[
Abstract
This paper addresses the integral energy fluxes in natural and controlled turbulent channel flows, where active skinfriction drag reduction techniques allow a more efficient use of the available power. We study whether the increased efficiency shows any general trend in how energy is dissipated by the mean velocity field (mean dissipation) and by the fluctuating velocity field (turbulent dissipation).
Direct Numerical Simulations (DNS) of different control strategies are performed at Constant Power Input (CPI), so that at statistical equilibrium each flow (either uncontrolled or controlled by different means) has the same power input, hence the same global energy flux and, by definition, the same total energy dissipation rate. The simulations reveal that changes in mean and turbulent energy dissipation rates can be of either sign in a successfully controlled flow.
A quantitative description of these changes is made possible by a new decomposition of the total dissipation, stemming from an extended Reynolds decomposition, where the mean velocity is split into a laminar component and a deviation from it. Thanks to the analytical expressions of the laminar quantities, exact relationships are derived that link the achieved flow rate increase and all energy fluxes in the flow system with two wallnormal integrals of the Reynolds shear stress and the Reynolds number. The dependence of the energy fluxes on the Reynolds number is elucidated with a simple model in which the controldependent changes of the Reynolds shear stress are accounted for via a modification of the mean velocity profile. The physical meaning of the energy fluxes stemming from the new decomposition unveils their interrelations and connection to flow control, so that a clear target for flow control can be identified.
]Global energy fluxes
in fullydeveloped turbulent channels
with flow control
D.Gatti, A.Cimarelli, Y.Hasegawa, B.Frohnapfel & M.Quadrio] DAVIDEGATTI ^{†}^{†}thanks: Email address for correspondence: davide.gatti@kit.edu, ANDREACIMARELLI, YOSUKEHASEGAWA, BETTINAFROHNAPFEL ANDMAURIZIOQUADRIO ^{†}^{†}thanks: Mercator Fellow at Karlsruhe Institute of Technology
1 Introduction
Drag reduction in turbulent flows, and in particular skinfriction drag reduction in wallbounded turbulent flows, is attracting scholars since the early days of fluid mechanics. Its goal is achieving a more energetically efficient flow system, yielding large economical savings and emission reductions for the global transport sector. Nowadays, a number of techniques exist, either passive or active and at various levels of technical readiness, which promise to successfully reduce turbulent friction drag. Its successful reduction is not only entailed with practical significance, but also helpful to advance our understanding of the physics of nearwall turbulence. The main question addressed in this paper is: How are the global energy fluxes (i.e. power input, transfer rates between mean and turbulent kinetic energy, and energy dissipation rates) modified in a fluid system where drag reduction enables increased energetic efficiency?
From an energy standpoint, an incompressible turbulent flow, however complex, can be thought of as a dissipative system that requires external energy to operate: homogeneous isotropic turbulence is not sustained without some external energy input, and a fluid in a duct only flows when a pump is present or a pressure gradient is established by means of external forces. The simplest wallbounded flow is the plane Poiseuille flow, contained between two parallel, indefinitely large walls. The flux of energy through this system is described, at the highest level of compactness, by the timeaveraged rate at which energy enters the system (i.e. the pumping power and possibly an extra control power if active flow control is considered), which must equal, at statistical equilibrium, the timeaveraged rate at which energy is dissipated via viscous mechanisms (i.e. the total viscous dissipation).
If one adopts the conventional Reynolds decomposition of the velocity vector into a timeaveraged and a fluctuating field, pumping energy is associated with the mean flow alone, while energy dissipation rate is split into one part associated with the dissipation of the timeaveraged velocity field, and another part associated with the fluctuating field. Hence:
When applied to the kinetic energy of the flow, the Reynolds decomposition separates the kinetic energy of the mean flow (MKE) and the turbulent kinetic energy (TKE). Since these two energies are decoupled, a compact energetic description of a channel flow is obtained by means of the socalled energy box (Quadrio 2011; Ricco et al. 2012). Though simplified by the volume and timeaveraging, the energybox description still highlights the energy transfer process from the mean to the fluctuating field, embodied by the production of turbulent kinetic energy (acting as a sink for MKE, but as a source for TKE).
The energy box for a channel flow with control is schematically depicted in Fig. 1. The control system is assumed to introduce spatial or temporal velocity fluctuations, thus supplying energy to TKE ( in Fig. 1 enters the TKE box). Of course, control strategies do exist (see for instance Sumitani & Kasagi 1995; Xu et al. 2007) which directly supply energy to the mean field; the present approach could be easily modified to account for it. Under the assumption that feeds TKE it holds that:
We want to understand whether an improvement of energetic efficiency via flow control is connected to generally valid changes in one of the energydissipating quantities, e.g. an increase/decrease of . If this is the case, maximising/minimising could become the control objective in smart control loops that aim at energetic savings through skinfriction drag reduction; and a substantial hint would be available for the development of RANS turbulence models capable of indirectly representing dragreduction techniques. In literature, a number of studies discuss the behaviour of in drag reduced flows. Some report a decrease of (Dimitropoulos et al. 2001; Ge et al. 2013; Agostini et al. 2014), while others an increase of (Ricco et al. 2012; De Angelis et al. 2005).
