2.1. Background method formulation in terms of auxiliary functionals

For shear flows governed by (2.1), the quantity that has most often been bounded using the background method is the mean dissipation. We let angle brackets denote an average over the spatial domain $\varOmega$ and let an overbar denote an infinite-time average, so the dimensionless dissipation averaged over the volume and infinite time is

(2.7) \begin{align} \overline {\big \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\big \rangle }=\lim _{T\to \infty }\frac {1}{T}\int _0^T\frac {1}{\varGamma _x\varGamma _y}\int _{\varOmega }|\boldsymbol{\nabla }\boldsymbol{u}|^2\,{\mathrm{d}}\boldsymbol{x}\,{\mathrm{d}}t, \end{align}

where $\varGamma _x\varGamma _y$ is the volume of the dimensionless domain and, if the infinite-time limit is not well defined, one can take the limit supremum. The dimensionless quantity (2.7) is the same one used in previous works such as that of Plasting & Kerswell (Reference Plasting and Kerswell2003). For physical interpretation in § 4, we examine this dissipation’s ratio to its laminar value since this ratio does not depend on the chosen non-dimensionalisation. The average dissipation is related a priori to the average rate of energy input by body and boundary forces, as explained for the examples of Wallefe flow and Couette flow in § 4.

Whether one seeks upper or lower bounds depends on the model. Often, it is easy to show that the dissipation among all flows is bounded below or above by the laminar dissipation, in which case, it remains only to find upper or lower bounds, respectively. For concreteness, our exposition in this section considers upper bounds on mean dissipation. Section 2.4 summarises what changes in the case of lower bounds. The background method can bound averages of other linear or quadratic integrals also, such as kinetic energy, by straightforward modifications to the formulations given here.

Our goal is to derive bounds on (2.7) that apply to all solutions of the governing equations (2.1) subject to boundary conditions, regardless of the initial conditions. We give the derivation in terms of a so-called auxiliary functional $V$ . Such functionals have not been explicitly used in the background method literature until recently, but they are implicit in all such arguments, as explained by Chernyshenko (Reference Chernyshenko2022) and Fantuzzi et al. (Reference Fantuzzi, Arslan and Wynn2022). An auxiliary functional $V[\boldsymbol{w}]$ maps a divergence-free, time-independent vector field $\boldsymbol{w}(\boldsymbol{x})$ to a real number. All past applications of the background method to planar shear flows are equivalent to choosing $V[\boldsymbol{w}]$ that is a quadratic spatial integral of the form

(2.8) \begin{align} V[\boldsymbol{w}]=\textit{Re}\left \langle \frac {a}{2}|\boldsymbol{w}|^2- \zeta \hat {\boldsymbol{x}}\boldsymbol{\cdot }\boldsymbol{w}\right \rangle \! , \end{align}

where the functional is defined by choosing the ‘balance parameter’ $a$ and the ‘background field’ $\zeta (z)\hat {\boldsymbol{x}}$ . The quantity $({1}/{a})\zeta$ is called the background field in the notation of most prior works (cf. Chernyshenko Reference Chernyshenko2022), but it is useful for what follows to define notation for $\zeta$ rather than for $({1}/{a})\zeta$ because then, $a$ and $\zeta$ appear linearly in $V$ . Generalisations of (2.8) that go beyond quadratic integrals have the potential to give stronger results, as they do for the Kuramoto–Sivashinsky equation (Goluskin & Fantuzzi Reference Goluskin and Fantuzzi2019), but the present work concerns the background method and thus only $V$ of the form (2.8). There is no advantage to considering a background field of more general form than $\zeta (z)\hat {\boldsymbol{x}}$ , as proved in Appendix A.1 using symmetry arguments. To enable integration by parts later, the background field is admissible only if it is continuous and piecewise smooth, and if it satisfies the same boundary conditions as $\boldsymbol{v}(\boldsymbol{x},t)$ .

The definition of $V$ does not involve time, but one obtains a scalar-valued function of time by considering $V[\boldsymbol{v}(\boldsymbol{x},t)]$ , where $\boldsymbol{v}$ solves (2.4). For choices of $V$ that lead to finite bounds on mean dissipation, it can be shown that $V[\boldsymbol{v}(\boldsymbol{x},t)]$ remains bounded as $t\to \infty$ for any admissible initial condition $\boldsymbol{v}(\boldsymbol{x},0)$ (Doering & Constantin Reference Doering and Constantin1994). All such $V$ satisfy the identity

(2.9) \begin{align} \overline {\frac {\mathrm{d}}{{\mathrm{d}}t}V[\boldsymbol{v}(\boldsymbol{x},t)]}=\lim _{T\to \infty }\frac {1}{T}\left (V[\boldsymbol{v}(\boldsymbol{x},T)]-V[\boldsymbol{v}(\boldsymbol{x},0)]\right )=0, \end{align}

so the time-averaged dissipation $\overline {\langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\rangle }$ that we want to bound is equal to $\overline {\langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\rangle +({\mathrm{d}}/{{\mathrm{d}}t})V}$ . Moreover, one can find an expression equal to $({\mathrm{d}}/{{\mathrm{d}}t})V[\boldsymbol{v}]$ without explicit time-dependence:

(2.10) \begin{align} \frac {\mathrm{d}}{{\mathrm{d}}t}V[\boldsymbol{v}(\boldsymbol{x},t)] &= \textit{Re}\left \langle (a\boldsymbol{v}-\zeta \hat {\boldsymbol{x}})\boldsymbol{\cdot }\partial _t\boldsymbol{v}\right \rangle \\[-10pt] \nonumber \end{align}

(2.11) \begin{align} &=\textit{Re}\left \langle (a\boldsymbol{v}-\zeta \hat {\boldsymbol{x}})\boldsymbol{\cdot }\left ( {-}\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{v} - U\partial _x\boldsymbol{v} - U^{\prime}v_3\hat {\boldsymbol{x}} -\boldsymbol{\nabla }p + \frac {1}{\textit{Re}}{\nabla} ^2\boldsymbol{v} \right )\right \rangle \\[-10pt] \nonumber \end{align}

(2.12) \begin{align} &= \big \langle {-}a|\boldsymbol{\nabla }\boldsymbol{v}|^2+\zeta^{\prime}\partial _zv_1-\textit{Re}\left (aU+\zeta \right )^{\prime}v_1v_3\big \rangle . \\[9pt] \nonumber\end{align}

Equation (2.10) is derived by moving the time derivative inside the volume integral, and (2.11) replaces $\partial _t\boldsymbol{v}$ according to (2.4). Equation (2.12) follows from integration by parts, recalling that we require $\zeta (z)\hat {\boldsymbol{x}}$ to satisfy the same boundary conditions as $\boldsymbol{v}$ , which may be stress-free or no-slip at each boundary. Finally, we find a useful expression with the same time average as the dissipation:

(2.13) \begin{align} \overline {\langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\rangle } &=\overline {\big \langle {U}^{\prime2}+2U^{\prime}\partial _zv_1+|\boldsymbol{\nabla }\boldsymbol{v}|^2 \big\rangle } \\[-10pt] \nonumber \end{align}

(2.14) \begin{align} &= \overline {\big\langle {U}^{\prime2}+2U^{\prime}\partial _zv_1+|\boldsymbol{\nabla }\boldsymbol{v}|^2 \big\rangle +\frac {\textrm{d}}{\textrm{d}t}V[\boldsymbol{v}(\boldsymbol{x},t)]} \\[-10pt] \nonumber \end{align}

(2.15) \begin{align} &= \overline {\mathcal{Q}[\boldsymbol{v}(\boldsymbol{x},t)]}, \\[9pt] \nonumber \end{align}

where the functional $\mathcal{Q}$ is defined as

(2.16) \begin{align} \mathcal{Q}[\boldsymbol{w}]=\big \langle {U}^{\prime2}+(2U+\zeta )^{\prime}\partial _zw_1-(a-1)|\boldsymbol{\nabla }\boldsymbol{w}|^2-\textit{Re}(aU+\zeta )^{\prime}w_1w_3\big \rangle \end{align}

for any time-independent, divergence-free vector field $\boldsymbol{w}=(w_1,w_2,w_3)$ that obeys the same boundary conditions as $\boldsymbol{v}$ . The equality (2.13) follows from substituting $\boldsymbol{u}=U\hat {\boldsymbol{x}}+\boldsymbol{v}$ , the next equality uses (2.9), and then applying (2.12) gives the expression for $\mathcal{Q}[\boldsymbol{w}]$ .

