1. Introduction
In many flows of practical interest, a common goal is to reduce drag. This is because frictional drag is the main culprit constraining speed and efficiency, and also contributes to wear. Notable victims of drag include airplanes, ships and fluid-carrying pipelines. Accordingly, significant effort has been – and is being – put forth to develop flow control strategies to reduce drag.
Among the many forms of flow control, here we focus on two: transpiration (blowing/suction) and spanwise wall motion. A number of prior studies have demonstrated the ability to achieve drag reduction when using transpiration (Choi, Moin & Kim Reference Choi, Moin and Kim1994; Lee et al. Reference Lee, Kim, Babcock and Goodman1997; Bewley, Moin & Temam Reference Bewley, Moin and Temam2001; Min et al. Reference Min, Kang, Speyer and Kim2006; Quadrio, Floryan & Luchini Reference Quadrio, Floryan and Luchini2007; Lieu, Moarref & Jovanović Reference Lieu, Moarref and Jovanović2010; Mamori, Iwamoto & Murata Reference Mamori, Iwamoto and Murata2014; Gómez et al. Reference Gómez, Blackburn, Rudman, Sharma and McKeon2016; Koganezawa et al. Reference Koganezawa, Mitsuishi, Shimura, Iwamoto, Mamori and Murata2019; Han & Huang Reference Han and Huang2020; Park & Choi Reference Park and Choi2020; Jiao & Floryan Reference Jiao and Floryan2021a,Reference Jiao and Floryanb) or spanwise wall oscillations (Jung, Mangiavacchi & Akhavan Reference Jung, Mangiavacchi and Akhavan1992; Choi & Graham Reference Choi and Graham1998; Choi, Xu & Sung Reference Choi, Xu and Sung2002; Quadrio & Ricco Reference Quadrio and Ricco2004; Ricco & Quadrio Reference Ricco and Quadrio2008; Quadrio, Ricco & Viotti Reference Quadrio, Ricco and Viotti2009; Viotti, Quadrio & Luchini Reference Viotti, Quadrio and Luchini2009; Auteri et al. Reference Auteri, Baron, Belan, Campanardi and Quadrio2010; Yakeno, Hasegawa & Kasagi Reference Yakeno, Hasegawa and Kasagi2014; Gatti & Quadrio Reference Gatti and Quadrio2016; Meysonnat et al. Reference Meysonnat, Roggenkamp, Li, Roidl and Schröder2016; Bird, Santer & Morrison Reference Bird, Santer and Morrison2018; Skote, Mishra & Wu Reference Skote, Mishra and Wu2019; Yao, Chen & Hussain Reference Yao, Chen and Hussain2019; Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021; Ricco, Skote & Leschziner Reference Ricco, Skote and Leschziner2021), even attaining sustained sub-laminar levels of drag (Min et al. Reference Min, Kang, Speyer and Kim2006; Jiao & Floryan Reference Jiao and Floryan2021a,Reference Jiao and Floryanb). Reducing drag saves energy that would otherwise be lost to friction, but the control input requires energy. Thus, despite reducing drag, controlling a flow may increase the energy expenditure on balance.
It is worthwhile asking whether controlling a flow can confer a net energetic benefit, and whether there are any fundamental limits to how much one stands to gain. Bewley (Reference Bewley2009) and Fukagata, Sugiyama & Kasagi (Reference Fukagata, Sugiyama and Kasagi2009) provide a partial answer. In their work, they prove that for pressure-driven flow, the energetic cost of transpiration is always greater than or equal to the energy saved due to drag reduction below the laminar level, for any distribution of transpiration (Bewley showed this for channel flow, while Fukagata et al. showed this for a duct with arbitrary constant-shape cross-section and also included the effects of an arbitrary body force). In other words, no matter the spatiotemporal pattern of transpiration used, or level of drag reduction attained – even if sub-laminar – the uncontrolled laminar flow requires the least net energy. This is an important result: it rigorously establishes that the optimal control solution, from an energetic standpoint, is to laminarize the flow. (As an exception, Fukagata et al. (Reference Fukagata, Sugiyama and Kasagi2009) raise the possibility that transpiration in a duct with varying cross-sectional shape may reduce net energy requirements, although it has not yet been demonstrated.)
