2.1. Basic equations, statistically steady channel flow

Poiseuille channel flow refers to flow between stationary infinite parallel plates driven by a constant mean pressure gradient $A$ . We define our coordinates so that the mean pressure decreases in the $x$ -direction and the boundary plates are perpendicular to the $z$ axis at $z=\pm z_{0}$ . We use the spatial average within each constant- $z$ plane (denoted by an overline) to perform a Reynolds decomposition. In the following, the term ‘mean’ will refer to this spatial average. For the fluid velocity field $\boldsymbol{u}$ we have $\boldsymbol{u}=\boldsymbol{U}+\boldsymbol{v}$ where $\boldsymbol{U}=\overline{\boldsymbol{u}}$ is the mean flow and $\boldsymbol{v}$ is the fluctuating flow. By definition, $A\boldsymbol{x}=-\boldsymbol{{\rm\nabla}}\overline{p}$ where $p$ is the pressure field and $\boldsymbol{x}$ is the $x$ -direction basis vector.

The NSE is given by

where ${\it\rho}$ is the mass density and ${\it\mu}$ is the dynamic viscosity. We only consider incompressible flows, in which case conservation of mass implies

The boundary conditions are that $\boldsymbol{u}$ vanishes on the boundary plates ( $\boldsymbol{u}=0$ at $z=\pm z_{0}$ ) and that $\boldsymbol{u}$ and $p$ are bounded as $x$ and $y$ tend to $\pm \infty$ . The Reynolds number is defined by

where angle brackets denote the volume average, $U=|\boldsymbol{U}|$ and ${\it\nu}={\it\mu}/{\it\rho}$ .

Energy balance of the fluid is given by $\langle \boldsymbol{u}\boldsymbol{\cdot }(2.1)\rangle$ , which yields

where

is the total kinetic energy of the fluid (we use the notation $u^{2}=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{u}$ for the inner product),

is the work done by the pressure field on the fluid and

is the total viscous dissipation of kinetic energy. The total dissipation can be decomposed into the mean flow dissipation

and the fluctuating flow dissipation

We now consider statistically steady flow conditions. A flow is called statistically steady when its mean quantities are constant in time. For statistically steady channel flow we assume as usual that only the axial component of $\boldsymbol{U}$ is non-vanishing, i.e.  $\boldsymbol{U}=U\boldsymbol{x}$ and that $U=U(z)$ is symmetric about $z=0$ . Thus from (2.4),

Reynolds-averaging equation (2.1) and integrating with respect to $z$ , we obtain the statistically steady channel flow mean stress equation

where $\boldsymbol{v}=(u,v,w)$ and $U^{\prime }=\text{d}U/\text{d}z$ . Evaluating (2.11) at the boundaries $z=\pm z_{0}$ yields

where $U_{b}^{\prime }=U^{\prime }(-z_{0})=-U^{\prime }(+z_{0})$ is the boundary gradient and $U_{{\it\tau}}$ is the ‘friction velocity’ which defines the friction Reynolds number $R_{{\it\tau}}=z_{0}U_{{\it\tau}}/{\it\nu}$ . Taken together, (2.8)–(2.10) and (2.12) imply that $D$ , $D_{m}$ and $D_{f}$ are all functions of $U(z)$ alone, and do not depend explicitly on $\boldsymbol{v}$ .

2.2. Malkus’s constraints

We now discuss Malkus’s choice of admissible candidate flows, which defines the domain over which the flow quantities $Q=E,I,D_{f},D_{m},D$ are maximised.

Since the objective of Malkus’s variational procedure is to model measured statistically steady flows, all candidate flows are assumed to have mean velocity profiles of the form $\boldsymbol{U}=U(z)\boldsymbol{x}$ and steady state expressions are used for the flow quantities $Q$ . Furthermore, different candidate flows are distinguished by their mean profiles $U(z)$ only. Four constraints are then imposed on $U(z)$ :

1. (i) Concavity of $U$ : $U(z)$ can have no inflection points, so that $U^{\prime \prime }<0$ (Malkus Reference Malkus1956). This constraint is implemented with the Fejér series representation

   (2.13) $$\begin{eqnarray}U^{\prime \prime }=-F^{\ast }F,\quad F=\mathop{\sum }_{k=0}^{\infty }F_{k}\text{e}^{\text{i}{\rm\pi}k(z_{0}+z)/z_{0}},\end{eqnarray}$$
   where $F^{\ast }$ is the complex conjugate of $F$ and the symmetry of $U(z)$ about $z=0$ is imposed by requiring that the $F_{k}$ are real (Malkus & Smith Reference Malkus and Smith1989). The primary motivation for this constraint is Rayleigh’s inflection point theorem, which states that inviscid parallel flow is unstable to infinitesimal disturbances if its velocity profile has any inflection points (Drazin & Reid Reference Drazin and Reid1981). Although nothing analogous to Rayleigh’s theorem has been shown for viscous fluids, the concavity constraint is also supported by the concavity of measured statistically steady profiles (Malkus & Smith Reference Malkus and Smith1989).