Since the time average $\overline {\mathcal{Q}[\boldsymbol{v}]}$ is bounded above by the maximum of $\mathcal{Q}[\boldsymbol{w}]$ over all possible $\boldsymbol{w}$ , (2.15) implies

(2.17) \begin{align} \overline {\big\langle |\boldsymbol{\nabla }\boldsymbol{u}|^2 \big\rangle } \leqslant \max _{\boldsymbol{w}\in \mathcal{H}_{3D}}\mathcal{Q}[\boldsymbol{w}] \end{align}

for any $a$ and any admissible $\zeta$ , where $\mathcal{H}_{3D}$ is the class of 3-D divergence-free vector fields $\boldsymbol{w}(\boldsymbol{x})$ that satisfy the same boundary conditions (2.5) or (2.6) as $\boldsymbol{v}(\boldsymbol{x},t)$ does. The right-hand maximum in (2.17) can be finite or infinite, depending on $a$ and $\zeta$ . When the maximum is finite, it can be computed numerically or bounded above analytically. One naturally wants to choose $a$ and $\zeta$ to make the resulting upper bound as small as possible. At each fixed $ \textit{Re}$ , the upper bound on dissipation that is optimal within the framework of the background method is the solution to a min–max problem,

(2.18) \begin{align} \overline {\big \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\big \rangle }\leqslant \min _{\substack {a,\,\zeta }}\max _{\boldsymbol{w}\in \mathcal{H}_{3\text{D}}}\mathcal{Q}[\boldsymbol{w}], \end{align}

where the function class over which $\zeta$ is minimised is such that $\zeta \hat {\boldsymbol{x}}\in \mathcal{H}_{3\text{D}}$ . In §§ 2.2 and 2.3, we give three more formulations that are equivalent to (2.18) and are useful for different purposes.

The optimal background method formulation (2.18) is a more precise version of (1.1) for the case of planar shear flows. Since the inner maximisation gives an infinite value for some $a$ and $\zeta$ , strictly speaking, the minimum and maximum should be called an infimum and supremum, respectively, but we use the notation $\min$ and $\max$ throughout. Provided these values are finite, we assume that all maxima and minima are attained, meaning that there exist $a$ , $\zeta$ and $\boldsymbol{w}$ for which $\mathcal{Q}[\boldsymbol{w}]$ is equal to its min–max in (2.18). Such attainment relies on optimising $\zeta$ and $\boldsymbol{w}$ over sufficiently large function spaces (Evans Reference Evans2022), but the exact choice of these spaces is beyond our scope.

2.2. Mean and mean-free decomposition of $\mathcal{Q}$

The maximisation over $\boldsymbol{w}$ in (2.17) can be decoupled into maximisations over the planar mean of $\boldsymbol{w}$ and over its remaining mean-free part. This decoupling leads to a variational problem familiar from energy stability analysis, eventually allowing us to apply the theorem of Busse (Reference Busse1972). It also reveals that finite values of the upper bound (2.17) depend on $a$ and $\zeta$ , but not on $ \textit{Re}$ , even though the functional $\mathcal{Q}$ being maximised depends also on $ \textit{Re}$ . For fixed $a$ and $\zeta$ , the maximum will assume a constant value at sufficiently small $ \textit{Re}$ , and for larger $ \textit{Re}$ , the right-hand side of (2.17) will be infinite.

We decompose the divergence-free vector field $\boldsymbol{w}$ as

(2.19) \begin{align} \boldsymbol{w}(\boldsymbol{x})=\boldsymbol{F}(z)+\dot {\boldsymbol{w}}(\boldsymbol{x}), \end{align}

where $\boldsymbol{F}$ is the mean of $\boldsymbol{w}$ over the periodic $x$ and $y$ directions, and $\dot {\boldsymbol{w}}$ is the remaining mean-free part. We denote the components of the mean as $\boldsymbol{F}=(F_1,F_2,0)$ , where the zero wall-normal component follows from impenetrability of the walls and incompressibility. With $\boldsymbol{w}$ so decomposed, the $\mathcal{Q}$ functional defined by (2.16) decouples into functionals of $\boldsymbol{F}$ and of $\dot {\boldsymbol{w}}$ ,

(2.20) \begin{align} \mathcal{Q}[\boldsymbol{F}+\dot {\boldsymbol{w}}]=\big \langle {U}^{\prime2}\big \rangle +\mathcal{F}[\boldsymbol{F}]+\mathcal{E}[\dot {\boldsymbol{w}}], \end{align}

where

(2.21) \begin{align} \mathcal{F}[\boldsymbol{F}] &= \big \langle (2U+\zeta )^{\prime}F^{\prime}_1-(a-1)\big ({F^{\prime2}_1}+{F^{\prime2}_2}\big )\big \rangle , \\[-10pt] \nonumber \end{align}

(2.22) \begin{align} \mathcal{E}[\dot {\boldsymbol{w}}] &= \big \langle -(a-1)|\boldsymbol{\nabla }\dot {\boldsymbol{w}}|^2-\textit{Re}\,(aU+\zeta )^{\prime}\dot {w}_1\dot {w}_3\big \rangle . \\[9pt] \nonumber \end{align}

To find the maximum of $\mathcal{Q}[\boldsymbol{w}]$ , we can separately maximise $\mathcal{F}[\boldsymbol{F}]$ and $\mathcal{E}[\dot {\boldsymbol{w}}]$ .

The functional $\mathcal{F}[\boldsymbol{F}]$ has a finite maximum only if $a\gt 1$ , so we assume this hereafter. Since the $F_2$ term in (2.21) is non-positive and independent of other terms, the maximiser $\boldsymbol{F}^*$ that attains the maximum of $\mathcal{F}$ must have $F_{2}^{\prime*}=0$ . With no-slip boundaries, this implies $F^*_2=0$ and with stress-free boundaries, we let $F^*_2=0$ to fix the reference frame. The Euler–Lagrange equation for $F_1$ then gives $F_1^*=(2U+\zeta)/[2(a-1)]$ . Thus,

(2.23) \begin{align} \max _{\boldsymbol{w}\in \mathcal{H}_{3\text{D}}}\mathcal{Q}[\boldsymbol{w}] = \max _{\dot {\boldsymbol{w}}\in \dot {\mathcal{H}}_{3\text{D}}}\mathcal{Q}[\boldsymbol{F}^*+\dot {\boldsymbol{w}}]= \frac {1}{a-1}\left \langle a{U}^{\prime2}+U^{\prime}\zeta^{\prime}+\frac 14{\zeta}^{\prime2}\right \rangle + \max _{\dot {\boldsymbol{w}}\in \dot {\mathcal{H}}_{3\text{D}}}\mathcal{E}[\dot {\boldsymbol{w}}], \end{align}

where $\dot {\mathcal{H}}_{3\text{D}}$ is the mean-free subspace of $\mathcal{H}_{3\text{D}}$ . Henceforth, we do not write $\dot {\boldsymbol{w}}$ ; fields denoted by $\boldsymbol{w}$ may be mean-free or not, depending on their function spaces.