In this work, we generalize the result of Bewley (Reference Bewley2009) and Fukagata et al. (Reference Fukagata, Sugiyama and Kasagi2009), showing that the same fundamental limit on energy holds not only for pressure-driven flows, but also for shear-driven and mixed pressure- and shear-driven flows. We also show that the same fundamental limit exists when the control takes the form of arbitrary spanwise wall motion instead of transpiration. Finally, and perhaps most interestingly, we raise the possibility of beating this fundamental limit by combining transpiration with spanwise wall motion. That is, we show that it may be possible to attain sustained sub-laminar energy expenditure in a controlled flow.
2. Derivation of a fundamental limit on energy savings
Consider constant-density flow in a straight channel bounded at the top and bottom by walls. The bottom wall moves with a constant velocity 
$U_{{bot}} \boldsymbol {i}$, the top wall moves with a constant velocity 
$U_{{top}} \boldsymbol {i}$, and we impose a pressure gradient 
$P_x \boldsymbol {i}$, where 
$\boldsymbol {i}$ is the unit vector in the streamwise direction. The flow satisfies the continuity and Navier–Stokes equations:
$$\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{gather}$$
$$\begin{gather}\rho \left( \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} \right) ={-}\boldsymbol{\nabla} p + \mu \nabla^2 \boldsymbol{u} - P_x \boldsymbol{i}, \end{gather}$$
on the domain 
$\varOmega = \{ (x, y, z) \in [0, L_x] \times [-h, h] \times [0, L_z] \}$ with boundary 
$\partial \varOmega$, sketched in figure 1. Above, 
${\boldsymbol {u} = (u, v, w)}$ is the velocity field, 
$p$ is the pressure field, 
$\rho$ is the density and 
$\mu$ is the dynamic viscosity. The pressure gradient is constant in space but may depend on time, adjusted such that the bulk velocity
\begin{equation} U_B = \frac{1}{2 L_x h L_z} \int_0^{L_x} \int_{{-}h}^{h} \int_0^{L_z} u \,{\rm d}\kern0.06em x \,{\rm d} y \,{\rm d} z \end{equation}
is constant. The flow is periodic in the streamwise (
$x$) and spanwise (
$z$) directions. When 
${U_{{bot}} = U_{{top}} = 0}$, we have pressure-driven Poiseuille flow. When 
$P_x = 0$, we have shear-driven Couette flow. Otherwise, we have a mixed Couette–Poiseuille flow driven by shear and pressure.
Sketch of domain and uncontrolled laminar flow.
At the walls, located at 
$y = \pm h$, we apply control in two forms: transpiration and spanwise wall motion. We allow these controls to have arbitrary spatial and temporal distributions. An implication of the flow being periodic in 
$x$ and 
$z$ is that the net mass flux through the walls must be zero.
The uncontrolled laminar flow has a velocity field
\begin{equation} \boldsymbol{u}_L = u_L \boldsymbol{i} = \left[ \frac{P_x (y^2 - h^2)}{2 \mu} + \frac{(U_{{top}} - U_{{bot}}) y}{2h} + \frac{U_{{top}} + U_{{bot}}}{2} \right] \boldsymbol{i}, \end{equation}
and a total pressure field 
$P_x x$. Work is done to maintain the bulk velocity and to move the walls against forces. The question we address is whether net energy can be saved relative to the uncontrolled laminar flow by controlling the flow at the walls, accounting for the energy expenditure of the control.