2. (ii) A smallest spatial scale in $U(z)$ : there is a largest $k_{0}$ , which truncates the series representation for $F$

   (2.14) $$\begin{eqnarray}F=\mathop{\sum }_{k=0}^{k_{0}}F_{k}\text{e}^{\text{i}{\rm\pi}k(z_{0}+z)/z_{0}}.\end{eqnarray}$$
   The determination of $k_{0}$ is based on the physical picture that the boundary layer experiences a repeating process of collapse followed by diffusive ‘healing’ (Malkus Reference Malkus2003). It is assumed that the length scale associated with this process is the marginally stable displacement thickness of the Blasius boundary layer. This can be used to derive a wavenumber associated with a given boundary shear or imposed pressure gradient
   (2.15) $$\begin{eqnarray}k_{0}={\it\gamma}R_{{\it\tau}}={\it\gamma}z_{0}\sqrt{\frac{U_{b}^{\prime }}{{\it\nu}}}={\it\gamma}z_{0}\sqrt{\frac{Az_{0}}{{\it\mu}{\it\nu}}},\end{eqnarray}$$
   where ${\it\gamma}$ was estimated as ${\it\gamma}\approx 0.2$ (Malkus Reference Malkus2003). Malkus & Smith (Reference Malkus and Smith1989) found that ${\it\gamma}\approx 0.18$ was needed to get realistic mean velocity profiles from maximising $E$ (specifically, figure 12 in Malkus & Smith Reference Malkus and Smith1989 shows ${\it\gamma}=0.174$ and ${\it\gamma}=0.183$ ). In the following we use the latter ${\it\gamma}$ values for ease of comparison with Malkus & Smith (Reference Malkus and Smith1989).

3. (iii) Mean stress balance at the boundaries $z=\pm z_{0}$ : by evaluating (2.11) and its derivative at the boundaries $z=\pm z_{0}$ , we obtain the following conditions that any candidate mean velocity profile $U(z)$ must satisfy at the boundaries:

   (2.16a,b ) $$\begin{eqnarray}U_{b}^{\prime }=A\frac{z_{0}}{{\it\mu}}\quad \text{and}\quad U_{b}^{\prime \prime }=-\frac{U_{b}^{\prime }}{z_{0}}.\end{eqnarray}$$
   Note that the boundary conditions on $U(z)$ itself do not enter the variational problem because all flow quantities $Q$ depend only on derivatives of $U$ due to constraint (iv). Although (2.11) (and incompressibility) also implies $U_{b}^{\prime \prime \prime }=0$ , this condition is redundant because it is identically satisfied by the series representation (2.14).

4. (iv) Fixed $R$ : all candidate flows have the same $R$ , which, from (2.3) implies that $\langle U\rangle$ is fixed.

Notice that only some consequences of statistical steadiness are imposed as a constraint on the admissible $U$ . For example, the mean stress equation (2.11) is only invoked at the boundaries in constraint (iii), and to obtain steady state expressions for the flow quantities $Q$ . It is therefore possible that some candidate profiles $U$ do not satisfy (2.11) in the channel for any physical $\boldsymbol{v}$ . Thus, while Malkus’s candidate profiles are consistent with statistical steadiness in some crucial respects, notably the statistically steady mean flow form $\boldsymbol{U}=U(z)\boldsymbol{x}$ , they need not satisfy (2.11), let alone correspond to statistically steady solutions of the NSE.

Malkus’s variational procedure may now be formally stated as follows. Since $D$ , $D_{m}$ and $D_{f}$ are functions of $U$ (§ 2.1), the flow quantities $Q=Q(U)=E,I,D_{f},D_{m},D$ are also functions of $U$ . Malkus’s variational procedure is to maximise $Q(U)$ with respect to mean velocity profiles $U(z)$ satisfying constraints (i)–(iv).

2.3. Maximum $E,I$ and $D_{f}$

Here we summarise how Malkus & Smith (Reference Malkus and Smith1989) calculated mean channel flow velocity profiles by maximising $E$ , $I=D_{f}/D_{m}$ and $D_{f}$ subject to constraints (i)–(iv), and provide some additional clarifications of the procedure used.

It is convenient to implement the maximisation of the flow quantities $Q=E,I,D_{f}$ in two steps. The first step is to fix $k_{0}$ and use constraint (i) and the truncated expression for $F$ in (2.14) to express $Q=Q(U)=Q(F_{0},\dots ,F_{k_{0}})$ as a function of the series coefficients $F_{k}~(k=0,\dots ,k_{0})$ , and maximise $Q$ with respect to the $F_{k}$ , which is a $(k_{0}+1)$ -dimensional optimisation problem. Let $Q^{max}(k_{0})$ denote the maximum $Q$ obtained in this way at each $k_{0}$ . The second step is to find the $k_{0}$ that maximises $Q^{max}(k_{0})$ .