The maximum in (2.23) is finite – thus giving an upper bound on dissipation – if and only if the $\mathcal{E}$ functional is non-positive for all mean-free fields. If $\mathcal{E}$ is non-positive, its maximum of zero is attained by $\boldsymbol{w}=\boldsymbol 0$ . Otherwise, its maximum is infinity since all terms in $\mathcal{E}$ are quadratic; any $\boldsymbol{w}$ for which $\mathcal{E}[\boldsymbol{w}]\gt 0$ can be scaled to make $\mathcal{E}$ arbitrarily large. With this observation, minimising (2.23) over $a$ and $\zeta$ gives our second formulation of the optimal background method bound,

(2.24) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\leqslant \min _{\substack {a\gt 1 \\ \zeta (z)}} \,\frac {1}{a-1}\left \langle a{U}^{\prime2}+U^{\prime}\zeta^{\prime}+\frac 14{\zeta}^{\prime2}\right \rangle \quad \text{such that}\quad \mathcal{E}[\boldsymbol{w}]\leqslant 0 \quad \forall \,\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}. \end{align}

The constrained minimisation in the second formulation (2.24) is equivalent to the unconstrained min–max in the first formulation (2.18). The $\mathcal{E}\leqslant 0$ constraint in (2.24) is called the spectral constraint because, as shown in the next subsection, it can be formulated in terms of the spectrum of a linear eigenproblem.

2.3. Spectral constraint

We now derive two more ways to formulate the spectral constraint in (2.24), thus giving a third and fourth formulation of the optimal background method problem. Both reformulations are familiar from energy stability analysis of shear flows. To see the connection to energy stability, note that non-positivity of the $\mathcal{E}$ functional defined by (2.22) is equivalent to

(2.25) \begin{align} -\big \langle |\boldsymbol{\nabla }\boldsymbol{w}|^2\big \rangle - \textit{Re} \left \langle hw_1w_3\right \rangle \leqslant 0 \quad \forall \,\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}, \end{align}

where we have divided by the positive quantity $a-1$ and let

(2.26) \begin{align} h(z) = \frac {1}{a-1}\left [aU^{\prime}(z)+\zeta^{\prime}(z)\right ]\!. \end{align}

The condition (2.25) is precisely the energy stability condition for a laminar flow whose derivative is $h(z)$ rather than $U^{\prime}(z)$ . For such a flow, the time derivative of the perturbation energy $\langle |\boldsymbol{v}|^2\rangle$ is non-positive for all admissible $\boldsymbol{v}(\boldsymbol{x},t)$ if and only if (2.25) holds. In other words, the spectral constraint of the background method for a flow with laminar shear $U^{\prime}$ is exactly the energy stability constraint for a flow with laminar shear $h$ . Just as laminar flows are energy stable only below a certain $ \textit{Re}$ value that depends on $U$ , the spectral constraint is satisfied only below a certain $ \textit{Re}$ that depends on $U$ , $a$ and $\zeta$ .

One way to reformulate the spectral constraint is to rearrange (2.25) as an inequality for $ \textit{Re}$ . Note that the first term of (2.25) is negative definite, while the second is sign-indefinite. In fact, for any field $[w_1,w_2,w_3](x,y,z)$ giving certain values for first and second terms in (2.25), the field $[-w_1,w_2,w_3](-x,y,z)$ gives the same first term but negates the second. Therefore, taking the absolute value of the second term in (2.25) gives an equivalent condition,

(2.27) \begin{align} -\big \langle |\boldsymbol{\nabla }\boldsymbol{w}|^2\big \rangle + \textit{Re} \left |\left \langle hw_1w_3\right \rangle \right | \leqslant 0 \quad \forall \,\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}. \end{align}

In turn, this is equivalent to

(2.28) \begin{align} \textit{Re} \leqslant \min _{\boldsymbol{w}\in \dot {\mathcal{H}}_{3D}}\mathcal{R}[\boldsymbol{w}], \end{align}

where

(2.29) \begin{align} \mathcal{R}[\boldsymbol{w}]=\frac {\big \langle |\boldsymbol{\nabla }\boldsymbol{w}|^2\big \rangle }{\left |\left \langle hw_1w_3\right \rangle \right |}. \end{align}

Expressing the spectral constraint in (2.24) using $\mathcal{R}$ gives our third formulation of the optimal background method,

(2.30) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\leqslant \min _{\substack {a\gt 1 \\ \zeta (z)}} \,\frac {1}{a-1}\left \langle a{U}^{\prime2}+U^{\prime}\zeta^{\prime}+\frac 14{\zeta}^{\prime2}\right \rangle \quad \text{such that}\quad \textit{Re}\leqslant \min _{\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}}\mathcal{R}[\boldsymbol{w}]. \end{align}

For fixed $a$ and $\zeta$ , the background method gives the right-hand integral in (2.24) as an upper bound if $ \textit{Re}$ satisfies (2.28) and it gives no finite bound at larger $ \textit{Re}$ . The energy stability criterion is commonly expressed as (2.28) for a flow with laminar shear profile $h$ . This is the formulation of energy stability used throughout the analysis of Busse (Reference Busse1972), which underlies our criterion derived in § 3. We thus express the spectral constraint as (2.28) throughout § 3.

Another reformulation of the spectral constraint is in terms of the spectrum of a linear eigenproblem. Since $\mathcal{E}[\boldsymbol{w}]\leqslant 0$ holds if and only if it holds when $\boldsymbol{w}$ is scaled to satisfy $\langle |\boldsymbol{w}|^2\rangle =1$ , the spectral constraint is equivalent to

(2.31) \begin{align} \min _{\substack {\boldsymbol{w}\in \dot {\mathcal{H}}_{3D}\\ \langle |\boldsymbol{w}|^2\rangle =1}}\left (-\mathcal{E}[\boldsymbol{w}]\right )\geqslant 0. \end{align}

We have negated $\mathcal{E}$ to obtain a non-negativity condition for consistency with prior works. The normalisation constraint on $\boldsymbol{w}$ has been added so that the left-hand minimum is negative but still finite when the spectral constraint is violated. Then, whether or not the spectral constraint holds, the variational problem (2.31) has mean-free minimisers that satisfy its Euler–Lagrange equations,

(2.32) \begin{align} -(a-1){\nabla} ^2\boldsymbol{w}+\frac 12\textit{Re}(aU+\zeta )^{\prime}[w_3\hat {\boldsymbol{x}}+w_1\hat {\boldsymbol{z}}]+\boldsymbol{\nabla }p=\lambda \boldsymbol{w}, \quad \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{w}=0, \end{align}

where $2p(\boldsymbol{x})$ is a Lagrange multiplier enforcing incompressibility and $\lambda$ is a Lagrange multiplier enforcing $\langle |\boldsymbol{w}|^2\rangle =1$ that acts as an eigenvalue in (2.32). The spectral constraint is equivalent to all eigenvalues of (2.32) being non-negative – that is, to this linear eigenproblem having a non-negative spectrum.

To simplify implementation of the eigenproblem (2.32), one can Fourier transform in the periodic directions. This gives an eigenproblem that is an ordinary differential equation in $z$ for each fixed wavevector $\boldsymbol{k}=(j,k)$ , where $j$ and $k$ are the streamwise and spanwise wavenumbers, respectively. For each $\boldsymbol{k}$ ,

(2.33) \begin{align} \begin{split} -(a-1)\left (\frac {\mathrm{d}^2}{{\mathrm{d}}z^2}-j^2-k^2\right ) \hat {w}_1+\frac 12 \textit{Re} (aU+\zeta )^{\prime} \hat {w}_3+ij\hat {p}&=\lambda \hat {w}_1, \\ -(a-1)\left (\frac {\mathrm{d}^2}{{\mathrm{d}}z^2}-j^2-k^2\right ) \hat {w}_2+ik\hat {p}&=\lambda \hat {w}_2, \\ -(a-1)\left (\frac {\mathrm{d}^2}{{\mathrm{d}}z^2}-j^2-k^2\right ) \hat {w}_3+\frac 12 \textit{Re} (aU+\zeta )^{\prime}\hat w_1+\frac {\mathrm{d} }{{\mathrm{d}}z}\hat {p}&=\lambda \hat {w}_3, \\ ij\hat {w}_1+ik\hat {w}_2+\frac {\mathrm{d}}{{\mathrm{d}}z}\hat {w}_3&=0, \end{split} \end{align}

where $i$ is the imaginary unit, the Fourier transforms $\hat {\boldsymbol{w}}(z)$ and $\hat {p}(z)$ are complex in general, all $\lambda$ are real, and $({\mathrm{d}}/{{\mathrm{d}}z})$ is the ordinary $z$ -derivative operator. The spectral constraint is equivalent to (2.33) having a non-negative spectrum of admissible $\boldsymbol{k}$ . This gives our fourth formulation of the optimal background method,

