1. Introduction
Transverse near-wall forcing as a means to mitigate skin-friction drag in turbulent flows has garnered significant attention, owing to its potential for substantial environmental and economic benefits (Quadrio Reference Quadrio2011; Ricco, Skote & Leschziner Reference Ricco, Skote and Leschziner2021). After the seminal work on spanwise wall oscillations by Jung, Mangiavacchi & Akhavan (Reference Jung, Mangiavacchi and Akhavan1992), three decades of research efforts have led to important progress; however, several crucial factors still hinder the deployment of spanwise forcing in technological settings. While devising viable and efficient implementations of the typically idealised near-wall forcing is the major challenge, additional concerns exist, including the decreasing effectiveness of drag reduction with increasing Reynolds numbers (
$Re$
).
To date, the Reynolds number dependence of skin-friction drag reduction has mostly been studied in the context of streamwise-travelling waves of spanwise wall velocity (StTW, Quadrio, Ricco & Viotti Reference Quadrio, Ricco and Viotti2009), a specific form of transverse forcing characterised by its comparatively large potential for drag reduction with moderate energy expenditure. StTW are described by
\begin{equation} w_w (x,t) = A \sin \left ( \kappa x - \omega t \right ), \end{equation}
where
$w_w$
is the spanwise (
$z$
) velocity component at the wall,
$A$
is the maximum wall velocity and thus a measure of the amplitude of the spanwise forcing,
$\kappa$
is the streamwise wavenumber,
$\omega$
is the angular frequency, and
$x$
and
$t$
are the streamwise coordinate and the time. The forcing, sketched in figure 1, consists of streamwise-modulated waves of spanwise velocity at the wall, with wavelength
$\lambda = 2\pi / \kappa$
and period
$T = 2\pi / \omega$
. The waves travel along the streamwise direction with phase speed
$c=\omega /\kappa$
, either downstream (
$c\gt 0$
) or upstream (
$c\lt 0$
) with respect to the mean-flow direction. The forcing described by (1.1) includes the two special cases of spatially uniform spanwise wall oscillations (Quadrio & Ricco Reference Quadrio and Ricco2004) for
$\kappa =0$
and steady waves (Viotti, Quadrio & Luchini Reference Viotti, Quadrio and Luchini2009) for
$\omega =0$
. With the appropriate set of control parameters, StTW have been shown to yield considerable drag reduction in a series of numerical experiments regarding channel and pipe flows (Quadrio et al. Reference Quadrio, Ricco and Viotti2009; Gatti & Quadrio Reference Gatti and Quadrio2013; Hurst, Yang & Chung Reference Hurst, Yang and Chung2014; Gatti & Quadrio Reference Gatti and Quadrio2016; Liu et al. Reference Liu, Zhu, Bao, Zhou and Han2022; Rouhi et al. Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023; Gallorini & Quadrio Reference Gallorini and Quadrio2024) and boundary layers (Skote et al. Reference Skote, Schlatter and Wu2015; Skote Reference Skote2022), as well as in laboratory experiments (Auteri et al. Reference Auteri, Baron, Belan, Campanardi and Quadrio2010; Bird, Santer & Morrison Reference Bird, Santer and Morrison2018; Chandran et al. Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023). In addition to canonical flows, including the compressible and supersonic regimes (Gattere et al. Reference Gattere, Zanolini, Gatti, Bernardini and Quadrio2024), StTW have been applied to more complex flows ranging from channels with curved walls (Banchetti, Luchini & Quadrio Reference Banchetti, Luchini and Quadrio2020) to rough boundary layers (Deshpande, Kidanemariam & Marusic Reference Deshpande, Kidanemariam and Marusic2024) (although restricted to spatially uniform spanwise wall oscillation) and transonic aerofoils with shock waves (Quadrio et al. Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022), showing that local skin-friction drag reduction can be exploited to also reduce the pressure component of the aerodynamic drag.
Understanding how the Reynolds number affects drag reduction by StTW is a particularly challenging goal for three main reasons. First, a sufficiently wide portion of a huge parameter space must be explored, which, even in simple canonical flows, includes the four parameters
$\left \{ A, \kappa , \omega ; Re \right \}$
, and poses a great challenge to numerical and laboratory experiments.
A second complication is the choice of an appropriate figure of merit for drag reduction. Typically, the drag reduction rate
$\mathcal{R}$
is defined as
\begin{equation} \mathcal{R} = 1 - \frac {C_f}{C_{f_{\; 0}}} \, , \end{equation}
i.e. as the control-induced relative change of the skin-friction coefficient
$C_f$
(Kasagi et al. Reference Kasagi, Hasegawa, Fukagata and Eckhardt2009). In (1.2) and in the remainder of this manuscript, the subscript ‘
$0$
’ denotes quantities measured in the reference uncontrolled flow. Specifically,
$C_f$
is defined as
$C_f = 2 \tau _x / (\rho U_b^2$
);
$\tau _x$
is the mean streamwise wall shear stress,
$U_b$
the bulk velocity and
$\rho$
the fluid density. However, as observed by Gatti & Quadrio (Reference Gatti and Quadrio2016), the quantity
$\mathcal{R}$
defined by (1.2) is inherently
$Re$
-dependent, owing to the
$Re$
dependence of
$C_f$
and
$C_{f_{\; 0}}$
. This is long known to be the case for the flow over rough surfaces (Nikuradse Reference Nikuradse1933; Jiménez Reference Jiménez2004), as well as for other flow control techniques relying on near-wall turbulence manipulation such as riblets (Luchini Reference Luchini, Désidéri, Hirsch, Le Tallec, Oñate, Pandolfi, Périaux and Stein1996; Spalart & McLean Reference Spalart and McLean2011). Choosing a figure of merit which eliminates this trivial dependency on the Reynolds number is crucial to describe properly the
$Re$
effect on drag reduction.
Figure 1. Schematic of a turbulent open channel flow actuated with streamwise-travelling waves of spanwise wall velocity with amplitude
$A$
, streamwise wavenumber
$\kappa$
and angular frequency
$\omega$
. Here,
$\lambda$
is the streamwise wavelength;
$c$
is the wave phase speed; and
$L_x$
,
$L_y=h$
and
$L_z$
are the dimensions of the computational domain in the streamwise, wall-normal and spanwise direction, respectively.
Third, the wall-shear stress generally differs in the reference (
$\tau _{x_0}$
) and controlled (
$\tau _x$
) channel flows, unless they are driven by the same pressure gradient (as done for example by Ricco et al. Reference Ricco, Ottonelli, Hasegawa and Quadrio2012); hence, the viscous scaling becomes ambiguous. As noted by Quadrio (Reference Quadrio2011), this results in two possible viscous normalisations of the controlled flow: the first, denoted with the superscript ‘
$+$
’, relies on the reference friction velocity
$u_{\tau _0} = \sqrt {\tau _{x_0} / \rho }$
; the second, denoted with the superscript ‘
$\ast$
’, is based on the actual friction velocity
$u_\tau = \sqrt {\tau _x / \rho }$
. Similarly, two different friction Reynolds numbers,
$Re_{\tau _0}={u_{\tau _0} h} / \nu$
and
$Re_\tau ={u_\tau h} / \nu$
, can be defined depending on the choice of the friction velocity. Here,
$h$
describes the half-height of a channel or the depth of an open channel, and
$\nu$
is the fluid kinematic viscosity. While the actual viscous scaling is the only sensible choice for the drag-reduced flow (Gatti & Quadrio Reference Gatti and Quadrio2016), the reference scaling is necessary when the wall friction of the drag-reduced flow is not known yet.
