12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

ROUGH-WALL HEAT TRANSFER AT MODERATE PRANDTL NUMBERS:TOWARDS RECONCILING THE DIVERSE MODEL PREDICTIONSKevin Zhong1 , Nicholas Hutchins1 , and Daniel Chung1
1: Department of Mechanical Engineering, University of Melbourne, VIC 3010, Australia,
kevin.zhong@student.unimelb.edu.au

12

ABSTRACTANSYS CFX
wall function

In the fully rough regime, a rough-wall heat transfer coefficient, e.g. the roughness Stanton number, takes the form
Stk ∼ (k+ )−p Pr−m , but proposed values for p and m vary, giving diverse predictions. Here, k+ is the viscous-scaled roughness size and Pr is the Prandtl number. To help clarify this, we
conduct direct numerical simulations of forced convection in arough-wall channel for Pr = 0.5, 1.0, and 2.0, sweeping fromtransitional to fully rough conditions by varying k+ . Minimal
channels are used, which resolve the roughness sublayer at affordable cost enabling an extensive parametric study. In predicting the mean heat transfer, we find that our data favours theso-called surface renewal phenomenology of Brutsaert (1975),
which predicts p = 1/4, m = 1/2 over the Reynolds–Analogytype phenomenologies, which predict p = 1/2 for the fullyrough regime. For the transitionally rough regime, conventional understanding argues that turbulence retains a form similar to that of a smooth wall flow except for an offset (Luchini,1996). We show in our low-k+ data that these ideas, althoughoriginally describing momentum transfer, can also generalize
well to turbulent heat transfer.

8

MacDonald et al. (2019)4

Brutsaert (1975)

Dipprey & Sabersky(1963)

Owen & Thomson (1963)

0
DNS: Present-4
101

ANSYS FLUENT
wall function
Yaglom & Kader (1974)

102

103

104

Figure 1. The disparate predictions by heat transfer models
at Pr = 1.0 for heat-transfer augmentation ∆Θ + in the fullyrough regime (ks+ & 70). The models (solid lines) are tuned to
fit the trends of the high-ks+ DNS (markers).

INTRODUCTION

plan area, density ρ, specificheat capacity c p and globalpfriction velocity Uτ ≡ τw /ρ where τw is the wall shear
stress. The + superscripts denote normalizations on Θτ , Uτ ,and ν e.g. z+ ≡ zUτ /ν. The formulation of (1) represents
the effects of roughness through a heat transfer augmentation∆Θ + (k+ , Pr) relative to the smooth-wall intercept Aθ , and depends on the roughness Reynolds number k+ ≡ kUτ /ν where
k is a representative roughness size. Equation (2) adopts theroughness Stanton number Stk based on z0 coordinates (Kays& Crawford, 1993), where z0 is defined by the logarithmic
mean velocity profile U + = (1/κ) log[(z − d)/z0 ] with κ ≈ 0.4(Brutsaert, 1982). Equation (3) uses the g-function definedon the equivalent sand-grain roughness ks (Dipprey & Sabersky, 1963). Models for the fully rough regime usually have
+ p m
the form St−1k ∼ g ∼ (k ) Pr , which have seen diverse proposals for the exponents p = 0.20–0.50, m = 0.44–0.80, both
empirically and phenomenologically motivated (Chung et al.,2021). The correct values to take for p and m remain ambiguous (Li et al., 2020) and this difficulty is illustrated in figure 1 which shows fully rough models for Pr = 1.0. Eachmodel has coefficients which can be tuned to fit the high-ks+
direct numerical simulation (DNS) data, making inspection ofmodel trends alone insufficient in discerning the correct values to take for the exponents. Moreover, figure 1 presents datafor only a single working fluid (i.e. fixed Pr = 1.0) and thus
does not inform us as to how well the models generalize to arbitrary working fluids. Rather than compare model trends, a