(2.34) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\leqslant \min _{\substack {a\gt 1 \\ \zeta (z)}} \,\frac {1}{a-1}\left \langle a{U}^{\prime2}+U^{\prime}\zeta^{\prime}+\frac 14{\zeta}^{\prime2}\right \rangle \quad \text{such that}\quad \lambda \geqslant 0 \text{ in (2.33)}\quad \forall \,\boldsymbol{k}\in K, \end{align}

where $K$ is the set of admissible wavevectors $\boldsymbol{k}=(j,k)$ . It suffices for $K$ to include only non-negative $j$ and $k$ ; adding constraints for $(-j,k)$ , $(j,-k)$ or $(-j,-k)$ would be redundant. The velocity is mean-free in $x$ and $y$ , so $(0,0)\notin K$ . For bounds to apply for all possible periods $\varGamma _x$ and $\varGamma _y$ , the spectrum of (2.34) must be non-negative for all other non-negative pairs $(j,k)$ . For bounds to apply to flows with fixed $\varGamma _x$ and $\varGamma _y$ , the spectrum of (2.34) must be non-negative only for $j$ and $k$ that are integer multiples of $2\pi /\varGamma _x$ and $2\pi /\varGamma _y$ . Enforcing the spectral constraint for only 2.5-D fields amounts to including only $(0,k)\in K$ ; this is what Plasting & Kerswell (Reference Plasting and Kerswell2003) did when computing optimal bounds for Couette flow.

The main question of our present work – whether bounds computed over 2.5-D and 3-D fields coincide – will have the same answer for all four equivalent formulations of the optimal background method in (2.18), (2.24), (2.30) and (2.34). The same is true for suboptimal bounds found by maximising over $\boldsymbol{w}$ at fixed $a$ and $\zeta$ . Our computations in § 4 implement the fourth formulation (2.34) to find the optimal $a$ and $\zeta$ under the assumption of 2.5-D optimisers, meaning all wavevectors in $K$ have the form $(0,k)$ . One can confirm a posteriori that these bounds apply to 3-D flows by directly checking that the spectral constraint in (2.34) holds also for $(j,k)$ with $j\neq 0$ . In some cases, however, direct checking of the 3-D spectral constraint can be avoided by using the criterion derived in § 3.

2.4. Lower bounds

The four formulations of the background method derived in §§ 2.1 to 2.3 require only slight modification to give lower bounds on dissipation rather that upper bounds. In the first formulation (2.18), we simply switch the role of minimisation and maximisation to find

(2.35) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\geqslant \max _{\substack {a,\,\zeta }}\min _{\boldsymbol{w}\in \mathcal{H}_{3D}}\mathcal{Q}[\boldsymbol{w}]. \end{align}

Decomposition of $\boldsymbol{w}$ into its mean and mean-free parts as in § 2.2 leads to the lower bound version of the second formulation

(2.36) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\geqslant \max _{\substack {a\lt 1 \\ \zeta (z)}} \,\frac {1}{a-1}\left \langle a{U}^{\prime2}+U^{\prime}\zeta^{\prime}+\frac 14{\zeta}^{\prime2}\right \rangle \quad \text{such that}\quad \mathcal{E}[\boldsymbol{w}]\geqslant 0 \quad \forall \,\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}. \end{align}

This differs from its analogue (2.24) for upper bounds only in that $\mathcal{E}$ must have the opposite sign, and finite bounds require $a\lt 1$ rather than $a\gt 1$ . Dividing the expression (2.22) for $\mathcal{E}$ by the positive quantity $1-a$ , as opposed to $a-1$ in the upper bound case, and giving the second term its worst-case sign as in § 2.3, we find that non-negativity of $\mathcal{E}[\boldsymbol{w}]$ is equivalent to

(2.37) \begin{align} \big \langle |\boldsymbol{\nabla }\boldsymbol{w}|^2\big \rangle - \textit{Re} \left |\left \langle hw_1w_3\right \rangle \right | \geqslant 0 \quad \forall \,\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}, \end{align}

where $h(z)$ is defined by (2.26) as for upper bounds. Rearranging (2.37) as an upper bound on $ \textit{Re}$ , we find that the spectral constraint is identical in the upper and lower bound cases, so the lower bound version of the third formulation is

(2.38) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\geqslant \max _{\substack {a\lt 1 \\ \zeta (z)}} \,\frac {1}{a-1}\left \langle a{U}^{\prime2}+U^{\prime}\zeta^{\prime}+\frac 14{\zeta}^{\prime2}\right \rangle \quad \text{such that}\quad \textit{Re}\leqslant \min _{\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}}\mathcal{R}[\boldsymbol{w}]. \end{align}

The fourth formulation is the same as (2.38), except the spectral constraint takes its eigenvalue form as in (2.34). In § 3, theoretical results apply to both upper and lower bounds because the spectral constraint takes the same form in both cases.

3.1. Assumed symmetry of the governing equations

Throughout § 3, we assume that the governing model is invariant under $180^\circ$ rotation about a spanwise axis,

(3.1) \begin{align} [u_1,u_2,u_3](x,y,z,t)\mapsto [-u_1,u_2,-u_3](-x,y,-z,t). \end{align}

That is, we assume that the left-hand side of (3.1) satisfies the governing equation (2.1) and boundary conditions if and only if the right-hand side satisfies them. This requires that any body forcing $f(z)$ must be odd about the $z=0$ midplane. It also requires the boundary conditions to be odd, meaning that if $u_1$ values are fixed at $z=\pm 1/2$ , then they must be negations of each other and likewise, if $\partial _zu_1$ values are fixed. The flow itself need not be invariant under (3.1).

For all shear flow models invariant under (3.1), the laminar flow profile $U(z)$ is odd about the midplane, and (2.4) and the boundary conditions governing perturbations $\boldsymbol{v}(\boldsymbol{x},t)$ are also invariant under (3.1). For shear flows with this symmetry, we always restrict to background profiles $\zeta (z)$ that are odd because this cannot worsen the eventual upper bound, as shown in Appendix A.2.

3.2. Busse’s criterion

For our present purpose, we consider the third formulation of the optimal background method bound, where the spectral constraint requires that $ \textit{Re}$ is no larger than the minimum of $\mathcal{R}[\boldsymbol{w}]$ . This constraint is identical in the upper and lower bounding formulations of (2.30) and (2.38). If this minimum is the same over 2.5-D and 3-D fields for given $\zeta$ and $a$ , then in any formulation of the background method, it suffices to consider 2.5-D fields. In particular, we aim to compute the optimal $\zeta$ and $a$ using 2.5-D fields and then verify that the resulting bounds are valid also for 3-D fields.

The minimum of $\mathcal{R}[\boldsymbol{w}]$ in (2.30) is exactly the critical $ \textit{Re}$ value of energy stability for a flow whose laminar profile is $(aU+\zeta)/(a-1)$ rather than $U$ . This follows from the relationship between the spectral constraint and energy stability that is described in the first paragraph of § 2.3. Thus, our present question about the background method amounts to asking whether the critical $ \textit{Re}$ of energy stability for the laminar profile $(aU+\zeta)/(a-1)$ is the minimum of $\mathcal{R}[\boldsymbol{w}]$ over 2.5-D fields or whether 3-D fields would give a smaller minimum. Busse (Reference Busse1972) derived a criterion for drawing exactly this conclusion about the energy stability problem for models with the symmetry (3.1). That criterion is directly applicable here since we are interested in energy stability of the laminar profile $(aU+\zeta)/(a-1)$ , which has the required symmetry. In particular, this profile is odd and its derivative $h$ is even, because the symmetry (3.1) ensures that $U$ is odd and lets us restrict to odd $\zeta$ .