Gatti & Quadrio (Reference Gatti and Quadrio2016), indicated also as GQ16 hereinafter, circumvented these difficulties by designing a campaign of several thousands of direct numerical simulation (DNS) runs of turbulent channel flows. Inspired by similar studies on rough walls (see for example Leonardi et al. Reference Leonardi, Orlandi, Djenidi and Antonia2015), they limited the otherwise prohibitive computational cost by choosing relatively small computational domains (Jiménez & Moin Reference Jiménez and Moin1991; Flores & Jiménez Reference Flores and Jiménez2010) for most of the study. At the expense of a residual domain-size dependence of the results, which cancels out in large part when observing the difference between controlled and uncontrolled flows, GQ16 generated a large dataset, along with a more limited number of simulations in wider domains to verify the accuracy of the results. This approach enabled not only the inspection of a large portion of the
$\left \{ A, \kappa , \omega \right \}$
-space at
$Re_{\tau _0} = 200$
and 1000, but also the transfer of the dataset between viscous ‘
$+$
’ and ‘
$\ast$
’ units via interpolation, allowing to assess the results in both scalings. Thanks to their comprehensive database (available as Supplementary Material to their paper), Gatti & Quadrio (Reference Gatti and Quadrio2016) challenged the then-current belief that skin-friction drag reduction was bound to decrease quickly with
$Re$
. They demonstrated that the drag reduction effect by spanwise forcing becomes in fact constant with
$Re$
, provided that it is not expressed via
$\mathcal{R}$
(1.2), that is per se
$Re$
-dependent, but through the Reynolds number invariant parameter
$\Delta B^*$
. The quantity
$\Delta B^*$
expresses the main effect of the StTW, which is to induce a change of the additive constant in the logarithmic law for the mean velocity profile
\begin{equation} U^*(y^*) = \frac {1}{k} \ln y^* + B_0^* + \Delta B^* \, , \end{equation}
where
$k$
is the von Kármán constant,
$B_0^*$
is the additive constant in the reference channel flow and
$B^*=B_0^*+\Delta B^*$
is the additive constant of the controlled flow. The independence of
$\Delta B^*$
upon
$Re$
is a common feature of all turbulence manipulations whose action is confined to the near-wall region. In these cases, the outer turbulence simply reacts to a wall layer with different drag (Gatti et al. Reference Gatti, Stroh, Frohnapfel and Hasegawa2018), as well known, for instance, in the context of drag-reducing riblets (Luchini Reference Luchini, Désidéri, Hirsch, Le Tallec, Oñate, Pandolfi, Périaux and Stein1996; Garcia-Mayoral & Jiménez Reference Garcia-Mayoral and Jiménez2011; Spalart & McLean Reference Spalart and McLean2011) and drag-increasing roughness (Clauser Reference Clauser1954; Hama Reference Hama1954).
Under the assumption that
$\Delta B^*$
is a function of the control parameters
$\left \{ A^\ast , \kappa ^\ast , \omega ^\ast \right \}$
, but not of the Reynolds number, Gatti & Quadrio (Reference Gatti and Quadrio2016) derived the following modified friction relation (hereinafter called GQ model):
\begin{equation} \Delta B^* = \sqrt {\frac {2}{C_{f_{\; 0}}}} \left [ (1-\mathcal{R})^{-1/2} -1 \right ] - \frac {1}{2 k}\ln \left (1 - \mathcal{R} \right ) \, , \end{equation}
where the
$Re$
dependence is not explicit, but rather embedded in
$C_{f_{\; 0}}$
. Provided the function
$\Delta B^* (A^\ast , \kappa ^\ast , \omega ^\ast )$
is measured at a sufficiently large
$Re$
for the log law in (1.3) to hold, the GQ model predicts
$\mathcal{R}$
at any arbitrary value of
$Re$
. According to (1.4),
$\mathcal{R}$
is always expected to decrease with
$Re$
for any combination of the control parameters, but at much lower rate than suggested by previous studies (Touber & Leschziner Reference Touber and Leschziner2012; Gatti & Quadrio Reference Gatti and Quadrio2013; Hurst et al. Reference Hurst, Yang and Chung2014), so that significant drag reduction can be still achieved at Reynolds numbers typical of technological applications. For instance, for StTW, GQ16 estimated possible drag reduction of 30 % with
$A^+=12$
at
$Re_{\tau _0}=10^5$
.
The GQ16 study is affected by two limitations. First,
$Re_{\tau _0}=1000$
, the largest
$Re$
considered in their study, may still be not enough for
$\Delta B^*$
to become completely
$Re$
-independent: GQ16 suggested that at least
$Re_{\tau _0}=2000$
should be considered. Second, the small residual effect of the restricted computational box sizes on the quantification of
$\mathcal{R}$
could, in principle, bias the extrapolation to higher
$Re$
. Nonetheless, the GQ model passed validation tests against previous (Touber & Leschziner Reference Touber and Leschziner2012; Hurst et al. Reference Hurst, Yang and Chung2014) and later literature data. For instance, Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) employed large eddy simulation (LES) to study drag reduction by StTW in open channel flows at
$Re_{\tau _0}=945$
and
$Re_{\tau _0}=4000$
. They explored the parameter space within the range
$\kappa ^+ \in [0.002, 0.02 ]$
and
$\omega ^+ \in [-0.2, 0.2 ]$
, at fixed
$A^+=12$
. This is to be compared with
$\kappa ^+ \in [0,0.05]$
and
$\omega ^+ \in [-0.5, 1]$
addressed by Gatti & Quadrio (Reference Gatti and Quadrio2016), who also considered various amplitudes
$A^+ \in [2, 20]$
. The study of Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) is however limited by the use of LES, in which part of the small-scale turbulence physics involved in drag reduction is modelled, and by the domain size (
$L_x = 2.04h$
,
$L_z = 0.63h$
at
$Re_{\tau _0}=4000$
), which is comparable to the restricted domain size (
$L_x = 1.35h$
,
$L_z = 0.69h$
at
$Re_{\tau _0}=1000$
) considered by Gatti & Quadrio (Reference Gatti and Quadrio2016), despite the larger
$Re_{\tau _0}$
. Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) confirmed that the GQ model predicts very well their drag reduction data, with deviations of the order of 2 %, for all StTW control parameters sufficiently far from those yielding drag increase.
Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) and Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) studied drag reduction via backward-travelling (
$c\lt 0$
) StTW. Their experimental study was carried out in a zero pressure gradient turbulent boundary layer up to the largest values of
$Re$
investigated so far,
$Re_\tau = 15\,000$
. By extending to the plane geometry the actuation strategy used by Auteri et al. (Reference Auteri, Baron, Belan, Campanardi and Quadrio2010) in a cylindrical pipe, they implemented the ideal forcing of (1.1) by dividing a portion of the wall into a series of forty-eight slats, each
$5\, \mathrm{cm}$
long, so that each six consecutive slats constitute a single wavelength with fixed
$\lambda = 0.3\, \mathrm{m}$
. The slats move in the spanwise direction at a fixed half-stroke
$d$
, resulting in a frequency-dependent maximum spanwise velocity
$A = \omega d$
. As a consequence, in those experiments, the amplitude and period of the oscillations could not be varied independently. With
$d$
and
$\lambda$
constant in physical units, the range of investigated parameters shifts towards smaller
$\kappa ^+$
,
$\omega ^+$
and
$A^+$
as
$Re_{\tau _0}$
increases. The authors observed, for the first time,
$\mathcal{R}$
to increase with
$Re$
(see figure 3(e) of Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), and explained it by the particularly slow time scale
$T^+ = 2 \pi / \omega ^+ \lt -350$
of their forcing, which was meant to target the large inertial, outer-scaled structures of turbulence (Deshpande et al. Reference Deshpande, Chandran, Smits and Marusic2023), whose importance increases with
$Re$
.