As a result of constraints (ii) and (iv), the first step of this procedure (maximisation of $E,I$ and $D_{f}$ with $k_{0}$ fixed) is equivalent to minimising $D_{m}$ with $k_{0}$ fixed. To see this, note first that $I=D_{f}/D_{m}$ can be written in the form

and that $E=DI$ and $D_{f}=D-D_{m}$ . Then, (2.12) and (2.15) together with constraint (iv) (fixed $\langle U\rangle$ ) imply that $D$ is fixed when $k_{0}$ is fixed. Thus, when $k_{0}$ is fixed $E$ , $I$ and $D_{f}$ are all decreasing functions of $D_{m}$ , so that maximising any of the former implies minimising $D_{m}$ . The differences between maximising $E$ , $I$ and $D_{f}$ emerge in the second step (maximisation with respect to $k_{0}$ ), since (as we will see) $E^{max}(k_{0})$ , $I^{max}(k_{0})$ and $D_{f}^{max}(k_{0})$ are maximal at different values of $k_{0}$ .

Minimisation of $D_{m}$ for fixed $k_{0}$ subject to constraints (i)–(iv) is implemented as follows. For a given fixed $k_{0}$ , $D_{m}={\it\mu}\langle U^{\prime 2}\rangle$ is expressed as a function of the series coefficients $F_{k}~(k=1,\dots ,k_{0})$ using (2.13) and (2.14). We then solve the Euler–Lagrange equation for the stationary points in $D_{m}$ with respect to variations in the $F_{k}$ , subject to (2.15), (2.16) and fixed $\langle U\rangle$ . For fixed $k_{0}$ , (2.15) and (2.16) imply the boundary conditions

The Euler–Lagrange equation for the stationary points in $D_{m}$ is then given by

where the variation ${\it\delta}$ is taken with respect to the $F_{k}$ , ${\it\lambda}_{1}$ and ${\it\lambda}_{2}$ are the Lagrange multipliers corresponding to the boundary conditions (2.18) and ${\it\lambda}_{3}$ is the Lagrange multiplier corresponding to fixed $\langle U\rangle$ . Note that only constraints (ii) and (iv) are explicitly expressed as constraints with corresponding Lagrange multipliers in the Euler–Lagrange equation (2.19). Constraints (i) and (ii) are implicit in the expression for (2.19) in terms of the $F_{k}$ and in (2.18). In total, (2.18) and (2.19) and constraint (iv) form a system of $k_{0}+4$ coupled third-order polynomial equations for the $k_{0}+4$ unknowns $F_{0},\dots ,F_{k_{0}}$ , ${\it\lambda}_{1}$ , ${\it\lambda}_{2}$ and ${\it\lambda}_{3}$ (see appendix A for details). This system of equations is relatively easy to solve on an ordinary desktop computer with Newton’s method even when $k_{0}$ is on the order of $10^{2}$ . In general there are multiple solutions (Malkus & Smith Reference Malkus and Smith1989). The solution with the smallest $D_{m}$ is selected (Malkus & Smith Reference Malkus and Smith1989).

Figure 1 shows the curves $E^{max}(k_{0})$ , $I^{max}(k_{0})$ and $D_{f}^{max}(k_{0})$ for $R=10^{4}$ , indicating the different optimal values of $k_{0}$ . Figure 2 shows that the corresponding optimal mean velocity profiles have significantly different values of $R_{{\it\tau}}$ , and have quite different shapes near the centre of the channel ( $\log ((z_{0}+z)/z_{0})=0$ ).

Figure 1.

$E^{max}(k_{0})$ (——), $I^{max}(k_{0})$ (- - - -) and $D_{f}^{max}(k_{0})$ ( $\cdots \cdots$ ) versus $k_{0}$ for $R=10^{4}$ , ${\it\gamma}=0.174$ . Each curve is normalised to its maximum value.

Figure 2.

Maximum $I$ (- - - -), $E$ (——) and $D_{f}$ ( $\cdots \cdots$ ) mean velocity profiles $U(z)$ for $R=10^{4}$ and ${\it\gamma}=0.174$ , normalised to the peak of the maximum- $E$ profile. The corresponding values of $R_{{\it\tau}}$ are 345 ( $k_{0}=60$ ), 523 ( $k_{0}=91$ ) and 615 ( $k_{0}=107$ ) respectively.

Figure 3.

Maximum- $E$ (——) mean velocity profiles $U(z)$ compared with DNS data ( $\circ$   $\circ$   $\circ$ ) for $R=10^{4}$ (Lee & Moser Reference Lee and Moser2014). Maximum- $E$ profiles shown for ${\it\gamma}\approx 0.174$ (predicted $R_{{\it\tau}}\approx 523$ ) and ${\it\gamma}\approx 0.183$ (predicted $R_{{\it\tau}}\approx 542$ ); for the DNS data, $R_{{\it\tau}}\approx 543$ . The vertical axis is in friction velocity $U_{{\it\tau}}$ units and $z^{+}=R_{{\it\tau}}(z+z_{0})/z_{0}$ .

