# A Polynomial Framework for Design of Drag Reducing Periodic Two-dimensional
Textured Surfaces

Shabnam Raayai-Ardakani<EMAIL_ADDRESS>Rowland Institute, Harvard
University, 100 Edwin H. Land Blvd., Cambridge, 02142, MA, USA

###### Abstract

Periodic and symmetric two-dimensional textures with various cross-sectional
profiles have been employed to improve and optimize the physical response of
the surfaces such as drag force, superhydrophobicity, and adhesion. While the
effect of the height and spacing of the textures have been extensively
studied, the effect of the shape of the textures has only been considered in
qualitative manners. Here, a polynomial framework is proposed to
mathematically define the cross-sectional profiles of the textures and offer a
quantitative measure for comparing the physical response of the textured
surfaces with various cross-sectional profiles. As a case study, textured
surfaces designed with this framework are tested for their hydrodynamic
frictional response in the cylindrical Couette flow regime in Taylor- Couette
flows. With the reduction in torque as the objective, experimental and
numerical results confirm that textures with height-to-half-spacing of lower
and equal to unity with concave profiles offer a lower torque compared to both
smooth surfaces and triangular textures. In addition, across multiple
polynomial orders, textures defined by second order polynomials offer a wide
range of responses, eliminating the need for considering polynomials of higher
orders and complexity. While the case study here is focused on the laminar
flow regime and the frictional torque, the same type of analysis can be
applied to other surface properties and physical responses as well.

###### keywords:

Textured Surfaces, Riblets, Taylor-Couette Flows, Drag Reduction

††journal:

## Introduction

Textures are well documented for their ability to alter and regulate physical
response of surfaces, both in the natural and engineering settings; Drag
reducing effect of ribs on the denticles of fast swimming shark species [1, 2,
3] as well as bio-inspired riblet-covered surfaces [4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18], super-hydrophobicity, super-repellency, and self
cleaning of Colocasia esculenta, Euphorbia myrsinites, Nelumbo nucifera
(Lotus), and Salvinia leaves, packed with convex papillose cells which are
covered by three-dimensional wax crystal [19, 20, 21, 22, 23, 24], slippery
and insect trapping mechanism of the concave textures on the rim (peristome)
of Nepenthes pitcher plants [25, 26, 27, 28], which can also act as superior
fog harvesters [29], as well as excellent adhesion of textures on the feet of
gecko and some artopods [30, 31, 24] and the textured gecko-inspired
engineered adhesives [32, 33, 34, 35, 36] are but a few examples.

Textures come in random and ordered configurations, and the characteristics of
the profiles and patterns are described with a variety of naming conventions.
Terms such as roughness, waviness, and lay [37] in manufacturing, dimples and
seams in sports [38, 39], or wrinkles, dimples, and creases in patterns formed
with soft materials [40, 41, 42] are a few of the words commonly used to
describe the corresponding topographies. In dealing with random roughness
under flow conditions, additional characteristics, such as the roughness
topography, local peak-to-trough heights, max peak-to-trough height, or sand
grain roughness measures are also considered [43].

One group of idealized and ordered engineering textures are two-dimensional
(2D) textures; these are simplified models of natural rib-like textures (shark
skin, butterfly wings, rice leaves), and have been widely considered for the
purposes of drag reduction [44] and anisotropic surface responses [45, 46].
Such 2D ordered textures are comprised of a unit element of texture that is
symmetric and repeated in a periodic pattern (Fig. 1(a)) and unit elements
ranging from triangular (also known as V-groove or saw-tooth) [8, 4, 47, 5,
14, 12], semi-circular [48, 49, 7], rectangular [50, 51, 52, 53, 54], families
of sine waves [55], and trapezoidal [56, 57, 54] have been considered before.
Due to the difference in the shapes of the textures, various physical
dimensions (height, spacing, radius of curvature, etc.) have been used to
compare the frictional drag response of the textures; height and spacing of
the texture elements are universally applicable to any periodic 2D texture,
but other geometric features (corner angles, the radii of curvatures, etc.)
are shape-specific and not expandable to other shapes, and descriptive names
only offer the means for qualitative comparison of the performance among the
variety of textures.

Figure 1: (a) Three dimensional schematic of a periodic, symmetric, 2D texture
on a rigid solid wall; (b) Projection of the 2D profile on the
($x^{*}$-$n^{*})$ plane. The horizontal distance between consecutive peaks
($P_{i}$ and $P_{i+1}$) and troughs ($T_{i}$ and $T_{i+1}$) is the spacing,
$\lambda$, and the vertical distance between peaks and troughs is $A$. The
profile is $\lambda$-periodic, and symmetric about lines of
$x^{*}=j\lambda/2$, where $j\in\mathbb{N}$. One example of a symmetry line is
presented with a dashed-dotted line at $x^{*}=\lambda/2$.

Here, a polynomial framework is proposed to quantitatively define the shape of
the textures, and allow for an objective comparison among the various families
of grooves. While the scope of this work is focused on textures in
hydrodynamic applications to optimize the frictional drag force, the
mathematical classification of the textures can be expanded to other
scientific applications as well. Here, the performance of the surfaces is
evaluated using experimental and numerical analysis of the textures in a
Taylor-Couette (TC) flow [58, 9, 10] especially focusing on the frictional
torque (equivalent to drag in cylindrical coordinate system) response of the
surfaces in cylindrical Couette flow regime (CCF) as the key measure of
comparison.

## Methods

### Experimental procedure

The experiments were conducted using a custom-designed and built Taylor-
Couette (TC) cell (Fig. 2) attached to a stress-controlled rheometer (TA
Discovery Hybrid Rheometer 2). The radius ratio of the TC cell is kept at
$\eta=R_{\rm i}/R_{\rm o}=0.65$ and the height ratio $\Gamma=L/(R_{\rm
o}-R_{\rm i})=5.1$ where $R_{\rm o}$ and $R_{\rm i}$ are the radii of the
outer cylinder and the inner cylinder respectively, and $L$ is the length of
the rotor (Fig. 2). Here, the outer cylinder is kept stationary ($\Omega_{\rm
o}=0$), the inner cylinder rotates at a rotation rate of $\Omega_{\rm i}$, and
the torque experienced by the rotor, $\cal{T}_{\rm total}$, is measured via
the sensors in the rheometer. Further details of the design of a similar TC
cell can be found in Ref. [10].

Textured covered rotors were 3D printed via stereo-lithography (Formlabs Form3
3D printer and photopolymer resin). The design of the full rotors comprise of
the inner rotor (which is in contact with the working fluid during the
experiments) and a sleeve with a tapped hole that allows for the rotors to be
installed on the rheometer and be connected to the sensors (Fig. 2). The
outermost radius of the textured cylinders (i.e. texture peak) is at $R_{\rm
i}$ and the smallest radius of the textures is located at the trough at
$R_{\rm i}-A$, where $A$ is the height of the textures (Fig. 1(b)).

The effect of the gap at the closed bottom of the rotor is subtracted from the
measured torque using the procedure outlined in Refs. [10, 9]; in summary, the
distance between the bottom of the rotor and the TC cell ($h_{\rm b}$) is
calibrated using

${\cal T}_{\rm bottom}=\mu\dfrac{\pi R_{\rm i}^{4}}{2h_{\rm b}}\Omega_{\rm i}$
(1)

and set to $h_{\rm b}=340\ \mu{\rm m}$, and using Eq. (1) the excess torque
from this bottom effect is then subtracted from all measurements (i.e. ${\cal
T}={\cal T}_{\rm total}-{\cal T}_{\rm bottom}$).

Figure 2: Image of the experimental Taylor-Couette setup, attached to a
stress-controlled rheometer on the left, and a schematic of the setup, on the
right.