The observed variations of and are strongly linked to the way the comparison between the uncontrolled and controlled flow is carried out (Frohnapfel et al. 2012). If the two flows are compared by enforcing the same mean pressure gradient (CPG), flow control results in an increase of the flow rate; as a result, the controlled flow is driven with a larger pumping power input, and increases. In contrast, if one compares two flows at constant flow rate (CFR), the effect of flow control on is undefined. In fact, drag reduction in the controlled flow yields a decrease of the pressure gradient and thus a decrease of (given by the product of pressure gradient and flow rate). may increase, decrease or not vary at all depending on the magnitude of .
The Constant Power Input (CPI) condition has been recently proposed (Hasegawa et al. 2014; Quadrio et al. 2016) as an alternative approach for flow control. With CPI, the simulation is set up in such a way that the total power input to the flow system is kept constant. A flow control strategy successfully improves the energetic efficiency of the system when the bulk mean velocity is increased compared to a natural channel flow at the same given total power. Since active flow control requires a certain (often nonnegligible) amount of power, the total power is kept constant across different cases, and therefore remains constant. This scenario therefore allows studying changes in (or ) in a more natural way.
CPI is adopted in the present paper to understand how the energy fluxes in a turbulent channel flow are modified when an increase in energetic efficiency is realised through skinfriction drag reduction. It will be shown that, even in the welldefined CPI setting, changes of and cannot be related to improvements in energetic efficiency. Indeed, and will be expressed in terms of two integrals of the wallnormal Reynolds shear stress distribution and the Reynolds number. Therefore, changes of and are fully determined by the modification in the Reynolds shear stress distribution, in combination with the fraction of power spent for control. We will describe an alternative decomposition of the total viscous dissipation which facilitates the derivation of new analytical results and the physical interpretation of flow control effects onto global energy fluxes and dissipations.
2 Problem description
We consider the incompressible, fullydeveloped flow of a viscous fluid with density , dynamic viscosity and kinematic viscosity between two plane parallel walls located apart. , and denote the streamwise, wallnormal and spanwise coordinates. The corresponding components of the velocity vector are , and , and the static pressure is . Throughout the paper, all dimensional quantities are indicated with an asterisk. Nondimensionalization in viscous (inner) units, i.e. by the kinematic viscosity and the friction velocity based upon the wall shear stress of the reference flow, is denoted by the superscript +. In all other instances quantities are nondimensionalized by the channel halfheight and the characteristic velocity , used in the definition of the Reynolds number , employed in CPI as presented in the following subsection. Angular brackets indicate temporal and spatial averaging across the statistically homogeneous directions. The Reynolds decomposition for a generic quantity is readily defined as . The average across the whole fluid domain and in time is denoted by square brackets as .
2.1 The CPI approach
In the present work we adopt the constant power input (CPI) framework introduced by Hasegawa et al. (2014): the flow is driven through the channel by a constant total power . The relative importance of the control power over the pumping power is expressed by the quantity:
which represents the fraction of the total power used by the control, and is therefore zero for the canonical reference flow.
While the control power necessarily depends upon the control strategy of choice, the pumping power per unit wetted area is given by:
(2.0) 
where is the mean streamwise pressure gradient, and is the bulk velocity.
Hasegawa et al. (2014) introduce the velocity , i.e. the bulk velocity of a laminar flow driven by the pumping pumper , as the appropriate characteristic velocity in the CPI approach. This choice is justified by the theoretical argument (Bewley 2009; Fukagata et al. 2009) that a laminar flow maximises the bulk velocity for a given . The ultimate goal of flow control under CPI is to increase the ratio towards the theoretical upper limit .
At CPI the corresponding powerbased Reynolds number is given by:
which leads directly to the dimensionless power input per unit wetted area:
(2.0) 
Hence, in CPI a constant implies working at a constant , just like CPG corresponds to constant and CFR to a constant . In case of active control, the available pumping power is:
(2.0) 
, and are connected through the dimensionless form of Eq. (2.0):
(2.0) 
which highlights once more that represents the total power and combines viscous and bulk velocity scales.