Figure 4.

Maximum- $E$ (——) mean velocity profiles $U(z)$ compared with DNS data ( $\circ$   $\circ$   $\circ$ ) for $R=2\times 10^{4}$ (Lee & Moser Reference Lee and Moser2014). Maximum- $E$ profiles shown for ${\it\gamma}\approx 0.174$ (predicted $R_{{\it\tau}}\approx 972$ ) and ${\it\gamma}\approx 0.183$ (predicted $R_{{\it\tau}}\approx 1013$ ); for the DNS data, $R_{{\it\tau}}\approx 1001$ . Same axis units as figure 3.

2.4. Comparison of maximum- $E$ profiles with DNS data

In this section we compare the maximum- $E$ mean velocity profiles obtained in the previous section with recent DNS data (Lee & Moser Reference Lee and Moser2014).

Malkus & Smith (Reference Malkus and Smith1989) were the first to show that maximum- $E$ profiles are qualitatively realistic: they exhibit a viscous sublayer near the boundary, a velocity defect law near the channel centre, and a logarithmic overlap region between the boundary and the centre (Malkus & Smith Reference Malkus and Smith1989, figure 9). Malkus & Smith (Reference Malkus and Smith1989) also showed that maximum- $E$ profiles agree quantitatively with channel flow measurements for $R\approx 2.5\times 10^{4}$ (Malkus & Smith Reference Malkus and Smith1989, figure 12). However, due to computational limitations the latter maximum- $E$ profiles (at $R\approx 2.5\times 10^{4}$ ) were obtained using a parametric approximation for the $F_{k}$ rather than solving the full variational problem (2.19). Furthermore, this was their only quantitative comparison (one set of measurements at one Reynolds number). Our comparison does not use the approximation scheme of Malkus & Smith (Reference Malkus and Smith1989), and is the first comparison of maximum- $E$ profiles with DNS data that we are aware of.

Figures 3 and 4 show maximum- $E$ profiles and corresponding DNS data for $R=10^{4}$ and $R=2\times 10^{4}$ respectively. Both figures show excellent agreement between maximum- $E$ and DNS profiles over most of the channel width, although the maximum- $E$ profiles exhibit a somewhat sharper transition from the viscous sublayer to the logarithmic sublayer ( $\log (z^{+})\approx 1$ ) than the DNS data. These results agree with figure 12 in Malkus & Smith (Reference Malkus and Smith1989), confirming that maximising $E$ subject to constraints (i)–(iv) yields realistic mean velocity profiles up to $R=2.5\times 10^{4}$ .

Although it would be interesting to check whether the above agreement extends to higher $R$ , particularly given that Lee & Moser (Reference Lee and Moser2014) calculated mean velocity profiles for $R=1.25\times 10^{5}~(R_{{\it\tau}}\approx 5200)$ , computing maximum- $E$ profiles for the corresponding value of $k_{0}={\it\gamma}R_{{\it\tau}}\approx 10^{3}$ is computationally very challenging without using the parametric approximation for the $F_{k}$ proposed in Malkus & Smith (Reference Malkus and Smith1989). Since this approximation appears to be essentially empirical (no theoretical basis was provided), we do not feel justified in using it. Nevertheless, it seems likely that maximum- $E$ profiles will be at least broadly consistent with measurements or simulations at higher values of $R$ because the qualitative features of maximum- $R$ profiles seem to be very robust to different choices of $R$ .

2.5. Maximum $D_{m}$ and $D$ and the smallest spatial scale

Having reviewed the maximisation of $E$ , $I$ and $D_{f}$ by Malkus & Smith (Reference Malkus and Smith1989) and verified the agreement between maximum- $E$ mean velocity profiles and DNS data, we now consider the maximisation of $D_{m}$ and $D$ . We also clarify the manner in which a smallest spatial scale emerges from constraint (ii), which is not obvious from (2.15).

Maximisation of $D_{m}$ subject to constraints (i)–(iv) has the following intuitive interpretation. Constraint (i) states that $U$ is concave. It follows that $U^{\prime }$ is larger near the boundary than away from it, and so $D_{m}\propto \int U^{\prime 2}\text{d}z$ is dominated by a boundary region contribution. Thus maximising $D_{m}$ amounts to maximising $U^{\prime }$ in the boundary region. On the other hand, constraint (iv) fixes the area under $U(z)$ . Increasing the boundary shear will therefore tend to flatten $U$ within the channel. As shown in appendix B and discussed further below, constraint (ii) imposes an upper bound on the boundary gradient $U_{b}^{\prime }\leqslant U^{\prime \ast }$ , where $z_{0}U^{\prime \ast }$ is large relative to the average velocity $\langle U\rangle$ . Consequently, maximum $D_{m}$ velocity profiles will tend to be ‘flat-topped’, that is, they will have a large boundary gradient $U^{\prime \ast }$ over a small boundary layer of width ${\it\delta}\approx \langle U\rangle /U^{\prime \ast }$ (i.e.  ${\it\delta}/z_{0}\ll 1$ ), whereas away from the boundaries they will be approximately constant $U(-z_{0}+{\it\delta}<z<z_{0}-{\it\delta})\approx \langle U\rangle$ . Indeed, it can be shown analytically that, when subject to constraints (i), (iv) and the upper bound condition $U_{b}^{\prime }\leqslant U^{\prime \ast }$ , maximising $D_{m}$ is equivalent to maximising $U_{b}^{\prime }$ (i.e. $U_{b}^{\prime }\rightarrow U^{\prime \ast }$ ) and minimising the shear away from the boundary leading to flat-topped profiles (appendix C).