In engineering and environmental applications, we often
encounter wall-bounded turbulent flows involving the transport of passive scalars such as temperature or contaminants.
The wall is seldom smooth in practice and is instead rough,causing drag and heat transfer enhancements. Heat transfer ata solid surface occurs through molecular conduction which depends on the thermal diffusivity of the fluid α and is accounted
for through the molecular Prandtl number Pr ≡ ν/α where νis the kinematic viscosity. In turbulent heat transfer, the influence of a rough-wall can be represented through intercepts tothe logarithmic temperature profile, Θ + (Chung et al., 2021):
Θ+ ≡Kays & Crawford (1993)
DNS: Peeters &Sandham (2019)Θ
1=log[(z − d)+ ] + Aθ (Pr) − ∆Θ + (k+ , Pr) (1)
Θτ
κθ
1
+
=
log[(z − d)/z0 ] + St−1(2)
k (k , Pr)κθ
1=log[(z − d)/ks ] + g(k+ , Pr)(3)κθ

where Θ (z − d) is the mean temperature relative to the wall
which varies with wall-normal distance, z, and d represents a
virtual-origin offset such that z − d measures the wall-normal
distance from the virtual origin plane (Chung et al., 2021).
Here, κθ ≈ 0.46 and Θτ ≡ hqw i/(ρc pUτ ) is the friction temperature with hqw i being the global heat flux defined on the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)sizes follow from MacDonald et al. (2017). Namely, Lx ≥max(3Ly , 1000ν/Uτ , λ ) and Ly ≥ max(100ν/Uτ , k/0.4, 2λ ).
The latter condition of Ly ≥ 2λ has been modified from MacDonald et al. (2017) so as to ensure the unphysical region,zc , lies above the roughness sublayer, which is approximately
zr ≈ λ /2 for our present sinusoids (Chan et al., 2018). Further smooth-wall minimal channel DNSs were also conducted
at matched Reτ and domain sizes with those outlined in table 1 to obtain domain-size-independent measures of the logintercepts in (1–3). For these log-intercept measurements to berobust, we must measure from the virtual origin by accounting for a shift, d, (Chung et al., 2021). Presently, we havedefined d to be the upwards displacement from the sinusoid
midplane. For low-k+ , d can be understood as a displacementof smooth-wall turbulence, which can be obtained by shifting
the Reynolds shear stress profiles to collapse with a smoothwall profile (Garcı́a-Mayoral et al., 2019). This approachhas been adopted for our k+ = [5.5, 11, 22] cases and yieldedd/k = [0.18, 0.15, 0.36]. At higher-k+ , this method may nolonger hold as the near-wall turbulent structure differs greatly
from that of a smooth wall (Jiménez, 2004). Our approach here
has been to perform an ad-hoc tuning of d to give a best-fit to
the logarithmic slopes, κ ≈ 0.4, κθ ≈ 0.46 which yielded theempirical fit d/k = −(k+ /365)2 + (k+ /213) + 0.27 and is fitted to our k+ = [33, 40, 56, 111] cases. Measures for e.g. ∆Θ +are then taken to be the difference between the smooth walland rough wall temperature profiles evaluated at the minimal+++
channel critical height: ∆Θ + ≡ Θsmooth(z+c ) −Θrough (zc ) andare provided in table 1. We also measured the roughness functions at various heights above the roughness sublayer thickness, zr ≈ λ /2 and below zc , which resulted in negligible variations for ∆Θ + no greater than 0.3.