2.2 Numerical details
The following discussion relies upon analytic as well as numerical results. For the latter, three DNS of turbulent channels have been produced on purpose under the CPI condition. The value of , kept constant across all cases, is , corresponding in the uncontrolled (reference) case to and . It is worth recalling that the different forcing strategies CFR, CPG and CPI lead to essentially identical turbulence statistics for wall friction, despite the fact that they produce different dynamical systems, i.e. the temporal behaviour of the spacemean (instantaneous) streamwise velocity and the spacemean (instantaneous) pressure gradient are different (Quadrio et al. 2016). As discussed before, the choice of the forcing strategy becomes important when two (or more) flows are to be compared with each other. Since the present work is concerned with changes in energy fluxes between a ”natural” turbulent flow and drag reduced ones, we choose to compare flows at CPI, such that they all possess the same global energy flux.
The employed DNS solver is that by Luchini & Quadrio (2006), which uses mixed spatial discretisation (Fourier series in the homogeneous directions, and compact, fourthorder explicit finite differences in the wallnormal direction). The computational domain has streamwise length of and spanwise length . The number of Fourier modes is in the streamwise direction and in the spanwise direction; the number of points in the wallnormal direction is , unevenly spaced in order to decrease the grid size near the walls. The corresponding spatial resolution in the homogeneous directions is and (dealiasing with the 3/2 rule is used); the wallnormal resolution increases from near the walls to at the centreline.
The equations of motion are advanced in time with a partially implicit approach, with an explicit threesubsteps lowstorage Runge–Kutta scheme combined with an implicit Crank–Nicolson scheme for the viscous terms. The size of the time step is , corresponding to an average value of CFL of 1.1.
2.3 Control strategies
Two drag reducing flow control strategies are considered, namely the spanwiseoscillating wall or OW (Jung et al. 1992), and the opposition control based on the wallnormal velocity component or VC (Choi et al. 1994). The OW control induces a spanwise wall movement resulting in a spanwise wall velocity distribution given by
where is the amplitude of the oscillation, and its period. VC control produces a distributed blowing and suction with the wallnormal velocity component at the wall opposing the same component in a wallparallel plane at a prescribed wall distance , according to
Both control techniques are active, with the difference that OW requires a significant amount of energy to operate while the required control power for VC is marginal.
The control parameters are selected so that both techniques work around their optimum in the CPI sense, i.e. they achieve the maximum increase of bulk velocity with respect to the value in the reference uncontrolled channel. This corresponds for OW to and , or and in viscous units. For VC, the sensing plane is placed at or . Various quantities characterising the three simulations are summarised in table 1. To determine the values of , the control power per unit wetted area is computed for OW according to Quadrio & Ricco (2004) as , where is the spanwise wall shear. for the VC case is defined following Choi et al. (1994) as , where is the pressure at the wall. The control is allowed to (locally or instantaneously) extract power from the flow, as the interest of the present work resides in the energy budget of the flow. Moreover, is kept constant on a timeaveraged sense, i.e. is given by the total power minus the timeaveraged value of .
The calculations start from an initial condition where the flow has already reached statistical equilibrium for the specific controlled case, and are advanced for further time units (corresponding to about 25000 viscous time units). During this time 200 flow fields are written to disk for the VC case; for the OW case, 200 flow fields are saved at 8 different phases of the oscillation, for a total of 1600 flow fields.
Case  

Ref  6500  199.7  3177  0  0.4887  1.000 
OW  6500  186.9  3267  0.09777  0.5026  1.028 
VC  6500  190.5  3474  0.00350  0.5345  1.094 
It should be noted that the drag reduction rate , i.e. the traditional figure of merit for flow control, is not a measure of improved energetic efficiency. , defined as the percentage change of the skin friction coefficient , is 17.2% in the OW case, and 23.9% in the VC case. These values are very similar (but not identical, since the comparison is made under a different condition) to the values of obtained at CFR at the condition of maximum net energy saving rate (Quadrio & Ricco 2004; Stroh et al. 2012). Appendix A discusses the choices of figure of merit for evaluating flow control at CPI, and describes the link between and , the figure of merit used in the present work.
3 Energy boxes
The energy box as introduced by Ricco et al. (2012) is used to visualise the global balance of kinetic energy. The energy box for the uncontrolled reference case is shown in Fig. 2, with numerical values for the energy fluxes scaled with the power input . The MKE is either dissipated directly by the mean flow via the volume integral of the timeaveraged viscous dissipation related to the mean velocity profile
or transferred to the TKE box via the production term
where
(3.0) 
indicates the wallnormal distribution of the Reynolds shear stress. The TKE is either produced by , and dissipated by the volume integral of the turbulent dissipation:
where repeated indices imply summation.
Owing to statistical equilibrium, and . The balance errors, due to the finite averaging time, are extremely small (less than ) and demonstrate the quality of the dataset. It is interesting to mention that the simulations are carried out with a CPG condition for the spanwise component of the momentum equation, so that the power input to the spanwise part of the MKE is identically zero. However, given the finite averaging time, an extremely small residual spanwise mean velocity profile exists, which causes the related mean dissipation not to be identically zero, but still fully negligible (less than ).