Since maximising $D_{m}$ is equivalent to maximising $U_{b}^{\prime }$ , it is also equivalent to maximising the total dissipation $D\propto U_{b}^{\prime }R$ (since constraint (iv) fixes $R$ ). In turn, maximising $D$ for fixed $R$ is equivalent to minimising $R$ for fixed $U_{b}^{\prime }$ (both subject to constraints (i)–(iii)) because the minimum in $R$ at each $U_{b}^{\prime }$ is monotonically increasing with $U_{b}^{\prime }$ (Malkus Reference Malkus1956). Again it can be seen that this will result in flat-topped profiles: when subject to concavity, fixed boundary shear and a minimum boundary layer length scale, minimising $R$ (which is proportional to the area under $U(z)$ ) implies making $U(z)$ as flat as possible away from the immediate vicinity of the boundary. Malkus & Smith (Reference Malkus and Smith1989) solved the latter problem using an Euler–Lagrange equation for minimum $\langle U\rangle$ given $k_{0}$ , analogous to (2.19).

Given that maximum- $D$ and maximum- $D_{m}$ profiles are equivalent, we could simply use the aforementioned method of Malkus & Smith (Reference Malkus and Smith1989) to obtain maximum- $D$ profiles in order to compare maximum- $D_{m}$ and maximum- $E$ , - $I$ and - $D_{f}$ profiles. However, in the interests of clarifying the connection between maximising $D$ and $D_{m}$ on the one hand, and $E$ , $I$ , and $D_{f}$ on the other, we now show that the minimum- $D_{m}$ (for fixed $k_{0}$ ) solutions of (2.19) can be used to maximise $D_{m}$ rather than working with two different Euler–Lagrange equations.

In § 2.3 we noted that maximising $E$ , $I$ and $D_{f}$ can be implemented in a two step procedure where the first step involves minimising $D_{m}$ with respect to the $F_{k}$ with $k_{0}$ fixed. In contrast, the analogous first step for $D_{m}$ is to maximise $D_{m}$ with $k_{0}$ fixed. It would therefore seem that the two problems are unrelated; indeed, in general the maximum in $D_{m}$ for fixed $k_{0}$ is not even a stationary point (as can be seen from the analysis in appendix B) and so the solutions of (2.19) will not include the $F_{k}$ that maximise $D_{m}$ . However, as noted above, maximising $D_{m}$ selects the maximum possible boundary gradient $U^{\prime \ast }$ permitted by constraint (ii). The value of $k_{0}$ corresponding to this upper bound $U^{\prime \ast }$ is given from (2.15) by

This upper bound occurs because the series representation (2.14) cannot accurately represent the high-shear boundary region if $U_{b}^{\prime }>U^{\prime \ast }$ because of the length-scale restrictions implied by (2.15) (see appendix B for details). Thus the set of candidate flows vanishes for $k_{0}>k_{0}^{\ast }$ , i.e. it is not possible to simultaneously satisfy the boundary conditions (2.18) and constraints (i)–(iv) if $k_{0}>k_{0}^{\ast }$ . Consequently for $k_{0}$ near $k_{0}^{\ast }$ , the set of candidate flows becomes so small that maximising and minimising $D_{m}$ are approximately equivalent. We may therefore maximise $D_{m}$ by first minimising $D_{m}$ at each $k_{0}$ using (2.19), and then increasing $k_{0}$ until it is no longer possible to find solutions. The maximum- $D_{m}$ profile is then given by the minimum- $D_{m}$ profile occurring at the greatest possible $k_{0}$ . When applying this method, we ensured that a failure to find solutions was not due to poor initialisation of Newton’s method by using the minimum- $D_{m}$ solution for $k_{0}-1$ to generate an initial condition for the minimisation of $D_{m}$ at $k_{0}$ .