Starting from (2.2), take an inner product with the velocity vector and integrate over the domain to arrive at
\begin{align} &\frac{\rho}{2} \int_\varOmega \frac{\partial}{\partial t} ( \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} ) \,{\rm d} V + \rho \int_\varOmega \boldsymbol{u} \boldsymbol{\cdot} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ) \,{\rm d} V \nonumber\\ &\quad ={-} \int_\varOmega \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} p \,{\rm d} V + \mu \int_\varOmega \boldsymbol{u} \boldsymbol{\cdot} \nabla^2 \boldsymbol{u} \,{\rm d} V - \int_\varOmega u P_x \,{\rm d} V. \end{align}The convective, pressure and viscous terms can all be simplified, and we proceed to do so one by one.
By applying vector identities, continuity and the divergence theorem, the convective term can be rewritten as
\begin{equation} \int_\varOmega \boldsymbol{u} \boldsymbol{\cdot} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ) \,{\rm d} V = \oint_{\partial \varOmega} \frac{1}{2} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u}) (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A. \end{equation}By applying a vector identity, continuity and the divergence theorem, the pressure term can be rewritten as
\begin{equation} \int_\varOmega \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} p \,{\rm d} V = \oint_{\partial \varOmega} p \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A. \end{equation}By applying vector identities and the divergence theorem, the viscous term can be rewritten as
\begin{equation} \int_\varOmega \boldsymbol{u} \boldsymbol{\cdot} \nabla^2 \boldsymbol{u} \,{\rm d} V = \oint_{\partial \varOmega} \boldsymbol{u} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A - \int_\varOmega \boldsymbol{\nabla} \boldsymbol{u} \boldsymbol{:} \boldsymbol{\nabla} \boldsymbol{u} \,{\rm d} V. \end{equation}With these simplifications, (2.5) becomes
\begin{align} & \frac{\rho}{2} \int_\varOmega \frac{\partial}{\partial t} ( \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} ) \,{\rm d} V + \frac{\rho}{2} \oint_{\partial \varOmega} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u}) (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A \nonumber\\ &\quad ={-} \oint_{\partial \varOmega} p \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A + \mu \oint_{\partial \varOmega} \boldsymbol{u} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A - \mu \int_\varOmega \boldsymbol{\nabla} \boldsymbol{u} \boldsymbol{:} \boldsymbol{\nabla} \boldsymbol{u} \,{\rm d} V - \int_\varOmega u P_x \,{\rm d} V, \end{align}which can be written as
\begin{align} &\frac{{\rm d}}{{\rm d} t} \frac{\rho}{2} \Vert \boldsymbol{u} \Vert^2 + \mu \Vert \boldsymbol{\nabla} \boldsymbol{u} \Vert^2 + \oint_{\partial \varOmega} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A + 2 L_x h L_z U_B P_x\nonumber\\ &\quad = \mu \oint_{\partial \varOmega} \boldsymbol{u} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A, \end{align}
with the norms for vector and tensor fields defined implicitly. Due to the periodic boundary conditions, only the quantities at the walls contribute to the integrals along the boundary 
$\partial \varOmega$. Therefore, (2.10) simplifies to
\begin{align} &\frac{{\rm d}}{{\rm d} t} \frac{\rho}{2} \Vert \boldsymbol{u} \Vert^2 + \mu \Vert \boldsymbol{\nabla} \boldsymbol{u} \Vert^2 + \int_{{walls}} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A + 2 L_x h L_z U_B P_x \nonumber\\ &\quad = \mu \int_{{walls}} \boldsymbol{u} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A. \end{align}In order to maintain the motion of the walls, forces must be applied to them to balance friction and momentum flux due to transpiration. Performing a control volume analysis on the top and bottom walls reveals that the required force on each wall is
\begin{equation} \boldsymbol{F}_{{top/bot}} = \int_{{top/bot}} \rho \boldsymbol{u} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}_i) \,{\rm d} A - \int_{{top/bot}} \mu (\boldsymbol{\nabla} \boldsymbol{u} + \boldsymbol{\nabla} \boldsymbol{u}^\text{T}) \boldsymbol{\cdot} \boldsymbol{n}_i \,{\rm d} A. \end{equation}
Note that we have written the unit normal as 
$\boldsymbol {n}_i$ to distinguish it from the unit normal used previously; they are related by 
$\boldsymbol {n}_i = - \boldsymbol {n}$. The first term is due to momentum flux from transpiration, and the second term is due to viscous forces.