(b)
Flow

Figure 2. (a) Channel flow schematic (not to scale). The
channel half-height h is measured from the sinusoid midplane.
(b) The present 3D sinusoids have amplitude k and fixed wavelength λ = 7.1k.

more robust approach is instead, to test the physical assumptions which underpin these models against high-fidelity DNS
data. The difficulty in this approach however, lies in the stringent computational cost associated with sweeping the (k+ , Pr)parameter space. Our present work will circumvent these difficulties through the use of minimal channels, which can accurately resolve the roughness sublayer at affordable cost andthereby measure the various log-intercepts in equations (1–3)(MacDonald et al., 2019). With a comprehensive set of DNSs
available, we scrutinise some of the phenomenologies describing fully rough heat transfer. These phenomenologies fall into
two categories: the surface renewal phenomenology of Brutsaert (1975) which proposes p = 1/4, m = 1/2 and ReynoldsAnalogy-type phenomenologies which propose p = 1/2 with
m = 2/3 or m = 3/4 (Owen & Thomson, 1963).
In the transitionally rough regime, characterized by lowto-moderate values of k+ , our conventional understanding of
this regime is one where the near-wall turbulence retains aform similar to that of a smooth wall, except shifted by an
offset (Luchini, 1996). We will show in our low-k+ DNS thatsome of these concepts can generalize well to the case of turbulent heat transfer.

THE FULLY ROUGH REGIME
In figure 3(a), we present measurements of the gfunction in (3) against ks+ coordinates where ks+ = 2.7k+
for our present 3D sinusoids, obtained by collapsing fullyrough drag measurements with the Nikuradse curve: ∆U + =(1/0.4) log(ks+ ) − 3.5 (not shown) (Nikuradse, 1933). The
data is shown alongside the Pr = 0.7 DNS of MacDonald
et al. (2019) for identical 3D sinusoids (note that the present
data have a different ks /k due to different virtual origin treatments), and the Pr = 1.2–5.9 data of Dipprey & Sabersky(1963) for close-packed granular roughness. In the fully roughregime, both roughness types favour p = 1/4 more closelythan p = 1/2 in g ∼ (k+ ) p ∼ (ks+ ) p . The empirical fits forg ∼ (ks+ ) p Prm in figure 3(b) yield p = 0.20, m = 0.40, similar to values of Dipprey & Sabersky (1963) and are close
to the analytical p = 1/4, m = 1/2 proposals of Brutsaert(1975). We observe in figure 3(c) that plotting St−1k ∼ g against
1/4 Pr1/2 is able to collapse the fully rough data onto a(z+)
0single asymptote. Here, a switch is made from ks to z0 whereks /z0 ≈ 30 in the fully rough regime, and z+
0 ≈ 0.135 for asmooth wall. Figure 3(c) then provides an overall view forheat transfer, from smooth to fully rough under a variety ofworking fluids. In figure 3(d), we show the roughness function ∆Θ + , which appears to plateau at ks+ ≈ 100 before beginning a gradual decrease. It is unclear whether this continual decrease of ∆Θ + towards a negative value will eventuallyoccur, as this would imply a reduction in heat transfer relative to a smooth wall, which stands in contrast to conventional
understanding of rough-wall heat transfer (Kays & Crawford,
1993). The observed ∆Θ + behaviour is also in contrast to dragwhich increases with ks+ according to the fully rough asymp-

DIRECT NUMERICAL SIMULATIONS
We consider a channel flow as illustrated in figure 2(a)and denote (x,y,z) as the streamwise, spanwise, and wallnormal directions. We solve the incompressible Navier–Stokes
equations and the passive-scalar equation for the fluid temperature, θ using a fourth-order finite-difference method. The
flow is forced at a constant mass flux, while the temperatureis driven through a streamwise-linearly-varying wall temperature gradient. The roughness is a 3D sinusoid with semiamplitude k and wavelength λ = 7.1k (figure 2b), defined byzw (x, y) = k cos(2πx/λ ) cos(2πy/λ ) and is mirrored on the
top and bottom walls of the channel (figure 2). No-slip andzero fluid–wall temperature contrast (θ = 0) conditions are
imposed through an immersed-boundary method. Our study
spans transitional cases at k+ ≈ 5.5 towards the fully roughregime at k+ ≈ 111 for Prandtl numbers, Pr = 0.5, 1.0, all documented in table 1. The friction Reynolds number varies fromReτ ≡ hUτ /ν = 395–2000 and the blockage ratio is h/k ≥ 18.
Minimal channels are used, which truncate the domain lengths
Lx × Ly whilst resolving the roughness sublayer and for allcases remain below the unphysical flow-region z = zc ≈ 0.4Ly(MacDonald et al., 2019), thus enabling an extensive setof 21 total rough-wall DNSs. Prescriptions for the domain