Figure 5 shows a maximum- $D_{m}$ profile obtained in this way, along with the corresponding maximum- $D$ profile obtained by minimising $R$ at different values of $U_{b}^{\prime }$ . These profiles are almost identical, confirming the equivalence of maximising $D_{m}$ and maximising $D$ (the small discrepancies between them arise due to the discreteness of $k_{0}$ which precludes exact agreement of their Reynolds numbers, $R=10^{4}$ and $R\approx 9985$ respectively). They are also indeed flat-topped (note that the logarithmic horizontal axis dramatically enlarges the $z=-z_{0}$ boundary region). Shown for comparison is the maximum- $E$ profile from figure 2, which we already saw exhibits a velocity defect law, in contrast to the flat-topped profile.

Figure 5.

Maximum- $E$ (——) and - $D_{m}$ (– ⋅ – ⋅ –) mean velocity profiles for $R=10^{4}$ , as well as the maximum- $D$ profile for $R\approx 9985$ ( $\cdots \cdots$ ). For the maximum- $D$ and - $D_{m}$ profiles, $k_{0}=123$ and $R_{{\it\tau}}\approx 707$ . All profiles normalised to the peak of the maximum- $E$ profile and ${\it\gamma}=0.174$ .

We conclude that maximising $D_{m}$ subject to constraints (i)–(iv) yields mean flow profiles inconsistent with maximising $E$ and empirical/numerical data, despite claims to the contrary (Malkus Reference Malkus2003). Maximising $D_{m}$ is equivalent to maximising $D$ , which amounts to selecting the profile with the greatest boundary shear consistent with (2.15) in combination with the constraints of concavity and fixed $R$ .

3.1. Malkus’s argument for maximum $D_{m}$

Malkus’s argument for maximising $D_{m}$ (Malkus Reference Malkus2003) is based on the following nonlinear instability criterion (Malkus Reference Malkus1996, Reference Malkus2003). Consider some statistically steady solution to the NSE $\boldsymbol{u}=\boldsymbol{U}+\boldsymbol{v}$ . Statistical steadiness implies that the volume-averaged mean and fluctuating flow kinetic energies (KE) ${\it\rho}\langle U^{2}\rangle /2$ and ${\it\rho}\langle v^{2}\rangle /2$ are constant in time. Now suppose that $\boldsymbol{u}$ is subjected to a finite amplitude perturbation ${\rm\Delta}\boldsymbol{v}$ where $\overline{{\rm\Delta}\boldsymbol{v}}=0$ , resulting in a new flow $\boldsymbol{u}+{\rm\Delta}\boldsymbol{v}$ . Then, according to Malkus’s instability criterion, $\boldsymbol{u}$ is regarded as unstable with respect to ${\rm\Delta}\boldsymbol{v}$ when the volume-averaged fluctuation KE of the perturbed flow grows with time at the initial instant, i.e. $\partial \langle (v+{\rm\Delta}v)^{2}\rangle /\partial t>0$ .

Malkus then examined conditions for the instability of a statistically steady candidate flow $\boldsymbol{u}_{0}=\boldsymbol{U}_{0}+\boldsymbol{v}_{0}$ with respect to a restricted class of perturbations of the form ${\rm\Delta}\boldsymbol{v}=c\boldsymbol{v}_{1}$ where $c$ is a constant and $\boldsymbol{u}_{1}=\boldsymbol{U}_{1}+\boldsymbol{v}_{1}$ is another statistically steady candidate flow. Candidate flows are assumed to satisfy the mean stress equation (2.11) and constraint (iv) (i.e. $\langle U_{0}\rangle =\langle U_{1}\rangle$ ). He showed that if $D_{m}(\boldsymbol{U}_{1})>D_{m}(\boldsymbol{U}_{0})$ (where $D_{m}(\boldsymbol{U})$ denotes the viscous dissipation of the mean flow $\boldsymbol{U}$ ) then $\boldsymbol{u}_{0}$ is unstable (in the above sense) with respect to this class of perturbations. In other words, the existence of a candidate mean flow $\boldsymbol{U}_{1}$ with $D_{m}(\boldsymbol{U}_{1})>D_{m}(\boldsymbol{U}_{0})$ is sufficient for the instability of $\boldsymbol{u}_{0}$ .

However, the restricted class of perturbations of the form $c\boldsymbol{v}_{1}$ may not be important stability determinants in actual flows since they may rarely occur spontaneously. To address this, Malkus (Reference Malkus1996) assumed that the restricted class of perturbations is representative of the perturbations that can occur in actual flows. Then, instability with respect to the restricted class also indicates instability with respect to physically relevant disturbances. It follows that candidate mean flows with the greatest $D_{m}$ will be the only stable ones, and thus the only ones observed in nature.

3.2. The concept of statistical stability and the limitations of Malkus’s argument

Before we discuss the limitations of Malkus’s argument it is useful to refine what we mean by statistical stability. Imagine an experiment in which only two statistically steady flows are possible: the first is unstable with respect to a large class of perturbations, and the second is unstable with respect to a much smaller class of perturbations. Then, even though both flows are unstable in the strict dynamical sense (i.e. with respect to at least one perturbation), the reproducible time-averaged fluid state observed in the experiment will be close to the second flow state. This is because the fluid is continually experiencing new disturbances, and so will be quickly driven out of the first flow state, whereas the second flow state can persist for some time before one of its instabilities arises. In this sense, the second flow is statistically more stable than the first.