At the present value of Reynolds number, i.e. , of is dissipated by the mean field through , while the remaining is converted into TKE through and eventually dissipated through . The relative importance of and is a function of the Reynolds number, with the latter becoming dominant at higher (Laadhari 2007; Abe & Antonia 2016).
Figure 3 shows the energy boxes for the VC and OW cases. As previously noted, in both cases enters the system via the TKE box. While this is obvious for the zeronet mass flux VC, for OW within the classic Reynolds decomposition we do not distinguish the temporallyfluctuating quasilaminar Stokes layer induced by wall oscillations (Quadrio & Ricco 2004) from the chaotic turbulent fluctuations, but an alternative approach would be possible. In fact, for OW can enter the MKE box for the timevarying but coherent spanwise velocity component, as done by Ricco et al. (2012) and Touber & Leschziner (2012), who adopted a triple decomposition for the velocity field. Our procedure can be easily modified to allow for the alternative approach without implications on the final findings.
The production term appears to consistently decrease when control is exerted, either because (VC case) increases at nearly unchanged , or because (OW case) decreases but decreases even more, owing to the power requirements of the control action. The fraction of total power dissipated via and exhibits opposite trends for the two types of control. For VC only marginally decreases, because is negligible; hence increases by 0.054 compared to the uncontrolled flow, and decreases by the same amount. On the other hand, for the OW case, for which is higher, the fraction of dissipated by the mean flow decreases by 0.044.
These opposite trends are largely related to the different amount of available for the two controls, and thus reflect the different control abilities to use efficiently. In both cases the ratio increases, indicating that a larger part of the available pumping power is dissipated by the mean flow. However, is not a suitable indicator for energetically efficient flow control. One can imagine extreme cases in which increases because almost all available power is spent to run an highlyinefficient dragreducing control which produces a small flow rate with the remaining small .
In order to quantitatively link the anticipated flow rate increase for a fixed to the energy dissipation mechanisms and energy transfer rates, an enabling step involves giving prominence to the ultimate flow control target, i.e. reaching the laminar state. This will be achieved in the next Section, where an extension of the usual Reynolds decomposition is introduced.
4 Extending the Reynolds decomposition
The most successful control would relaminarise the flow, thereby maximising the flow rate for the available pumping power. The control target, i.e. the achievement of a laminar flow, can be exposed in every quantity of interest by further decomposing the mean velocity in the usual Reynolds decomposition as:
where is the laminar parabolic profile yielding the same , and is the deviation of the actual mean velocity profile from . Fig. 4 shows the profiles of , and for the reference case. By definition, the and profiles possess the same bulk velocity, hence has zero average, i.e.
Such decomposition is inspired by the work of Eckhardt et al. (2007), which discusses Taylor–Couette and Rayleigh–Bénard flows. They introduce a “convective” (or “wind”) dissipation rate, defined as the difference between the total dissipation rate and the dissipation rate of a laminar flow. It is worth noting that the present decomposition is as arbitrary as the classic Reynolds decomposition: the sum still amounts to the usual mean velocity profile. However, in contrast to , which is not a solution of the Navier–Stokes equation, is a possible state of the flow, occuring when a dragreducing control achieves complete relaminarization. The fact that is analytically known is a key property that enables the following analysis, in which kinetic energy transfer rates will be decomposed into separate components, which can in turn be related analytically to the Reynolds shear stress.
4.1 Three mean momentum equations
As usual, the governing equation for the mean velocity profile is
(4.0) 
where is the negative Reynolds shear stress (3.0), and corresponds to the negative mean streamwise pressure gradient. The governing equation for the laminar component is obtained by setting in Eq. (4.0):
(4.0) 
where is the streamwise pressure gradient required to achieve the same bulk mean velocity in a laminar flow.
One obtains the governing equation for by subtracting Eq. (4.0) from Eq. (4.0):
(4.0) 
where is the increment of the pressure gradient due to the deviation of the mean velocity from a laminar parabolic one with the same .
Obviously, the wallnormal distribution of Reynolds shear stress is essential to determine the mean velocity profile and the resulting control performance, as confirmed by Eqs. (4.0) and (4.0). In the following, mathematical expressions relating all budget terms in Fig. 1 to and are derived. In order to do so, the integrals of the above momentum equations need first to be expressed in terms of .
4.2 Triple integration for : the FIK identity under CPI
The original FIK identity was derived at CFR by Fukagata et al. (2002), and Marusic et al. (2007) provided an analogous expression for CPG. In both cases, a weighted integral of the Reynolds shear stress, i.e. , describes either the additional drag generated by turbulence (at CFR) or the flow rate decrease due to turbulence (at CPG). For CPI, the influence of turbulence on the resulting flow rate can be captured based on the same weighted integral of as shown in the following.