The criterion of Busse is most naturally stated using poloidal–toroidal variables, on which its derivation relies. Any divergence-free and mean-free $\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}$ in the present geometry can be represented as

(3.2) \begin{align} \boldsymbol{w}=\boldsymbol{\nabla }\times \boldsymbol{\nabla }\times (\varphi \hat {\boldsymbol{z}})+\boldsymbol{\nabla }\times (\psi \hat {\boldsymbol{z}}), \end{align}

where $\varphi (\boldsymbol{x})$ and $\psi (\boldsymbol{x})$ are the poloidal and toroidal potentials, respectively. These potentials are uniquely determined by $\boldsymbol{w}$ , up to arbitrary additive constants. See Schmitt & Von Wahl (Reference Schmitt and Von Wahl1992) for a proof that this decomposition is always possible with the present geometry and boundary conditions. In these variables, no-slip conditions (2.5) at both boundaries are

(3.3) \begin{align} \varphi ,\:\partial _z\varphi ,\:\psi =0\quad \text{at}\quad z=\pm \frac {1}{2}, \end{align}

and stress-free conditions (2.6) at both boundaries are

(3.4) \begin{align} \varphi ,\:\partial _z^2\varphi ,\:\partial _z\psi =0\quad \text{at}\quad z=\pm \frac {1}{2}. \end{align}

In terms of $\varphi$ and $\psi$ , the $\mathcal{R}$ functional defined by (2.29) can be expressed as

(3.5) \begin{align} \mathcal{R}[\boldsymbol{w}]=\frac {\left \langle |\hat {\boldsymbol{z}}\times \boldsymbol{\nabla }{\nabla} ^2\varphi |^2+|\boldsymbol{\nabla }\times \boldsymbol{\nabla }\times (\psi \hat {\boldsymbol{z}})|^2\right \rangle }{\left |\left \langle h\,{\nabla} ^2_2\varphi \left (\partial _y\psi +\partial _{xz}\varphi \right )\right \rangle \right |}, \end{align}

where ${\nabla} ^2_2=\partial _x^2+\partial _y^2$ denotes the Laplacian operator in only the periodic directions. The derivation of (3.5) from (2.29) has used the identity $\langle |\boldsymbol{\nabla }\boldsymbol{w}|^2\rangle =\langle |\boldsymbol{\nabla }\times \boldsymbol{w}|^2\rangle$ that holds for both no-slip and stress-free boundaries.

The criterion of Busse (Reference Busse1972) gives a sufficient condition for the $\mathcal{R}[\boldsymbol{w}]$ functional defined by (2.29), or by (3.5) in poloidal–toroidal variables, to have the same minimum over 2.5-D and 3-D fields. It applies in the present geometry if $h(z)$ is an even function and if the minimisations over $\boldsymbol{w}$ allow for all spanwise or streamwise periods. The condition can be checked by computing minima of $\mathcal{R}[\boldsymbol{w}]$ over three different lower-dimensional subspaces to find the following values:

(3.6) \begin{align} R_e=\min _{\substack {\boldsymbol{w}\in \dot {\mathcal{H}}_{2.5\text{D}}\\ \varphi ,\psi \text{ even in }z}}\mathcal{R}[\boldsymbol{w}], \quad \, R_o=\min _{\substack {\boldsymbol{w}\in \dot {\mathcal{H}}_{2.5\text{D}}\\ \varphi ,\psi \text{ odd in }z}}\mathcal{R}[\boldsymbol{w}], \quad \, R_\varphi =\min _{\substack {\boldsymbol{w}\in \dot {\mathcal{H}}_{2\text{D}}\\ \psi =0}}\mathcal{R}[\boldsymbol{w}]. \end{align}

The theorem of Busse (Reference Busse1972) can be stated as

(3.7) \begin{align} \text{If}\quad \frac {1}{R_e^2}\geqslant \frac {1}{R_o^2}+\frac {1}{R_\varphi ^2}, \quad \text{then}\ \min _{\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}}\mathcal{R}[\boldsymbol{w}] = R_e. \end{align}

The proof of this statement is agnostic to whether $h=U^{\prime}$ , as in Busse’s context of energy stability, or $h=(aU'+\zeta')/(a-1)$ , as in the spectral constraint of the background method. Observe that $R_e$ and $R_o$ are computed over different subspaces of 2.5-D fields $\boldsymbol{w}(y,z)$ , and $R_\varphi$ is computed over a subspace of 2-D fields $\boldsymbol{w}(x,z)$ . It is explained in Appendix C.2 why Busse’s proof of (3.7) requires that the minimisations over 2.5-D and 2-D fields admit all spanwise and streamwise periods, respectively. The inequality in (3.7) is what we refer to as ‘Busse’s criterion’. Later, we use an equivalent form of Busse’s criterion in which the inequality is rearranged as

(3.8) \begin{align} \chi \leqslant 1, \quad \text{where}\ \chi = R_e \left (\frac {1}{R_o^2}+\frac {1}{R_\varphi ^2}\right )^{1/2}. \end{align}

A proof of (3.7) is given in Appendix C, where we follow the same approach as Busse (Reference Busse1972) with some details added or changed for clarity. At present, we explain only the last part of the argument. The desired equality in (3.7) certainly holds as an inequality,

(3.9) \begin{align} \min _{\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}}\mathcal{R}[\boldsymbol{w}] \leqslant \min _{\substack {\boldsymbol{w}\in \dot {\mathcal{H}}_{2.5\text{D}}\\ \varphi ,\psi \text{ even in }z}} \mathcal{R}[\boldsymbol{w}]\equiv R_e, \end{align}

since the space of $\boldsymbol{w}$ for the right-hand minimisation is contained in the space for the left-hand minimisation. With a much longer argument, it is shown in Appendix C that the minimum of interest over 3-D fields is bounded below by

(3.10) \begin{align} \min _{\boldsymbol{w}\in \dot {\mathcal{H}}_{3D}}\mathcal{R}[\boldsymbol{w}]\geqslant \min \left \{R_e,\sqrt \beta R_o,\sqrt {1-\beta }R_\varphi \right \} \end{align}

for every $\beta \in (0,1)$ . The inequality opposite to (3.9) holds, thereby giving equality, if $R_e$ is the smallest of the three values on the right-hand side of (3.10). There exists a choice of $\beta \in (0,1)$ for which $R_e$ is the smallest value, meaning $R_e\leqslant \sqrt {\beta }R_o$ and $R_e\leqslant \sqrt {1-\beta }R_\varphi$ , if and only if the inequality in (3.7) holds. (One can choose $\beta =R_e^2/R_o^2$ .) Therefore, Busse’s criterion (3.7) indeed follows once the inequality (3.7) is proved in Appendix C.

The version of Busse’s theorem formulated in (3.7) aims to conclude that the minimum of $\mathcal{R}$ over 3-D fields is attained by 2.5-D fields whose poloidal and toroidal potentials are even in $z$ . This is indeed the case for various $h(z)$ arising in energy stability analysis and for some of the $h(z)$ arising in our background method computations of § 4. A criterion for the minimum to be attained by 2.5-D fields whose potentials are instead odd in $z$ can be derived by switching the roles of $R_e$ and $R_o$ in the proof of Appendix C. This gives the following:

(3.11) \begin{align} \text{If}\quad \frac {1}{R_o^2}\geqslant \frac {1}{R_e^2}+\frac {1}{R_\varphi ^2}, \quad \text{then}\ \min _{\boldsymbol{w}\in \dot {\mathcal{H}}_{3\text{D}}}\mathcal{R}[\boldsymbol{w}] = R_o. \end{align}

The only difference between (3.11) and (3.7) is swapping oddness and evenness. We are not aware of examples where the minimum of $\mathcal{R}$ over 3-D fields coincides with $R_o$ rather than $R_e$ , but we have not ruled them out.

