# Large Eddy Simulations of Fully-Developed Turbulent Flows Over Additively
Manufactured Rough Surfaces

Himani Garg  Lei Wang  Guillaume Sahut  Christer Fureby
<EMAIL_ADDRESS>1Lund University, Department of Energy Sciences ,
P.O. Box 118, SE-22100 Lund, Sweden

###### Abstract

## Abstract

In the last decade, with the growing demand for efficient and more sustainable
products that reduce our CO2 footprint, progresses in Additive Manufacturing
(AM) have paved the way for optimized heat exchangers, whose disruptive design
will heavily depend on predictive numerical simulations. Typical AM rough
surfaces show limited resemblance to the artificially constructed rough
surfaces that have been the basis of most prior fundamental research on
turbulent flow over rough walls. Hence, current wall models used in steady and
unsteady three-dimensional (3D) Navier-Stokes simulations do not consider such
characteristics. Therefore, a high-fidelity Large Eddy Simulation (LES)
database is built to develop and assess novel wall models for AM. This article
investigates the flow in rough pipes built from the surfaces created using AM
techniques at Siemens based on Nickel Alloy IN939 material. We developed a
code to generate the desired rough pipes from scanned planar surfaces. We
performed high-fidelity LES of turbulent rough pipe flows at Reynolds number,
$Re=11,700$, to reveal the influence of roughness parameters on turbulence,
mainly the average roughness height and the Effective Slope. The equivalent
sand-grain roughnesses, $k_{s}$, of the present AM rough surfaces are
predicted using the Colebrook correlation. The main contributors to the skin
friction coefficient are found to be turbulence and drag forces. In the
present study, the existence of a logarithmic layer is marked even for high
values of $k_{s}$. The mean flow, the velocity fluctuations, and the Reynolds
stresses show turbulence’s strong dependence on the roughness topography.
Profiles of turbulence statistics are compared by introducing an effective
wall-normal distance defined as zero-plane displacement. The effective
distance collapses the shear stresses and the velocity fluctuations outside
the roughness sublayer; thus, Townsend’s similarity of the streamwise mean
velocity is marked for the present roughnesses. Furthermore, a mixed scaling
is introduced to improve the collapse of turbulence statistics in the
roughness sublayer. In addition, an attempt to investigate surface roughness’s
impact on flow physics using the acquired LES results based on quadrant
analysis of the Reynolds shear stresses and anisotropy of turbulence is made.

Turbulence, Large Eddy Simulations, Additively manufactured roughness, Wall-
bounded flows

##  I Introduction

With the growing demand for efficient and more sustainable products that
reduce our CO2 footprint, there is an urgent need for a new generation of
cooling components that can deliver superior thermal performance while
minimizing the cost, size, manufacturing time, and weight. Micro-components
and even micro-devices are increasingly used in numerous industrial
applications from biomedicine to fuel cells. Most of these devices have
microfluidic systems with various functionalities, from process to cooling.
Unfortunately, research on micro- and mini-channel heat transfer have yet to
realize its full potential. The market penetration has been limited due to the
high-pressure drop associated with narrow flow channels, flow maldistribution
issues, and high manufacturing costs.

One way to increase heat transfer performance is to increase the surface area
to volume ratio using micro- and mini-channels. Additive Manufacturing (AM)
offers possibilities to produce completely novel designs, which are not
feasible or economically viable with traditional subtractive manufacturing.
Metal-based AM has been a promising technology in the design of advanced
cooling strategies for gas turbines. For internal cooling, the goal is to
maximize the heat transfer coefficient while minimizing the pressure drop to
ensure cooling air is used most efficiently. However, a major challenge
arising when using AM for producing metal parts is the large surface roughness
resulting from the layer-by-layer buildup process. The effects of surface
roughness in the cooling channels are twofold. Firstly, it augments the
convective heat transfer by disturbing the boundary layer. Secondly, it
produces a significant increase in pressure loss, which may reduce the cooling
air’s mass flow rate. More importantly, the high-pressure burnt gas may rush
into the cooling passage and destroy the hot components. Therefore, it is
critical to understand and accurately predict the roughness effects on the
flow and heat transfer in additively manufactured channels, which constitutes
the main motivation behind this work.

One of the fundamental questions is how to quantify the effects of surface
roughness on the mean velocity profiles. Hama [1] and Clauser [2] showed that
wall roughness usually gives rise to significant form drag and skin friction,
increasing turbulence and leading to a downward shift of the mean velocity
profile, commonly known as the roughness function. Almost a century ago, the
most famous investigation regarding the effect of roughness on the skin
friction coefficient in rough-wall turbulence was carried out by Nikuradse
[3]. In his seminal paper, a large number of measurements of a pressure drop
in pipes roughened by homogeneous sand grains revealed that if the Reynolds
number is sufficiently high, the skin friction coefficient only depends on an
equivalent sand-grain roughness height, $k_{s}$. His work was further extended
by Colebrook  _et al._ [4] and Moody [5], related the pressure drop in a pipe
to the relative roughness (ratio of roughness height to pipe diameter) and
Reynolds number. Results by Allen  _et al._ [6] dictate that the pressure drop
in the Moody chart is overestimated in transitionally rough regimes for honed
and commercial steel pipes. This clearly suggests that the Colebrook roughness
function used in the formulation of the Moody chart may not be relevant to a
wide range of roughness of engineering interest, according to Flack and
Schultz [7]. Nevertheless, the Moody chart is a powerful tool for assessing
the wall roughness effect on pressure loss as long as the equivalent roughness
to the actual roughness is known a priori. However, in terms of additively
manufactured roughness, the equivalent sand-grain roughness height is usually
unknown. Consequently, estimating equivalent roughness to predict skin
friction is a daunting task, and many studies have dedicated their efforts to
exploring the correlation to predict the equivalent sand-grain roughness
height based on the existing roughness topology [8; 9; 10; 11]. For the
prediction of the equivalent sand-grain roughness height, Dirling [12] and
later on, Van Rij, Belnap, and Ligrani [11] proposed a roughness parameter
including a roughness density and a shape parameter. They reported that the
suggested correlation could also apply to walls of non-uniform, three-
dimensional roughness with irregular geometries. Furthermore, this correlation
was reported to provide an adequate estimate of real turbine blade roughness
by Bons [13]. Moreover, based on statistical measures of surface elevation,
several kinds of correlations for the equivalent sand-grain roughness height
have been proposed [14; 7; 15].

Townsend [16] wall similarity hypothesis states that, at a sufficiently high
Reynolds number, turbulent structures located above the roughness sublayer are
independent of wall roughness and viscosity. The roughness sublayer, i.e., the
region located directly above the roughness and extending approximately to
$5k_{s}$ from the wall, in which the roughness length scales directly impact
the turbulent motions, has been a subject of in-depth interest. Experimental
investigations by Raupach, Antonia, and Rajagopalan [17] and Flack, Schultz,
and Shapiro [18] demonstrated strong experimental support for outer-layer
similarity in the turbulence structure over smooth and rough walls with
regular roughnesses. Jiménez [19] stated that the validity of outer-layer
similarity depends on the relative roughness of the flow, $k/\delta$, where
$k$ is the roughness height and $\delta$ is the boundary layer thickness. If
the roughness height is small compared to the boundary layer thickness,
$k/\delta<0.025$, the effect of the roughness is confined to the inner layer,
and the wall similarity hypothesis will hold. If, on the other hand, the
roughness height is large compared to the boundary layer thickness,
$k/\delta>0.025$, roughness effects on the turbulence may extend across the
entire boundary layer, and the concept of wall similarity will be invalid.