With this picture in mind, let us consider the two main assumptions of Malkus’s argument. These are his nonlinear stability criterion, and the heuristic assumption that stability with respect to a hypothetical restricted class of eddy perturbations represents general stability.

Malkus’s stability criterion is different from the conventional formulation of dynamical stability, which is defined in terms of asymptotic (time tending to infinity) convergence of disturbed and undisturbed flows (Joseph Reference Joseph1976). In general an initial growth in the amplitude of a disturbance does not imply that the disturbance will persist at some later time (Joseph Reference Joseph1976). However Malkus’s criterion is much better suited to the evaluation of statistical stability. It is appropriate that the initial effect of a disturbance should be the primary focus because new disturbances will occur long before the asymptotic behaviour of any particular disturbance becomes relevant. We therefore regard Malkus’s criterion as reasonable and adopt a modified version of it in § 3.3.

On the other hand, the assumption that the restricted class of perturbations of the form $c\boldsymbol{v}_{1}$ is representative of physically relevant perturbations is more controversial. In particular, it leads directly to the conclusion that maximum- $D_{m}$ flows are dynamically stable (i.e. not unstable with respect to any physically relevant perturbations). This is a strong claim which runs counter to the intuition that turbulent flow is inherently sensitive to initial conditions (in terms of the detailed eddy dynamics). Malkus himself regarded this conclusion as unlikely (Malkus Reference Malkus1996). However, as illustrated by the above example, a flow (or class of flows) need not be dynamically stable in order for it to be statistically stable. In § 3.3 we will consider perturbations more generally in terms of their properties under time-reversal.

3.3. Outline of a more general statistical stability argument

We now give the outline of a more general statistical stability argument. The primary goal is to provide a physically motivated statistical stability argument which, although not mathematically rigorous, nevertheless recognises the limitations of Malkus’s argument discussed in § 3.2 and explicitly spells out the assumptions that it uses. We hope that this will serve as a starting point for more rigorous treatments.

Following the definition of statistical stability discussed in § 3.2, to prove that a flow quantity $Q$ is maximised we must show that instabilities will be statistically more likely to increase $Q$ than to decrease $Q$ .

We suppose that the mean pressure gradient $A$ is held fixed in a Poiseuille channel flow experiment and consider the evolution of a general disturbance $\boldsymbol{u}\rightarrow \boldsymbol{u}+{\rm\Delta}\boldsymbol{u}$ where ${\rm\Delta}\boldsymbol{u}={\rm\Delta}\boldsymbol{U}+{\rm\Delta}\boldsymbol{v}$ and $\overline{{\rm\Delta}\boldsymbol{v}}=0$ . These disturbances represent deviations of actual experimental flows from the theoretical idealisation of an environmentally isolated flow with perfectly rigid, infinite boundaries. For example they could represent pressure fluctuations due to environmental noise, small imperfections of the plates, or small wave-packets generated at the entrance of the channel.

From (2.4) the difference in the work done on the disturbed and original flows is given by

and from (2.7) the corresponding difference in total dissipations is given by

where ${\it\alpha}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})=2{\it\mu}\langle \boldsymbol{{\rm\nabla}}\boldsymbol{u}:\boldsymbol{{\rm\nabla}}({\rm\Delta}\boldsymbol{u})\rangle$ . Combining these results, the initial growth rate of the kinetic energy difference between the disturbed and original flows is

We can exploit the symmetry properties of (3.3), specifically ${\it\alpha}(\boldsymbol{u},-{\rm\Delta}\boldsymbol{u})=-{\it\alpha}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})$ and $D(-{\rm\Delta}\boldsymbol{u})=D({\rm\Delta}\boldsymbol{u})$ , to show that the class of disturbances with ${\rm\Delta}\dot{K}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})<0$ is larger than the class with ${\rm\Delta}\dot{K}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})>0$ . It seems reasonable to expect that if ${\rm\Delta}\boldsymbol{u}$ is a possible disturbance then so is $-{\rm\Delta}\boldsymbol{u}$ ; simply reverse the direction of the small local forcing of the fluid that created ${\rm\Delta}\boldsymbol{u}$ . Now, if ${\rm\Delta}\boldsymbol{u}$ is some disturbance for which ${\rm\Delta}\dot{K}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})>0$ , then

Thus, for any disturbance that causes the disturbed KE to increase there corresponds one that causes it to decrease. In contrast, for ${\rm\Delta}\boldsymbol{u}$ with ${\rm\Delta}\dot{K}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})<0$ we can have ${\rm\Delta}\dot{K}(\boldsymbol{u},-{\rm\Delta}\boldsymbol{u})<0$ if $|{\rm\Delta}\dot{K}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})|<2D({\rm\Delta}\boldsymbol{u})$ . Denoting the proportion of all disturbances with ${\rm\Delta}\dot{K}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})<0$ by $p^{-}$ and with ${\rm\Delta}\dot{K}(\boldsymbol{u},{\rm\Delta}\boldsymbol{u})>0$ by $p^{+}$ , we therefore have