Despite the promising results, these studies also have shortcomings. With
$d$
and
$\lambda$
constant in physical units, which is unavoidable in laboratory experiments, the control parameters could not be kept constant in either ‘
$+$
’ or ‘
$*$
’ viscous units while varying
$Re_{\tau _0}$
. In particular, the fixed wavelength leads to a
$\kappa ^+$
that decreases with
$Re$
. Furthermore, the effect of
$\omega$
and
$A$
cannot be addressed separately. This precludes the investigation of the full space of the control parameters: for instance, large values of
$\omega ^+$
at low
$A^+$
cannot be tested. Lastly, the key observation that
$\mathcal{R}$
increases with
$Re$
relies on the joint observation of low-
$Re$
LES data by Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) obtained in an open channel flow and high-
$Re$
experimental data by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) in a boundary layer, thus bringing together different methods and flow configurations.
The present research fills these gaps in the existing literature by leveraging a novel DNS dataset of turbulent open channel flow to accurately quantify the Reynolds number effects on the drag-reducing performance of StTW. The computational domain adopted in the present simulations is large enough to properly account for all relevant scales of turbulence, including the large inertial scales. The considered Reynolds numbers, ranging from
$Re_{\tau _0}=1000$
to
$Re_{\tau _0}=6000$
, are large enough to minimise the low-
$Re$
effects, matching some of the experimental data points by Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023). The dataset is further designed to address the Reynolds number scaling of drag reduction in both viscous and outer units independently, by considering the same flow configuration and by using the same numerical method for all
$Re$
.
The paper is organised as follows. After this Introduction, § 2 describes the computational procedure and the simulation parameters used to produce the DNS dataset. In § 3, the effect of the Reynolds number is analysed in terms of both drag reduction and power budgets, and compared with existing literature. Finally, concluding arguments are given in § 4.
2. Methods and procedures
A new DNS dataset of incompressible turbulent open-channel flows (see figure 1) is used to study the effect of the Reynolds number on the reduction of the turbulent friction drag achieved by StTW. The open channel flow, i.e. half a channel flow with a symmetry boundary condition at the centreplane, is considered here to reduce the computational cost without affecting the drag reduction results; indeed, it was often used in the past, including e.g. the similar studies by Yao, Chen & Hussain (Reference Yao, Chen and Hussain2022), Pirozzoli (Reference Pirozzoli2023) and Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023). The StTW are applied as a wall boundary condition for the spanwise velocity component after (1.1). Periodic boundary conditions are applied in the homogeneous streamwise and spanwise directions, no-slip and no-penetration boundary conditions are used for the longitudinal and wall-normal components at the bottom wall; free slip is used at the top boundary. The computational set-up is identical to that in the study of Pirozzoli (Reference Pirozzoli2023), in which open-channel flow was studied in the absence of flow control. The solver relies on the classical fractional step method with second-order finite differences on a staggered grid (Orlandi Reference Orlandi2006). The Poisson equation resulting from the divergence-free condition is efficiently solved via Fourier expansion in the periodic directions (Kim & Moin Reference Kim and Moin1985). The governing equations are advanced in time starting from the initial condition of a statistically stationary, uncontrolled turbulent open channel flow by means of a hybrid third-order, low-storage Runge–Kutta algorithm, whereby the diffusive terms are handled implicitly. Statistical averaging, indicated hereinafter as
$\langle \cdot \rangle$
, implies averaging in time and along the two homogeneous directions.
Four sets of simulations, whose details are listed in table 1, are run at prescribed values of the bulk Reynolds number
$Re_b = U_b h / \nu$
; the bulk velocity is kept constant at every time step as described by Quadrio, Frohnapfel & Hasegawa (Reference Quadrio, Frohnapfel and Hasegawa2016). Each set comprises one reference simulation, in which the wall is steady, and a variable number of cases with StTW at different values of
$\left \{A, \kappa , \omega \right \}$
. In the following, we will refer to each simulation set via its (nominal) value of
$Re_{\tau _0}$
; the actual values of
$Re_\tau$
vary throughout simulations of each set, as a consequence of the wall actuation at constant
$U_b$
.
Table 1. Details of the direct numerical simulations of open channel flows (including domain size and discretisation) modified by StTW, grouped in sets of
$N_{{cases}}$
simulations performed at a constant value of bulk Reynolds number
$Re_b = U_b h / \nu$
. The last column indicates the colour and symbol employed in the following figures to represent each set of simulations.
All DNS investigations are carried out in a domain with
$L_x = 6 \pi h$
and
$L_z = 2 \pi h$
, which is much larger than what has been adopted by Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) and GQ16 at similar values of
$Re$
, but a bit smaller than the domain used by Yao et al. (Reference Yao, Chen and Hussain2022). Whereas weak longitudinal eddies may be not resolved, a box sensitivity study carried out by Pirozzoli (Reference Pirozzoli2023) showed that the practical impact on the leading-order flow statistics and on the spanwise spectra is extremely small.
Figure 2. Statistics of streamwise velocity fluctuations for the reference simulation at
$Re_{\tau _0} = 6000$
: (a) spanwise premultiplied spectra
$k_z^+ \phi _{uu}^+$
; (b) streamwise variance
$\langle uu \rangle ^+$
with its large-scale
$\langle uu \rangle _L^+$
and small-scale
$\langle uu \rangle _S^+$
contributions. Large scales are defined as those for which
$2 \pi / k_z \gt 0.5h$
.
Figure 2 indeed supports the adequacy of the present computational box by analysing the streamwise velocity fluctuations of the reference open channel flow at
$Re_{\tau _0}=6000$
, i.e. the largest Reynolds number considered in the present study. Figure 2(a) shows the spanwise pre-multiplied spectrum
$k_z^+ \phi _{uu}^+$
, where
$k_z$
is the spanwise wavenumber and
$\phi _{uu}$
is a component of the velocity spectrum tensor, with a clear outer peak visible at
$\lambda _z \approx h$
. Figure 2(b) shows the variance
$\langle uu \rangle ^+$
of the streamwise velocity, split into the large-scale
$\langle uu \rangle ^+_L$
and small-scale
$\langle uu \rangle ^+_S$
contributions. The large-scale contribution is obtained by integrating the spectrum only for wavelengths
$\lambda _z \gt 0.5h$
as suggested by Bernardini & Pirozzoli (Reference Bernardini and Pirozzoli2011), Dogan et al. (Reference Dogan, Örlü, Gatti, Vinuesa and Schlatter2019) and Yao et al. (Reference Yao, Chen and Hussain2022). With this definition, the large-scale fluctuations are responsible for 12 % of the total variance in the vicinity of the wall, and for as much as 85 % at the free-slip surface. Moreover, it should be noted that the longest travelling wave that we have tested at the highest Reynolds number (
$Re_{\tau _0}=6000$
) is fourteen times shorter than the domain length, thus allowing subharmonic effects, if present, to be properly resolved.