On the other hand, numerical simulations have been recognized as a powerful
tool for detailing fundamental flow physics over rough surfaces with different
topologies. Both Direct Numerical Simulation (DNS) and Large Eddy Simulation
(LES) for flows over spanwise extended transverse ribs [20; 21; 22; 23; 24]
and three-dimensional roughness with regular arrangement [25; 26; 27], such as
cubes, cylinders, transverse wedges, and longitudinal wedges with the
staggered arrangement, have been performed. The progress in computer
technology also enabled researchers to perform DNS of turbulence over rough
surfaces with complex geometries, including irregular and random surfaces [28;
29; 30; 31; 32; 33] and real, scanned surfaces [34; 35; 36]. In particular,
Napoli, Armenio, and De Marchis [28] considered two-dimensional irregular
corrugated walls to discuss the influence of surface slope on turbulence. In
their study, the roughness function for the rough surfaces with an Effective
Slope ($ES$) lower than a certain threshold increased with $ES$. However, the
dependence on $ES$ became weaker for rough surfaces with $ES$ higher than the
threshold. A few systematic experimental [37] and numerical [38; 32]
investigations further confirmed the dependence of turbulence on $ES$. In
addition, Bhaganagar and Chau [31] found from their DNS study on flow over
three-dimensional irregular rough surfaces that the skewness and the kurtosis
could be effective for characterizing the roughness function. In their study,
the spacing between the roughness elements was not sensitive to the roughness
function for irregular rough walls. The importance of the skewness has also
been reported by DNS studies of surfaces with randomly distributed semi-
ellipsoid/cone roughness [32] and realistic rough surfaces [30].

A great deal of effort has been made to examine the relation between roughness
topological parameters and an increase in skin friction. However, the surface
roughness produced by the AM technique is characterized by spatial non-
uniformity, which is affected by a variety of parameters such as profile
curvature, layer thickness, laser power, sample orientation, metallic
composition, and particle size. Therefore, AM roughness is considered to be
quite different from regular, random, and artificial roughnesses. To date,
very few experimental and numerical studies have been carried out to
investigate the convective heat transfer properties in mini-channels made with
AM [39; 40; 41; 42; 43; 44]. Stimpson  _et al._ [39, 40] focused on
characterizing the morphology and quantifying the roughness of micro-channel
surfaces made with Direct Metal Laser Sintering (DMLS) to develop correlations
that relate the physical roughness specifications to the effect of the
roughness on the flow friction and heat transfer. In addition, Snyder and
Thole [42] successfully demonstrated the concept of tailoring surface
roughness using AM process parameters to improve the performance of a generic
micro-channel cooling design. Very recently, the first DNS study of flows in
AM-made channels has been published by Altland  _et al._ [45]. However, their
focus was more placed on super-rough channels where large-scale roughness size
is comparable to the channel height. The flow in such super-rough channels is
more strongly influenced by the roughness on both surfaces and therefore lacks
streamwise and spanwise homogeneity everywhere in the channel. This study is
very interesting but does not align with the aim of the present study, where
we focus on AM-made surfaces of generic nature with small-scale roughness.

Despite these dedicated initial efforts to understand AM roughness, all these
studies mainly focus on heat transfer and pressure drop measurement,
documenting significant augmentation of skin friction factor and Nusselt
number. How AM roughness impacts the internal flow behavior compared to
artificial, random, and regular roughnesses is still an open question. In most
of these studies, the provided correlations compare average roughness height,
$R_{a}$, with $k_{s}$, which is optimal for some engineering applications but
can not be generalized. This choice could be because, in experiments, the
measurement of $R_{a}$ is straightforward, whereas the measurement of higher-
order statistics is quite cumbersome. Furthermore, based on previous
literature, one expects higher-order moments to correlate better with $k_{s}$
than $R_{a}$. Therefore, how well AM roughness scales with other statistical
measurements of roughness, such as standard deviation ($R_{q}$), skewness
($s_{k}$), kurtosis ($k_{u}$), and $ES$ remains to be understood. Overall,
there is a large gap in understanding the dynamical impact of AM roughness on
turbulent flow structures, such as mean flow, turbulent fluctuations, shear
stresses, and secondary flow motions. We need to find out whether the
logarithmic layer survives and, even if so, how well it scales with equivalent
sand-grain roughness. Finally, the performance of well-established LES
subgrid-scale models for these types of roughnesses is unknown. To answer
these research questions and fill the gap in our understanding, we conduct
Wall-Resolved LES (WRLES) in additively manufactured rough channels. A
systematic study of the relation between the roughness characteristics and the
equivalent roughness is carried out to reveal how roughness characteristics
affect the equivalent roughness. As of the writing of this article (to the
best of the authors’ knowledge), this is the first time LES has been used to
simulate a turbulent additively-manufactured rough wall pipe flow with grid-
conforming three-dimensional roughness elements.

##  II LES methodology

In LES, an attempt is made to capture the large-scale unsteady motions which
carry the bulk of the mass, momentum, and energy in a flow. The aim is to
approximately resolve the Taylor scales, the length scales on which these
_resolved-scale_ motions occur depending primarily on the local grid
resolution. Small-scale motions that occur on length scales smaller than a
given filtering length, typically taken as the local grid spacing, cannot be
captured, and their effects on the resolved-scale motions must be modeled. The
filtered incompressible Navier-Stokes Equations (NSE) are given by

$\displaystyle\dfrac{\partial\overline{u}_{i}}{\partial x_{i}}$
$\displaystyle=0,$ (1) $\displaystyle\dfrac{\partial\overline{u}_{i}}{\partial
t}+\dfrac{\partial}{\partial x_{j}}(\overline{u}_{i}\overline{u}_{j})$
$\displaystyle=-\dfrac{1}{\rho}\dfrac{\partial\overline{p}}{\partial
x_{i}}-\dfrac{\partial\tau^{\text{sgs}}_{ij}}{\partial
x_{j}}+\frac{\partial}{\partial
x_{j}}\left[\nu\left(\frac{\partial\overline{u}_{i}}{\partial
x_{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\right)\right],$ (2)

where $\overline{{u}}_{i}$ is the filtered velocity, $\overline{p}$, the
filtered pressure, $\nu$, the kinematic viscosity, and
$\tau^{\text{sgs}}_{ij}=\overline{u_{i}u_{j}}-\overline{u_{i}}\,\overline{u_{j}}$,
the subgrid-scale stress tensor.

In WRLES approach used in this study, a subgrid model is used for the subgrid
stress tensor combined with a specific model to represent the proximity to the
wall [46]. The most frequently used subgrid models are eddy-viscosity-based
models. For the present study, we adopted the Wall-Adapting Local Eddy
viscosity (WALE) model [47]. In WRLES the aim is to resolve the large-scale,
energy-containing flow structures in the outer boundary layer while modeling
the flow structures in the inner layer. To this end, an algebraic subgrid wall
model based on Spalding’s law-of-the-wall [48] is used to compute the subgrid
viscosity, $\nu_{\text{sgs}}$, in the inner wall layer to provide a smooth,
consistent flow profile adjacent to the wall.

##  III Numerical methods

OpenFOAM version 7 (OF7) is used to simulate the turbulent pipe flow with AM
rough walls. The NSE are solved using the finite volume method with a second-
order cell-centered discretization scheme for convective and diffusive fluxes.
Time stepping is performed using the second-order accurate implicit Adams-
Bashforth method [49] with a maximum Courant–Friedrichs–Lewy (CFL) number of 1
for numerical stability. The pimpleFoam solver is used in which the pressure-
momentum system is decoupled using the Pressure Implicit with Splitting of
Operator (PISO) algorithm for LES [50]. The resulting pressure equation is
solved using a Generalized Geometric-Algebraic Multigrid (GAMG) method [51]
with a Diagonal Incomplete Cholesky (DIC) smoother [52]. Finally, velocity is
solved using a Preconditioned BiConjugate Gradient (PBiCG) solver [53] with a
Diagonal Incomplete Lower-Upper (DILU) preconditioning method [52].

##  IV Simulation setup

###  IV.1 Geometry creation: development of a dedicated open-source code

The AM rough surfaces used in this study are made of Nickel Alloy IN939 and
provided by Siemens. They have been obtained from planar rough surfaces
scanned with a light microscope and written in STereoLithography (STL) file
format. Figure 1 shows the transformation of a rough plane into a rough pipe.
This operation consists of three steps: (1) mirroring the surface along the
rotation axis ($x$-axis in Fig. 1) so that the two surface borders can be
merged easily after rotation (see cylinder closing line in Fig. 1(a)); (2),
applying a rotation to all points of the STL file around the rotation axis;
and (3), ensuring that the cylinder is properly closed along its main axis, as
shown in Fig. 1(b).