3.3. Application of Busse’s criterion to the background method

We now describe how the criterion in § 3.2 can be used to ensure that 2.5-D background method computations apply to 3-D flows for any model with the symmetry (3.1). A calculation over 2.5-D fields at chosen $ \textit{Re}$ gives $a$ and $\zeta$ . For this combination of $ \textit{Re}$ , $a$ and $\zeta$ , one must confirm that the extremum of $\mathcal{Q}$ is the same over 3-D fields as over 2.5-D fields. In both the upper and lower bounding cases, this is equivalent to $\mathcal{R}$ having the same minimum over 3-D and 2.5-D fields. The latter can be verified using the criterion of § 3.2 as part of the following procedure.

1. (i) Fix $ \textit{Re}$ . Compute the optimal background method bound over 2.5-D velocity fields whose potentials $\varphi$ and $\psi$ are even in $z$ .

2. (ii) For $h(z)$ defined by (2.26) using the optimal $a$ and $\zeta (z)$ found in step (i), compute $R_o$ and $R_\varphi$ by solving the minimisation problems that define them in (3.6). There is no need to compute $R_e$ because $R_e=\textit{Re}$ by construction.

3. (iii) Evaluate $\chi$ by the formula in (3.8). If $\chi \leqslant 1$ , the bound computed in step (i) coincides with the optimal background method bound over 3-D velocity fields.

Details of our own implementation of this procedure are described in § 4 and Appendix B. For step (i), our optimal background method computation is based on the fourth formulation (2.34). If the criterion in step (iii) fails to hold, meaning $\chi \gt 1$ , one cannot yet conclude that the bound from step (i) coincides with the optimal background method bound over 3-D fields. An option in this case is to directly check that the $a$ and $\zeta$ found in step (i) satisfy the 3-D spectral constraint, which would show that the bound indeed holds for 3-D flows. Another option, which is potentially simpler but makes the bound at least slightly worse, is described in the next subsection.

3.4. Restricting to background profiles that satisfy Busse’s criteria

If an optimal background method computation over 2.5-D fields yields $a$ and $\zeta$ for which the $\chi \leqslant 1$ criterion fails, one can repeat the bounding computation with the criterion added as a constraint. This latter 2.5-D computation yields $a$ and $\zeta$ for which computations over 2.5-D and 3-D fields must coincide; thus, it gives a bound that must apply to 3-D flows. We formulate a slightly different version of Busse’s criterion to enforce as a constraint, in which $\beta$ appears linearly rather than inside of square roots.

Recall that the desired coincidence of 2.5-D and 3-D optima follows if $R_e$ is the smallest of the three quantities on the right-hand side of (3.10). In other words, we need there to exist $\beta \in (0,1)$ such that $R_e\leqslant \sqrt \beta R_o$ and $R_e\leqslant \sqrt {1-\beta }R_\varphi$ . The analysis in Appendix C.1 leading to (3.10) also gives, along the way, a very similar criterion in which $\beta$ appears linearly. In particular, the minimum of $\mathcal{R}[\boldsymbol{w}]$ over 3-D fields coincides with $R_e$ if

(3.12) \begin{align} R_e\leqslant \min _{\substack {\boldsymbol{w}\in \dot {\mathcal{H}}_{2.5\text{D}}\\ \varphi ,\psi \text{ odd in }z}}\frac {N_o[\boldsymbol{w}]}{\left |D_o[\boldsymbol{w}]\right |} \quad \text{and} \quad R_e\leqslant \min _{\substack {\boldsymbol{w}\in \dot {\mathcal{H}}_{2\text{D}}\\ \psi =0}}\frac {N_\varphi [\boldsymbol{w}]}{\left |D_\varphi [\boldsymbol{w}]\right |}, \end{align}

where the numerator and denominator functionals are defined in (C3) and (C4) of Appendix C.1. Rearranging the criterion (3.12) gives an equivalent constraint,

(3.13) \begin{align} \begin{split} N_o[\boldsymbol{w}]-\textit{Re}\,D_o[\boldsymbol{w}]\geqslant 0& \quad \forall \,\boldsymbol{w}\in \dot {\mathcal{H}}_{2.5\text{D}}\ \text{ with }\varphi ,\psi \text{ odd in }z, \\ N_\varphi [\boldsymbol{w}]-\textit{Re}\,D_\varphi [\boldsymbol{w}]\geqslant 0& \quad \forall \,\boldsymbol{w}\in \dot {\mathcal{H}}_{2\text{D}}\ \text{ with }\psi =0, \end{split} \end{align}

where $\beta$ appears linearly in $N_o$ and $N_\varphi$ . To get (3.13) from (3.12), we have used the fact that $R_e=\textit{Re}$ in the present context of the background method (cf. § 3.3), and we have removed the absolute values on $D_o$ and $D_\varphi$ by the same reasoning that precedes (2.27).

When carrying out an optimal background method computation over 2.5-D fields, one can include the two additional constraints in (3.13), and optimise over $\beta \in (0,1)$ along with $a$ and $\zeta$ . Simultaneous optimisation of these parameters is not convex because $\beta$ and $a$ multiply each other, but the global optimum can still be found using a branch-and-bound algorithm, as described further in Appendix B.3. The resulting bounds will be guaranteed to apply to 3-D flows. In cases where the $\chi \leqslant 1$ criterion would have been violated without these additional constraints, the constraints lead to bounds that are at least slightly worse because $a$ and $\zeta$ have been further constrained such that $\chi =1$ . Our numerical implementation of these additional constraints is described in Appendix B.3, and results for Couette flow are reported in § 4.2.

4.1. Waleffe flow

The Waleffe flow configuration (Waleffe Reference Waleffe1997) is a version of Kolmogorov flow with stress-free walls and a forcing profile that is half a period of a sine function. Following Waleffe, we choose the dimensional velocity scale $\mathsf{U}$ to be the root-mean-squared velocity of the laminar flow, but for the length scale we have chosen the layer’s full thickness $d$ rather than its half-thickness. In the resulting dimensionless (2.1), the forcing profile is $f(z)=\textit{Re}^{-1}\pi ^2\sqrt {2}\sin (\pi z)$ and the laminar profile is $U(z)=\sqrt {2}\sin (\pi z)$ . The dimensionless laminar flow has dissipation ${\langle {U}^{\prime2}\rangle }=\pi ^2$ .

In Waleffe flow, as in channel flow driven by a fixed pressure gradient (Constantin & Doering Reference Constantin and Doering1995), dissipation is maximised by its laminar value. This can be seen by choosing $a=2$ and $\zeta =2U$ , so that $\mathcal{Q}$ defined by (2.16) becomes $\mathcal{Q}[\boldsymbol{w}]= \langle {U}^{\prime2} \rangle - \langle |\boldsymbol{\nabla }\boldsymbol{w}|^2 \rangle$ . Maximising over $\boldsymbol{w}$ gives

(4.1) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\leqslant \langle {U}^{\prime2} \rangle , \end{align}

which must be the optimal upper bound (2.18) because it is saturated by the laminar state. Note also that the total velocity field $\boldsymbol{u}$ obeys (2.1) and its kinetic energy evolves as

(4.2) \begin{align} \frac {\mathrm{d}}{{\mathrm{d}}t}\frac 12\big \langle |\boldsymbol{u}|^2\big \rangle = -\frac {1}{\textit{Re}}\big \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\big \rangle + \langle f u_1\rangle . \end{align}

The left-hand side of (4.2) time-averages to zero, as explained preceding (2.9), so the mean dissipation is balanced by the mean work performed by the force $f(z)$ ,

(4.3) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle } = \textit{Re}\,\overline {\langle f u_1\rangle }. \end{align}

An interpretation of why both quantities are maximised by the laminar state is that this flow perfectly aligns the flow direction with the force direction. Here, we report our bounds on mean dissipation in terms of the ratio

(4.4) \begin{align} \varepsilon = \frac {1}{\textit{Re}}\frac {\big \langle {U}^{\prime2}\big \rangle }{\,\overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\,} =\frac {1}{\textit{Re}}\frac {\big \langle fU\big \rangle }{\,\overline {\left \langle fu_1\right \rangle }\,}, \end{align}

which is a kind of friction coefficient. Such a ratio is unaffected by whether or how $\boldsymbol{u}$ has been non-dimensionalised. The laminar upper bound (4.1) on mean dissipation gives the lower bound $({1}/{\textit{Re}})\leqslant \varepsilon$ , and the lower bounds on mean dissipation that we report now give upper bounds on $\varepsilon$ .