First, Eq. (4.0) is integrated in the direction from to , yielding
One more integration in results in
Finally, applying a third integration from to and integrating the rightmost term by parts provides the equation for the bulk mean velocity in the form
(4.0) 
Here, is defined as the weighted integral of :
(4.0) 
Eq. (4.0) nicely shows that also for CPI the term containing can be understood as the decrease of due to turbulence, since the laminar flow rate is given by . However, at CPI the streamwise pressure gradient changes with a change in . Therefore, in order to remove from Eq. (4.0), we multiply it by :
(4.0) 
and note that the quantity is the pumping power input per unit area, which under the CPI condition is known by Eq. (2.0). Substitution of this relation into Eq. (4.0) leads to
(4.0) 
which can be solved to yield:
(4.0)  
In a different form, this equation was already derived by Hasegawa et al. (2014) (see their Eq.(3.8) at p.99); it provides the relationship between and , and is the FIK identity expressed for the CPI condition. Just like in the corresponding expressions for CFR and CPG, the Reynolds shear stresses appear in (4.0) only via their weighted integral .
4.3 Triple integration for
Integrating Eq. (4.0) twice in the wall normal direction yields:
(4.0) 
One more integration from to results in the expression for :
(4.0) 
4.4 Triple integration for
Integrating Eq. (4.0) twice in the wallnormal direction results in
(4.0) 
By integrating once more from to , one obtains
Hence,
(4.0) 
5 Extending the energy box
Having extended the Reynolds decomposition to account for the laminar component in the mean flow, we can now leverage it to redesign the energy box description of the channel flow. So far, expressions for the integrals of the corresponding velocity components containing the weighted integral of the Reynolds shear stresses and the Reynolds number have been derived. In this section, every energy flux is related to the profile, eventually obtaining an extended version of the energy box.
The turbulent production term can be written as
where
(5.0) 
is the turbulent production due to the laminar component and
the one related to the deviation component. Note that is proportional to , the weighted integral of that also appears in the FIK identity. Following the previously discussed interpretation of in terms of “losses” associated with the presence of turbulence, can be understood as the fraction of “wasted” by turbulence. Similarly, can be interpreted as the consequence of the existence of the Reynolds shear stresses, implying a deviation of the mean profile from the laminar profile.
Along the same lines, the mean dissipation can be decomposed as follows:
(5.0)  
The last term vanishes since
the last integral being zero because is a constant which can be taken out of the integral, and . Therefore, the dissipation of the mean field is expressed as the sum
where
is the energy dissipation rate associated with the laminar component, and
is the additional dissipation associated with the deviation component. The sum of and is the share of wastefully dissipated by turbulence, i.e. not used for creating a flow rate, for which only is necessary. Fig. 5 shows the wallnormal distribution of the integrands of (5.0). The advantage of decomposing into and is that possesses a unique relationship with . Therefore, the only remaining issue is to understand how changes in flows with increased energetic efficiency.
We can also derive an interesting relationship between and . Integrating Eq. (4.0) in the direction yields
(5.0) 
Multiplying by and integrating in , the following equation is obtained:
The lefthandside is zero, because
Hence, the following identity is obtained
(5.0) 
which emphasizes that the production is always a source for MKE and a sink for TKE. This is reasonable, since the laminar profile is the one with minimum energy dissipation for a given flow rate. Therefore, any deviation from the laminar profile corresponds to additional energy losses, expressed by .
Moreover, Eq. (5.0) indicates that both and cancel each other in the MKE budget, which becomes:
(5.0) 
In summary, the original energy box sketched in Fig. 1 can now be extended as shown in Fig. 6. The dissipation of the mean velocity and the turbulent production are decomposed into their contributions from the laminar profile and the deviation from the laminar profile. The turbulent production due to is always negative, so that the arrow is pointing in the opposite direction, i.e. from TKE to MKE.
5.1 Energy fluxes as a function of
Every energy flux appearing in Fig. 6 is expressed as a function of the Reynolds shear stresses and of the Reynolds number in the following.
5.1.1 Input powers and
By definition, the pumping and control powers are expressed by:
(5.0) 
5.1.2 Laminar dissipation
Integration of Eq. (4.0) in leads to
Multiplying by and integrating from to , results in
Substituting with Eq. (4.0) on the lefthandside gives
(5.0) 
where Eq. (4.0) has been used for the final expression. Using Eqs. (4.0) and (4.0) yields:
(5.0)  
Note that enters this expression only through its weighted integral .
5.1.3 Laminar production
The turbulence production related to the laminar component (which is indeed an oxymoron) is given by
(5.0)  
Again, depends on via only. We note that can assume negative values in the rare cases when which corresponds to an inverse sign of the Reynolds shear stress. This is e.g. achieved by streamwise travelling waves of blowing and suction (Min et al. 2006). In this control case sublaminar drag is reported at CFR which would correspond to at CPI, see Eq. (5.0). It should be kept in mind that, by definition, it is impossible to obtain ; i.e. the controlled flow cannot produce a flow rate higher than the one achieved when all available power is spent for pumping a laminar flow.