(a)

(b)

Figure 1: Sketch of 1(a), a planar rough surface mirrored along the surface
mirroring line and 1(b), the same surface wrapped around the direction of the
surface mirroring line where superimposed points of the cylinder closing line
are merged, e.g., points $A$ and $B$, and points $C$ and $D$. Prior to bending
(point (2)), the two smallest sides of the surface have been flattened using
Blender to ease the use of periodic boundary conditions.

Since point (1) is relatively standard, we will now focus on points (2) and
(3). Two options have been considered for the creation of rough cylinders.
First, one can easily bend a planar surface to produce a cylinder using
commercial or open-source computer-aided design software. However, point (3)
may be quite time-consuming. Indeed, the operation of merging points is
typically achieved using an absolute threshold distance value such that two
points whose separating distance is lower than the threshold value are
considered to be merged. The immediate problem here is the difficulty, or even
sometimes the impossibility, of determining an appropriate threshold value. To
properly accomplish this operation, one then needs to make many attempts in a
trial and error approach and carefully inspect the whole closing line to
ensure both that no hole remains and that not too many points have been
merged. Another concern about time consumption emerges when this procedure has
to be performed a significant number of times, e.g., to gather statistics.
Such repetitive actions then become tedious.

To reduce time consumption in the generation of these rough cylinders, we have
developed a specific open-source code written in Fortran and available on
GitHub111https://github.com/CoffeeDynamics/STLRoughPipes. This code is run in
a Unix command-line interpreter and currently takes as arguments the path to
the planar rough STL file, a tolerance factor to merge the points along the
closing line of the cylinder, and an optional roughness factor to rescale the
pipe surface roughness. For all surfaces used in this study, a tolerance
factor of $0.002$ led to properly closed cylinders. This tolerance factor is
inversely proportional to the surface resolution and hence requires particular
attention. The ability to specify it in the command line constitutes a
valuable ingredient of this code by speeding up the process of building rough
cylinders from rough planar surfaces. Furthermore, the possibility to rescale
the pipe surface roughness from the command line enables fast parametric
studies of the influence of the roughness height for a given surface
topography. The code produces a cylinder with a rough surface in STL format,
as shown in Fig. 1(b), as well as some statistical quantities (e.g., average
surface roughness, arithmetic mean deviation, skewness, and kurtosis, as will
be detailed in Table 1, Section V). It can then be invoked in a loop over STL
files to process a large number of surfaces at once.

###  IV.2 Initial and boundary conditions, dimensionless numbers

The AM-roughened pipe flow is simulated at a fixed $Re_{b}=11,700$, where
$Re_{b}=U_{b}D/\nu$ is the bulk Reynolds number of the flow, with $U_{b}$
denoting the bulk velocity, $D$, the diameter of the circumscribed cylinder,
and $\nu$, the kinematic viscosity. The flow is assumed to have constant
material properties. In addition to $U_{b}$, $u_{\tau}$ is introduced in
connection to wall-bounded flows, calculated a posteriori based on the
pressure gradient, where $u_{\tau}=\sqrt{0.25D\Delta P/(\rho L_{x})}$ is the
friction velocity scale and $\Delta P/L_{x}$ is the pressure gradient. Based
on the computed value of $u_{\tau}$, the turbulent friction Reynolds number,
$Re_{\tau}=u_{\tau}D/(2\nu)$ is defined, yielding the four values
$Re_{\tau}=448$, $524$, $615$, and $875$ used in this study. In addition, the
roughness Reynolds number, $k_{s}^{+}$, based on $k_{s}$, and defined as
$k_{s}^{+}={k_{s}u_{\tau}}/{\nu}$ is also considered, resulting in
$k_{s}^{+}=17,44,83$, and $422$. This will be visited later. Note that for
roughened-wall flows, $u_{\tau}$ and the skin friction coefficient, $f_{d}$,
are no longer composed solely of the skin-friction drag but actually reflect
the total wall drag which is composed of pressure and viscous drag components.
Here, $L_{x}$ denotes the length of the computational domain. The axial,
radial, and azimuthal directions are represented by $x$, $r$, and $\theta$,
and the corresponding velocity components, by $u_{x}$, $u_{r}$, and
$u_{\theta}$. The effective wall-normal distance, i.e., along the radial
direction, is denoted by $r=r^{0}-d$, where $r^{0}$ is the pipe radius and $d$
is the zero-plane displacement, this will be visited later in Sec. V.2. Here,
$u_{x}^{\prime}$, $u_{r}^{\prime}$, and $u_{\theta}^{\prime}$ are the velocity
fluctuations in the streamwise, radial, and azimuthal directions,
respectively. Since the fully-developed turbulent pipe flow with rough walls
considered here is homogeneous in the streamwise direction, periodic boundary
conditions are also imposed in this direction. No-slip boundary conditions are
imposed at the pipe wall for all velocity components, whereas zero-Neumann
boundary condition is used for the pressure. An additional force term is
introduced into the NSE to mimic the effect of the pressure gradient in the
context of periodic boundary conditions in the streamwise direction. The
magnitude of this force is determined by the bulk velocity. For the present
study, we chose the pipe length to equal roughly eight times the pipe
diameter, i.e., $L_{x}=8D$, a sufficiently large ratio to ensure that all
statistics considered here are free from periodicity effects. In addition, a
mesh of roughly $2$ million cells has been used for all cases, and this choice
is motivated by the detailed mesh sensitivity study performed at
$Re_{b}=11,700$ for the smooth pipe flows in [54].

##  V Results and discussion

###  V.1 Surface roughness parameters and characterization

(a)

(b)

(c)

(d)

Figure 2: Visualization of AM rough surface height map, $y(x,z)$. The 3D
scanned surfaces are extracted from the measurement of 3D printed
microchannels at Siemens. Note that the actual scanned surfaces were larger,
$2\times 2$ mm. Due to the unavailability of high resolution, only a subpart
of the existing surfaces is considered in this study, roughly $2\times 0.5$ mm
(before mirroring). While moving from (a) to (d), the sampled surface is
roughly the same, but the roughness height increases.

Generally, the surface morphology comprises a series of geometrical
irregularities randomly span over a smooth surface. Surface roughness
characterization is essential in studying many fundamental problems, such as
friction and heat transfer, from an engineering point of view. The real
surface geometry is so complex that a finite number of parameters cannot
describe it completely. For the present study, we consider four additively
manufactured rough surfaces that share roughly the same topography but show
different roughness heights, see Fig. 2. We adopted numerous statistical
surface parameters to characterize surface roughness [55; 56]. Roughness
parameters can be calculated either on the profile of a sampling line (two-
dimensional (2D) forms) or a sampling area (3D forms). We consider the whole
surface in the present study, and the 3D roughness parameters are then
calculated. The different parameters adopted are summarized in Table 1. We
also adopt one of the extensive parameters called Effective Slope ($ES$) [28]
to distinguish between roughness and waviness surface types, given by

$ES=\frac{1}{L_{x}L_{z}}\int_{0}^{L_{z}}\int_{0}^{L_{x}}\left|\frac{\partial
y(x,z)}{\partial x}\right|dxdz,$ (3)

where $y(x,z)$ is the roughness amplitude, also known as the height function,
while $L_{x}$ and $L_{z}$ are the sampling lengths in the streamwise and
spanwise directions, respectively. $ES$ is defined as the average value of the
slope magnitude of the roughness.

Roughness parameters for the cases presented here and obtained with the newly
developed code mentioned in Sec. IV.1 are reported in Table 2. For the current
study, the values of $s_{k}$ and $k_{u}$ are fixed for all rough surfaces. The
critical value of $ES$ for considered rough surfaces is mostly found to be
much smaller than the critical value of $ES=0.35$ [30]. Forooghi  _et al._
[32] showed that, for lower values of $ES$, the surface slope becomes the main
controlling factor and suppresses the impact of the topographical parameters
listed in Table 1. Once the simulations reached convergence, we extracted the
values of $\Delta P/L_{x}$, $u_{\tau}$, $f_{d}$, and eventually $k_{s}$ for
all roughnesses. These values are documented in Table 3.