Next, we define an effective traction vector:
\begin{equation} \boldsymbol{t} := \rho \boldsymbol{u} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}_i) - \mu (\boldsymbol{\nabla} \boldsymbol{u} + \boldsymbol{\nabla} \boldsymbol{u}^\text{T}) \boldsymbol{\cdot} \boldsymbol{n}_i. \end{equation}We rewrite the forces on the walls as
\begin{equation} \boldsymbol{F}_{{top/bot}} = \int_{{top/bot}} \boldsymbol{t} \,{\rm d} A. \end{equation}The rate of work done in order to move the walls is
\begin{equation} \dot W_{{top/bot}} = \int_{{top/bot}} \boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{t} \,{\rm d} A. \end{equation}
Note that 
$\boldsymbol {u}_{{wall}}$ differs from the fluid velocity at the walls by
\begin{equation} \boldsymbol{u}_{{wall}} = \boldsymbol{u} - (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \boldsymbol{n}. \end{equation}
It does not include the velocity component normal to the walls so that 
$\dot W_{{top}}$ and 
$\dot W_{{bot}}$ only account for the rate of work associated with the motion of the walls themselves. After some algebraic manipulation, the rate of work done in order to maintain both walls’ motions, 
$\dot W_{{walls}} = \dot W_{{top}} + \dot W_{{bot}}$, is
\begin{equation} \dot W_{{walls}} = \int_{{walls}} [ \rho ( \boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{u} ) ( \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}_i ) - \mu \boldsymbol{n}_i \boldsymbol{\cdot} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ) - \mu \boldsymbol{u}_{{wall}} \boldsymbol{\cdot} ( \boldsymbol{n}_i \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ) ] \,{\rm d} A. \end{equation}Subtracting (2.17) from (2.11) and rearranging terms yields
\begin{align} &\frac{{\rm d}}{{\rm d} t} \frac{\rho}{2} \Vert \boldsymbol{u} \Vert^2 + \mu \Vert \boldsymbol{\nabla} \boldsymbol{u} \Vert^2 + \int_{{walls}} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A + 2 L_x h L_z U_B P_x \nonumber\\ &\quad = \rho \int_{{walls}} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{u}) (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A - \mu \int_{{walls}} \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A + \dot W_{{walls}} \nonumber\\ &\qquad + \mu \int_{{walls}} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A. \end{align}
Note that 
$\boldsymbol {u}_{{wall}} \boldsymbol {\cdot } \boldsymbol {u} = \boldsymbol {u}_{{wall}} \boldsymbol {\cdot } \boldsymbol {u}_{{wall}}$.