5.1.4 Production and dissipation
Using Eq. (5.0) one obtains
(5.0)  
where Eq. (4.0) is used and is defined as
(5.0) 
Hence, is again expressed as a function of , but in addition to the weighted integral the term also appears, corresponding to the integral of .
According to (5.0), the turbulent production is obtained as
(5.0) 
Therefore, . While can switch sign, is always positive which is consistent with the fact that any deviation from the laminar profile corresponds to energetic losses.
5.1.5 Turbulent dissipation
5.2 The extended energy box
Figure 7 shows the extended energy box for the reference flow without drag reduction. At this low value of the Reynolds number, at which is known (Laadhari 2007) to overwhelm , this decomposition highlights that is about one fourth of the total power, and is comparable to . The share of that is not being used to produce a flow rate (it should be remembered that the only velocity profile contributing to the flow rate is ) is . This power depends on only via , and can be considered as power wasted to produce turbulence; here it is approximately 76% of the total power, a fraction that is expected to increase with .
How these fluxes vary with can be examined by resorting to empirical formulas expressing how and change with ; such formulas are for example discussed by Abe & Antonia (2016). Appendix B reports this analysis, which leads to a new and possibly improved relationship between and .
Case  

Ref  1.000  0.239  0.350  0.761  0.350    0.410 
VC  0.996  0.286  0.357  0.711  0.357  0.004  0.356 
OW  0.902  0.253  0.292  0.694  0.292  0.098  0.454 
Figure 8 shows the extended energy box for the considered control cases. The numerical values of all fluxes are summarised in Table 2.
Owing to the CPI approach, is decreased when a positive is present for active control. Being proportional to , increases when the control is successful, leading to an increased flow rate. However, the increase of is much larger for VC than for OW, for which the ability to increase is hindered by the reduced pumping power available under CPI.
The term is observed to decrease in both the controlled cases. In fact, the change of is prescribed by the power budget of the left panel of the extended energy box, which reads
since . Equivalently, this budget can be rewritten by using (5.0), (2.0) and (5.0) as:
In the present context must decrease when active control is on, and must increase when control is successful, so that their difference must decrease. This result has the interesting implication that the product must decrease as well, which is not obvious since increases while decreases. This implication is related to the flow rate increase at CPI being bounded by , whereas can theoretically even drop to negative values, as mentioned in §5.1.3.
The sign change of (and ) is however undefined. In fact, in our two examples slightly increases for VC but decreases for OW. This is a consequence of the presence, see e.g. Eq. (5.0), of both and in their definition, but with opposite signs, and highlights differences in between the two controlled cases. This observation reveals that a reduction of is not sufficient for achieving energetically successful flow control, despite being always detrimental to achieving higher flow rate. In the present case, for instance, OW reduces , while VC fails in doing so.
As a consequence, also the sign of the last flux is in general undetermined, and in fact in our two examples decreases for VC and increases for OW. However, in the specific OW case, we know (Quadrio & Ricco 2011) that the wall oscillations generate a spanwise Stokes flow that, even in a turbulent channel flow, closely resembles the laminar Stokes solution. Hence, nearly the whole (the precise figure is 97% in the present case) is dissipated directly by the spanwise Stokes layer, instead of the smallscale fluctuating turbulent field. If the contribution of the Stokes layer is removed, decreases in the OW case, too.
5.3 Relationship between flow rate increase, and
All the terms featuring in the extended energy box have been related to the profile of the Reynolds shear stresses. We can then return to our original goal, and discuss how the dissipation rates of the mean and fluctuating fields are affected by flow control techniques intended to increase flow rate. Thanks to the CPI constraint, the sum is always unchanged, so that considering one of the two terms is sufficient. From the simple energy box sketched in Fig. 1, the global energy balance indicates that:
Using Eq. (5.0) and (5.0), The dissipation is expressed by
(5.0)  
which only contains , , and .
Although the flow rate increase is uniquely determined by as shown by Eq. (4.0), both and additionally involve . Its definition (4.0) shows that is the correlation between and , whereas Eq. (5.0) shows that is the L2 norm of . As long as the distribution is unspecified, and are unknown. Therefore, the relationship between flow rate increase and the changes in and is also unknown — knowing it would be equivalent to solving the closure problem of turbulence.