Table 1: Summary of roughness parameters based on roughness height, $y$, with continuous and discrete formulations, where $L_{x}$ is the surface length, $n$, the number of samples, and $p$, the probability density function. Parameter |  Description | Continuous formulation | Discrete formulation
---|---|---|---
$R_{a}$ |  Arithmetic average height | $\frac{1}{L_{x}}\int_{0}^{L_{x}}y(x)dx$ | $\frac{1}{n}\sum_{i=1}^{n}y_{i}$
$R_{q}$ |  Root-mean-square roughness height | $\sqrt{\frac{1}{L_{x}}\int_{0}^{L_{x}}\left\\{y(x)-R_{a}\right\\}^{2}dx}$ | $\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left\\{y_{i}-R_{a}\right\\}^{2}}$
$R_{v}$ |  Maximum depth of valleys below the mean line, within the sampling surface | $\left|\min y_{i}\right|+R_{a}$ | $\left|\min y_{i}\right|+R_{a}$
$R_{p}$ |  Maximum height of peaks above the mean line, within the sampling surface | $\max y_{i}-R_{a}$ | $\max y_{i}-R_{a}$
$R_{\text{max}}$ |  Maximum height between the highest peak and the deepest valley of the profile | $R_{v}+R_{p}$ | $R_{v}+R_{p}$
$s_{k}$ |  Skewness | $\frac{1}{R_{q}^{3}}\int_{-\infty}^{+\infty}\left\\{y-R_{a}\right\\}^{3}p(y)dy$ | $\frac{1}{nR_{q}^{3}}\sum_{i=1}^{n}\left\\{y_{i}-R_{a}\right\\}^{3}$
$k_{u}$ |  Kurtosis | $\frac{1}{R_{q}^{4}}\int_{-\infty}^{+\infty}\left\\{y-R_{a}\right\\}^{4}p(y)dy$ | $\frac{1}{nR_{q}^{4}}\sum_{i=1}^{n}\left\\{y_{i}-R_{a}\right\\}^{4}$
${ES}$ |  Effective Slope based on height map $y(x,z)$ | $\frac{1}{L_{x}L_{z}}\int_{0}^{L_{z}}\int_{0}^{L_{x}}\left|\frac{\partial y(x,z)}{\partial x}\right|dxdz$ | Centered difference scheme
Table 2: Statistical quantities for the four rough surfaces considered. The descriptions and analytical expressions are given in Table 1. Case |  $R_{a}$ [m] |  $R_{q}$ [m] |  $R_{v}$ [m] |  $R_{p}$ [m] |  $R_{\text{max}}$ [m] |  $s_{k}$ [-] |  $k_{u}$ [-] |  ${ES}$ [-]
---|---|---|---|---|---|---|---|---
C1 | 1.594e-3 | 2.819e-3 | 1.217e-2 | 1.418e-2 | 2.636e-2 | 0.599 | 6.107 | 0.102
C2 | 1.992e-3 | 3.524e-3 | 1.522e-2 | 1.773e-2 | 3.295e-2 | 0.599 | 6.107 | 0.124
C3 | 2.630e-3 | 4.652e-3 | 2.009e-2 | 2.340e-2 | 4.349e-2 | 0.599 | 6.107 | 0.157
C4 | 3.984e-3 | 7.048e-3 | 3.043e-2 | 3.546e-2 | 6.589e-2 | 0.599 | 6.107 | 0.216
Table 3: Estimation of pressure gradient, $\Delta P/L_{x}$, friction velocity, $u_{\tau}$, skin friction factor, $f_{d}$, equivalent sand-grain roughness height, $k_{s}$, equivalent sand-grain roughness height normalized by $R_{a}$ and $R_{q}$, and equivalent sand-grain roughness height, $k_{s}^{ES}$, employing a correlation based on $ES$, from LES simulations. The value of the pressure gradient is extracted at the end of each iteration and averaged over long time series. Case |  $\Delta P/L_{x}$ [Pa.m-1] |  $u_{\tau}$ [m/s] |  $f_{d}$ [-] |  $k_{s}$ [m] |  $k_{s}/R_{a}$ [-] |  $k_{s}/R_{q}$ [-] |  $k_{s}^{ES}$ [m]
---|---|---|---|---|---|---|---
C1 | 0.068 | 0.067 | 0.051 | 0.005 | 3.201 | 1.806 | 0.004
C2 | 0.087 | 0.077 | 0.069 | 0.011 | 5.760 | 3.253 | 0.006
C3 | 0.106 | 0.087 | 0.087 | 0.019 | 7.262 | 4.106 | 0.008
C4 | 0.194 | 0.119 | 0.180 | 0.071 | 17.874 | 10.103 | 0.014

###  V.2 Mean flow profiles

Figure 3: Cross-sectional view of the mean streamwise velocity, normalized by
the friction velocity, $u_{\tau}$, for different rough surfaces: (a) C1, (b)
C2, (c) C3, and (d) C4.

All the statistics obtained in this study have been averaged in time and along
the streamwise and azimuthal directions. Moreover, due to the symmetry between
the pipe wall’s lower and upper parts, the statistics are symmetrically
averaged with respect to the mid-plane of the pipe.

One of the most important effects of the wall roughness on the turbulent flow
is the reduction of the streamwise velocity with respect to smooth-wall
conditions, with the consequent modification of the friction coefficient. In
the log region, at a sufficient distance from the roughness elements, the
stream-wise velocity profile can be expressed as

$\left<u_{x}\right>^{+}=\frac{1}{k}\text{log}(y^{+})+C_{s}-\Delta U^{+},$ (4)

where $\left<u_{x}\right>^{+}$ is the non-dimensional time-averaged streamwise
velocity, $k$, the von Kármán constant, $y^{+}$, the non-dimensional wall-
normal distance, $C_{s}$, a constant equal to $5.2-5.5$ for channel flows
depending on the Reynolds number, and $\Delta U^{+}$, the roughness function.
The first two terms of the right-hand side come from the expression valid for
a smooth wall. The value of $\Delta U^{+}$ is representative of the downward
shift of the velocity profile in the wall-normal direction due to the increase
of resistance induced by the wall roughness. The value of the additive
constant $C_{s}=5.2$ is used in the present case.

(a)

(b)

(c)

Figure 4: Comparison of LES results for rough and smooth pipe flows. (a) Mean
streamwise velocity $\left<{u_{x}}\right>^{+}$ as a function of $r^{+}$. The
dashed black line presents DNS data at $Re_{\tau}=361$. The dashed pink and
green lines represent the universal behavior of the turbulent velocity
profile. (b) Mean streamwise velocity normalized by centerline velocity as a
function of $r/R$. (c) Mean streamwise defect velocity normalized by
$u_{\tau}$ as a function of $r/R$.

Firstly, in Fig. 3, we show a cross-sectional view of the mean streamwise
velocity, normalized by $u_{\tau}$ for different rough surfaces, C1, C2, C3,
and C4. Qualitatively speaking, when normalized by $u_{\tau}$, the downward
shift of the streamwise velocity is clearly visible with increasing roughness.
For detailed quantitative analysis, the non-dimensional numerical streamwise
mean velocity profiles for different roughnesses are shown in semi-logarithmic
plots in Fig. 4 (a). The non-dimensional wall distance, $r^{+}$, is measured
from the plane at which the total drag acts, i.e., shifting the velocity
profiles for rough surfaces by removing the negative velocities, a method
frequently referred to as zero-plane displacement. In the same plot, we also
provide the numerical results of mean streamwise velocity profiles for LES
[54] and DNS [57] of smooth pipe flow at the same bulk Reynolds
($Re_{b}=11,700$) for reference. We observe that the surface roughness shifts
the logarithmic profile downwards; the shift is measured by $\Delta U^{+}$,
indicating a shortage in momentum compared to smooth walls. The roughness
heights of the considered surfaces are a significant fraction of the boundary
layer thickness, resulting in large enough roughness functions. In literature,
similar roughness functions are obtained for the sand-grain roughness at
approximately the same Reynolds number, even though the roughness heights
differ substantially [7; 58; 35; 59; 60; 61; 62; 63]. This indicates that the
roughness height alone is not sufficient to scale the momentum deficit
resulting from surface roughness.

In Fig. 4 (b), we also plot the mean velocity profiles in the outer units. We
observe a significant departure of the rough pipe flow profiles from the one
over the smooth surface close to the wall region and in the overlap region,
$r/R<0.4$. The more significant the roughness, the larger the shift from the
mean flow profile of the smooth surface. In the outer region, irrespective of
the roughness heights, all profiles collapse.