The working fluid is a mixture of 70% Glycerol (Sigma Aldrich BioXtra
$\geqslant 99\%$) and 30% de-ionized water (volumetric ratio) and the
viscosity of the working fluid is measured at the beginning and the end of the
experimental procedure as a function of temperature. To account for any affect
of temperature on the viscosity of the working fluid, throughout the
experiment temperature of the fluid and the surrounding environment is
measured in 60 second intervals using a two channel thermometer (Digi-sense
Tracable Memory-loc Datalogging Thermometer) and the fluid temperature is used
to adjust the viscosity of the working fluid during data analysis. The maximum
temperature change recorded in either channel of the thermometer, during each
experimental procedure is less than 0.5-0.8${}^{\circ}\ {\rm C}$ (viscosity
variation within $2\%$ of the average viscosity), and while this variation is
accounted for in the analysis, turns out to have a negligible effect on the
results. (This can also be shown using the Nahme-Griffith number [9].) The
density of the mixture is calculated using the formula given in Ref. [59].

The rotation rates, and torque, as well as fluid properties are used to
calculate the Reynolds number and the dimensionless torque (following the
definition from references [60, 61]) using

${\rm Re}_{d}=\dfrac{\rho(R_{\rm o}-R_{\rm i})R_{\rm i}\Omega_{\rm i}}{\mu}$
(2)

and

$G=\dfrac{\cal{T}}{4\pi\rho{\nu}^{2}L}.$ (3)

where $\rho$, and $\mu$ are the fluid density and dynamic viscosity
respectively, and $\nu=\mu/\rho$.

For the purpose of comparison, a smooth rotor with inner radius of $R_{\rm
i}$, and length $L$ is also tested with the same procedure. The measured
dimensionless torque for the smooth rotor $G_{0}$ (after subtraction of the
bottom effect) is on average within $0.6\%$ of the theoretical torque

$G_{\rm th}=\dfrac{\eta}{(1-\eta^{2})(1-\eta)}{\rm Re}_{d}$ (4)

in CCF regime within $10\leqslant{\rm Re}_{d}\leqslant 40$ and within $2\%$ of
the theoretical torque within $40<{\rm Re}_{d}\leqslant 60$ (average). The
measured $G_{0}$ is used throughout the text to normalize the torque of the
rest of the samples. These measured torques and Reynolds numbers were used to
identify the onset of transition to the Taylor vortex flow, ${\rm Re}_{d,{\rm
tr}}=64$, following the method described in Refs. [60, 61].

The experimental procedure used here is based on a set of 60-step “peak hold”
procedures were at each step a constant rotation rate is imposed on the inner
rotor for a span of 15 seconds and torque is read five times during these 15
seconds and an average of these five readings is used for the analysis
($\cal{T_{\rm total}}$). This “peak hold” is repeated for velocities
corresponding to CCF regime (i.e. Reynolds numbers less than ${\rm Re}_{d,{\rm
tr}}=64$). Measurements for the textured surfaces didn’t present any sign of
transition happening earlier than ${\rm Re}_{d}=64$. Each set of experiment is
then repeated 6 times and the results are presented here as mean $\pm$ one
standard deviation.

### Numerical simulations

For comparison, direct numerical simulation of the CCF over textured surfaces
are conducted using the finite volume package, OpenFOAM. The simulations are
performed for one single unit of texture, in axisymmetric, steady-state
conditions, using the SIMPLE algorithm, with periodic boundary conditions at
$x^{*}=0$, and $x^{*}=\lambda$. Here the diffusion terms are discretized with
a second-order central difference, and the convective terms are discretized
with a second-order central difference with correction for the non-
orthogonality of the mesh faces due to the textures.

## Results

### Periodic, symmetric 2D textures

A 2D periodic texture (Fig. 1(a)), consists of symmetric unit elements
(between a peak-trough-peak group, $P_{i}$, $T_{i}$, and $P_{i+1}$ \- Fig.
1(b)) where the streamwise and spanwise directions are denoted by $s^{*}$ and
$x^{*}$ and the normal to the two directions is denoted by $n^{*}$ (all
starred variables are dimensioned). The unit of texture has a spacing of
$\lambda$ (i.e.
$(x^{*}_{{}_{P_{i+1}}}-x^{*}_{{}_{P_{i}}})=(x^{*}_{{}_{T_{i+1}}}-x^{*}_{{}_{T_{i}}})=\lambda$)
and height of $A$ (i.e. $(n^{*}_{{}_{P_{i}}}-n^{*}_{{}_{T_{i}}})=A$), and half
of the symmetric profile of the wall in between $P_{\rm i}$ and $T_{\rm i}$ is
defined by a function $n^{*}_{\rm w}=f^{*}(x^{*})$. Due to the symmetry, all
directions are non-dimensionalized by half of the wavelength
($s=s^{*}/(\lambda/2)$, $x=x^{*}/(\lambda/2)$, $n=n^{*}/(\lambda/2)$), and the
height-to-half-spacing ratio is denoted by ${\cal{R}}=A/(\lambda/2)$ (i.e.
$(x_{{}_{T_{i}}}-x_{{}_{P_{i}}})=1$, and $(n_{{}_{P_{i}}}-n_{{}_{T_{i}}})=\cal
R$). Here, it is assumed that all peaks and troughs are located at $n=n_{0}$
and $n=n_{0}-{\cal R}$ respectively (where $n_{0}$ is an arbitrary point of
reference and can be shifted to any choice of reference point and coordinate
system) and for all the cases considered here, the domain located above the
texture profile ($n>n_{\rm w}$) is filled with fluid and the domain below this
boundary ($n<n_{\rm w}$) is a rigid wall.

With this setting, for textures with the same peak and trough locations,
function $f$ is the key to distinguishing between textures of different
shapes. For example, in the simplest form, one can set $f$ to be a line
connecting $P_{i}$ and $T_{i}$, in the form of

$n_{\rm w}=-{\cal{R}}x+n_{0}\ \ \ \ {\rm for}\ \ \ \ 0\leqslant x\leqslant 1$
(5)

where ${\cal R}$, or the height-to-half-spacing ratio is also the magnitude of
the slope of the function $f$ (in $0<x<1$). This functional form returns the
triangular or V-groove textures that have been considered extensively in
literature [8, 4, 47, 5, 14, 12]. In a similar manner, $n_{\rm w}=n_{0}$
returns the case of a smooth reference surface.

### Concavity and convexity

Figure 3: (a) Family of curved textures, defined by second order polynomials
with height-to-half-spacing of $\cal R$, and $-{\cal
R}\leqslant\kappa_{2}\leqslant{\cal R}$. (b) Profiles of two unit textures on
the surface of 3D printed rotors covered with second order textures with
${\cal R}=1.50$ and $-1.50\leqslant\kappa_{2}\leqslant 1.50$. Figure 4: (a)
Dimensionless torque experienced by textured rotors with profiles defined by
second order polynomials, with ${\cal R}=0.50$, $1.00$, and $1.50$, and
$\lambda/d=0.13$, and $0.26$ as a function of $\kappa_{2}$. Torque is
normalized by the torque experienced by the smooth rotor. (b) Contours of
dimensionless torque of textured rotors with a profile defined by second order
polynomials as a function of $\cal R$, and $\kappa_{2}$, normalized by the
torque of a smooth rotor, $G_{0}$, calculated numerically, for grooves with
$\lambda/d=0.13$ (top), and $\lambda/d=0.26$ (bottom). Sample lines from the
contour plots of $G/G_{0}$ for select height-to-half-spacings are plotted in
Fig. 5 for additional clarity. Figure 5: Line plots of torque normalized by
the torque of a smooth rotor for select height-to-half-spacing ratios as a
function of $\kappa_{2}$, and $\lambda/d=0.13$, and $0.26$, corresponding to
the contour plots in Fig. 4(b).