Now take the time-average of (2.18), where the time-average of a function 
$f$ is defined by
\begin{equation} \langle f \rangle := \lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T f(t) \,{\rm d} t. \end{equation}
Assuming 
$\Vert \boldsymbol {u} \Vert ^2$ remains bounded, time-averaging yields
\begin{align} & \mu \langle \Vert \boldsymbol{\nabla} \boldsymbol{u} \Vert^2 \rangle + \left\langle \int_{{walls}} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A \right\rangle + 2 L_x h L_z U_B \langle P_x \rangle \nonumber\\ &\quad = \rho \left \langle \int_{{walls}} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{u}_{{wall}}) (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A \right \rangle - \mu \left \langle \int_{{walls}} \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A \right \rangle + \langle \dot W_{{walls}} \rangle \nonumber\\ &\qquad + \mu \left \langle \int_{{walls}} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A \right \rangle. \end{align} The norm of the velocity gradient can be rewritten. Let 
$\boldsymbol {u} = \boldsymbol {u}_L + \boldsymbol {u}'$, where 
$\boldsymbol {u}_L$ is the uncontrolled laminar flow from (2.4), and 
$\boldsymbol {u}' = (u', v', w')$ is the deviation from it. Substituting this decomposition into the expression for the norm of the velocity gradient gives
\begin{equation} \Vert \boldsymbol{\nabla} \boldsymbol{u} \Vert^2 = \Vert \boldsymbol{\nabla} \boldsymbol{u}_L \Vert^2 + \Vert \boldsymbol{\nabla} \boldsymbol{u}' \Vert^2 + 2 \int_\varOmega \boldsymbol{\nabla} \boldsymbol{u}_L \boldsymbol{:} \boldsymbol{\nabla} \boldsymbol{u}' \,{\rm d} V. \end{equation}
Since the uncontrolled laminar flow has only one component, and it is a function of only 
$y$, the last term can be rewritten as
\begin{equation} \int_\varOmega \boldsymbol{\nabla} \boldsymbol{u}_L \boldsymbol{:} \boldsymbol{\nabla} \boldsymbol{u}' \,{\rm d} V ={-} \frac{{\rm d}^2 u_L}{{\rm d} y^2} \left.\int_\varOmega u' \,{\rm d} V + \int_0^{L_z} \int_0^{L_x} \frac{{\rm d} u_L}{{\rm d} y} u' \right|_{y ={-}h}^{h} \,{\rm d}\kern0.06em x \,{\rm d} z, \end{equation}
where we have used integration by parts and the fact that 
$u_L$ is quadratic in 
$y$. The first term is zero since the uncontrolled laminar flow and the controlled flow have the same bulk flow in the 
$x$ direction. The second term is zero since 
$u' \equiv 0$ on the walls (because the uncontrolled laminar flow and controlled flow have the same boundary condition for the 
$x$ component), making the entire expression in (2.22) equal to zero. Thus, the norm of the velocity gradient is equal to the sum of that of the uncontrolled laminar flow and that of the deviation from the uncontrolled laminar flow; in this sense, the two velocity gradients may be thought of as being orthogonal. Substituting (2.21)–(2.22) into (2.20) gives
\begin{align} & \mu \Vert \boldsymbol{\nabla} \boldsymbol{u}_L \Vert^2 + \mu \langle \Vert \boldsymbol{\nabla} \boldsymbol{u}' \Vert^2 \rangle + \left \langle \int_{{walls}} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A \right \rangle + 2 L_x h L_z U_B \langle P_x \rangle \nonumber\\ &\quad = \rho \left \langle \int_{{walls}} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{u}_{{wall}}) (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A \right \rangle - \mu \left \langle \int_{{walls}} \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A \right \rangle + \langle \dot W_{{walls}} \rangle \nonumber\\ &\qquad + \mu \left \langle \int_{{walls}} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A \right \rangle. \end{align}Applying (2.23) to the uncontrolled laminar flow yields
\begin{equation} \mu \Vert \boldsymbol{\nabla} \boldsymbol{u}_L \Vert^2 + 2 L_x h L_z U_B P_{x,L} = \dot W_{{walls},L}, \end{equation}
where a subscript 
$L$ denotes laminar quantities. The third, fifth and last terms in (2.23) do not appear in (2.24) because there is no flow normal to the walls when there is no control; the sixth term does not appear because the laminar flow is rectilinear. Subtracting (2.24) from (2.23) gives