One can however consider the space of possible controls, and assess how and change in controlled flows, by exploring a range of values for and parametrized changes of the profile of the Reynolds shear stresses of the uncontrolled flow. Parametrising , however, is quite an arbitrary step. Fig. 9 (left) shows how changes from the reference flow in the VC and OW controlled cases. The changes are different in nature: the OW profile is mostly rescaled while the VC profile behaves differently near the wall, with a secondary zero point in the flow (the position of the socalled virtual wall, Choi et al. 1994). However, in both cases the controlinduced modifications of yield very similar effects upon , as illustrated in Fig. 9 on the right. The controlled flows exhibit an upward shift (in actual viscous units) of the logarithmic portion of the mean velocity profile, defined as
(5.0) 
in which is the von Kármán constant and an additive constant. For the OW and VC cases the measured value of is 2.03 and 2.76 respectively.
We thus devise a procedure that uses the approximate analytical form of the mean velocity profile proposed by Spalding (1961) as a baseline over which the parameter is varied within the range to to mimic the effect of wallbased flow control. The log layer constant are set to the values of and as proposed by Luchini (2018). (The same work reviews the existing approximations for and proposes an elegant semiempiric formula, which restores universality of the log layer constants among different canonical internal flows. Unfortunately, this expression could not be adopted here as it does not easily accommodate a variable . However, neither the specific expression of nor the way changes of the log layer constants are implemented affect the following discussion.) is imposed, which corresponds to for the reference channel (). At CPI for this value of , the range of corresponds to values of between 0.724 to 1.218.
For each , the value of is adjusted such that condition (2.0) is satisfied. The distribution of is then computed according to Eq. (4.0), from which and are evaluated and then eventually inserted into formula (5.0). This is certainly a simplified and approximated procedure, which is not expected to hold for any type of control technique. However, it is general enough to be valid for the two control strategies considered in the present work, which are representative of wallbased control.
Figure 10 depicts the changes in and predicted via this procedure. The vertical axis shows the flow rate that occurs in presence of a shift at CPI, and the horizontal axis is the control effort . The DNS results of the two flow control techniques discussed before, namely OW and VC, are also included in the plot. The region where corresponds to successful control under CPI. The model nicely visualizes and generalizes (within the simplifications indicated above) the result obtained for OW and VC: when active control techniques with variable are considered at CPI, the way energy is dissipated in an energetically more efficient system does not imply a unique trend in (or ). However, along the vertical axis (, passive control) an increase of (or a decrease of ) is always related to a successful control, although, strictly speaking, this is only true for the present model.
In the lower half of the plots in figure 10 the flow rate achieved at constant decreases. The line marks flow states where flow control does not modify and hence the mean velocity profile. Here the control effect is only a reduction of . In the region above the line but below , the control increases the flow rate compared to a canonical channel driven at , i.e. it interacts positively with turbulence and reduces . However, owing to its power cost, it does not go above , corresponding to flow control techniques that at CFR produce drag reduction but no net energy savings. Lastly, below the line the control effect on turbulence is detrimental. Such a scenario is observed for example for the travelling waves of spanwise wall velocity (Quadrio et al. 2009) in their dragincreasing regime, and at is typical of most rough surfaces.
Figure 11 (left) shows the distribution of , i.e. the fraction of total power wastefully dissipated by turbulence, as a function of and . It can be seen that in contrast to or this quantity does not depend on , but a decrease of is directly linked to an increase of , the indicator of successful control.
The simple model presented above can easily be extended to higher Reynolds numbers in order to verify the robustness of the present evidence against changes of . In Fig.11 the same range of is considered at the increased Reynolds number of , which yields for a reference channel. This value of is large enough for the various budget terms to reach their asymptotic behaviour (see Appendix B). Since the link between and the induced change in flow rate is dependent, the range corresponds to smaller variations of flow rate at this larger , with values of ranging between 0.745 and 1.158. This reflects the known decrease of control efficiency with increasing Reynolds number for ”similar” wallbased controls (Gatti & Quadrio 2016). The corresponding model results reveal that, also at high Reynolds number, changes in or can be of either sign. Figure 11 (right) shows the distribution of at the high Reynolds number, that remains independent from , thus confirming that has a consistent trend for energetically efficient flow control at CPI, whereas alone can either increase or decrease. When comparing the left and right part of Fig.11 it is also interesting to note that the relative change in for the controlled flows is two orders of magnitude smaller, indicating that at high Reynolds number basically all pumping power is wasted by turbulence (see Appendix B).
6 Concluding remarks
The present work has considered how flow control aimed at improving the energetic efficiency of turbulent channel flows changes the energy fluxes, especially the energy dissipation rates and of the mean and fluctuating velocity fields. Such an analysis is difficult when based on data obtained at CFR or CPG, where flow control brings about an inherent change of the total power input: the total viscous dissipation rate of kinetic energy changes whenever control is applied. Therefore, the Constant Power Input (CPI) concept has been selected to compare energy transfer rates between controlled and uncontrolled flow. With CPI the total viscous dissipation rate is fixed, and the success of control implies an increase of the flow rate for the same value of , which is equivalent to the total power entering the flow system. In order to keep the total power constant, the power consumed by the considered active flow control techniques is subtracted from the available pumping power .