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Reτ

Pr

h/k

k+

Lx+

Ly+

Nx

Ny

Nz

∆x+

∆y+

∆z+
b

∆zt+

∆U +

∆Θ +

395

0.5

72

5.5

1091

156

768

128

420

1.42

1.22

0.26

5.55

1.5

0.3

395

1.0

72

5.5

1091

156

768

128

420

1.42

1.22

0.26

5.55

1.5

0.8

395

0.5

72

5.5

1091

156

768

128

560

1.42

1.22

0.21

3.89

1.5

1.9

395

0.5

36

11

1013

156

384

64

520

2.64

2.43

0.26

5.45

4.1

1.2

395

1.0

36

11

1013

156

384

64

520

2.64

2.43

0.26

5.45

4.1

2.9

395

1.0

36

11

1013

156

384

64

640

2.64

2.43

0.23

3.96

4.0

6.3

395

0.5

18

22

1091

312

192

64

660

5.68

4.87

0.28

5.54

6.7

2.0

395

1.0

18

22

1091

312

192

64

660

5.68

4.87

0.28

5.54

6.8

4.7

395

2.0

18

22

1091

312

192

64

660

4.26

2.43

0.24

3.76

6.9

8.9

590

0.5

18

33

1396

465

324

108

920

4.31

4.31

0.29

6.34

8.0

2.6

590

1.0

18

33

1396

465

324

108

920

4.31

4.31

0.29

6.34

8.0

5.0

590

2.0

18

33

1396

465

456

152

1120

3.06

3.06

0.25

5.00

8.0

9.0

720

0.5

18

40

1704

568

384

128

1180

4.44

4.44

0.29

5.59

8.4

2.7

720

1.0

18

40

1704

568

384

128

1180

4.44

4.44

0.29

5.59

8.6

5.2

720

2.0

18

40

1704

568

468

156

1360

3.64

3.64

0.25

5.12

8.5

9.1

1000

0.5

18

56

2367

789

576

192

1560

4.11

4.11

0.29

6.58

9.3

2.8

1000

1.0

18

56

2367

789

576

192

1560

4.11

4.11

0.29

6.58

9.6

5.2

1000

2.0

18

56

2367

789

512

384

1860

4.62

2.05

0.24

5.39

8.7

8.7

2000

0.5

18

111

4733

1578

1080

360

2700

4.38

4.38

0.29

10.6

10.9

2.8

2000

1.0

18

111

4733

1578

1080

360

2700

4.38

4.38

0.29

10.6

10.8

4.7

2000

2.0

18

111

4733

1578

1024

768

3200

4.62

2.05

0.25

8.58

11.0

7.1

Table 1. Table of simulations for the present study. Nx , Ny , and Nz are the number of grid points in the streamwise, spanwise, andwall-normal directions, with uniform grid spacings ∆x+ and ∆y+ , while the wall-normal grid spacing is given by the (constant) grid+
spacing below the roughness crests ∆z+b and the spacing at the channel centreline ∆zt .
tote: ∆U + = (1/0.4) log(ks+ ) − 3.5 , owing to the dominanceof pressure drag, a phenomena which has no analogue in heattransfer (Owen & Thomson, 1963). The velocity-temperature
dissimilarity in the fully rough regime is visualized in figure
4, which shows that the surface area is covered predominantly
by recirculated (u+ < 0) flow which mixes the temperature,with the well-mixedness improving for higher Pr. The significant temperature variations are confined primarily to the local
conductive sublayer thickness, which conforms to the roughness topography shape as noted by MacDonald et al. (2019)and becomes thinner with increasing Pr. The conductive sublayer is observed to be thinnest in windward faces, where
fluid impinges on the rough surface creating sharper velocity and temperature gradients and thereby, regions of higher
local heat-transfer (Peeters & Sandham, 2019). As noted in
Leonardi et al. (2015); Forooghi et al. (2018), at leeward faces
where low-speed, stagnant fluid resides, a well-mixed regionof temperature with small temperature gradients tends to form,
thereby producing the thicker conductive sublayers observedin these regions (figure 4d– f ).