When considering a curved (or non-linear) profile, $f$, for each unit element
of the texture, after an average slope, $\cal R$, (i.e. height-to-half-space
ratio), concavity becomes a second geometric feature that is considered. With
concavity defined as $d^{2}n_{\rm w}/dx^{2}>0$ (and vice versa convexity as
$d^{2}n_{\rm w}/dx^{2}<0$) the simplest function representing a curved texture
has a constant second derivative, resulting in a second order polynomial of
the form $n_{\rm w}=a_{2}x^{2}+a_{1}x+a_{0}$ where $2a_{2}$ is the constant
second derivative and $a_{1}$ and $a_{0}$ are found based on $\cal R$, and the
choice of $n_{0}$. If the constant second derivative is given as
$2\kappa_{2}$, then $f$ is written as

$n_{\rm w}=\kappa_{2}x^{2}+(-\kappa_{2}-{\cal{R}})x+n_{0}\ \ \ \ \ \ \
0\leqslant x\leqslant 1.$ (6)

Figure 6: Distribution of the angular velocity ($u_{\theta}/r^{*}$) normalized
by the angular velocity of the inner rotor ($\Omega_{\rm i}$) for textures
with ${\cal R}=1.00$, and $-1\leqslant\kappa_{2}\leqslant 1$. Angular velocity
of a smooth case is presented on the left for comparison.

For textures where the peaks and troughs are the only global (and local)
extrema, the profile of the textures, $f$, needs to be locally monotonic
between each peak-trough segment and this results in $\kappa_{2}$ being
bounded by the limits $-{\cal{R}}\leqslant\kappa_{2}\leqslant{\cal{R}}$ (Fig.
3(a)). The definition of the profile as written in Eq. (6) decomposes the
effect of the texture geometry into two terms identified by the height-to-
half-spacing ratio, $\cal R$, and $\kappa_{2}$, as a curvature parameter.
Therefore, if $k_{2}=0$ (i.e. $d^{2}n_{\rm w}/dx^{2}=0$), $f$ returns the
profile of a triangular texture (Eq. (5)). At a constant $\cal R$, varying
$\kappa_{2}$ from its lowest ($\kappa_{2}=-{\cal R}$) to highest
($\kappa_{2}={\cal R}$) limits, results in a series of textures spanning from
convex ($\kappa_{2}<0$) to concave ($\kappa_{2}>0$) profiles (Fig. 3).

Experimental and numerical evidence show that varying $\kappa_{2}$ directly
affects the frictional response of the flow in the laminar regime. 3D printed
texture-covered inner rotors with ${\cal R}=0.50$, $1.00$, and $1.50$ and
various values of $\kappa_{2}$ (Fig. 3(b)) are tested in the CCF regime and
the frictional torque experienced by the rotors ($\cal T$) is measured as a
function of the rotation rate, $\Omega_{\rm i}$, of the inner rotors. In this
cylindrical setting, $r$, $\theta$, and $x$ are the the respective normal,
streamwise, and spanwise directions. The measurements (Fig. 4(a)) are
presented in terms of dimensionless torque, $G$, normalized by reference
$G_{0}$ of a smooth rotor, as a function of $\cal R$, $\kappa_{2}$, and the
Reynolds number of the flow, ${\rm Re}_{d}$.

While independent of the Reynolds number (similar to the previous reports for
CCF [9, 10]), for all samples (Fig. 4(a)), decreasing $\kappa_{2}$ from zero
(i.e. moving from a triangular profile toward a convex profile) results in an
increase in the torque compared to that of the V-grooves ($\kappa_{2}=0$).
However, for ${\cal R}=0.50$ and $1.00$ increasing $\kappa_{2}$ from zero
(moving from triangular to a concave texture) results in a reduction in the
measured torque with respect to the triangular texture. This is while for
${\cal R}=1.50$, the same increase in $\kappa_{2}$ does not offer a
substantial change. These trends are similar to two aspects of previous
reports of tests using curved textures in turbulent flow in wind tunnel;
First, between triangular, and semi-circular textures in concave and convex
format, concave textures reduce the drag force the most; triangular textures
present less reduction compared to the concave ones, and convex ones do not
show any difference compared with a smooth wall [8]. Second, increasing the
radius of curvature of troughs of grooves results in enhancements in the total
drag reduction achieved, while increasing the radius of the curvature at the
peaks, results in an increase in the total measured drag force [7]. On the
other hand, the current measurements also confirm that (similar to triangular
textures [9, 10]) employing a larger wavelength results in a lower level of
torque on the rotors and the introduction of $\kappa_{2}$ does not affect this
trend.

Changes in torque as a function of $\kappa_{2}$ are dependent on $\cal R$ of
the grooves (as presented in the contours of Fig. 4(b) and corresponding line
plots in Fig. 5); while confirming the trends presented in Fig. 4(a), direct
numerical simulations of textures with $0\leqslant{\cal{R}}\leqslant 3$ and
$\lambda/d=0.13$ and $\lambda/d=0.26$ show that for profiles with
${\cal{R}}\leqslant 1.00$, going from a convex to a concave profile offers a
decrease in the measured torque; textured rotors with $1.00<{\cal R}\leqslant
2.00$ (similar to ${\cal R}=1.50$ in Fig. 4(a)), do not experience a
substantial change in the torque for $0\leqslant\kappa_{2}\leqslant{{\cal R}}$
while moving toward a convex profile results in an increase in the torque
(compared with the V-grooves); and lastly for cases with ${\cal R}>2.00$, the
trend slowly reverses where convex profiles experience frictional torques
lower than the case of a triangular groove and all concave profiles increase
the torque.

Variation in $G$ as a function of $\kappa_{2}$ can by assessed by examining
the distribution of angular velocity and consequently the wall shear stress at
the textured wall. Contours of the angular velocity of the flow,
$u_{\theta}/r$, (Fig. 6) show that in comparison to a smooth rotor, the iso-
lines inside the grooves are curved, mimicking a pattern dictated by the shape
of the texture at the boundary (but not exactly the same shape) and as one
moves further away from the boundary (Fig. 6 at $r^{*}/R_{\rm o}\gtrapprox
0.68$) the iso-lines return to being parallel to the $x$ direction. The effect
of $\kappa_{2}$ is especially visible at peaks and troughs where for all the
textures, the iso-lines are squeezed close to each other at the peaks, but at
the troughs the lines are pushed further apart.

In laminar flows, such as the case of the CCF here, viscous diffusion plays
the key role in the development of the flow. With the diffusion occurring in
the wall-normal direction and the wall shear stress being proportional to the
velocity gradient in the wall-normal direction, employing a coordinate
transformation from the generic ($x-n$) plane to a curvilinear, orthogonal
coordinate system of ($\eta-\xi$) where the iso-$\xi$ and iso-$\eta$ lines are
respectively parallel and perpendicular to the textured boundary, offers
additional insight.

Figure 7: Boundaries of the domain mapped between (a) ($x-n$) plane and (b)
($\eta-\xi$) plane resulting in the curvilinear orthogonal coordinate system
for an example triangular texture.  Figure 8: Iso-lines of curvi-linear,
orthogonal coordinate system ($\eta-\xi$), perpendicular to the texture
boundary for textures with ${\cal R}=1.00$, and
$-1.00\leqslant\kappa_{2}\leqslant 1.00$. Locations along the iso-$\xi$ with
negative and positive curvature have been marked with blue squares and red
circles respectively.