For turbulent flow over smooth and rough surfaces, the wall similarity
hypothesis, an extension of Townsend’s Reynolds number similarity hypothesis,
states that for sufficiently high $Re$ outside the viscous or roughness
sublayer, the turbulent motion in the boundary layer is independent of the
wall roughness and the viscosity, except for the wall shear stresses,
$\tau_{w}$, and the boundary layer thickness, $\delta$ [16; 17]. This means
that the outer region is thought to be unaffected by roughness, as reported in
review papers [17; 19; 64]. This further implies that the mean flow for both
smooth and rough walls obeys the universal velocity defect law in the overlap
and outer regions of the boundary layer, which can be expressed as

$U_{0}^{+}-\left<u_{x}\right>^{+}=f\left(\dfrac{r}{R}\right).$ (5)

In an attempt to verify this statement’s validity for AM rough surfaces, we
plot the defect velocity of non-zero pressure gradient turbulent flow over
different rough and smooth surfaces in Fig. 4 (c). The mean flow similarity
can be observed by overlapping the smooth and rough walls when plotted in
velocity-defect form for $r/R>0.25$. This substantiates that the outer layer
similarity may be maintained in the present 3D printed rough surfaces with
relatively high roughness height, $\delta/R_{a}=37-84$. Furthermore, the
observation of the similarity for the rough surfaces whose roughness height is
a significant fraction of the boundary layer thickness is in good accordance
with the observations by experimental [65], and DNS [38] studies. Further
evidence of wall similarity has also been provided in the review paper by
Flack and Schultz [66], evaluating the similarity hypothesis against the data
of several studies with a range of roughness, from where similarity is
expected to hold $(\delta/R_{a}=110)$ to significantly rougher surfaces
($\delta/R_{a}=16$). Note that the range of roughness considered in the
present study is relatively much higher than the ones considered by Flack,
Schultz, and Connelly [65] but is comparable to the ones considered by Flack
and Schultz Flack and Schultz [66].

###  V.3 Roughness function

In the literature, the velocity shift and roughness function, as a result of
wall similarity, are considered to be proportional to $\log k_{s}^{+}$ and are
independent of wall distance [67; 68]. Several correlations have been
suggested for roughness function based on the collapse, either to traditional
Nikuradse data [3] or Colebrook et al. [4] data. (For a fully rough regime,
the Nikuradse data and Colebrook data should collapse, their difference is
just in the transitional regime. For the present analysis, we have adopted the
general form of the Colebrook-type roughness function, given by

$\Delta U^{+}=2.44\log{\left(1+0.26k_{s}^{+}\right)}.$ (6)

One can also find a correlation for the Nikuradse roughness function in [69].
The value of $k_{s}^{+}$ is unknown a priori in our present study. The value
of $k_{s}^{+}$ been computed by extracting the value of $\Delta U^{+}$ from
Fig. 4(a) and utilizing the Eq. 6. An attempt to estimate the value of $k_{s}$
using the more recent correlation proposed by Marchis et al. [70] based on
$k_{a}$ and $ES$. But the results were inconsistent due to relatively small
values of $ES$ for the present roughnesses and hence not included in this
study. This point deserves a separate discussion and will be discussed in
future investigations.

In Fig. 5(a), we plot the relationship between the roughness function and the
roughness Reynolds number, $k_{s}^{+}$, based on equivalent sand-grain
roughness height, for a wide range of rough surfaces. For comparison, uniform
sand-grain roughness results by Nikuradse [3] and Colebrook [4], rough-wall
pipe flow results by Shockling et al. [71], and rough-wall channel flow
results by MacDonald et al. [72] and Peeters and Sandham [73] are presented.

(a)

(b)

Figure 5: Roughness function as a function of (a) $k_{s}^{+}$ and (b) $ES$.

Two flow regimes can be observed in Fig. 5 (a). Firstly, for $k_{s}^{+}=17$,
the flow becomes transitionally rough. In this regime, the viscosity can no
longer dampen out the turbulent eddies created by the rough elements, and
hence the form drag on the rough elements as well as the viscous drag. This
indicates that the rough surface C1 with $k_{s}^{+}\approx 17$ is roughly in
the transition regime. However, this contrasts with some other works, where
the surfaces display Nikuradse-type behavior, tending toward the hydraulically
smooth flow regime at finite $k_{s}^{+}$ with slight inflectional behavior in
the transition regime. This could be linked to the fact that in the present
case, $k_{s}$ is based on Colebrook roughness function formulation rather than
Nikuradse’s, so a departure in the transition regime is seen. Secondly, with a
further increase of $k_{s}^{+}>17$, the roughness function reaches a linear
asymptote, also known as a fully rough regime. For $k_{s}^{+}>17$, the C2, C3,
and C4 roughnesses strictly follow the Colebrook-type roughness function. In
this regime, the skin friction coefficient is independent of the Reynolds
number, and form drag on the roughness elements is the dominant mechanism
responsible for the momentum deficit.

Surprisingly, the results indicate that the range of the transitionally rough
regime is quite narrower, $k_{s}^{+}<25$. This contrasts with Nikuradse’s that
the fully rough regime exists for $k_{s}^{+}\geq 70$ for mono-dispersed close-
packed sand-grain roughness. However, Ligrani and Moffat [74] pointed out that
$k_{s}^{+}$ signifying both the onset of roughness effects and the beginning
of the fully rough regime depends strongly on the roughness type and
uniformity. Their rough surfaces, consisting of uniform close-packed spheres,
displayed a fairly narrow transitionally rough regime, $15<k_{s}^{+}<55$,
while Nikuradse’s uniform, close-packed sand grains, had a wider
transitionally rough regime, $5<k_{s}^{+}<70$. Overall, the agreement between
the present data sets and the literature is excellent. This concurs with the
findings by Schultz and Myers [75], who showed that the roughness function for
a given roughness obtained through different means yields similar results.

Usually, for dense roughness regimes, the roughness function does not scale
with $k_{s}^{+}$. Hence, an additional factor, $ES$, is considered for scaling
turbulent flows for all roughness regimes. In Fig. 5 (b), we show the
variation of the present roughness function, $\Delta U^{+}$, with $ES$. For
comparison, we also plot different types of roughnesses available in the
literature, i.e., systematically varied rough surfaces with positive and
negative skewness [76], smoothly corrugated walls with zero skewness [77], 2D
and 3D roughness, as well as 2D waves [70], irregular roughness with a range
of positive and negative skewness [30], and irregular surface roughness [28].

In Fig. 5 (b), for the dense roughness regime, $ES>0.35$, $\Delta U^{+}$
decreases with increasing $ES$, which is attributed to a reduction in the
Reynolds shear stress in a near wall region that is being pushed away from the
rough wall. The comparison also shows that for irregular roughness, $\Delta
U^{+}$ is more significant for the surfaces with positive $s_{k}$ than for the
ones with negative $s_{k}$. However, the calculated values of $ES$ are rather
small ($<0.35$) in comparison to the one observed in the literature,
indicating that the present roughnesses might fall into a wavy roughness
regime. Nevertheless, the results show the opposite: $\Delta U^{+}$ does not
scale with $ES$. For $ES<0.35$, the rough surfaces yield a higher value of
$\Delta U^{+}$. This raises the question of whether these AM surfaces are in a
wavy regime. To answer, we estimated the wavelength of the surface elevation,
$\lambda=m_{0}/m_{1}$, where
$m_{0}=\int_{0}^{+\infty}E_{z^{\prime}z^{\prime}}dk$ and
$m_{1}=\int_{0}^{+\infty}kE_{z^{\prime}z^{\prime}}dk$ are zeroth- and first-
order moments of power spectral density, $E_{z^{\prime}z^{\prime}}$, of the
surface elevation and $k$ is the wave number. The obtained value of $\lambda$
is much smaller than $R$, which is a signature of sparsely packed and isolated
roughness elements. This confirms that our AM roughnesses are not wavy but in
a very sparsely packed roughness regime. We think that yielding a higher
roughness function value, despite a smaller value of $ES$, could be a critical
feature of AM rough surfaces. More investigations on AM surfaces are needed to
validate this hypothesis.