THE TRANSITIONALLY ROUGH REGIME
In the fully rough regime, we observed in figure 4 thatthe conductive sublayer appears as a thin layer relative to theroughness size and conforms to the roughness topography. In
the transitionally rough regime however, where k+ is lowto-moderate, the roughness size k can be comparable to the
conductive sublayer thickness such that the conductive sublayer no longer conforms to the roughness topography likein the fully rough regime. Instead, as illustrated in figure
5, the near-wall flow of low-k+ regimes retain a quiescentstate akin to that observed in the viscous–conductive regionsof a smooth wall flow. Here, we follow the idea of Luchini(1996), arguing that the heat transfer augmentation, ∆Θ + , inthese regimes can be expressed through a displacement perceived by the mean temperature, `Θ and a displacement of
the turbulence d, which, when accounted for, can collapse themean temperature with a smooth wall profile. The full procedure is illustrated in figure 6. Here, the displacement d,
as elaborated in Garcı́a-Mayoral et al. (2019) can be determined by collapsing the Reynolds shear stress profiles withsmooth wall profiles and here, we have assumed that the displacement of the turbulence perceived by both the momentum3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

(a)

102

1/4

101

Pr = 0.7 (MHC19)

(c)

101

102

30

103

104

Close-packed granular type20

Pr = 2.0
Pr = 1.0
Pr = 0.5(Present)

1/2

100

(b) 40(DS63)
Pr = 5.9
Pr = 4.4
Pr = 2.8
Pr = 1.2

10

3D sinusoids

5
100

105

15

101

102

103

(d) 1510thTransitional

Smo
o

5

h

ug

o
yr

ll
Fu

Pr = 2.0

10Pr = 1.0

50

Pr = 0.5
-5

104

0

0.5

1

1.5

2

2.5

0
101

3

102

Pr = 0.7(MHC19)103

Figure 3. Log-intercept measurements in equations (1–3). (a,b) Log-intercept g(ks+ , Pr) for present 3D sinusoids compared against
the Pr = 1.2–5.9 close-packed granular type roughness of Dipprey & Sabersky (1963), with empirical fits given in (b). Also included
are the Pr = 0.7 DNS of MacDonald et al. (2019) for identical 3D sinusoids. (c) The roughness Stanton number Stk on z0 coordinates,+ 1/4 1/2
presenting the behaviours of heat transfer from smooth (cθ ≈ 11.3) to fully rough, where the St−1
Pr scaling of Brutsaertk ∼ (z0 )(1975) predicts the fully rough data well. The square markers are smooth-wall DNS. (d) Heat transfer augmentation ∆Θ + for thepresent 3D sinusoids DNS. The error bars for high-ks+ correspond virtual origin measurements at the sinusoid midplane and crests.
300
200
100
0
-100
300
200
100
0
-100

Pr = 0.5(a)Pr = 1.0

Flow

(d)

0

400

-5

0

800

(c)

(e)

(f)

0

5

Pr = 2.0

(b)