The spatial resolution of the simulations is designed based on the criteria discussed by Pirozzoli & Orlandi (Reference Pirozzoli and Orlandi2021). In particular, the collocation points are distributed in the wall-normal direction
$y$
so that approximately thirty points are placed within
$y^+ \leqslant 40$
, with the first grid point at
$y^+ \lt 0.1$
. The mesh is stretched in the outer wall layer with the mesh spacing proportional to the local Kolmogorov length scale, which there varies as
$\eta ^+ \approx 0.8 (y^+)^{1/4}$
(Jiménez Reference Jiménez2018). A mild refinement towards the free surface is used to resolve the thin layer in which the top boundary condition dampens the wall-normal velocity fluctuations. The grid resolution in the wall-parallel directions is set to
$\Delta x^+ \approx 8.5$
and
$\Delta z^+ \approx 4.0$
for all the flow cases. Note that the resolution becomes finer in actual viscous units in all cases with drag reduction.
Figure 3. Portion of the parameter space spanned in the present study overlaid to the drag reduction map by GQ16 computed at
$A^+=5$
. Each symbol corresponds to one simulation at the Reynolds number encoded by its shape/colour, as described by the legend.
Figure 3 shows at a glance the range of the StTW parameters addressed in the present study for the simulation sets at
$Re_{\tau _0}= \left \{1000, 2000, 3000, 6000 \right \}$
. This is the widest range of
$Re$
considered so far in numerical simulations with spanwise wall forcing.
The portion of the
$ \{\kappa ^+, \omega ^+ \}$
-space spanned in the present study is smaller than that addressed in GQ16. In fact, we limit ourselves to considering
$\kappa ^+ \leqslant 0.02$
and
$ | \omega ^+ | \leqslant 0.1$
, which is now known to be the most interesting part of the parameter space, where the maxima of drag reduction
$\mathcal{R}$
and net saving
$\mathcal{S}$
are expected.
The control parameters have been selected according to the following guiding principles.
-
(i) The intent to further scrutinise the validity of the results by GQ16, obtained in constrained computational domains, led us to consider a wider portion of the StTW parameter space at
$Re_{\tau _0}=1000$
, the highest value considered in their study. -
(ii) GQ16 observed that
$\Delta B^*$
may still retain residual dependence on
$Re$
at their highest value of
$Re_{\tau _0}=1000$
and suggested that at least
$Re_{\tau _0}=2000$
is needed for an
$Re$
-independent measure. Therefore, the same region of the parameter space considered in point (i) is also considered at
$Re_{\tau _0}=2000$
. -
(iii) Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) reported for the first time a drag reduction that increases with
$Re$
for small values of
$\kappa ^+$
and
$\omega ^+$
, in particular for
$\kappa ^+=0.0008$
(i.e.
$\lambda ^+ \approx 8000$
),
$\omega ^+=-0.0105$
(i.e.
$T^+ \approx -600$
) and
$A^+ \approx 5$
(in fact, their
$A^+$
varies slightly across the
$Re$
range), as shown in figure 3(e) of their paper. We have added this combination of
$ \{ \kappa ^+, \omega ^+ \}$
to all simulations sets to verify the increase of
$\mathcal{R}$
with
$Re$
. This is one of the two controlled cases we have carried out at
$Re_{\tau _0}=6000$
. The second case, with
$\kappa ^+=0.0014$
,
$\omega ^+=-0.009$
and
$A^+=2.5$
, matches exactly one of the cases considered experimentally by Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023), at the same value of
$Re_{\tau _0}=6000$
. -
(iv) All controlled simulations are performed at
$A^+=5$
for two reasons: first, this value of
$A^+$
is representative of the amplitude range in the experiments by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) for the case discussed in point (iii); second, this value is close to
$A^+ \approx 6$
at which GQ16 measured the maximum of net power saving
$\mathcal{S}$
. By adopting this value of
$A^+$
, we can verify whether positive
$\mathcal{S}$
can also be achieved at higher
$Re$
.
This results in the set of control parameters shown in figure 3, and listed in tables 2, 3, 4 and 5 of Appendix A together with the main results. As will be clarified in the following, understanding the
$Re$
dependence of
$\mathcal{R}$
and
$\mathcal{S}$
requires accurate estimation of the mean wall friction, which we guarantee by monitoring statistical uncertainty via the method described by Russo & Luchini (Reference Russo and Luchini2017), as shown in figures 6 and 9. Statistics are accumulated for at least
$10 h / u_{\tau _0}$
time units after the initial transient, during which the control leads the flow towards a reduced level of drag.
3. Results
The outcomes of the present study are presented following the guiding principles outlined in § 2. First, we present drag reduction maps at
$Re_{\tau _0}=1000$
and 2000, and use them to provide ultimate validation of the GQ16 results. Second, we evaluate
$\Delta B^\ast$
at
$Re_{\tau _0} = 2000$
and verify the
$Re$
independence of this drag reduction metric. Third, drag reduction is reported up to
$Re_{\tau _0}=6000$
for the same actuation parameters for which Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) observed drag reduction increase with
$Re$
. Finally, the possibility to achieve net power savings at high
$Re$
is discussed.
3.1. Maps of
$\mathcal{R}$
: validity of the results by GQ16
Figure 4 compares the present drag reduction results at
$Re_{\tau _0}=1000$
and
$Re_{\tau _0}=2000$
with the data by GQ16, which need to be transferred to the present values of
$Re_{\tau _0}$
. The procedure involves starting from their
$\mathcal{R}$
and
$C_{f_{\; 0}}$
data, then using the GQ model ((1.4) with
$k=0.39$
; GQ16 showed that the specific value of
$k$
in the range
$0.385$
–
$0.4$
does not significantly affect the results) to compute
$\Delta B^\ast$
. The resulting cloud of
$\Delta B^\ast$
data points at discrete
$ \{ A^+, \kappa ^+, \omega ^+\}$
values is linearly interpolated on a Cartesian grid spanning the
$\{ \kappa ^+, \omega ^+ \}$
space at the value of
$A^+=5$
considered in the present study. Finally,
$\Delta B^\ast$
is again converted back to
$\mathcal{R}$
values via the GQ model, now with the values of
$C_{f_{\; 0}}$
corresponding to
$Re_{\tau _0}=1000$
and
$Re_{\tau _0}=2000$
.
Figure 4. Maps of drag reduction (
$\mathcal{R}$
) as a function of actuation parameters (
$\omega ^+$
,
$\kappa ^+$
), at (a)
$Re_{\tau _0}=1000$
and (b)
$Re_{\tau _0}=2000$
. The colourmap, the contour lines and symbols coloured after table 1 refer to the present data, whereas the black contour lines and symbols refer to the data by GQ16, which at
$Re_{\tau _0}=2000$
are obtained from extrapolation through GQ model (1.4). The contour lines are every 5 % of
$\mathcal{R}$
, dashed lines mark the
$\mathcal{R}=0$
iso-line.