\begin{align} & \mu \langle \Vert \boldsymbol{\nabla} \boldsymbol{u}' \Vert^2 \rangle + \left \langle \int_{{walls}} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A \right \rangle + 2 L_x h L_z U_B ( \langle P_x \rangle - P_{x,L} ) \nonumber\\ &\quad = \rho \left \langle \int_{{walls}} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{u}_{{wall}}) (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A \right \rangle - \mu \left \langle \int_{{walls}} \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A \right \rangle \nonumber\\ &\qquad + \langle \dot W_{{walls}} \rangle - \dot W_{{walls},L} + \mu \left \langle \int_{{walls}} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A \right \rangle. \end{align} Some of the terms in (2.25) can be simplified. Putting ourselves in the frame of reference where 
$U_{bot} = -U_{top}$, it follows that 
$\boldsymbol {u}_{{wall}} \boldsymbol {\cdot } \boldsymbol {u}_{{wall}} = U_{top}^2 + w^2$, where 
$w$ is the spanwise component of the velocity, and the integral in the fourth term becomes
\begin{equation} U_{top}^2 \int_{{walls}} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A + \int_{{walls}} w^2 (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A = \int_{{walls}} w^2 (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A, \end{equation}since the net mass flux through the walls must be zero. The fifth term in (2.25) simplifies to
\begin{equation} \int_{{walls}} \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{u}_{{wall}} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A = \left.\int_0^{L_z} \int_0^{L_x} v \frac{\partial v}{\partial y} \right|_{y ={-}h}^{h} \,{\rm d}\kern0.06em x \,{\rm d} z, \end{equation}
where we have used integration by parts, continuity and periodicity of the flow in 
$x$ and 
$z$. The last term in (2.25) simplifies to
\begin{equation} \int_{{walls}} (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \boldsymbol{n} \boldsymbol{\cdot} (\boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u}) \,{\rm d} A = \left.\int_0^{L_z} \int_0^{L_x} v \frac{\partial v}{\partial y} \right|_{y ={-}h}^h \,{\rm d}\kern0.06em x \,{\rm d} z, \end{equation}which is the same as in (2.27). These two terms cancel, and (2.25) simplifies to
\begin{align} & \mu \langle \Vert \boldsymbol{\nabla} \boldsymbol{u}' \Vert^2 \rangle + \left \langle \int_{{walls}} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A \right \rangle + 2 L_x h L_z U_B ( \langle P_x \rangle - P_{x,L} ) \nonumber\\ &\quad = \rho \left \langle \int_{{walls}} w^2 (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A \right \rangle + \langle \dot W_{{walls}} \rangle - \dot W_{{walls},L}. \end{align}Finally, we rearrange terms to arrive at
\begin{align} & - 2 L_x h L_z U_B \langle P_x \rangle + \langle \dot W_{{walls}} \rangle - \left \langle \int_{{walls}} \left( p + \frac{\rho}{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right) \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n} \,{\rm d} A \right \rangle \nonumber\\ &\quad - ( - 2L_x h L_z U_B P_{x,L} + \dot W_{{walls},L} ) \nonumber\\ &\qquad = \mu \langle \Vert \boldsymbol{\nabla} \boldsymbol{u}' \Vert^2 \rangle - \rho \left \langle \int_{{walls}} w^2 (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A \right \rangle. \end{align}The left-hand side is the difference in power needed to maintain the controlled flow (first line) and the uncontrolled laminar flow (second line). This includes the power needed to apply the pressure gradient, the power needed to maintain the motion of the walls, and the power needed to apply transpiration, which includes the rate of work done against pressure and the rate of injection of kinetic energy into the flow.
Note that (2.30) also holds for open channel flow, as in the simulations by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021), so our ensuing discussion may also hold some relevance for boundary-layer flows, as in the accompanying experiments of Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021). However, we have not proven that such a relation holds for boundary-layer flows. Indeed, the relevance to boundary-layer flows is complicated by the fact that they are spatially developing (see Ricco & Skote (Reference Ricco and Skote2022) for an example of how spatially developing flows may fundamentally differ in this type of study).