###  V.4 Turbulence intensities in the outer scaling

(a)

(b)

(c)

Figure 6: (a) Axial turbulence intensities, ${u_{x}}^{+}_{\text{rms}}$, (b)
turbulent kinetic energy, TKE+, and (c) turbulent shear stresses,
$R_{u_{x}^{\prime}u_{r}^{\prime}}^{+}$, for different rough surfaces, shown as
a function of outer scaling, $r/R$. The results for smooth pipe flow for WALE-
LES [54] (red points) and DNS [57] (dashed line), at $Re_{b}=11,700$, are also
plotted for reference.

We have measured the intensity of the velocity fluctuations, a lowest-order
indicator of the rough wall’s influence on other flow parts. We show here the
most important ones, streamwise velocity fluctuation profiles for different
roughnesses in Fig. 6 (a), turbulent kinetic energy in Fig. 6 (b), in outer
scaling. The profiles for the wall-normal and azimuthal velocity fluctuations
are not shown here, as the effect of roughness is weaker on these quantities.
For comparison, we also show the profiles for smooth pipe flows at the same
bulk Reynolds number, $Re_{b}=11,700$, obtained from LES [54] and DNS [57].

As $k_{s}^{+}$ increases and the form drag dominates over the viscous scale,
the near-wall peak in the streamwise velocity fluctuations, ${u_{x}}_{rms}$,
disappears, and the profile develops a maximum in the logarithmic layer around
$r/R=0.05-0.2$. This outer peak is probably related to the plateau found in
that region for high Reynolds number flows over smooth walls [78]. Also, in
the outer region, all the present roughnesses except C4 collapse very well for
$r/R>0.3$, with reasonable numerical uncertainty. These small mismatches
between the smooth pipe flow and rough surface flow in the outer scale could
be attributed to the statistical averaging. Furthermore, for the C4 case, the
effect of flow blockage may be significant, resulting in higher form drag, and
an eventually slight departure from Townsend’s similarity is observed.
However, overall these results support Townsend’s Reynolds number similarity
hypothesis that the flow is insensitive to the wall boundary condition except
for the role it plays in setting the outer scale and velocity scales.

In addition, we also plot the turbulent Reynolds shear stress profiles in Fig.
6 (c). The profiles are normalized by $u_{\tau}^{2}$. As demonstrated earlier,
with an increase in $k_{s}^{+}$ from C1 to C4 cases, the peak of shear
stresses not only diminishes but also shifts outward, thanks to the high form
drag contribution. A pretty good collapse of all rough surfaces except C4 for
$r/R>0.3$ has been observed. The results are relatively close to the LES
smooth-wall case with little uncertainties, which could be linked to the
approximation of friction velocity estimate. This once again demonstrates the
outer scale similarity, as noticed in previous works. Note that, in the C4
case, where $R/R_{a}<50$, while for C1 and C3 cases, $R/R_{a}>50$ which belong
to the roughness regime, as mentioned by Jiménez [79]. This shows that there
is probably a transition between C3 and C4 rough surfaces where AM roughnesses
differ quite a lot in terms of their behavior and could drastically influence
heat transfer properties.

We are further interested in scaling streamwise velocity fluctuation profiles
with inner scale, i.e., $r/R_{a}$, based on the average roughness height,
$R_{a}$. We show the comparison of ${u_{x}}_{rms}$, normalized by $u_{\tau}$,
as a function of $r/R_{a}$ in Fig. 7 (a). With increased $k_{s}^{+}$ values,
the peak value close to the wall diminishes and shifts inwards. All profiles
develop a maximum in the range of $r/R_{a}=8-10$. This inward shift is
attributed to the yielding of the higher value of $u_{\tau}$, despite the
relatively small value of the bulk Reynolds number, a typical characteristic
of roughness. Nevertheless, one point that stands out here is that the
roughness profiles do not fully collapse in the inner region, which implies
that $r/R_{a}$ is not an accurate scaling for roughness.

(a)

(b)

Figure 7: Streamwise turbulence velocity fluctuations, ${u_{x}}_{\text{rms}}$,
shown as a function of inner scaling based on average roughness height,
$r/R_{a}$, for different rough surfaces, and normalized by (a) $u_{\tau}$ and
(b) $\sqrt{u_{\tau}U_{\text{0}}}$, a new mixed scaling introduced here.

This requires introducing another scaling related to roughness in the
normalization of ${u_{x}}_{rms}$. Following the proposition by De Graaff and
Eaton [78], we introduced a new scaling, referred to as mixed scaling from
hereon, $\left<...\right>^{m}=\left<...\right>/(\sqrt{u_{\tau}U_{0}})$, in the
normalization of ${u_{x}}_{rms}$, for which results are shown in Fig. 7 (b).
For AM roughnesses, this scaling also works reasonably well, but there is an
additional decrease in the peak value with increasing $k_{s}^{+}$, probably
linked to the slight over-approximation of $u_{\tau}$. The presence of the
freestream velocity, $U_{0}$, in the scaling of the streamwise fluctuations is
usually associated with the effect of the inactive motions postulated by
Townsend [16]. However, the details of how they affect the streamwise
fluctuations away from the wall layer are poorly understood.

Figure 8 shows the profile of radial velocity fluctuations as a function of
the inner scale, based on $R_{a}$. The roughness effect is weaker for this
quantity, as the profiles are pretty close to each other, irrespective of the
roughness. Consistently with previous results, the scaling with $R_{a}$ shows
a good collapse in the inner-wall region for all roughnesses. The slight
discrepancy could be attributed to the statistical averaging uncertainty.
These results further show that there is no need to introduce mixed scaling,
which implies the negligible contribution of the inactive motions to this
quantity, as postulated by Townsend [16].

Figure 8: Profiles of root-mean-square of radial velocity fluctuations,
${u_{r}^{+}}_{\text{rms}}$, normalized by $u_{\tau}$ are shown as functions of
inner scaling based on average roughness height, $r/R_{a}$, for different
rough surfaces.

Similarly, the turbulent Reynolds shear stresses normalized by the friction
velocity squared as a function of $r/R_{a}$ are presented in Fig. 9. As
expected, the shear stress scales well in the inner wall region when plotted
as a function of $r/R_{a}$. The peak value for different roughnesses increases
with decreasing $k_{s}^{+}$ due to the scaling involved in
$R^{+}_{u_{x}^{\prime}u_{r}^{\prime}}$. Indeed, the larger the roughness, the
larger the form drag and hence $u_{\tau}$. To end this discussion, we finally
show the plots of turbulent kinetic energy for different scalings in Fig. 10.
The turbulent kinetic energy is mainly dominated by the streamwise velocity
and shows very similar behavior. The results clearly scale very well in the
inner wall region once the mixed scaling is introduced.

Figure 9: Turbulence shear stresses, $R^{+}_{u_{x}^{\prime}u_{r}^{\prime}}$,
normalized by $u_{\tau}^{2}$ are shown as functions of inner scaling based on
average roughness height, $r/R_{a}$, for different rough surfaces.

(a)

(b)

Figure 10: Turbulence kinetic energy, TKE, expressed in terms of inner scaling
based on average roughness height, $r/R_{a}$, for different rough surfaces,
normalized by (a) $u_{\tau}^{2}$ and (b) $u_{\tau}U_{0}$, a new mixed scaling
introduced here.

(a)

(b)

Figure 11: Turbulent kinetic energy production as a function of inner scale
units based on (a) average roughness, $r/R_{a}$, (b) root-mean-square
roughness scale units, $r/R_{q}$. Here,
$P=-R_{u_{x}^{\prime}u_{r}^{\prime}}\partial\left<u_{x}^{+}\right>/\partial
r^{+}$ is evaluated using the law of the wall-wake for the mean velocity, for
zero-pressure gradient boundary layers.

###  V.5 Turbulent kinetic energy production

In both DNS and experiments of smooth-wall flows, it has been shown that for
low Reynolds numbers, the peak of kinetic energy production mainly occurs
within the viscous buffer layer region, at a wall-normal distance, $r^{+}$, of
roughly $12$. However, at high Reynolds numbers, the significant contribution
to the bulk turbulence production comes from the logarithmic region. It raises
a similar interesting question in the case of AM roughnesses. Therefore, in
the present simulations, we looked into the contribution to the bulk
turbulence production. In Fig. 11, we show the production term,
$P=-R_{u_{x}^{\prime}u_{r}^{\prime}}\partial\left<u_{x}^{+}\right>/\partial
r^{+}$, on a semi-logarithmic scale. In Fig. 11(a-b), we show the results for
the production term but scaled with the average roughness height, $R_{a}$, and
the root-mean-square value of roughness height, $R_{q}$. Irrespective of the
scaling factor, we observe that all curves collapse into one with slight
uncertainties. In comparison to smooth-wall pipe flow, the maximum value of P
for the present roughnesses is smaller and roughly one half of smooth pipe
flows [54]. Also, the peak value increases in the inner layer and shifts
inward for all roughnesses. The results depict that the primary contribution
to bulk production comes from the near-wall region at low $k_{s}^{+}$, and the
logarithmic region dominates at adequately higher $k_{s}^{+}$.