The comparison shows excellent agreement between the two datasets. This finding suggests very weak sensitivity of StTW actuation on the flow geometry (open channel versus closed plane channel) and further strengthens the reliability of the GQ16 data. In fact, due to their limited domain size, GQ16 had no data for
$0 \lt \kappa ^+ \lt 0.005$
, but even there, the new data compare very well with the GQ16 map. The maximum difference between the present and GQ16 datasets evaluated across the interpolated maps shown in figure 4 is only 2.5 %, and the standard deviation is 0.8 %. The agreement shows that no measurable direct effect of large-scale turbulent structures on
$\mathcal{R}$
exists at these values of
$Re_{\tau _0}$
other than their possible contribution to
$C_{f_{\; 0}}$
, which is already accounted for by the GQ model.
3.2. Maps of
$\Delta B^\ast$
: validity of the GQ model
The GQ model relies on the hypothesis that, provided
$Re$
is high enough for the logarithmic law (1.3) to describe well the mean velocity profile, the quantity
$\Delta B^*$
is a function of the control parameters only, and thus independent of the Reynolds number. This hypothesis is here tested using the
$\Delta B^*$
maps for the DNS set at
$Re_{\tau _0}=1000$
and
$2000$
. The maps are generated by applying the GQ model with the corresponding values of
$C_f$
,
$C_{f_{\; 0}}$
and
$\mathcal{R}$
. The results, reported in figure 5, show maximum change of
$\Delta B^*$
across
$Re$
of only 0.36, with standard deviation 0.10. These values can be considered quite small, given that the maximum statistical uncertainty on the change of
$\Delta B^*$
at 95 % confidence level is 0.24 across the map of figure 5 and the mean absolute value is 0.17. This result thus confirms that the drag reduction effect barely changes with
$Re$
, once it is expressed in terms of
$\Delta B^*$
.
Figure 5. Maps of
$\Delta B^\ast$
as a function of actuation parameters (
$\omega ^+$
,
$\kappa ^+$
) at
$Re_{\tau _0}=1000$
(
) and
$Re_{\tau _0}=2000$
(
). The symbols are coloured after table 1 and show the parameters of each simulation underlying the map interpolation shown in the figure. Contours are shown in unit intervals, the dashed lines marking the
$\Delta B^\ast =0$
iso-line.
This additionally indicates that
$Re_{\tau _0}=1000$
is sufficient to obtain a reasonably
$Re$
-independent estimate of
$\Delta B^*$
. This observation is also supported by the good agreement between the GQ16 data at
$Re_{\tau _0} = 1000$
and the results by Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023) obtained up to
$Re_{\tau _0}=4000$
in relatively small domains.
3.3. Monotonicity of
$\mathcal{R}$
with
$Re$
Figure 6. Drag reduction rate (
$\mathcal{R}$
) as a function of the reference friction Reynolds number (
$Re_{\tau _0}$
) for backward-travelling wave with parameters
$A^+=5$
,
$\kappa ^+=0.00078$
and
$\omega ^+=-0.0105$
, close to the conditions considered by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), i.e.
$A^+ \approx 5$
,
$\kappa ^+ \approx 0.0008$
and
$\omega ^+ \approx -0.0105$
(in their laboratory experiment, the viscous-scaled parameters vary slightly with
$Re$
). The present results are denoted with coloured symbols (see table 1); experimental data by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) are black squares, while black circles denote their LES numerical data; the straight line is the prediction of the GQ model (1.4) corresponding to
$\Delta B^*=0.51$
and to the values of
$C_{f_{\; 0}}$
obtained from the uncontrolled simulations at the respective value of
$Re_{\tau _0}$
. The error bars have been determined as described in § 2, corresponding to a 95 % confidence level.
The GQ model predicts that
$\mathcal{R}$
decreases monotonically with
$Re$
; however, more slowly than the power-law decrease assumed in early studies (Choi, Xu & Sung Reference Choi, Xu and Sung2002; Quadrio & Ricco Reference Quadrio and Ricco2004; Touber & Leschziner Reference Touber and Leschziner2012). The decrease rate is less at higher
$Re$
and for smaller
$\mathcal{R}$
. Ample numerical and experimental evidence so far, including the results of the present study, support the predictions of the GQ model.
Contrasting evidence that
$\mathcal{R}$
may instead increase with
$Re$
has been recently provided from the combined laboratory and numerical efforts of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021). As shown in figure 3(e) of their paper, they found that
$\mathcal{R}$
obtained by backward-travelling waves at small values of
$\kappa ^+$
and
$\omega ^+$
, namely
$\kappa ^+=0.0008$
and
$\omega ^+=-0.0105$
, increases from 1.6 % at
$Re_{\tau _0}\approx 1000$
, as measured numerically in large eddy simulation (LES) of open channel flow, up to 13.1 % at
$Re_{\tau _0}\approx 12\,800$
, as measured experimentally in a turbulent boundary layer. Since the actuator employed in their experiments yields a wave with a frequency-dependent amplitude and constant wavelength in physical units (30 cm), those authors could not exactly maintain the same value of viscous-scaled control parameters across the considered Reynolds number range. Specifically, the amplitude increased from
$A^+=4.6$
at
$Re_\tau = 9000$
to
$A^+ = 5.7$
at
$Re_\tau = 12\,800$
(see table 1 of Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023)). Furthermore, although the original figure 3(e) of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) reports a constant value of
$\kappa ^+=0.0008$
at all
$Re$
, we cannot reconcile it with the actuator wavelength being fixed in physical units for the experimental points.
In the present work, we verify this contrasting evidence by studying the
$Re$
dependence of
$\mathcal{R}$
across the widest range of Reynolds number tested so far via DNS. For this purpose, we consider StTW actuation at
$Re_{\tau _0}=1000$
, 2000, 3000 and 6000, with control parameters selected to match as closely as possible those reported in figure 3(e) of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), namely
$\kappa ^+=0.00078$
and
$\omega ^+=-0.0104$
. The wave amplitude is set to
$A^+=5$
, midway between the range of variation in their experiments. Figure 6 compares our numerical results with the numerical and experimental results of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021). Our measurements still fit very well the prediction of the GQ model and confirm an overall decreasing trend of
$\mathcal{R}$
with
$Re$
.
To verify whether the differences observed in figure 6 are due to the different Reynolds number range considered here and by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), we advocate the work of Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023). Those authors extended the experimental database of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) with additional data points, some of which at
$Re_{\tau _0} \approx 6000$
, i.e. the highest Reynolds number considered in the present study. Hence, we have precisely reproduced their actuated flow case with
$ \{A^+, \omega ^+, \kappa ^+, Re_{\tau _0} \} = \left \{2.5, -0.009, 0.0014, 6000 \right \}$
, the remaining differences being the flow configuration (open channel versus boundary layer), as well as actuation details (ideal harmonic actuation in numerical simulation versus spatially discretised wave in experiment). This case also falls within the range of potential use for outer-scaled actuation according to Deshpande et al. (Reference Deshpande, Chandran, Smits and Marusic2023), due to the comparatively large actuation period
$T^+ = -700$
and wavelength
$\lambda ^+ \approx 4500$
, similar to the case presented in figure 6. A drag reduction of
$\mathcal{R} = 2.3\, \% \pm 1.1\, \%$
is measured here, to be compared with
$\mathcal{R} = 6\, \%$
measured experimentally by Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023). This finding hints at systematic differences between the present numerical simulations and the laboratory experiments of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) and Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023). We reiterate that this is possibly due to irreducible differences in the flow and wall actuation set-ups, or even to the extreme challenges posed by laboratory experiments targeting such complex drag reduction strategies. We will go back to this important issue in § 4. For the moment, the present data corroborate the expectation that
$\mathcal{R}$
decreases with
$Re$
at the rate predicted by the GQ model.