400

10

800

0

0

400

5

800

10

15

Figure 4. (a–c) Instantaneous streamwise velocity contours at k+ ≈ 111 with contour levels at u+ = 0, 4 (white, black) to highlightstagnant fluid regions and the slip velocity across the crests respectively. (d– f ) The corresponding temperature fields for Pr = 0.5–
2.0, where black lines trace instantaneous measurements of the local conductive sublayer, which tracks the regions where the local
temperature varies linearly.
layer: dΘ + /dz+ = Pr (figure 6b) (Kader, 1981). The framework of Luchini (1996) proposes that measuring from the turbulence origin (z − d)+ is what enables the rough-wall profiles to now follow the shape of a smooth-wall profile thatis situated at the turbulence origin, only offset by an amount++Θsmooth−Θrough= ∆Θ + which is assumed to be a constant that
propagates into the log region. As illustrated in figure 6(c), theand temperature fields are the same (i.e. dθ ≈ d). The mean
displacement location z = `Θ is measured relative to the sinusoid midplane and is defined as the distance where the mean
temperature reaches its wall value (Θ + = 0) based on an extrapolation of the mean temperature profile above the roughness and within the conductive sublayer such that the temperature gradient matches that of a smooth wall conductive sub-4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

20
10

Pr = 0.5

Pr = 1.0

(a)

Flow

Pr = 2.0

(b)

(c)

(e)

(f )

0
-10
20
10

(d)

0
-10
0

40

0

2

80

4

0

40

6

8

80

0

0

40

2

4

80

6

8

Figure 5. (a–c) Time-averaged velocity and (d– f ) temperature fields at k+ ≈ 5.5. The solid black lines overlay Stokes flow++
calculations, which are converted to an equivalent wall-unit scaling via u+
Stokes = [uStokes /(kStokes dU/dz|Stokes )]k , θStokes =+[θStokes /(kStokes dΘ /dz|Stokes )]Prk . The Stokes subscripts denote values from the Stokes flow calculations and the mean profile gradients dU/dz, dΘ /dz are measured in the homogeneous region above the roughness. These transformations allow for matching with the
+
+
+
+
+
viscous–conductive gradients in the DNS, i.e., dUStokes/dz+
Stokes = 1, dΘStokes /dzStokes = Pr with zStokes = (z/kStokes )k . The white+++lines locate the mean displacement origins of Stokes flow for (a–c) velocity, `u = (`u /k)k and (d– f ) temperature, Pr`θ = (`θ /k)Prk+ .

(a)

(b)

(c)
Rough

Virtual-origin plane
Crest

Rough
Smooth1

1

Mean height
Figure 6. Schematic illustration of the virtual-origin framework in low-k+ flows. (a) Reference locations of the present terminology.
+is(b) An example rough-wall profile measured from the roughness mean height, z+ , from which the mean temperature slip length `Θ+++measured. (c) The virtual-origin shift being applied, now demonstrating the prediction ∆Θ = Pr(`Θ − d ), which results from themean gradient constraint dΘ + /dz+ = Pr (compare gray triangles).
new location of the mean temperature origin is now (`Θ − d)+when measuring from (z − d)+ and thus, with both smoothand rough walls having matched gradients, dΘ + /dz+ = Pr,++− d + ).
− d + ) = Pr ⇐⇒ ∆Θ + = Pr(`Θ
predicts ∆Θ + /(`Θ

except from a Stokes flow. The mismatch `θ 6= `u implies dragand heat transfer augmentation caused by roughness differ
even in the Stokes-flow-limit. The difference can be ascribedto the presence of pressure drag in the momentum field and its
absence in the temperature field which can be inferred fromthe governing equations: −∇p + ∇2 u = 0, ∇2 θ = 0 where p is
the pressure. Presently, we find `U /`u ≈ 1.2, `Θ /`θ ≈ 1.1, andd/`v ≈ 0.51 (`v is the spanwise slip length, equal to `u for ourpresent roughness). The theory of Luchini (1996) predicts thatthese ratios should approach unity provided we are operating
in a viscous–conductive regime where k+ → 0. The non-unity
values of these ratios indicate that our present k+ ≈ 5.5 is already too large to apply Stokes-flow predictions and can beobserved visually in figure 5 by the mismatch of the Stokes
flow calculations (solid black lines) and the k+ ≈ 5.5 DNS(coloured contours). The mismatch manifests as a upward
shift relative to Stokes flow, suggesting that the advection introduced in the mean flow tends to be diverted away from thewall. This observation appears to be consistent with our measured `U /`u > 1, `Θ /`θ > 1. That is, the mean flow origin issituated higher than the one predicted by Stokes flow. Despite
the mismatch, our results in figure 7 suggest that a slip-lengthtype approach (Luchini, 1996) can still provide a valid avenue
for modelling transitionally-rough heat transfer.