We have computed optimal lower bounds on mean dissipation in Waleffe flow. These complement, but are not equivalent to, upper bounds from Rollin, Dubief & Doering (Reference Rollin, Dubief and Doering2011) on the ratio between dissipation and the $3/2$ power of kinetic energy for the same model. In our bounding computations, the domain is fixed to have streamwise and spanwise periods of $\varGamma _x=\varGamma _y=2\pi$ , which we have confirmed is large enough for bounds to closely approximate their large-domain limits (cf. the end of Appendix B.1) at all but the smallest $ \textit{Re}$ values. The criterion of § 3.3 based on Busse’s theorem can only be applied in the large-domain limit because part of the theorem’s proof in Appendix C.2 assumes that all wavenumbers are admissible in the periodic directions. For our fixed domain, the energy method shows that the laminar flow is globally stable if $ \textit{Re}\leqslant \textit{Re}_E\approx 6.88$ and $ \textit{Re}_E$ is also the value above which our upper bounds on dissipation depart from its laminar value. This energy stability value agrees with the approximation $ \textit{Re}_E\approx 7$ reported by Doering, Eckhardt & Schumacher (Reference Doering, Eckhardt and Schumacher2003) for the same fixed domain.

Figure 1(a) shows our optimal upper bounds on the friction coefficient $\varepsilon$ in 3-D Waleffe flow, which asymptote to roughly 0.145. These have been computed by bounding mean dissipation below at various fixed $ \textit{Re}$ values using the procedure of § 3.3. To compute $R_e$ in step (i) of this procedure, we use the formulation in Appendix B.1. For the $a$ and $\zeta$ found in step (i), values of $R_o$ and $R_\varphi$ are computed in step (ii) using the formulations in Appendix B.2. Figure 1(b) shows the values of $\chi$ calculated in step (iii) to check the $\chi \leqslant 1$ criterion (3.8). These values are far smaller than unity; therefore, the bounds in figure 1(a), which were computed over 2.5-D fields, indeed apply to 3-D flows. Furthermore, the $\chi$ values in figure 1(b) appear to approach a constant, suggesting that the $\chi \leqslant 1$ criterion holds for all $ \textit{Re}$ . If so, the optimal bound for 3-D flows at large $ \textit{Re}$ is the asymptote of the bounds in figure 1(a).

Figure 1. (a) Optimal lower bounds on mean dissipation in Waleffe flow, plotted as upper bounds () on the friction coefficient $\varepsilon$ defined for this model by (4.4), along with the optimal lower bounds () on $\varepsilon$ that take the laminar value $1/\textit{Re}$ . (b) Confirmation of the $\chi \leqslant 1$ criterion, which implies that the bounds apply to all 3-D flows despite being computed over 2.5-D velocity fields.

4.2. Plane Couette flow

Plane Couette flow has no body forcing and is driven by relative motion of the boundaries. We let the dimensionless boundary conditions be $\boldsymbol{u}=\pm (1/2)\hat {\boldsymbol{x}}$ at $z=\pm (1/2)$ , so the laminar profile is $U(z)=z$ . In contrast to Waleffe flow, the laminar state of Couette flow minimises the mean dissipation rather than maximising it. Indeed, the lower bound

(4.5) \begin{align} \big \langle {U}^{\prime2}\big \rangle \leqslant \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle } \end{align}

follows from (2.13), whose middle right-hand term is zero in Couette flow since it is equal to $-2\langle U^{\prime\prime}v_1\rangle$ and $U^{\prime\prime}=0$ here. In the context of the background method, this is the lower bound found with $a=0$ and $\zeta =0$ . The bound is sharp because it is saturated by the laminar state. Note also that the kinetic energy of $\boldsymbol{u}$ evolves as

(4.6) \begin{align} \frac {\mathrm{d}}{{\mathrm{d}}t}\frac {\textit{Re}}{2}\big \langle |\boldsymbol{u}|^2\big \rangle = -\big \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\big \rangle + \frac {1}{2}\frac {1}{\varGamma _x\varGamma _y}\int \left ( \partial _zu_1\big |_{z=\frac 12}+\partial _zu_1\big |_{z=-\frac 12} \right ) \,{\mathrm{d}}x\,{\mathrm{d}}y. \end{align}

The left-hand derivative vanishes in the infinite-time average, so mean dissipation is balanced by mean work by the boundaries on the flow,

(4.7) \begin{align} \overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle } = \frac {1}{\varGamma _x\varGamma _y}\overline {\int \frac 12 \left ( \partial _zu_1\big |_{z=\frac 12}+\partial _zu_1\big |_{z=-\frac 12} \right ) \,{\mathrm{d}}x\,{\mathrm{d}}y}. \end{align}

For Couette flow, we follow Plasting & Kerswell (Reference Plasting and Kerswell2003) and others in defining a friction coefficient as

(4.8) \begin{align} \varepsilon = \frac {1}{\textit{Re}}\frac {\overline {\left \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\right \rangle }\,}{\big \langle {U}^{\prime2}\big \rangle } = \frac {1}{\textit{Re}}\frac {\dfrac {1}{\varGamma _x\varGamma _y}\overline {\displaystyle \int \left (\partial _zu_1\big |_{z=\frac 12}+\partial _zu_1\big |_{z=-\frac 12} \right ) \,{\mathrm{d}}x\,{\mathrm{d}}y}\,}{U^{\prime}\big (\frac 12\big )+U^{\prime}\big (-\frac 12\big )}, \end{align}

which is a ratio unaffected by non-dimensionalisation. Note that the dissipation ratio defining $\varepsilon$ for Couette flow is inverse to the definition (4.4) for Waleffe flow, where the laminar value is in the numerator. The laminar lower bound (4.5) on mean dissipation in Couette flow gives the lower bound $({1}/{\textit{Re}})\leqslant \varepsilon$ , which is the same for Waleffe flow, and upper bounds on mean dissipation that we report later give upper bounds on $\varepsilon$ .

In our bounding computations, we fix streamwise and spanwise periods of $\varGamma _x=\varGamma _y=2\pi$ . As in Waleffe flow, this approximates the large-domain limit of bounds well. In this domain, the energy method guarantees global stability of the laminar flow when $ \textit{Re}\leqslant \textit{Re}_E\approx 82.74$ , above which our bounds depart from the laminar dissipation value. The energy stability threshold among all spanwise periods is $ \textit{Re}_E\approx 82.66$ , which occurs for $\varGamma _y\approx 2.016$ or a multiple thereof (Joseph Reference Joseph1976), so that is the $ \textit{Re}$ above which the bounds of Plasting & Kerswell (Reference Plasting and Kerswell2003) depart from the laminar value.

Figure 2(a) shows upper bounds on the friction coefficient $\varepsilon$ in Couette flow computed over 2.5-D velocity fields. The solid symbols are optimal bounds computed over 2.5-D fields in the same way as the bounds reported previously for Waleffe flow. These bounds do not reach large enough $ \textit{Re}$ to give a precise asymptote, but they are consistent with the asymptote of $\varepsilon \lesssim 0.008553$ estimated by Plasting & Kerswell (Reference Plasting and Kerswell2003) based on their computations for $ \textit{Re}$ up to $7\times 10^4$ . Because the implementation of Plasting & Kerswell was based on Euler–Lagrange equations, rather than SDPs, they were able to reach larger $ \textit{Re}$ and to enforce the spectral constraint for all spanwise wavenumbers. Nonetheless, our computations suffice to investigate the coincidence of bounds computed over 2.5-D and 3-D fields.