3.4. Net power savings at large values of
$Re$
Net power saving
$\mathcal{S}$
derives from the (positive or negative) balance between the power saved through drag-reducing control and the power required for wall actuation, hence,
\begin{equation} \mathcal{S} = \mathcal{R} - \frac {P_{\textit{in}}}{P_{p_{0}}} \, , \end{equation}
where
$P_{p_{0}}$
is the pumping power per unit wetted area in the uncontrolled case, which for constant
$U_b$
reads
\begin{equation} P_{p_{0}} = U_b \tau _{x_0} \, , \end{equation}
and
$P_{in}$
is the control input power per unit wetted area, expressed as
\begin{equation} P_{in} = \langle w_w \tau _z \rangle = \rho \nu \left . \left\langle w \frac {\partial w}{\partial y} \right\rangle \right |_w = \frac {\rho \nu }{2} \left . \frac {\mathrm{d}}{\mathrm{d} y} \langle w w \rangle \right |_{w}\, , \end{equation}
where
$\tau _z = \rho \nu (\partial w / \partial y)_w$
is the spanwise wall shear stress.
Similarly to what done for
$\mathcal{R}$
, the Reynolds number dependence of
$\mathcal{S}$
can also be predicted theoretically. Whereas
$\mathcal{R}$
is accurately expressed by the GQ model, the
$Re$
dependence of
$P_{in} / P_{p_{0}}$
can be easily expressed following Ricco & Quadrio (Reference Ricco and Quadrio2008), who noticed that this ratio is equivalent to
$P_{in}^+ / P_{p_{0}}^+$
. Since
$P_{in}^+$
is very well approximated by the power
$P_\ell ^+$
required to generate the laminar transverse Stokes layer (Quadrio & Ricco Reference Quadrio and Ricco2011; Gatti & Quadrio Reference Gatti and Quadrio2013) – which does not depend on
$Re$
if the viscous-scaled parameters are kept constant – the
$Re$
dependence of
$P_{in} / P_{p_{0}}$
comes only from
$P_{p_{0}}^+ = U_b^+ = \sqrt {2/C_{f_{\; 0}}}$
. By using the expression of
$P_{\ell }^+$
by Gatti & Quadrio (Reference Gatti and Quadrio2013), we thus obtain
\begin{equation} \frac {P_{in}}{P_{p_{0}}} \approx \frac {P_{\ell }^+}{U_b^+} = \frac {(A^+)^2 (\kappa ^+)^{1/3}}{2 U_b^+} {Re} \left [ \mathrm{e}^{\pi i / 6} \frac {{Ai}^\prime (\theta )}{{Ai}(\theta )} \right ] \, , \end{equation}
where
$i$
is the imaginary unit, Re indicates the real part of a complex number,
${Ai}$
is the Airy function of the first kind,
${Ai}^\prime$
its derivative and
$\theta = - \mathrm{e}^{\pi i / 6} (\kappa ^+)^{1/3} ( \omega ^+/\kappa ^+ + i\kappa ^+ )$
. Equation (3.4) shows that
$P_{in}^+ = U_b^+ P_{in} / P_{p_{0}} \approx P_\ell ^+$
is a Reynolds-independent quantity for StTW parameters sufficiently far from the region of drag increase, where the approximation
$P_{in}^+ \approx P_{\ell }^+$
is known to fail. As a result, it is sufficient to measure
$P_{in}^+$
at a given Reynolds number, or estimate it via
$P_{\ell }^+$
, to retrieve
$P_{in} / P_{p_{0}}$
at any Reynolds number, i.e. at any arbitrary
$U_b^+=\sqrt {2/C_{f_{\; 0}}}$
. Equation (3.4) shows that
$P_{in} / P_{p_{0}}$
decreases with
$Re$
as
$1/U_b^+$
, so that
$\mathcal{S}$
can in fact increase with
$Re$
, provided the normalised actuation power decays with
$Re$
faster than
$\mathcal{R}$
.
Figure 7. Maps of actuation power (
$P_{in}^+$
) as a function of the actuation parameters (
$\omega ^+$
,
$\kappa ^+$
) at
$Re_{\tau _0}=1000$
(
) and
$Re_{\tau _0}=2000$
(
). The symbols are coloured after table 1 and show the parameters of each simulation underlying the map interpolation shown in the figure. Data by GQ16 (
) and
$P_{in}^+$
from (3.4) (
) are also reported.
Figure 7 confirms that
$P_{in}^+$
is indeed constant with
$Re$
throughout the investigated parameter space, including the drag-increasing regime, where
$P_{in}^+$
and
$P_\ell ^+$
do differ and the former can only be measured empirically. The GQ16 dataset well aligns with the present data, the lacking information for
$0 \lt \kappa ^+ \leqslant 0.005$
notwithstanding.
Figure 8. Maps of net power saving (
$\mathcal{S}$
) as a function of the actuation parameters (
$\omega ^+$
,
$\kappa ^+$
) at
$Re_{\tau _0}=1000$
(
) and
$Re_{\tau _0}=2000$
(
). The symbols are coloured after table 1 and show the parameters of each simulation underlying the map interpolation shown in the figure. Contour lines are shown in intervals of 5 %, the dashed lines denoting the
$\mathcal{S}=0$
iso-line.
The net power saving at
$Re_{\tau _0}=1000$
and
$2000$
is reported in figure 8. Overall, the contours of
$\mathcal{S}$
do not change significantly, since degradation of
$\mathcal{R}$
is compensated by reduction of the actuation input power. Larger differences are observed for nearly optimal
$\mathcal{S}$
(see the
$\mathcal{S}=15\, \%$
iso-line in figure 8), in a region which shrinks and shifts towards higher
$\kappa ^+$
at higher
$Re$
. This can be explained by the stronger decay of
$\mathcal{R}$
in this region (as predicted by the GQ model due to larger
$\mathcal{R}$
) and by the comparatively small value of
$P_{in} / P_{p_{0}}$
, which causes
$\mathcal{S}$
to have similar
$Re$
dependence as
$\mathcal{R}$
.
GQ16 noticed that at
$Re_{\tau _0} \approx 1000$
and
$A^+=5.5$
, the locus of near-optimum net power saving (
$\mathcal{S} = 15\, \%$
) extends along the ridge of maximum
$\mathcal{R}$
between
$\kappa ^+ = 0.0085$
and
$0.04$
, the maximum being at
$ \{\omega ^+, \kappa ^+ \} = \left \{0.093, 0.026 \right \}$
. This implies that the point of maximum
$\mathcal{S}$
might reside outside of the parameter space considered in figure 8 for both Reynolds numbers under scrutiny here.
Figure 9. Net power saving (
$\mathcal{S}$
) as function of reference friction Reynolds number (
$Re_{\tau _0}$
) for backward-travelling waves with the same parameters considered by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021). The present data are indicated with coloured symbols (see table 1); experimental data by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) are black squares, while black circles denote their LES numerical data; the straight line is the theoretical prediction obtained by combining the GQ model (1.4) for
$\Delta B^\ast =0.51$
with (3.4) for
$P_{in}^+=1.1$
and the values of
$C_{f_{\; 0}}$
obtained from the uncontrolled simulations at the respective value of
$Re_{\tau _0}$
.