To find the curvilinear-orthogonal coordinate system corresponding to each of
the texture domains, inverse of an elliptical mapping between the $(x-n)$
coordinate and the new $(\eta-\xi)$ coordinates is employed (Fig. 7). The
forward map is defined by Laplace equations of the form

$\nabla^{2}\eta(x,n)=0$ (7)

$\nabla^{2}\xi(x,n)=0$ (8)

with boundary conditions of

$\dfrac{\partial\eta}{\partial h}\bigg{\rvert}_{B_{1}}=0\ \ \ \ \&\ \ \ \ \
\xi\bigg{\rvert}_{B_{1}}=0$ (9)

$\dfrac{\partial\xi}{\partial h}\bigg{\rvert}_{B_{2}}=0\ \ \ \ \&\ \ \ \ \
\eta\bigg{\rvert}_{B_{2}}=1$ (10)

$\dfrac{\partial\eta}{\partial h}\bigg{\rvert}_{B_{3}}=0\ \ \ \ \&\ \ \ \ \
\xi\bigg{\rvert}_{B_{3}}=1$ (11)

$\dfrac{\partial\xi}{\partial h}\bigg{\rvert}_{B_{4}}=0\ \ \ \ \&\ \ \ \ \
\eta\bigg{\rvert}_{B_{4}}=0$ (12)

where $B_{1}$ is the textured boundary and the rest of the boundaries are
numbered $B_{2}$-$B_{4}$ in the counter clockwise direction (Fig. 7), and $h$
is the local normal to the surface at each boundary. Thus the inverse of this
mapping [62] is calculated by solving Eq. (13), and Eq. (14)

$g_{{}_{22}}\dfrac{\partial^{2}x}{\partial\eta^{2}}-2g_{{}_{12}}\dfrac{\partial^{2}x}{\partial\eta\partial\xi}+g_{{}_{11}}\dfrac{\partial^{2}x}{\partial\xi^{2}}=0$
(13)

$g_{{}_{22}}\dfrac{\partial^{2}n}{\partial\eta^{2}}-2g_{{}_{12}}\dfrac{\partial^{2}n}{\partial\eta\partial\xi}+g_{{}_{11}}\dfrac{\partial^{2}n}{\partial\xi^{2}}=0$
(14)

where $g_{{}_{11}}$, $g_{{}_{12}}$, and $g_{{}_{22}}$ are the metric tensors
connecting the two coordinate systems

$g_{{}_{11}}=\left(\dfrac{\partial
x}{\partial\eta}\right)^{2}+\left(\dfrac{\partial n}{\partial\eta}\right)^{2}$
(15)

$g_{{}_{22}}=\left(\dfrac{\partial
x}{\partial\xi}\right)^{2}+\left(\dfrac{\partial n}{\partial\xi}\right)^{2}$
(16)

$g_{{}_{12}}=\left(\dfrac{\partial
x}{\partial\eta}\right)\left(\dfrac{\partial
x}{\partial\xi}\right)+\left(\dfrac{\partial
n}{\partial\eta}\right)\left(\dfrac{\partial n}{\partial\xi}\right).$ (17)

The equations are solved iteratively with a python script, using an algorithm
similar to the approach presented in [63] subject to the boundary conditions
in Eq. (9), Eq. (10), Eq. (11), and Eq. (12).

The trend in the iso-$\xi$ lines match the iso-velocity lines (in Fig. 6) very
well and close to the peaks of all the textures ($x\approx 0$ in Figs. 6 and
8), both iso-$\eta$ and iso-$\xi$ lines are squeezed close to each other,
while in the trough area ($x\approx 1$ in Figs. 6 and 8), the distances
between the iso-lines are more relaxed.

Figure 9: (a) Distribution of wall shear stress as a function of the spanwise
location, $x$, within each texture unit, for textures with second order
profiles, ${\cal R}=0.50$, $1.00$, and $1.50$, and $\lambda/d=0.26$, as a
function of $-{\cal R}\leqslant\kappa_{2}\leqslant{\cal R}$. (b) Location of
point of cross-over of the shear stress from $\tau>\tau_{{}_{0}}$ to
$\tau<\tau_{{}_{0}}$, as a function of $\kappa_{2}$ for textures with ${\cal
R}=0.50$, $1.00$, and $1.50$, and $\lambda/d=0.26$ within $0\leqslant
x\leqslant 1$. Figure 10: Wall shear stress distribution within one texture
element, point-wise normalized by the wall shear stress distribution of a
triangular texture, $\tau_{{}_{1}}$, as a function of the spanwise direction,
$x$, for textures defined with second order polynomials with height-to-half-
spacing values of ${\cal R}=0.50$, $1.00$, and $1.50$ and $\lambda/d=0.26$.
Figure 11: Variations in the average wall shear stress, $\overline{\tau}$,
normalized by the shear stress on a smooth rotor, $\tau_{{}_{0}}$ (right axis
of the plots), experienced by textures defined by second order polynomials as
a function of $\kappa_{2}$ with height-to-half-spacing ratios of ${\cal
R}=0.50$, $1.00$, and $1.50$ and $\lambda/d=0.26$. The variations in the
wetted surface area of the textures, normalized by the surface area of a
smooth rotor, $S_{0}$, is also shown as a function of $\kappa_{2}$ (left axis
of each plots).

Such variations in the iso-lines of the ($\eta-\xi$) coordinate system (Fig.
8) and the angular velocity distribution (Fig. 6) as a function of
$\kappa_{2}$ of these textures, directly results in a non-uniform velocity
gradient and thus a non-uniform shear stress distribution on the wall as shown
in Fig. 9(a) normalized by the shear stress experienced by a smooth rotor,
$\tau_{0}$. The squeezed iso-lines at the peaks and the spaced out iso-lines
in the valleys result in the shear stress at the peaks of all textures to be
larger than the shear stress experienced by a smooth rotor $\tau_{0}$ (line
$\tau/\tau_{0}=1.00$ in Fig. 9(a)), while in the troughs $\tau$ is lower than
$\tau_{0}$ (similar to previous reports for triangular grooves [5, 9, 64]).

The effect of the curvature of the texture, manifested in the $\kappa_{2}$
parameter, is visible in the local curvature of the iso-$\xi$ lines inside the
grooves (the first iso-$\xi$ line is the texture boundary). In all cases, the
iso-$\xi$ lines above the peaks have a negative local curvature (blue squares
in Fig. 8) and as one moves toward the valleys the curvature slowly increases
and turns positive (red circles in Fig. 8). As one moves from convex to
concave textures, the extent of the segment of the iso-$\xi$ line with
negative curvature slowly recedes and the portion of the iso-$\xi$ line with
positive curvature increases.

This variation in the local curvature of the iso-$\xi$ lines causes the
distribution of the velocity gradient and the wall shear stress at the riblet-
covered wall to span a range of below and above the shear stress on a smooth
rotor, $\tau_{{}_{0}}$, and for ${\cal R}\geqslant 0.50$ the points of cross-
over between $\tau>\tau_{{}_{0}}$ and $\tau<\tau_{0}$ takes place within
$0<x\leqslant 0.503$ (Fig. 9(b)). In addition, the reduction in the portion of
the iso-$\xi$ lines with negative curvature observed as $\kappa_{2}$ is
increased directly results in the $x_{\rm cross-over}$ to be shifted closer to
the peak of the grooves, and thus a larger segment of the textured wall
exhibits shear stress values lower than ${\tau}_{0}$. This trend directly
results in a decrease in the $G$ values for ${\cal R}\leqslant 1.00$ when
$\kappa_{2}$ is increased. In addition, a point-wise comparison of the wall
shear stress distribution of the textures with $\kappa_{2}\neq 0$ and the wall
shear stress of the triangular grooves, $\tau_{{}_{1}}$, (Fig. 10) shows that
a large segment of the wall shear of convex textures is placed higher than
that of ${\tau}_{{}_{1}}$ and for concave ones, a large portion of each half
of the texture has a wall shear stress lower than ${\tau}_{1}$. This
ultimately results in convex textures having a larger torque compared to the
triangular textures and for concave textures with ${\cal R}\leqslant 1.00$ to
have lower torques than the triangular ones. However, for cases with ${\cal
R}>1.00$ the level of reduction in the local shear stress compared with
${\tau}_{1}$ observed in the ${\tau}<{\tau}_{1}$ portion of the texture for
${\kappa}_{2}>0$, is nearly similar for all the concave textures, and while
the average wall shear stress has a decreasing trend, the total wetted surface
area of the texture is increased (Fig. 11), and thus no substantial difference
with respect to the triangular textures is seen.