###  V.6 Rough surface joint probability density functions

In this section, we analyze the axial and radial velocity fluctuations for a
fixed value of $r^{+}$ using the quadrant analysis approach [80], by the
desire to gain insight into the near-wall streaks. We divided the
($u_{x}^{\prime},u_{r}^{\prime}$) plane into four quadrants corresponding to
different sign combinations of the velocity fluctuations: two quadrants
contributing negatively to $\left<u_{x}^{\prime}u_{r}^{\prime}\right>$,
attributed to ejections (second quadrant, $Q_{2}$; $u_{x}^{\prime}<0$,
$u_{r}^{\prime}>0$) and sweeps (fourth quadrant, $Q_{4}$; $u_{x}^{\prime}>0$,
$u_{r}^{\prime}<0$), and two quadrants contributing positively to
$\left<u_{x}^{\prime}u_{r}^{\prime}\right>$, owing to outward (first quadrant,
$Q_{1}$; $u_{x}^{\prime}>0$, $u_{r}^{\prime}>0$) and inward (third quadrant,
$Q_{3}$; $u_{x}^{\prime}<0$, $u_{r}^{\prime}<0$) interactions.

(a)

(b)

(c)

(d)

(e)

Figure 12: Typical quadrant plots (JPDF) are shown for illustration purposes
between axial velocity fluctuations, $u_{x}^{\prime}$, and radial velocity
fluctuations, $u_{r}^{\prime}$, normalized by their root-mean-squares
($\sigma_{i^{\prime}}$) for different roughness Reynolds numbers at
$r^{+}=3\sqrt{k_{s}^{+}}$: (a) smooth surface, (b) C1, (c) C2, (d) C3, and (e)
C4. For each plot, the PDF of $u_{x}^{\prime}/\sigma_{u_{x}^{\prime}}$ and
$u_{r}^{\prime}/\sigma_{u_{r}^{\prime}}$ are shown on the top and right
corners, respectively.

In Fig. 12, we present the contour plots of the Joint Probability Density
Functions (JPDF) of velocity fluctuations, $J(u_{x}^{\prime},u_{r}^{\prime})$.
The JPDF are taken at the closest location to $r^{+}=3\sqrt{k_{s}^{+}}$
(indicative of the middle of the logarithmic region) for different
roughnesses, normalized by their local standard deviations,
$\sigma_{u_{x}^{\prime}}$, and $\sigma_{u_{r}^{\prime}}$. For comparison, the
JPDF of smooth-wall turbulent pipe flow is also shown. For all cases, the
isocontours of the JPDF exhibit an elliptical shape whose main axis crosses
quadrants $Q_{2}$ and $Q_{4}$. For the smooth surface case, the JPDF maximum
value is located in quadrant $Q_{4}$, and a long tail is observed in quadrant
$Q_{2}$. This suggests the highest number of
$\left\\{u_{x}^{\prime}u_{r}^{\prime}(t)<0\right\\}$ events, with, in
particular, $Q_{2}$ (ejections) being characterized by more extreme events,
i.e., with higher values of $|u_{x}^{\prime}u_{r}^{\prime}(t)|$. Overall the
smooth-wall results show a higher degree of homogeneity. In contrast, ejection
and sweep contributions for rough-wall flow exhibit differences in the form of
localized regions of more intense events, and $Q_{4}$ (sweeps) are
characterized by more extreme events. In Fig. 12, for each plot, normalized
Probability Density Functions (PDF) are also shown on the top
($u_{x}^{\prime}$) and right ($u_{r}^{\prime}$) corners, respectively.
Notably, the PDF of $u_{x}^{\prime}$ and $u_{r}^{\prime}$ are slightly
positively and negatively skewed for smooth surfaces, respectively. On the
other hand, for all roughnesses, we observe that the PDFs of $u_{x}^{\prime}$
and $u_{r}^{\prime}$ are largely positively skewed and slightly positively
skewed, respectively. This means high-speed streaks are dominant structures in
rough surface flows compared to smooth surface flows. Qualitatively, the
roughness has a negligible impact on the behavior of the results and delivers
a similar message. The contribution of ejections and sweeping events has also
been computed quantitatively, but it requires a separate long discussion and
hence is not included in this paper.

###  V.7 Roughness-induced secondary flow structures

One intriguing and practically significant manifestation of turbulence in
wall-bounded flows is the ability to induce the so-called secondary flows,
that is, systematized fluid motions in the plane perpendicular to the
streamwise direction. In turbulent regimes, the average velocity field
contains non-zero transverse components, and secondary flow occurs in non-
circular cross-section ducts related to inhomogeneities of the Reynolds
stresses. The spanwise inhomogeneity produces distinct gradients in the
turbulent Reynolds stress distribution and alters the local flow properties.
The development of secondary flow motions can also occur on a plane or
symmetrical wall-bounded flows if a local spanwise heterogeneity in wall
conditions exists, e.g., due to surface roughness.

(a)

(b)

(c)

(d)

Figure 13: Cross-sectional view of the instantaneous streamwise velocity
fluctuations field for different rough surfaces: 13(a) C1, 13(b) C2, 13(c) C3,
and 13(d) C4. Streamlines from the instantaneous flow field show the flow
development and vortical flow patterns attributed to wall roughness. In all
plots, the wall is colored with the friction velocity, $u_{\tau}$.

(a)

(b)

(c)

(d)

Figure 14: Cross-sectional view of the root-mean-square of the streamwise
velocity fluctuations field for different rough surfaces: 14(a) C1, 14(b) C2,
14(c) C3, and 14(d) C4. Streamlines obtained from the mean flow field show the
development of the flow and secondary flow patterns attributed to wall
roughness. In all plots, the wall is colored with the friction velocity,
$u_{\tau}$.

It is interesting to see how highly non-uniform and sparsely packed AM
roughness induces the secondary flow motions and if there are some common
features with regular or artificial roughness. We show the distribution of the
instantaneous and root-mean-square (rms) velocity fluctuations overlayed with
the secondary flow, depicted by in-plane streamlines for different rough
surfaces, in Figs. 13 (a)-(d) and 14 (a)-(d), respectively. In the four cases
of Fig. 13 (a)-(d), pronounced small-scale motions and counter-rotating pairs
of vortices are encountered, resulting in momentum transport by ejecting and
sweeping the fluid from and toward the wall. The size and frequency of these
small-scale features are strictly dependent on the location of the plane and
the time at which the snapshot is taken. From the instantaneous flow fields
and the overlayed streamlines, it is very difficult to see if any secondary
flow motion persists, and if so, then the impact of the roughness on the
secondary motions is unclear. Therefore we look at the distribution of the
streamwise rms velocity overlayed by the in-plane streamlines in Fig. 14
(a)-(d). Interestingly, out of four cases, pronounced secondary motion
patterns can be observed mainly for C3 and C4 roughnesses. In the case of C3
roughness (Fig. 14 (c)), the two main large-scale vortices originate from the
recessed wall area. Two additional small vortices are located in the top and
bottom regions of the wall, again close to the recessed area. On the other
hand, in the case of C4 roughness (Fig. 14 (d)), two main large-scale vortices
originated from the recessed area of the wall, but were seldomly pushed away
from the inner wall region. In addition, one can see four extra counter-
rotating pairs of small structures on each side of the slightly bigger large-
scale vortex. This is somewhat consistent with the results obtained by
Vanderwel and Ganapathisubramani [81]; Hwang and Lee [82]; Wangsawijaya  _et
al._ [83] for strip-type roughness and for ridge-type roughness, where the
upwelling and downwelling movements were indicated above the elevated and
recessed wall areas, respectively.