3. Discussion
Suppose we use only transpiration for control, as in the analyses of Bewley (Reference Bewley2009) and Fukagata et al. (Reference Fukagata, Sugiyama and Kasagi2009). Then 
$w \equiv 0$ at the walls, and the right-hand side of (2.30) is equal to 
$\mu \langle \Vert \boldsymbol {\nabla } \boldsymbol {u}' \Vert ^2 \rangle \ge 0$. Thus, the left-hand side, which is the difference in power needed to maintain the controlled flow and the uncontrolled laminar flow, is non-negative. In other words, the uncontrolled laminar flow requires the least net energy no matter the spatiotemporal distribution of the control. From an energetic standpoint, the optimal solution is to laminarize the flow. This holds whether the flow is driven by pressure, shear or any combination of the two, recovering the result of Bewley (Reference Bewley2009) and Fukagata et al. (Reference Fukagata, Sugiyama and Kasagi2009) as a special case.
Now suppose that we use only spanwise wall motion for control. Then 
$\boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {n} = 0$ at the walls, and the right-hand side of (2.30) is equal to 
$\mu \langle \Vert \boldsymbol {\nabla } \boldsymbol {u}' \Vert ^2 \rangle \ge 0$. We conclude again that the uncontrolled laminar flow requires the least net energy no matter the spatiotemporal distribution of the control.
Finally, consider the case where we simultaneously apply transpiration and spanwise motion at the walls. In this case, the last term in (2.30) is non-zero. We may interpret this term as the covariance between the square of the spanwise speed and the transpiration speed, that is, as the covariance between the two forms of control. Physically, this term originates from the work done on the walls to maintain their motions. Specifically, it is the work associated with the component of the external force that arises due to the momentum flux across the walls (the first terms on the right-hand sides of (2.12) and (2.13)). If this covariance is greater than the increase in the norm of the velocity gradient due to the controlled flow, then the controlled flow requires less net power than the uncontrolled laminar flow. Mathematically, the criterion for net energy savings relative to the uncontrolled laminar flow is
\begin{equation} \rho \left \langle \int_{{walls}} w^2 (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{n}) \,{\rm d} A \right \rangle > \mu \langle \Vert \boldsymbol{\nabla} \boldsymbol{u}' \Vert^2 \rangle. \end{equation}Physically, this criterion states that if the negative of the average work arising due to momentum flux is greater than the average additional spatial variation in the flow induced by control, then the sustained net energy expenditure is sub-laminar.
Sustained sub-laminar drag has previously been attained by passive means (Mohammadi & Floryan Reference Mohammadi and Floryan2013), implying sub-laminar energy expenditure. We are unaware, however, of any results demonstrating net energy savings relative to the uncontrolled laminar flow when using active flow control, making this possibility rather interesting (Fukagata et al. (Reference Fukagata, Sugiyama and Kasagi2009) raised the possibility of doing so by applying transpiration around a bump on a channel wall, but it was never demonstrated). Moreover, the criterion in (3.1) is constructive since the covariance term can be completely specified as it only contains control terms. For example, this criterion reveals a necessary condition for net energy savings: spanwise wall motion and transpiration must have spatiotemporal overlap. In particular, the spanwise wall motion must, on average, be greater in regions of suction than in regions of blowing. This condition is not sufficient, however, since it is unknown a priori whether the designed control induces additional dissipation greater than the covariance term.
Nevertheless, the criterion provides some insight. The increase in the norm of the velocity gradient induced by control contributes to increased dissipation relative to the uncontrolled flow. Since dissipation tends to be greatest near walls, the additional dissipation induced by the control will be dominated by contributions in the vicinity of the control. With both terms in (3.1) depending on near-wall or wall quantities, the criterion provides a path forward for rational design of the control.
It is important to note that the criterion in (3.1) is ideal in the sense that it considers all of the negative work to be recoverable. In a real system, the amount of negative work that can be recovered depends on the devices used to implement control. This is an important consideration in any physical flow control system. Nevertheless, our theoretical findings open the possibility of sustained sub-laminar energy expenditure, an important first step. Pursuing this possibility is certainly worthy of future work.
Acknowledgements
The author thanks Drs J.M. Floryan and A. Prosperetti for helpful feedback.
Declaration of interests
The author reports no conflict of interest.