The obtained results show that or undergo changes of either sign in a successfully controlled flow, depending on and the particular way in which the control strategies modify . Therefore, these quantities alone cannot meaningfully serve as objectives in the optimisation of active control techniques. The often accepted notion of a decreased with successful control (Jovanovic et al. 2005; Bannier et al. 2016) is shown to be true only for wallbased control strategies with negligible or no (passive control). This statement, however, follows from a model assuming that control acts at the wall, and results into the wellknown upward shift of the logarithmic portion of the mean velocity profile.
The introduced extended Reynolds decomposition splits the mean velocity profile into a laminar profile with the same flow rate, and a profile expressing the zerointegral deviation of the mean profile from . The turbulent energy production and the dissipation associated to the mean field can thus be expressed as sum of the contributions of the laminar and deviation components, i.e. and , with the single terms being analytically known. The extended energy box highlights that:

The laminar dissipation is the preferable way to dissipate energy, because it is the only dissipation related to a flow rate.

The laminar production corresponds to the fraction of the pumping power wasted by turbulence, and decreases with successful flow control. may become negative in extreme cases (when sublaminar drag is achieved), and this is seen mathematically since depends on , Eq. (5.0), and can become negative, see (4.0).

The production by deviation is the additional power required by a turbulent channel whose mean velocity profile deviates from the laminar one. That any such deviation is detrimental (Bewley 2009; Fukagata et al. 2009) is seen here by the fact that , i.e. a fraction of is used to produce mean kinetic energy which is not associated with any flow rate and eventually dissipated by viscosity. However, a reduction of is not a sufficient condition for successful flow control. For instance, (and hence ) may be zero also for nonzero if is satisfied, a condition which does not occur for canonical channels and has not been observed for the presently investigated flow control strategies.

While is the fraction of pumping power wasted to produce turbulence, is the fraction of total power wasted by turbulence, i.e. the fraction of the total available power not used to produce a flow rate. Therefore has to be minimised by control, while alone can undergo changes of either sign.
The concepts discussed in the present paper might foster interesting developments. For example the few existing dragreductionaware RANS turbulence models (e.g. Hassid & Poreh 1978; Mele et al. 2016) are often based upon minor modifications of (the model equation for) that produce the desired skinfriction reduction effect. The present work has made evident how this approach may lack generality even for passive control. In the flow control domain, knowing the role of the Reynolds stress through the integrals and allows a general parametrization of flow control strategies and their energetic effects.
Some limitations of the present work, for instance neglecting control strategies which introduce energy into the mean flow directly, can be remedied easily, while others are more challenging to overcome. Most notably, applying the extended Reynolds decomposition to complex flows is not trivial, as the laminar solution is unknown except for few idealised flow geometries (plane channel, pipe flow, etc.). In case of ducts with arbitrary crosssection, however, the Stokes solution can be adopted instead, as it is readily obtained and is known to yield minimal dissipation (hence power input) at CFR (Fukagata et al. 2009). However, we believe that the significance of the present work mainly resides in its ability to clarify, thanks to both the newlyderived relationships for and and the CPI setting, meaning and objectives of increased energetic efficiency based on skinfriction drag reduction techniques. Much like the FIK identity itself, the limitation of using such tools in the context of simple canonical parallel flows does not diminish their ability to effectively highlight different aspects of the complex physics of nearwall turbulence.
Acknowledgments
Support through the Deutsche Forschungsgemeinschaft (DFG) project FR2823/51 is gratefully acknowledged. Computing time has been provided by the computational resource ForHLR Phase I funded by the Ministry of Science, Research and the Arts, BadenWürttemberg and DFG. DG gratefully acknowledges the additional support of the Japan Society for the Promotion of Science. YH is supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) through the GrantinAid for Scientific Research (B) (No. 17H03170).
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A The figure of merit for flow control
The assessment of flow control strategies in the CPI framework requires the definition of a suitable figure of merit to quantify the control success. A sound figure of merit consistently exceeds a threshold value, known a priori, when flow control is successful. This simple requirement is satisfied by the ratio between the flow rate in the controlled and uncontrolled channel. The control is successful if , i.e. if flow control yields a higher flow rate than an uncontrolled channel at the same .
Other possible figures of merit do not fulfil the aforementioned requirement with CPI. Notably, the condition expressing ”less drag” is also verified by unsuccessful active control that wastes power without positively affecting turbulence, hence just lowering .
Also the drag reduction rate , defined as the relative change of skinfriction coefficient , i.e.
(A 0) 
is not a suitable figure of merit for successful control at CPI. In fact, it can be easily shown that exceeding a certain value implies successful flow control. However, this value is not known a priori and depends upon the unknown value of . This result can be simply obtained by rewriting as