### Inflection point

Figure 12: (a) Dimensionless torque experienced by textured rotors defined by
third order polynomials, with ${\cal R}=1.00$, $\kappa_{2}=0.00$, and
$\lambda/d=0.26$ as a function of $\kappa_{3}$. Torque is normalized by the
torque experienced by the smooth rotor. Images of samples are presented above
the corresponding points. (b) Dimensionless torque experienced by textured
rotors with third order profiles, with ${\cal R}=1.00$, $\lambda/d=0.26$, and
$\kappa_{3}=-3.00$, $-1.00$, and $1.00$, as a function of $\kappa_{2}$. Torque
is normalized by the torque experienced by the smooth rotor, $G_{0}$. Images
of the samples are presented above the corresponding points. (c) Contours of
dimensionless torque of textured rotors with a profile defined by third order
polynomials with ${\cal R}=1.00$ as a function of $\kappa_{3}$, and
$\kappa_{2}$, normalized by the torque of a smooth rotor, $G_{0}$, calculated
numerically, for grooves with $\lambda/d=0.26$.

After concavity, the next level of complexity is the possibility of the
profile $f$ to possess an inflection point and the simplest function for that
is a third order polynomial of the form $a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$.
For a third order polynomial, assuming $d^{3}n_{\rm w}/dx^{3}=6\kappa_{3}$,
and an average concavity (the second derivative of the function is no longer
constant) defined as

$\dfrac{dn_{\rm w}}{dx}\bigg{|}_{x=1^{-}}-\dfrac{dn_{\rm
w}}{dx}\bigg{|}_{x=0^{+}}=2\kappa_{2}$ (18)

and height-to-half-spacing ratio of $\cal R$, we have:

$n_{\rm
w}=\kappa_{3}x^{3}+\left(-\dfrac{3}{2}\kappa_{3}+\kappa_{2}\right)x^{2}+\left(\dfrac{\kappa_{3}}{2}-\kappa_{2}-{\cal
R}\right)x+n_{0},$ (19)

where at a constant $\cal R$, to keep the textures monotonic in $0\leqslant
x\leqslant 1$, the limits of $\kappa_{2}$ and $\kappa_{3}$ are defined by

$-\left({\cal
R}-\dfrac{{\kappa_{3}}}{2}\right)\leqslant\kappa_{2}\leqslant\left({\cal
R}-\dfrac{{\kappa_{3}}}{2}\right)$ (20)

for $-{\cal{R}}\leqslant\kappa_{3}\leqslant 2{\cal{R}}$ and

$-\sqrt{-3\kappa_{3}\left(\dfrac{\kappa_{3}}{4}+{\cal
R}\right)}\leqslant\kappa_{2}\leqslant\sqrt{-3\kappa_{3}\left(\dfrac{\kappa_{3}}{4}+{\cal
R}\right)}$ (21)

for $-4{\cal{R}}\leqslant\kappa_{3}\leqslant-{\cal{R}}$ (cross-sectional
images of 3D printed textures with $f$ defined by Eq. (19) as a function of
$\kappa_{2}$ and $\kappa_{3}$ are presented in Fig. 12(a) and 12(b)). Note
that if $\kappa_{3}=0$, then Eq. (19) returns back to Eq. (6), and thus this
definition encompasses all the textures defined by the lower order polynomials
as well. While all third order polynomials have one inflection point in
$\mathbb{R}$, it is not guaranteed that this point resides within the bounds
of $0\leqslant x\leqslant 1$ where profile $f$ is defined and for Eq. (19),
only profiles with $|\kappa_{2}/\kappa_{3}|<3/2$ have an inflection point in
$0\leqslant x\leqslant 1$.

For textures defined by third order polynomials with ${\cal R}=1$, and
$\lambda/d=0.26$, in the absence of mean concavity/convexity (i.e.
$\kappa_{2}=0$), addition of the $\kappa_{3}$ term
($-4\leqslant\kappa_{3}\leqslant 2$) does not offer a significant difference
in the torque compared to the triangular textures ($\kappa_{2}=\kappa_{3}=0$
\- Fig. 12(a)) and $G$ in the CCF regime (independent of the Reynolds number)
is only marginally affected by $\kappa_{3}$ as found by numerical simulations
and experimental results, with the limiting ends ($\kappa_{3}=-4$ and
$\kappa_{3}=2$) offering slightly larger torque values.

Figure 13: Distribution of wall shear stress for textures with ${\cal
R}=1.00$, $\lambda/2=0.26$, and $\kappa_{2}=0.00$, normalized by (a) the shear
stress of smooth rotor, $\tau_{0}$, and (b) point-wise normalized by the shear
stress of a triangular groove ($\kappa_{2}=\kappa_{3}=0$), $\tau_{1}$, as a
function of $\kappa_{3}$ and spanwise location, $x$. (Wall shear stress
results for the triangular textures have been additionally marked with dashed
lines.) (c) Distribution of wall shear stress for textures defined by third
order polynomials with ${\cal R}=1.00$, $\lambda/2=0.26$, and
$\kappa_{3}=-3.00$ as a function of $\kappa_{2}$ and spanwise location, $x$.
Figure 14: Iso-lines of curvilinear, orthogonal coordinate system $(\eta-\xi)$
inside grooves, defined by third order equation Eq. (19), with ${\cal R}=1$,
$\lambda/d=0.26$, and $\kappa_{2}=0$ as a function of $\kappa_{3}$. Red
circles and blue squares denotes segments of the iso-$\xi$ lines with positive
and negative curvature respectively.

This negligible difference in the response of textures defined by this group
of third order polynomials and that of a triangular profile
($\kappa_{3}=0.00$) is further assessed using the wall shear stress
distribution at the textured wall normalized by the shear stress of a smooth
rotor, $\tau_{0}$, and point-wise normalized by the shear stress of a
triangular textures, $\tau_{1}$, both as a function of the spanwise direction
(Figs. 13(a) and 13(b)). In all cases, the wall shear stress distribution
oscillates about the wall shear stress of the triangular groove
($\kappa_{3}=0.00$, marked by dashed lines on Figs. 13(a) and 13(b)); for
$\kappa_{3}<0$ the shear stress distribution twice crosses over the wall shear
stress of the triangular textures where along $x$ within $0<x<1$, it is
initially lower, then higher, and then lower than the wall shear stress of the
triangular grooves. For $\kappa_{3}>0$, the cross-over happens three times
where close to the peak, for a small portion, the wall shear stress is lower
than the triangular case, and then moves to higher, then lower, and ultimately
higher than the wall shear stress for the triangular textures. (Note that the
point closest to $x=1$ that seems like a cross-over in $\tau/\tau_{{}_{1}}$ in
Fig. 13(b) is due to the $\tau_{{}_{1}}$ changing sign close to the trough and
at this point $\tau/\tau_{{}_{1}}\rightarrow\infty$.) As a result of this
oscillation for both positive and negative $\kappa_{3}$, the changes in the
wall shear stress cancel each other out. In addition, while the average shear
stress of each of these textures is lower than that of a triangular texture
(Fig. 15(a)), the excess wetted surface area of the textures with
$\kappa_{3}\neq 0$ overcomes the decrease in the average shear stress. All
these result in ultimately a negligible difference in the total frictional
torque experienced by this family of textured walls as a function of
$\kappa_{3}$.

While the cross-over of the wall shear stress distributions between
$\tau>\tau_{1}.00$ and $\tau<\tau_{1}.00$ close to peaks and troughs are also
observed in the profiles of textures defined by polynomials of second order
(Figs. 9 and 10), the cross-over observed close to $x\approx 1/2$ is due to
the presence of the inflection point at $x=1/2$. The effect of the inflection
point is clear in the distribution and the curvature of the iso-$\xi$ lines
(Fig. 14) inside these grooves. For $\kappa_{3}<0$, the iso-$\xi$ lines are
clustered close to each other at the peaks, and as one moves along $x$, they
slowly become distant, until reaching close to the inflection point where the
lines cluster together and then again grow apart close to the troughs. As a
result of this, unlike the case of the profiles defined by second order
polynomials, close to the texture wall for cases with $\kappa_{3}<0.00$, a
portion of the iso-$\xi$ lines past the inflection point at $x=1/2$ has a
negative curvature, which also corresponds to the region with
$\tau/\tau_{1}>1.00$.