As done for the drag reduction in figure 6, the variation of
$\mathcal{S}$
with
$Re$
is shown in figure 9 for the same parameters considered by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021). Interestingly,
$\mathcal{S}$
is observed to increase with
$Re$
at this combination of parameters, essentially due to the shrinking of the negative
$P_{in} / P_{p_{0}}$
contribution and to the relatively constant
$\mathcal{R}$
. The increase of
$\mathcal{S}$
is compatible with the theoretical prediction that can be obtained by combining the GQ model of (1.4) with the prediction for
$P_{in} / P_{p_{0}}$
of (3.4). The differences between the present numerical database and the laboratory experiments of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), previously noted for
$\mathcal{R}$
, are confirmed here.
The present results enable a better understanding of the available literature data. For instance, by comparing the numerical data by Rouhi et al. (Reference Rouhi, Fu, Chandran, Zampiron, Smits and Marusic2023), which consider StTW at small wavelengths (due to the restricted domain size) and relatively large amplitude
$A^+ = 12$
and frequencies, with their experimental data, which consider backward-travelling waves at larger wavelengths but smaller amplitudes of
$A^+ \approx 5$
and frequencies, Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) conclude that mostly low-frequency forcing
$ | \omega ^+ | \lt 0.018$
is capable to achieve positive
$\mathcal{S}$
, despite the moderate values of
$\mathcal{R}$
. This conclusion is observed here to be an artefact of the comparison between StTW at different amplitudes: according to GQ16, it is known that already at
$Re_{\tau _0}=1000$
, no positive
$\mathcal{S}$
can be achieved via StTW for amplitudes
$A^+ \gtrapprox 14$
. The present data clearly show that the observation of GQ16 is valid also if smaller values of wavenumbers and frequencies are considered: the locus of maximum
$\mathcal{S}$
in the
$\left \{\omega , \kappa \right \}$
-space essentially coincides with that of maximum
$\mathcal{R}$
, and it shifts towards larger
$\left \{\omega , \kappa \right \}$
for increasing values of
$Re$
rather than to smaller ones if the comparison among various
$Re$
is performed at a constant value of
$A^+$
close to the optimal
$A^+ \approx 6$
identified by GQ16.
4. Concluding discussion
In the present work, we have addressed the Reynolds number dependence of skin-friction drag reduction induced by spanwise forcing, in terms of both drag reduction rate
$\mathcal{R}$
and net power saving
$\mathcal{S}$
. In particular, we have focused on streamwise-travelling waves of spanwise wall velocity (StTW, Quadrio et al. Reference Quadrio, Ricco and Viotti2009). A new database of high-fidelity direct numerical simulation (DNS) studies of turbulent open channel flow with and without StTW has been generated for
$Re_{\tau _0}=1000$
, 2000, 3000 and 6000. This is the widest Reynolds number range considered so far in numerical experiments with spanwise forcing and reduces the gap from the highest value of
$Re_{\tau _0}$
considered in analogous laboratory experiments (Chandran et al. Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) to a factor of 2.5.
The main outcome of the present study is to confirm the validity of the predictive model for drag reduction proposed by Gatti & Quadrio (Reference Gatti and Quadrio2016) and its underlying hypothesis. The present data corroborate the observation that the parameter
$\Delta B^*$
, which quantifies the control-induced velocity shift in actual viscous units ‘
$*$
’ at matched
$y^*$
with respect to the non-actuated flow is an
$Re$
-independent measure of drag reduction when the Reynolds number is sufficiently large for the logarithmic law to apply. We have shown that
$Re_{\tau _0} \gtrapprox 1000$
is sufficient for
$\Delta B^*$
to become nearly
$Re$
-independent, since no statistically significant differences have been measured between the
$Re_{\tau _0}=1000$
and
$Re_{\tau _0}=2000$
cases, for a wide range of actuation parameters, and up to
$Re_{\tau _0}=6000$
for one selected combination of actuation parameters.
This key result implies that drag reduction induced by StTW at a given combination of
$\{A^+, \omega ^+, \kappa ^+ \}$
is bound to monotonically decrease with the Reynolds number, at a rate that depends on
$\mathcal{R}$
itself and on (the inverse square root of) the skin-friction coefficient
$C_{f_{\; 0}}$
of the uncontrolled flow, as embodied in the GQ model; see (1.4). Fortunately, the decay rate is less severe than the power law
$\mathcal{R} \sim Re_{\tau _0}^{-0.2}$
suggested empirically in early studies on spanwise wall oscillations (Choi et al. Reference Choi, Xu and Sung2002; Touber & Leschziner Reference Touber and Leschziner2012), conveying that significant drag reduction can still be achieved at very high
$Re$
.
The increase of drag reduction with the Reynolds number, observed by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) with actuation parameters corresponding to the outer-scaled actuation, is not confirmed by our numerical experiments with
$ \{A^+, \omega ^+, \kappa ^+ \} = \{5,-0.0104,0.00078 \}$
in turbulent open channels. In contrast, the present results follow well the prediction of the GQ model and show a very mild decrease of
$\mathcal{R}$
with
$Re$
for these specific parameters. While the observation of
$\mathcal{R}$
increasing with
$Re$
is indeed surprising and unique in the literature, we can only speculate on the reasons behind this discrepancy.
On the one hand, the difference in the flow set-up considered here and by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) (open channel versus boundary layer) could affect the Reynolds number dependence of
$\mathcal{R}$
. In this respect, Skote (Reference Skote2014) applied StTW to numerical turbulent boundary layers at low
$Re$
and noted that the Kármán constant can increase in the presence of drag-reduction effects. This could affect the
$Re$
-dependency of
$\mathcal{R}$
, since the GQ model assumes constancy of
$k$
. However, this effect is only seen in the initial non-equilibrium state of the boundary layer, which was also corroborated by the experimental data of Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023).
On the other hand, Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) and later Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) implemented a spatially discrete form of the StTW, similarly to Auteri et al. (Reference Auteri, Baron, Belan, Campanardi and Quadrio2010), and synthesised harmonic waves by independently moving stripes with finite width. Auteri et al. (Reference Auteri, Baron, Belan, Campanardi and Quadrio2010) and, more recently, Gallorini & Quadrio (Reference Gallorini and Quadrio2024) addressed the effects of the wave discretisation on the achievable drag reduction. Owing to discretisation, the turbulent flow perceives a number of higher Fourier harmonics of the discrete piecewise-constant wave, as if multiple waves with different parameters were applied. As a result, quantitative comparison between the ideally continuous and piecewise-constant forcing, as performed by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) when comparing their numerical and experimental data, is not trivial, since some discrete waves far from the optimal forcing parameters can outperform the corresponding ideal sinusoidal waveform, whenever part of the harmonic content of the discrete wave falls in high-
$\mathcal{R}$
regions of the drag reduction map.