In figure 7, we demonstrate these shifts taking our lowest
k+ ≈ 5.5 as an example case. Shown first in figure 7(a) are
the mean temperature profiles measured from the mean roughness height, z+ , which evidently show a discrepancy between
the rough and smooth wall profiles. However, now measuring from the virtual-origin plane, (z − d)+ and accounting forthe temperature offset ∆Θ + = Pr(`Θ − d)+ , the collapse withthe smooth wall profiles improve (figure 7b). The collapse
is best for lower values of Pr, where the roughness elements
have not significantly altered the conductive sublayer (figure5). The analogous approach for velocity adopting the mean ve+
locity displacement, `Uand the same turbulence displacement++d which yields the drag estimate ∆U + = (`U− d + ) (Garcı́a+
Mayoral et al., 2019) is also shown in figure 7. Here, `U
is the+mean velocity analogue to `Θ and the viscous sublayer velocity gradient instead follows dU + /dz+ = 1. For k+ ≈ 5.5, wefind that `U /`Θ ≈ 1.5, independent of Pr. This value is similar
to the one we obtained from a Stokes flow calculation for ourpresent roughness, `u /`θ ≈ 1.4 where the lower-case u and θ
subscripts denote calculations of the same slip lengths `U , `Θ512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)

Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia.
We would also like to thank Prof. Detlef Lohse for providing feedback in this project and Mr. Jeremy Wong for helpful
discussions concerning virtual-origin shifts.