Similarly, for the rest of the third order textures with ${\cal R}=1$,
$\lambda/d=0.26$, $\kappa_{3}\neq 0.00$ (Figs. 12(b), 13(c), and 15(b)), the
trend in the measured torque in CCF regime is nearly insensitive to
$\kappa_{3}$ and is primarily dependent on $\kappa_{2}$. Similar to the case
of $\kappa_{2}=0.00$, the oscillation of the wall shear stress distribution of
the cases with $\kappa_{3}\neq 0.00$ around the wall shear stress distribution
of the texture with the same $\kappa_{2}$ and $\kappa_{3}=0.00$ results in
nearly similar levels of torque among textures with constant $\kappa_{2}$ and
different $\kappa_{3}$ terms. Also, gathering the results of the numerical
simulations of all third order textures into a contour plot of normalized
torque as a function of $\kappa_{2}$ and $\kappa_{3}$ (Fig. 12(c)) shows that
the changes are dominantly happening along the $\kappa_{2}$ axis and the
changes along the $\kappa_{3}$ axis are minor.

Figure 15: Variations in the average wall shear stress, $\overline{\tau}$,
normalized by the shear stress on a smooth rotor, $\tau_{{}_{0}}$ (right axis
of the plots), experienced by textures defined by third order polynomials with
${\cal R}=1.00$, $\lambda/d=0.26$, and (a) $\kappa_{2}=0.00$ as a function of
$\kappa_{3}$, and (b) $\kappa_{3}=-3.00$ as a function of $\kappa_{2}$ .The
variations in the wetted surface area of the textures, normalized by the
surface area of a smooth rotor, $S_{0}$, is also shown for both cases (left
axis of each plots).

### A generalized mathematical framework for construction of symmetric and
periodic 2D textures

The complexity of a texture profile is then extended by considering a
polynomial of $J$-th order. To generalize the definition, suppose we want to
generate the profile of a texture, $f$, in the form of a function
characterizing the local height of the texture in between each peak-trough
($P_{i}$-$T_{i}$) segment within $0\leqslant x\leqslant 1$ (Fig. 1(b)). Let
the profile of the textures to be of the form of a polynomial

$n_{\rm w}=\sum_{j=0}^{J}a_{j}x^{j}\ \ \ \ 0\leqslant x\leqslant 1$ (22)

where $J$ or the degree of polynomial is a measure of the geometric complexity
of the texture. Coefficients $a_{j}$ can take any value, however, here, the
domain of each of the coefficients is limited to ensure the all profiles pass
though the chosen $P_{i}$ and $T_{i}$; the profiles are continuous everywhere;
the derivatives of the profiles are continuous everywhere except at the peaks
and troughs of the textures; chosen peaks and troughs of the function are the
only global extrema of the function, and no local minima and/or maxima is
allowed within $0<x<1$.

While any family of functions can be a valid choice for mathematically
representing the shape of the profiles, polynomials not only have the ability
to capture some of the most popular texture profiles (as shown earlier) but
also allow for gradually increasing the complexity of the textures by changing
the degree $J$; Using Eq. (22) with $J=0$ (the zeroth order), $n_{w}=n_{0}$
returns the smooth reference surface (i.e. case of ${\cal{R}}=0$), and $J=1$
creates the V-groove (or triangular) profile with $a_{0}=n_{0}$ and
$a_{1}=-{\cal R}$ (Eq. (5)).

For the first order, height and spacing of the texture are sufficient to
uniquely define the profiles and the coefficients $a_{0}$ and $a_{1}$.
However, from the second order forward, there is no geometric guide dictating
the boundary conditions to define the value of the coefficients and variations
in values of $a_{j}$ ($j\geqslant 2$) results in more complicated texture
profiles, while keeping the wavelength and height-to-half-spacing constant. At
each $j$ level, the effect of $a_{j}x^{j}$ monomial can be decomposed into two
portions, one part is entirely responsible for the effect of $j$th order and
the other portion ensures that the profile passes through the given peaks and
troughs. Let’s assume that the portion of the $a_{j}$ coefficient responsible
for the effect of the level of complexity, $j$, is represented by the
parameter $\kappa_{j}$ (all $\kappa_{j}$ independent of each other), and is
defined as the difference of the ($j-1$)th derivative of the profile at the
(limits of) peak and trough normalized by $j!$. Thus for $2\leqslant
j\leqslant J$

$\dfrac{d^{(j-1)}n_{\rm
w}}{dx^{(j-1)}}\bigg{\rvert}_{x=1^{-}}-\dfrac{d^{(j-1)}n_{\rm
w}}{dx^{(j-1)}}\bigg{\rvert}_{x=0^{+}}=j!\ \kappa_{j}$ (23)

and for $j=1$

$n_{\rm w}\bigg{\rvert}_{x=1}-n_{\rm w}\bigg{\rvert}_{x=0}=1!\ \kappa_{1}.$
(24)

and $\kappa_{0}=n_{0}$ is entirely dependent on the choice of the reference
surface. As a result, coefficients $a_{j}$, can be decomposed into weighted
sums of the $\kappa_{{}_{l}}$ where $l\geqslant j$, separating the effect of
each order,

$a_{j}=\sum_{l=j}^{J}{\kappa}_{{}_{l}}m_{{}_{lj}}$ (25)

and ultimately collecting all $m_{{}_{lj}}$ coefficients in matrix
$\mathbf{\cal M}$ we can write $n_{\rm w}$ in the form

$n_{\rm
w}(x)=\begin{bmatrix}\kappa_{0}&\kappa_{1}&\cdots&\kappa_{{}_{J}}\end{bmatrix}\mathbf{\cal{M}}\begin{bmatrix}x^{0}\\\
x^{1}\\\ \vdots\\\ x^{J}\end{bmatrix}.$ (26)

Based on this definition, $\mathbf{\cal M}$ is a lower triangular matrix where
$m_{{}_{00}}=m_{{}_{11}}=\dots=m_{{}_{JJ}}=1$ and the rest of the coefficients
of each column can be calculated using Eq. (23) and Eq. (24). From the
boundary conditions, for all textures we have $\kappa_{1}=-{\cal R}$ and
$\kappa_{0}={\cal R}$, and the rest of $\kappa_{j}$ can take any value within
the physical ranges defined by the conditions described by the user.

The effect of this decomposition can be demonstrated using the two cases
studied earlier; a second order profile of the form of Eq. (6), is written as
the sum of $\kappa_{2}(x^{2}-x)$, and $-{\cal R}x+n_{0}$. Rewritten this way,
the second order profile is decomposed into the profile of a first order
texture (Eq. (5)) and a term of second order entirely dependent on
$\kappa_{2}$; the first monomial, $\kappa_{2}x^{2}$ adds curvature to the
profile and the second monomial $-\kappa_{2}x$ ensures that the addition of
the $\kappa_{2}x^{2}$ does not violate the requirement of the profile passing
through the designated peaks and troughs (Fig. 16(a)). Similarly, the third
order profile of Eq. (19) can be written as the sum of
$\kappa_{3}\left(x^{3}-{3}/{2}x^{2}+{1}/{2}x\right)$ (a third order part
identified by the $\kappa_{3}$) and Eq. (6) (Fig. 16(b)). Thus, this
decomposition, separates the effect of each level of complexity while also
including the effect of lower order terms within its definition. This results
in textures defined by lower order polynomials to be subsets of those defined
by higher orders and allowing for easier comparison across shapes (i.e.
setting $\kappa_{{}_{J}}=0$ returns the family of textures defined by order
$J-1$). Thus, the decomposition of the coefficients of the polynomials using
this definition of the $\kappa_{{}_{j}}$ allows for successive expansion of
the texture profiles one order at a time toward more complicated shapes, while
keeping the lower order ones among the possible variations.