Since the bounds represented by solid symbols in figure 2(a) were computed over 2.5-D fields, it remains to confirm that they apply to 3-D flows. First, we try to show this using the procedure of § 3.3, for which the computed bounds constitute step (i). The last step of the procedure gives the $\chi$ values shown in figure 2(b). The $\chi \leqslant 1$ condition is satisfied only when $ \textit{Re}\lesssim 254$ , so only at these small $ \textit{Re}$ does Busse’s criterion guarantee that the solid symbols in figure 2(a) are bounds for 3-D Couette flow. At larger $ \textit{Re}$ values, one option is to carry out 2.5-D bounding computations with additional constraints that enforce Busse’s criterion, as proposed in § 3.4. We have implemented these computations as described in Appendix B.3. The resulting bounds, which are guaranteed to apply to 3-D flows, appear as hollow symbols in figure 2(a). The asymptote of these bounds is roughly $\varepsilon \lesssim 0.009$ , as estimated by fitting the $c_i$ parameters in $\varepsilon \approx c_0+c_1\textit{Re}^{-c_2}$ to the hollow symbols. This asymptotic bound is worse than the value of 0.008553 from Plasting & Kerswell (Reference Plasting and Kerswell2003), but it is better than the previous best value of $0.0109$ from Nicodemus et al. (Reference Nicodemus, Grossmann and Holthaus1998a ).

Figure 2. (a) Optimal upper bounds on mean dissipation in Couette flow, computed over 2.5-D velocity fields with no additional constraints () and with constraints enforcing Busse’s criterion ( $\circ$ ). These are plotted as upper bounds on the friction coefficient $\varepsilon$ defined for this model by (4.8), along with the optimal lower bounds () on $\varepsilon$ that take the laminar value $1/\textit{Re}$ . The large- $ \textit{Re}$ asymptotes are approximately 0.0086 and 0.0097, respectively. (b) Values of $\chi$ for the bounds computed without enforcing Busse’s $\chi \leqslant 1$ criterion, which violate the criterion above $ \textit{Re}\approx 254$ .

We have confirmed that all solid symbols in figure 2(a) are indeed bounds for 3-D flows when $ \textit{Re}\gtrsim 254$ , despite the $\chi \leqslant 1$ criterion failing, by directly checking the spectral constraint for 3-D fields. One formulation of the spectral constraint requires the eigenproblem (2.33) to have only non-negative eigenvalues for all admissible wavevectors $\boldsymbol{k}=(j,k)$ . Since the spectrum of eigenvalues is real and bounded below, we need only find the minimum eigenvalue $\lambda _{\textit{min}}$ at each wavevector. With the present periods of $\varGamma _x=\varGamma _y=2\pi$ , the admissible $(j,k)$ include all pairs of non-negative integers except $(0,0)$ . The $\lambda _{\textit{min}}\geqslant 0$ constraints for $j=0$ are already enforced by the 2.5-D bounding computations, but the constraints for $j\geqslant 1$ remain to be checked a posteriori. At each $ \textit{Re}$ , with $a$ and $\zeta$ obtained from the 2.5-D bounding computation, we computed $\lambda _{\textit{min}}$ for various $(j,k)$ using the software Dedalus 3.0.2 (Burns et al. Reference Burns, Vasil, Oishi, Lecoanet and Brown2020). These computations implemented an eigenproblem that is equivalent to (2.33), but written using the wall-normal velocity $w_3$ and wall-normal vorticity $\eta =\partial _xw_1-\partial _yw_1$ , whose Fourier transforms $\hat w_3(z)$ and $\hat \eta (z)$ must satisfy

(4.9) \begin{align} \begin{split} 2\lambda \hat \eta +2\left (a-1\right )\left (\frac {\mathrm{d}^2}{\mathrm{d}z^2}-|\boldsymbol{k}|^2\right )\hat \eta +i\textit{Re}\left (aU+\zeta \right )^{\prime }k \hat {w}_3 &= 0, \\[4pt] 2\lambda \left (\frac {\mathrm{d}^2}{\mathrm{d}z^2}-|\boldsymbol{k}|^2\right ) \hat {w}_3 +2\left (a-1\right )\left (\frac {\mathrm{d}^2}{\mathrm{d}z^2}-|\boldsymbol{k}|^2\right )^2 \hat {w}_3 \hspace {40pt} \\[-2pt] +\,i\textit{Re}\Bigl [\bigl (2j\frac {\mathrm{d}}{\mathrm{d}z}\hat {w}_3+k\eta \bigr )\left (aU+\zeta \right )^{\prime } +j\hat {w}_3\left (aU+\zeta \right )^{\prime \prime }\Bigr ] &= 0. \end{split} \end{align}

The no-slip boundary conditions require that $\hat w_3=\hat w_3^{\prime}=\hat \eta =0$ at $z=\pm 1/2$ . For each $ \textit{Re}$ , the $\zeta (z)$ and $a$ in (4.9) come from the bounding computation over 2.5-D fields using QUINOPT. A Legendre basis was used to discretise $z$ in Dedalus to match the basis in which QUINOPT returns $\zeta (z)$ . At each $ \textit{Re}$ where we computed bounds, we then solved (4.9) to find $\lambda _{\textit{min}}(j,k)$ at all admissible pairs of $j\in [0,3]$ and $k\in [0,20]$ . Figure 3(a) shows the resulting $\lambda _{\textit{min}}$ values in the $ \textit{Re}=1000$ case. Here, $\lambda _{\textit{min}}=0$ when $(j,k)=(0,4)$ , $(0,5)$ , $(0,15)$ or $(0,16)$ , meaning these are the wavevectors at which the spectral constraint is active, which is consistent with figure 3 of Plasting & Kerswell (Reference Plasting and Kerswell2003). All other $\lambda _{\textit{min}}(j,k)$ are strictly positive. Our computations at other $ \textit{Re}$ give similar $\lambda _{\textit{min}}$ trends, including the fact that $\lambda _{\textit{min}}$ increases monotonically with $j$ at every $k$ .

Figure 3. (a) Minimum eigenvalues $\lambda _{\textit{min}}(j,k)$ of the spectral constraint eigenproblem (2.33) in the $ \textit{Re}=1000$ case, computed by solving (4.9). (b) Minimum of $\lambda _{\textit{min}}$ over streamwise wavenumbers $k$ at various $ \textit{Re}$ for spanwise wavenumbers $j=1,2,3$ .

For evidence that optimal bounds over 2.5-D and 3-D fields continue to coincide as $ \textit{Re}\to \infty$ , we aim to extrapolate the finding that $\lambda _{\textit{min}}(j,k)\gt 0$ when $j\geqslant 1$ . To do this, we minimise $\lambda _{\textit{min}}(j,k)$ over $k$ and examine the dependence on $ \textit{Re}$ . Figure 3(b) shows the results for $j=1,2,3$ . Each curve asymptotes to a strictly positive value as $ \textit{Re}$ is raised and the asymptotic value increases quickly with $j$ . If this trend continues, it justifies the claim of Plasting & Kerswell (Reference Plasting and Kerswell2003) that their 2.5-D bounds give the optimal bounds for 3-D Couette flow.

In the case of Couette flow, we can compare the computational cost of checking the 3-D spectral constraint with the cost of checking the $\chi \leqslant 1$ criterion. Both are less expensive than the first step of computing optimal bounds over 2.5-D fields. Checking the 3-D spectral constraint requires computing $\lambda _{\textit{min}}$ for various $(j,k)$ pairs, whereas checking the $\chi \leqslant 1$ criterion requires computing $R_o$ and $R_\varphi$ . With $ \textit{Re}=1000$ , for instance, the time to compute $\lambda _{\textit{min}}$ for a single $(j,k)$ pair using Dedalus was roughly half of the time needed to compute $R_o$ and $R_\varphi$ using QUINOPT, so checking the spectral constraint for any reasonable number of $(j,k)$ pairs would be significantly more expensive than checking our $\chi \leqslant 1$ criterion.