than the energy redistributed.

The current results shed some light on the role of the spatial inhomogeneity on the

generation of drag in rough-wall boundary layers. The spatial inhomogeneity, being de-

pendent on the roughness geometry, and in turn altering the energy redistribution and

strongly aﬀecting the Reynolds shear stress through its production, may provide the link

between geometry detail and the dynamics in the roughness sublayer. Furthermore, the

fact that the wake energy depends on the work of mean ﬂow against the form drag as

a source term suggests that the wake ﬁeld is more persistent compared to turbulence in

a non-equilibrium boundary layer: while the turbulence production relies on active ﬂuc-

tuations and the mean shear, the presence of the wake requires non-zero mean velocity

only. The persistent wake indicates strong interactions between the roughness sublayer

and the outer layer, insensitive to the variation of the outer-layer conditions.

Support from Hydro Qu´ebec, the Natural Science and Engineering Research Council of

Canada (NSERC), and the High Performance Computing Virtual Laboratory (HPCVL)

is acknowledged.

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