Figure 16: (a) Profile of an example texture, $f(x)$, defined by a second
order polynomial decomposed into terms identified by $x^{0}$ and $x$, and
$x^{2}$. Note that with $\kappa_{2}x^{2}$ term alone, the profile cannot
maintain the location of the peak and trough of the texture, and the
$-\kappa_{2}x$ monomial is required to satisfy this boundary condition. (b)
Profile of an example texture, $f(x)$, defined by a third order polynomial
decomposed into terms identified by $\kappa_{0}=n_{0}$ and $\kappa_{1}=-{\cal
R}$, $\kappa_{2}$, and $\kappa_{3}$. Figure 17: Torque experienced by textured
rotors with profiles defined by fourth order polynomials (Eq. (27)) as a
function of $\kappa_{4}$, normalized by the torque of a smooth rotor, $G_{0}$,
with ${\cal R}=1.00$, $\lambda/d=0.26$, and various values of $\kappa_{2}$ and
$\kappa_{3}$. Variations in the shape of the profiles as a function of all the
$\kappa$ parameters of each group are also shown below the plot. Figure 18:
Curvilinear orthogonal coordinates $(\eta-\xi)$ for textures defined by fourth
order polynomials with ${\cal R}=1.00$ (a) $\kappa_{2}=\pm 2.00$, and
$\kappa_{3}=-2.00$, and (b) $\kappa_{2}=0.00$, $\kappa_{3}=-2.00$ and $-3.00$
as a function of $\kappa_{4}$ values within their allowed limits.

Polynomials of order higher than 3, offer a wide range of possibilities to
design texture profiles, however, the increase in the complexity does not
offer a substantial difference in the hydrodynamic response of these textures
in the CCF when compared with the second order textures. Results of
experiments and simulations with textures defined by fourth order polynomials
of the form

$n_{\rm
w}=\kappa_{4}(x^{4}-2x^{3}+x^{2})+\kappa_{3}\left(x^{3}-\dfrac{3}{2}x^{2}+\dfrac{1}{2}x\right)+\kappa_{2}(x^{2}-x)-{\cal
R}x+n_{0}$ (27)

presented in Fig. 17, show that variations in the torque measured with such
textures are mainly dictated by $\kappa_{2}$, and as $\kappa_{2}$ value is
decreased, the effect of $\kappa_{4}$ becomes important only in offering a
wider variation in the torque, but the variation in the values of torque
recorded using textures with fourth order polynomials are within the same
range as the variations seen with the second order profiles. In textures with
a positive average concavity ($\kappa_{2}>0$) and a constant $\kappa_{3}$,
increase in $\kappa_{4}$ only offers a slight increase in the torque, and
analysis of the iso-$\xi$ lines of this sub-family of textures confirms the
expansion of a region with negative curvature around $x=1/2$ which results in
the slight increase in the torque (Fig. 18(a)). In the absence of average
concavity ($\kappa_{2}=0$), the response of the textures are entirely
dependent on the $\kappa_{4}$ parameter and independent of $\kappa_{3}$, and
as $\kappa_{4}$ is increased, the portion of the region close to the wall
where the iso-$\xi$ lines have a negative curvature increases. As a result,
within this sub-family of textures, increase in $\kappa_{4}$ results in an
increase in the torque (Figs. 17 and 18(b)). As one moves toward
$\kappa_{2}<0$ cases, the effect of $\kappa_{4}$ becomes more pronounced
allowing for a wider range of torque responses compared to the cases with
$\kappa_{2}\geqslant 0$ (Figs. 17 and 18(a)). Overall, in all cases, at
constant $\kappa_{2}$ and $\kappa_{3}$, textures with the lowest $\kappa_{4}$
offer the lowest torque of the sub-family, and increasing $\kappa_{4}$ results
in an increase in the portion of iso-$\xi$ close to the wall with negative
curvature which directly results in an increase in the total frictional torque
within each sub-family considered (Fig. 18).

### Families of extreme concave/convex textures

The definition of textures based on polynomials of higher orders allows us to
consider families of textures that constitute the two most extreme cases of
each order $J$: the case of textures that are either extremely concave defined
by $n_{\rm w}={\cal R}(1-x)^{J}+(n_{0}-{\cal R})$ or extremely convex given by
$n_{\rm w}=-{\cal R}x^{J}+n_{0}$. (Figs. 19(a) and 19(b) - $\kappa_{j}$
coefficients of these textures up to $J=10$, for ${\cal R}=1$, are listed in
Tbls. 2 and 2.) In the extreme concave case as $J$ is increased toward
infinity the shape of the texture resembles an idealized form of rectangular
(razor blade) textures [53, 57] and for the extreme convex ones as $J$ is
increased toward infinity, the profile asymptotically reaches back to the case
of a smooth surface. In the concave group, as the order $J$ of the polynomial
is increased, the portion of the texture in the vicinity of the trough is
smoothed more and more (Fig. 19(a)), and in the convex group, increase in the
order of the polynomial results in a wider smooth area at the peak of the
textures (Fig. 19(b)). While the increase in the polynomial order of these
textures requires additional $\kappa_{j}$ terms to define the profile, the
increase in the concavity/convexity of these profiles is directly manifest in
the increase in the magnitude of the $\kappa_{2}$ term of each profile (Tbls.
2 and 2).

Figure 19: Schematic of families of (a) extreme concave and (b) extreme convex
textures of orders 2-10 with arbitrary height-to-half-spacing of $\cal R$.
Triangular groove ($J=1$) is shown for reference. (c) Torque experienced by
extreme concave (triangular symbol) and extreme convex (rectangular symbol)
textures of ${\cal R}=1$, and $\lambda/d=0.26$, normalized by the torque of a
smooth surface. Torque values of a smooth surface and triangular texture
($J=0$ and $J=1$) are shown as reference. Results of the direct numerical
simulation of concave and convex textures up to $J=10$ are shown with dashed
and dotted lines respectively. Distribution of the wall shear stress
normalized by the wall shear stress of a smooth surface as a function of the
spanwise direction for (d) extreme concave and (e) extreme convex textures
with ${\cal R}=1.00$, and $\lambda/d=0.26$. The results for $J=1$ is presented
for reference.

In the case of extreme concave textures ($J\geqslant 2)$, with ${\cal
R}=1.00$, and $\lambda/d=0.26$, textures of order 2 and 3 have a nearly
similar torque response (Fig. 19(c)) which is less than the case of the
triangular textures and then beyond that as the order $J$ is increased, the
total frictional torque is also increased, and after $J=7$ it passes to higher
than the torque experienced by a triangular texture. On the extreme convex
side, for $J\geqslant 2$ as $J$ is increased (Fig. 19(c)), the torque is also
increased and is always higher than torque of a triangular texture and by
$J=7$ it crosses the torque of a smooth surface.