We verify this hypothesis by reproducing the cases of figure 6 (StTW at
$\{A^+, \omega ^+, \kappa ^+ \}$
=
$\left \{5, -0.0104, 0.00078 \right \}$
) up to
$Re_{\tau _0}=3000$
, this time employing discrete StTW instead of ideally continuous ones. Specifically, we discretise each wavelength into six pieceweise-constant streamwise stripes, similarly to the experiments by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021). Accordingly, the StTW boundary condition of (1.1) is replaced with its discrete counterpart, proposed by Gallorini & Quadrio (Reference Gallorini and Quadrio2024), as follows:
\begin{equation} w(x,t;s) = A \sin \left ( \omega t - \frac {2\pi i}{s} \right ) \quad \text{for} \; \frac {i}{s} \lambda \leqslant x \lt \frac {i+1}{s} \lambda , \quad s=6 \quad \text{and} \quad i \in \mathbb{N}. \end{equation}
With the results presented in figure 10, we cannot state that
$\mathcal{R}$
increases with
$Re$
beyond statistical uncertainty also for discrete waves. However, as previously hypothesised and confirmed by Gallorini & Quadrio (Reference Gallorini and Quadrio2024), discrete waves with non-optimal parameters can outperform ideal continuous ones in terms of
$\mathcal{R}$
. In the present case, the considered discrete waves achieve approximately 1 % higher
$\mathcal{R}$
, albeit at the cost of a significantly increased input power: at
$Re_{\tau _0}=3000$
,
$\mathcal{S}$
is −0.95 for the continuous wave but −8.50 for the discrete one.
Figure 10. Drag reduction rate (
$\mathcal{R}$
) as a function of the reference friction Reynolds number (
$Re_{\tau _0}$
) for backward-travelling wave with parameters
$A^+=5$
,
$\kappa ^+=0.00078$
and
$\omega ^+=-0.0105$
, close to the conditions considered by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), i.e.
$A^+ \approx 5$
,
$\kappa ^+ \approx 0.0008$
and
$\omega ^+ \approx -0.0105$
(in their laboratory experiment the viscous-scaled parameters vary slightly with
$Re$
). This is a replica of figure 6 with the addition of the light coloured squares: these points have been obtained with discrete StTW, in which each wavelength has been discretised in six piecewise-constant streamwise sectors, and at the parameters mentioned previously.
The conclusion drawn by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) that
$\mathcal{R}$
increases with
$Re$
is based on data obtained with different methods. Specifically, the low-
$Re$
data were obtained from LES of turbulent open channel flow in relatively small domains with continuous StTW applied at the wall, whereas the high-
$Re$
data were obtained from boundary layer experiments with discrete StTW. As demonstrated previously, differences in flow configurations, methodologies and associated uncertainties complicate direct comparisons. Moreover, comparing discrete waves at high
$Re$
with ideal sinusoidal waves at low
$Re$
may introduce significant bias in interpreting the
$Re$
dependence of
$\mathcal{R}$
, particularly if discrete waves achieve greater
$\mathcal{R}$
. Additionally, estimating the ideal power consumption of discrete StTW using the expression valid for the ideal waves (e.g. (3.4)), as done by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) and Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023), significantly overestimates
$\mathcal{S}$
and should be avoided.
Whereas the previous speculations remain to be verified in future studies, the present results support the claim that ideal StTW applied in turbulent open channels are neither expected nor observed to yield an increase of drag reduction with increasing
$Re$
for any combination of wave parameters that are kept constant in viscous units.
Lastly, we also confirm that the Reynolds number dependence of the net power saving
$\mathcal{S} = \mathcal{R} - P_{in} / P_{p_{0}}$
is in line with theoretical predictions. Whereas
$\mathcal{R}$
directly derives from the GQ model,
$P_{in} / P_{p_{0}}$
can be obtained directly from
$P_{in}^+ = U_b^+ P_{in} / P_{p_{0}}$
, which is known to be
$Re$
-independent (Gatti & Quadrio Reference Gatti and Quadrio2013). Interestingly, we have found that
$P_{in}^+$
does not change with
$Re$
throughout the drag-reduction map, not only in those regions where
$P_{in}^+$
is known to be well approximated by
$P_\ell ^+$
, i.e. the value obtained from the laminar generalised Stokes layer solution. In other words, the ideal viscous scaling of
$P_{in}^+$
is retained even close to the valley of drag increase, where turbulence is known to interact with the generalised Stokes layer generated by StTW actuation. This result, as already discussed by Gatti & Quadrio (Reference Gatti and Quadrio2013, Reference Gatti and Quadrio2016), has two main implications. First, in the portion of the StTW parameter space where
$\mathcal{S}$
is maximum,
$\mathcal{S}$
is dominated by
$\mathcal{R}$
and hence exhibits similar
$Re$
dependence; here,
$\mathcal{S}$
decreases with
$Re$
at a rate which is slightly less than
$\mathcal{R}$
. Second, for StTW parameters far from the optimum, both
$\mathcal{R}$
and
$P_{in}/ P_{p_{0}}$
contribute to
$\mathcal{S}$
. In this case, the normalised control cost may decrease with
$Re$
at a faster rate than
$\mathcal{R}$
, so that
$\mathcal{S}$
can actually increase with
$Re$
. However, this can occur only in regions of non-optimal values of
$\mathcal{S}$
. Hence, we argue that the observation by Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) that only low-frequency, low-wavenumber forcing can achieve positive
$\mathcal{S}$
at high
$Re$
may be an artefact due to the properties of their experimental set-up, in which the same region of the viscous-scaled parameter space cannot be spanned for different values of
$Re$
(see figure 11). Indeed, those authors can only achieve the optimal values of
$A^+$
at the highest values of
$Re$
, at which only low
$\omega ^+$
and
$\kappa ^+$
are possible owing to the small space and time scales of the turbulent flow. The more systematic scan of the StTW parameter space carried out in the present study shows that the loci of optimal
$\mathcal{S}$
and
$\mathcal{R}$
roughly coincide in the
$\{\omega ^+, \kappa ^+ \}$
plane.
Figure 11. Wavenumber (
$\kappa ^+$
), angular frequency (
$\omega ^+$
) and amplitude (
$A^+$
) for StTW actuation considered by Chandran et al. (Reference Chandran, Zampiron, Rouhi, Fu, Wine, Holloway, Smits and Marusic2023) for different values of
$Re_{\tau _0}$
. The lighter symbols show the projection of the data points onto the
$\{\omega ^+, \kappa ^+\}$
-plane.
Funding
This work was supported by the EuroHPC Joint Undertaking (JU) under grant EHPC-EXT-2022E01-054 on the Leonardo Booster machine at CINECA.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Dataset details
This appendix reports the combination of the StTW control parameters of the simulations performed to produce the present dataset, together with the main quantities of interest. Tables 2, 3, 4 and 5 are for
$Re_{\tau _0} = 1000$
,
$Re_{\tau _0} = 2000$
,
$Re_{\tau _0} = 3000$
and
$Re_{\tau _0} = 6000$
, respectively.
Table 2. List of the controlled simulations carried out at
$Re_{\tau _0} =1000$
. In the last case, denoted with the superscript
$^\ast$
, a discrete travelling wave has been imposed according to (4.1).
Table 3. List of the controlled simulations carried out at
$Re_{\tau _0} =2000$
. In the last case, denoted with the superscript
$^\ast$
, a discrete travelling wave has been imposed according to (4.1).
Table 4. List of the controlled simulations carried out at
$Re_{\tau _0} =3000$
. In the last case denoted with the superscript
$^\ast$
a discrete travelling waves has been imposed according to (4.1).
Table 5. List of the controlled simulations carried out at
$Re_{\tau _0} =6000$
.
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