REFERENCES
ANSYS Inc. 2011 ANSYS CFX-Solver Theory Guide.
ANSYS Inc. 2013 ANSYS Fluent Theory Guide.
Brutsaert, W. 1975 A theory for local evaporation (or heattransfer) from rough and smooth surfaces at ground level.
Water Resour. Res. 11 (4), 543–550.
Brutsaert, W. 1982 Evaporation into the atmosphere: theory,
history and applications, , vol. 1. Springer Science & Business Media.
Chan, L., MacDonald, M., Chung, D., Hutchins, N. & Ooi, A.
2018 Secondary motion in turbulent pipe flow with threedimensional roughness. J. Fluid Mech. 854, 5–33.
Chung, D., Hutchins, N., Schultz, M. P. & Flack, K. A. 2021
Predicting the drag of rough surfaces. Annu. Rev. Fluid
Mech. 53, 439–471.
Dipprey, D. F. & Sabersky, R. H. 1963 Heat and momentum
transfer in smooth and rough tubes at various Prandtl numbers. Int. J. Heat Mass Transf. 6 (5), 329–353.
Forooghi, P., Stripf, M. & Frohnapfel, B. 2018 A systematic
study of turbulent heat transfer over rough walls. Int. J. Heat
Mass Transf. 127, 1157–1168.
Garcı́a-Mayoral, R., de Segura, G. Gómez & Fairhall, C. T.
2019 The control of near-wall turbulence through surface
texturing. Fluid Dyn. Res. 51 (1), 011410.
Jiménez, J. 2004 Turbulent flows over rough walls. Annu. Rev.
Fluid Mech. 36, 173–196.
Kader, B. A. 1981 Temperature and concentration profiles infully turbulent boundary layers. Int. J. Heat Mass Transf.
24 (9), 1541–1544.
Kays, W. M. & Crawford, M. E. 1993 Convective heat andmass transfer, 3rd edn. McGraw-Hill.
Leonardi, S., Orlandi, P., Djenidi, L. & Antonia, R. A. 2015
Heat transfer in a turbulent channel flow with square bars or
circular rods on one wall. J. Fluid Mech. 776, 512–530.
Li, D., Rigden, A., Salvucci, G. & Liu, H. 2017 Reconciling the Reynolds number dependence of scalar roughness
length and laminar resistance. Geophys. Res. Lett. 44 (7),3193–3200.
Li, Q., Bou-Zeid, E., Grimmond, S., Zilitinkevich, S. & Katul,
G. 2020 Revisiting the relation between momentum andscalar roughness lengths of urban surfaces. Q. J. R. Meteorol. Soc. .
Luchini, P. 1996 Reducing the turbulent skin friction. In Computational Methods in Applied Sciences 1996, Proc. 3rd
ECCOMAS CFD Conference, pp. 465–470. Wiley.
MacDonald, M., Chung, D., Hutchins, N., Chan, L., Ooi, A.
& Garcı́a-Mayoral, R. 2017 The minimal-span channel forrough-wall turbulent flows. J. Fluid Mech. 816, 5–42.
MacDonald, M., Hutchins, N. & Chung, D. 2019 Roughness
effects in turbulent forced convection. J. Fluid Mech. 861,
138–162.
Nikuradse, J. 1933 Laws of flow in rough pipes. English translation published 1950. Tech. Rep. 1292. NACA Tech. Mem.
Owen, P. R. & Thomson, W. R. 1963 Heat transfer acrossrough surfaces. J. Fluid Mech. 15 (3), 321–334.
Peeters, J. W. R. & Sandham, N. D. 2019 Turbulent heat transfer in channels with irregular roughness. Int. J. Heat Mass
Transf. 138, 454–467.

(b)Figure 7. (a) Mean temperature (coloured) and velocity(black) profiles. Solid lines show profiles for the rough-wall(k+ = 5.5) above the roughness crests (z+ > 5.5). The velocity profiles are staggered by 10 on the y-axis for clarity.
The dashed lines are smooth wall profiles at matched Pr andthe vertical dashed line demarcates the physical region z+
c =0.4Ly+ of minimal channels. (b) The temperature profiles are++− d + ) and the velocity by (`U− d+)shifted upwards by Pr(`Θ+so that measuring from the virtual origin (z − d) improves
+
+
= 0).
collapse with the smooth wall profiles (d + = `U
= `Θ

CONCLUSIONS
We have conducted a set of rough-wall DNSs, systematically varying both the Prandtl number and roughness Reynolds
number k+ from the transitional to fully rough regime. Our
work addresses the ongoing ambiguity concerning fully roughheat transfer predictions, which typically have the form ofa scaling for a roughness heat transfer coefficient: Stk ∼(k+ )−p Pr−m , the exponents p and m being topics of contention (Li et al., 2017, 2020). We find that our data favours themodel of Brutsaert (1975), which proposes p = 1/4, m = 1/2.
At lower-k+ transitional regimes, the near-wall flow retains smooth wall characteristics, but shifted by an offset. Accounting for these shifts allows for the collapse of low-k+ temperature profiles with smooth walls and implies a possible avenue for modelling transitionally-rough heat transfer.

ACKNOWLEDGEMENTS
We acknowledge the support of the Australian ResearchCouncil Discovery Project DP200100969. This research was
supported by the Sustaining and strengthening merit-based
access to National Computational Infrastructure (NCI) LIEF
Grant (LE190100021) and facilitated by The University of
Melbourne. We also acknowledge the assistance and resources from the National Computational Infrastructure (NCI
Australia), an NCRIS enabled capability supported by theAustralian Government and the resources provided by the6