Table 1: ${\kappa}_{j}$ values for extreme concave textures for polynomial
orders 0-10 for ${{\cal R}=1}$

Polynomial Order | $\kappa_{0}$ | $\kappa_{1}=-{\cal R}$ | $\kappa_{2}$ | $\kappa_{3}$ |  $\kappa_{4}$, |  $\kappa_{5}$, |  $\kappa_{6}$, | $\kappa_{7}$ |  $\kappa_{8}$, |  $\kappa_{9}$, | $\kappa_{10}$
---|---|---|---|---|---|---|---|---|---|---|---
$J=0$ |  $n_{0}$-1 |  |  |  |  |  |  |  |  |  |
$J=1$ |  $n_{0}$-1 | -1 |  |  |  |  |  |  |  |  |
$J=2$ |  $n_{0}$-1 | -1 | 1 |  |  |  |  |  |  |  |
$J=3$ |  $n_{0}$-1 | -1 | 3/2 | -1 |  |  |  |  |  |  |
$J=4$ |  $n_{0}$-1 | -1 | 2 | -2 | 1 |  |  |  |  |  |
$J=5$ |  $n_{0}$-1 | -1 | 5/2 | -10/3 | 5/2 | -1 |  |  |  |  |
$J=6$ |  $n_{0}$-1 | -1 | 3 | -5 | 5 | -3 | 1 |  |  |  |
$J=7$ |  $n_{0}$-1 | -1 | 7/2 | -7 | 35/4 | -7 | 7/2 | -1 |  |  |
$J=8$ |  $n_{0}$-1 | -1 | 4 | -28/3 | 14 | -14 | 28/3 | -4 | 1 |  |
$J=9$ |  $n_{0}$-1 | -1 | 9/2 | -12 | 21 | -126/5 | 21 | -21 | 4.5 | -1 |
$J=10$ |  $n_{0}$-1 | -1 | 5 | -15 | 30 | -42 | 42 | -30 | 15 | -5 | 1

Table 2: ${\kappa}_{j}$ values for extreme convex textures for polynomial
orders 0-10 for ${{\cal R}=1}$

Polynomial Order | $\kappa_{0}$ | $\kappa_{1}=-{\cal R}$ | $\kappa_{2}$ | $\kappa_{3}$ |  $\kappa_{4}$, |  $\kappa_{5}$, |  $\kappa_{6}$, | $\kappa_{7}$ |  $\kappa_{8}$, |  $\kappa_{9}$, | $\kappa_{10}$
---|---|---|---|---|---|---|---|---|---|---|---
$J=0$ |  $n_{0}$-1 |  |  |  |  |  |  |  |  |  |
$J=1$ |  $n_{0}$-1 | -1 |  |  |  |  |  |  |  |  |
$J=2$ |  $n_{0}$-1 | -1 | 1 |  |  |  |  |  |  |  |
$J=3$ |  $n_{0}$-1 | -1 | -3/2 | -1 |  |  |  |  |  |  |
$J=4$ |  $n_{0}$-1 | -1 | -2 | -2 | -1 |  |  |  |  |  |
$J=5$ |  $n_{0}$-1 | -1 | -5/2 | -10/3 | -5/2 | -1 |  |  |  |  |
$J=6$ |  $n_{0}$-1 | -1 | -3 | -5 | -5 | -3 | -1 |  |  |  |
$J=7$ |  $n_{0}$-1 | -1 | -7/2 | -7 | -35/4 | -7 | -7/2 | -1 |  |  |
$J=8$ |  $n_{0}$-1 | -1 | -4 | -28/3 | -14 | -14 | -28/3 | -4 | -1 |  |
$J=9$ |  $n_{0}$-1 | -1 | -9/2 | -12 | -21 | -126/5 | -21 | -21 | -4.5 | -1 |
$J=10$ |  $n_{0}$-1 | -1 | -5 | -15 | -30 | -42 | -42 | -30 | -15 | -5 | -1

For the family of extreme concave textures ($J\geqslant 2$), at the trough,
the slope of the profiles come to zero and as $J$ is increased a larger
segment of the profile in the vicinity of $x=1$ has a near-zero slope. Reverse
of this trend applies to the peaks of extreme convex profiles where near the
peaks the segment of the profile with near-zero slope is increased as $J$ is
increased. This directly affects the wall shear stress distribution (Fig.
19(d)) where in the extreme concave cases for $J\geqslant 2$, the local shear
stress at the trough of the grooves consistently increases as $J$ is increased
and only for $J=2$, the local shear stress at the trough is minimum within
$0\leqslant x\leqslant 2$ and the minimum local wall shear stress for the rest
of the profiles is pushed away from the trough and closer to the peaks. Thus,
as order $J$ is increased the average shear stress within the texture profile
$\overline{\tau_{{}_{J}}}$ takes a decreasing trend (Fig. 20). However, as $J$
is increased, the increase in the wetted surface area results in the the total
frictional torque to be the lowest for orders 2 and 3, and then to increases
as $J$ is increased. On the extreme convex side, as the order is increased the
local wall shear stress at the peak is slowly reduced and asymptotically
reaches the value of the wall shear stress of a smooth surface (Fig. 19(e)).
However, due to the excess wetted surface area compared to the smooth surface,
for $J>7$ the total frictional torque is slightly larger than that of a smooth
surface. With average shear stress $\overline{\tau_{{}_{J}}}$ of the convex
textures being always larger than that of the concave textures (Fig. 20), even
with the decreasing trend (as $J$ is increased), with the increase in the
wetted surface area, torque experienced by convex textures takes a
monotonically increasing trend as the order is increased ($J>1$).

Figure 20: Total wetted surface area, $S_{{}_{J}}$ (left axis), and the
average shear stress, $\overline{\tau_{{}_{J}}}$ (right axis) of extreme
concave (triangular symbols) and convex (square symbols) textures as a
function of their order $J$ for $J\geqslant 1$. Wetted surface area is
normalized by the wetted surface area of a smooth reference rotor ($S_{0}$),
and the average shear stress is normalized by the shear stress on a smooth
rotor, $\tau_{0}$. It is clear that for $J=0$, $S_{{}_{J}}/S_{0}=1$ and
$\overline{\tau_{{}_{J}}}/\tau_{0}=1$ (not shown).

Ultimately, the comparison of the two extreme concave and convex textures with
each other and with the smooth wall and the triangular texture shows that
having a near smooth peak is more detrimental to the frictional torque
experienced by a texture than having a near smooth trough, and while not the
most ideal case, the extreme-concave textures can maintain a torque lower or
equal to the level of the triangular textures for $J<8$ while extreme convex
textures can barely maintain the reduction and are not well suited for the
purpose of drag reduction in the CCF regime. This can also be a reason for
natural textures on shark denticles [2, 65], and peristome of the pitcher
plants [25, 26, 27, 28] having a concave form resembling profiles of $J=2$ or
$J=3$, as opposed to triangular or convex shapes.

## Discussion & conclusion

The decomposition of a texture profile into the effects of the $\kappa_{j}$
parameters, allows for a more comprehensive understanding of the effect of the
geometry of the boundary on the physical response of the surface. Here, with a
focus on the hydrodynamic response of the surface in laminar flows, it is
clear that after the height-to-half-spacing, the $\kappa_{2}$ parameter,
defining the average level of concavity of a texture plays the most important
role in controlling the total frictional response of the surfaces. Textures
defined by second order polynomials expand the range of frictional responses
compared to the conventional V-grooves, but moving to third and fourth order
polynomials does not offer a visible improvement compared to those defined by
second order polynomials.

The proposed decomposition can be employed in reverse for any known texture of
given profile $n_{w}=g(x)$ by fitting a polynomial of degree $J$ using a
spline interpolation. If the coefficients of the $x^{j}$ in the spline are
$b_{j}$, then the values of $\kappa_{j}$ terms for this profile can be found
by solving the inverse of the linear system of equations in Eq. (25), in the
form ${\boldsymbol{\kappa}}=\mathbf{\cal{M}}^{-T}\mathbf{b}$ where
$\boldsymbol{\kappa}$ and $\mathbf{b}$ are the vectors consisting of
$\kappa_{j}$ terms and $b_{j}$ terms respectively.

It should be emphasized that the shape of polynomials at each degree enables
us to create textures with profiles of various levels of complexity and assign
quantitative labels to them. As a result of this framework, we are able to
move away from qualitative description of texture profiles and track the
effect of the geometry not only in terms of the key dimensions of the texture,
but also in terms of the dimensionless $\boldsymbol{\kappa}$ vector that
defines all the fine details of the shape of the profiles.

While this work focuses on the hydrodynamic response of surfaces, this
framework can easily be expanded to other physical problems where textured
surfaces play a role and the effect of each of $\kappa_{j}$ parameters can be
assessed based on the objectives of the problem of interest.

## Declaration of Competing Interest

Author declares no conflict of interest.

## Acknowledgements

This work was supported by the Rowland Fellows program at Harvard University.

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