In contrast, in the case of C2 roughness (Fig. 14 (b)), one can see the early
development of some secondary flow structures, but for C1 roughness (Fig. 14
(a)), only few inflection points are spotted. The origin of such inflexion
points remains unanswered in this study and demands future investigations.
Comparing these cases against the smooth surface pipe flow suggests that the
roughness height, eventually surface topography, is an important parameter for
secondary motion formation and destruction.

###  V.8 Effect of surface anisotropy on the anisotropy of the Reynolds stress
tensor

The net anisotropy of the Reynolds stresses is commonly quantified using the
second, $II_{b}$, and third, $III_{b}$, invariants of the normalized
anisotropy tensor, $b_{ij}$, given by [84]

$b_{ij}=\frac{\left<\overline{u^{\prime}_{i}u^{\prime}_{j}}\right>}{\left<\overline{u^{\prime}_{k}u^{\prime}_{k}}\right>}-\frac{1}{3}\delta_{ij}.$
(7)

The state of anisotropy can then be characterized with the two variables
$\eta$ and $\zeta$ defined as

$\eta^{2}=-\frac{1}{3}II_{b}$ (8)

and

$\zeta^{3}=-\frac{1}{2}III_{b}.$ (9)

All realizable states of the Reynolds stress tensor are contained within a
triangle in the $\left(\zeta,\eta\right)$ plane, the so-called Lumley
triangle. All special turbulence cases can be characterized by the two
invariants of $b_{ij}$ at the theoretical limits of the Lumley triangle.
Details on the different realizable states can be found in Garg et al. [54].
Lumley triangle plots for all rough surfaces are shown in Fig. 15. We also
show the Lumley triangle plot for smooth pipe flow as a reference. All cell-
centered values of $\zeta$ and $\eta$ are post-processed using a probability
density function (PDF) and represented with a colormap to match the wall-
normal distance. In the reference case of smooth-wall turbulent pipe flow, see
Fig. 15(a) very close to the wall, the flow is close to a two-component state
following the upper boundary of the Lumley triangle toward the one-component
state denoted by the upper right corner. The maximum anisotropy is reached for
$r^{+}\approx 8$. For $r^{+}>8$, the anisotropy decreases, and the
$(\zeta,\eta)$ curve follows a path close to the right boundary of the Lumley
triangle, indicating a close-to-axisymmetric, rod-like state and approaching,
but not reaching, the state of maximum isotropy at the bottom summit of the
Lumley triangle as $r^{+}$ increases.

(a)

(b)

(c)

(d)

(e)

Figure 15: Anisotropy-invariant mapping of turbulence in turbulent rough pipe
flow compiled from the present LES data at different roughness Reynolds
numbers: (a) smooth pipe flow, $k_{s}^{+}=0$, (b) $k_{s}^{+}=17.052$, (c)
$k_{s}^{+}=44.131$, (d) $k_{s}^{+}=83.084$, and (e) $k_{s}^{+}=422.45$. The
data points for each case are based on all cells in the domain and colored
with normalized wall distance values, $r^{+}$. Colormap varies from blue
(minimum) to red (maximum).

On the other hand, for the rough surfaces, the turbulent states are spread
throughout the Lumley triangle, with the exception of the plain strain
condition indicated by the dashed line, see Fig. 15(b-e). In the deepest
valleys of the surfaces, the Reynolds stress anisotropy tensor is close to a
strongly anisotropic, one-component case. With increasing $r^{+}$, results
follows a different path on the $(\zeta,\eta)$-map compared to the case for
smooth-wall pipe flow. For all rough surfaces (C1-C4), the flow approaches the
left side of the Lumley triangle, reaching an axisymmetric disk-like (ablate
spheroid) state at the roughness mean plane. At this location, the streamwise
and azimuthal Reynolds stresses are of comparable magnitude. The anisotropy
appears centered around the axisymmetric expansion and the two-component
limit, indicating that one stress component is either larger than the other
two or that the two components are similar in magnitude. An axisymmetric,
disk-like state of the Reynolds stress anisotropy tensor is typical of mixing
layers. Similar behavior has also been observed for turbulent flows over
transverse bar roughness [22], k-type roughness [85], and irregular roughness
[63]. Above the roughness mean plane, the path returns to the right side of
the triangle, i.e., close to an axisymmetric, rod-like (prolate spheroid)
state, and tracks the behavior of the smooth-wall case once the wall-normal
coordinate exceeds the maximum roughness height. Finally, the most commonly
occurring anisotropic states are axisymmetric expansion, one-component, two-
component, and two-component axisymmetric states, and the probability of a
certain turbulent state to occur increases with an increase in $k_{s}^{+}$.

##  VI Conclusions

We have presented a systematic numerical investigation of turbulent flows in
Additively Manufactured (AM) rough surfaces employing roughness-resolved high-
fidelity Large Eddy Simulations (LES) in OpenFOAM 7. The aim of the present
simulations is twofold. First, shedding light on the effect of AM roughness on
turbulent flow properties and highlighting the difference between existing
regular and artificial roughnesses utilizing various statistical measures.
Second, providing a rich database that will ease the development and
assessment of rough-wall models. The first challenging task in building this
database is the generation of rough pipes from scanned planar AM rough
surfaces and their characterization. To this end, we developed an open-source
code to generate pipes from planar surfaces and to characterize the surface
topographical properties. Four configurations of rough pipes from actual AM
surfaces with fixed skewness and kurtosis but different roughness height
distributions and effective slopes have been constructed and simulated.

By means of accurate spatial measurements, the numerical estimation of the
roughness function is obtained for all cases and utilized to approximate
equivalent sand-grain roughness height, $k_{s}$. The present AM rough surfaces
yield a more significant value of roughness function compared to the
literature results for regular and artificial roughnesses in a similar value
range of $k_{s}$ [71; 72; 73]. The significant impact of $k_{s}$ on the skin
friction coefficient is found through enhanced turbulence and drag forces. The
Effective Slope, $ES$, which is directly linked to the alignment of the wall
roughness with the flow, has a substantial impact on the flow topology, as
expected. However, the present AM surfaces with $ES<0.35$ yield a significant
value of roughness function, in contrast to regular and artificial roughnesses
[76; 77; 70; 30; 28]. This could be a typical feature of AM surfaces and needs
to be clarified in future investigations. In these cases, the existing
empirical correlations find their limits, and new correlations are required.
Furthermore, by defining an effective wall-normal distance, the collapse of
the first- and second-order turbulence statistics is observed outside the
roughness sublayer, validating Townsend’s hypothesis for the present AM
roughnesses. However, for one of the rough surfaces with $k_{s}/R_{a}=17.84$,
the validity of Townsend’s hypothesis is still questionable and needs future
detailed investigations. Finally, a mixed boundary condition is proposed to
improve axial fluctuation profiles’ collapse in the roughness sublayer. We
also attempted to understand, at least qualitatively, the impact of roughness
topography on the generation of the secondary flow motions. Unfortunately,
despite the abundance of results available for regular roughness, quantifying
the AM roughness impact on secondary flow motion was not feasible in the
current simulations. This calls for detailed future investigations of
different types of roughnesses. Lastly, we quantified the impact of roughness
topography on turbulence anisotropy and compared it with smooth pipe flows.
For the present rough surfaces, increasing $k_{s}$ results in high anisotropy.
Furthermore, all turbulent states, i.e., one-component, two-component, and
three-component states, existed with higher probabilities when increasing
$k_{s}$, consistently with literature [22; 85; 63].

We hope that the analysis reported in this work will help establish sound
methodologies to compare in detail numerical simulations of AM rough surface
flows to the available experimental and numerical results. This type of fine-
scale comparison appears crucial to us in improving the theoretical
understanding of the subject and to validate the related numerical
developments.

## Acknowledgement

The authors greatly appreciate the financial support provided by Vinnova under
project number "2020-04 529" and Senior lecturer Martin Andersson (funding
acquisition lead). Computer time was provided by the Swedish National
Infrastructure for Computing (SNIC), partially funded by the Swedish Research
Council through grant agreement no. 2018-05973. The authors also appreciate
Siemens for providing the scanned AM surfaces and helpful discussions.

## Author declarations

### Conflict of interest

The authors have no conflicts of interest to disclose.

### Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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