The initial growth or decay of ${\rm\Delta}K$ with time need not imply the initial growth or decay of the perturbation KE (i.e. in its amplitude), since ${\rm\Delta}\dot{K}$ is related to the perturbation KE growth rate by

In fact, if ${\rm\Delta}K<0$ , then a decrease in ${\rm\Delta}K~({\rm\Delta}\dot{K}<0)$ implies an increasing difference between the perturbed and unperturbed KE, possibly due to the initial growth of ${\rm\Delta}\boldsymbol{u}$ . On the other hand, if ${\rm\Delta}K>0$ then a decrease in ${\rm\Delta}K$ could signify the initial decay of ${\rm\Delta}\boldsymbol{u}$ .

We assume that disturbances with both ${\rm\Delta}K>0$ and ${\rm\Delta}\dot{K}<0$ or both ${\rm\Delta}K<0$ and ${\rm\Delta}\dot{K}>0$ (i.e. those that lead to an initial convergence of the perturbed and unperturbed KE) can be regarded as stable disturbances that return the flow to its undisturbed state. These are disregarded in our statistical stability analysis since only unstable disturbances are relevant for determining statistical stability (§ 3.2). Then there exist two classes of instabilities: disturbances with both ${\rm\Delta}K<0$ and ${\rm\Delta}\dot{K}<0$ , and disturbances with both ${\rm\Delta}K>0$ and ${\rm\Delta}\dot{K}>0$ . Denote the fraction of all possible disturbances which have ${\rm\Delta}K<0$ by $q^{-}$ and ${\rm\Delta}K>0$ by $q^{+}$ . We expect $q^{+}$ and $q^{-}$ to be of comparable magnitude, i.e. $q^{+}/q^{-}\approx 1$ because if $q^{+}$ was appreciably larger than $q^{-}$ , disturbances would be continually acting to increase the KE of the fluid and would therefore represent a significant source of energy.

Assuming further that there is no correlation between the ${\rm\Delta}K$ and ${\rm\Delta}\dot{K}$ of disturbances, the probability that a randomly selected disturbance will be in each of the instability classes is given respectively by $p^{-}q^{-}$ ( ${\rm\Delta}K<0$ and ${\rm\Delta}\dot{K}<0$ ) and $p^{+}q^{+}$ ( ${\rm\Delta}K>0$ and ${\rm\Delta}\dot{K}>0$ ). It follows that the relative likelihood of a randomly selected instability leading to a decrease in ${\rm\Delta}K$ versus an increase in ${\rm\Delta}K$ is

The interpretation of this result is that the majority of unstable disturbances of $\boldsymbol{u}$ will lead to a new flow with a smaller KE growth rate (or greater rate of KE decline). Since $\boldsymbol{u}$ is arbitrary we expect the fluid, which is being continually perturbed, to exhibit a decline in total kinetic energy ${\it\rho}\langle u^{2}\rangle /2$ and ultimately attain its minimum possible value. This suggests a principle of minimum total KE for fixed $A$ .

Note that the preceding argument relies on two properties specific to the KE: (a) the symmetry properties of the dynamical equation governing KE perturbations with respect to the transformation ${\rm\Delta}\boldsymbol{u}\rightarrow -{\rm\Delta}\boldsymbol{u}$ and (b) the close connection between the KE and the vector field norm $\langle u^{2}\rangle$ which connects the dynamical evolution of KE to the stability properties of perturbations. Thus the above argument cannot be directly replicated for different flow quantities other than the KE. However, it is obviously possible that the final conclusion of the above analysis would be different under different assumptions.

At first sight, a principle of minimum KE for fixed $A$ does not seem to have a clear connection to the principles suggested by Malkus for channel flow. However, it does suggest a principle of maximum $D$ for fixed $R$ , as initially proposed by Malkus (Reference Malkus1956). To see this, note that a decrease in total KE to its minimum will likely be associated with a decrease in both mean flow KE ${\it\rho}\langle U^{2}\rangle /2$ and fluctuating flow KE ${\it\rho}\langle v^{2}\rangle /2$ . Suppose that this decrease causes the mean flow KE to attain its minimum possible value. By assumption $A$ is fixed, which implies that the boundary gradient $U_{b}^{\prime }$ is also fixed for statistically steady flows, (2.12). Furthermore, the boundary layer will have some minimum possible average thickness determined by its marginal stability properties, and we assume that statistically steady mean flow profiles must be concave (see discussion of constraint (i)). Then minimising the mean flow KE ${\it\rho}\langle U^{2}\rangle /2$ (i.e. minimising the area under the curve $U(z)^{2}$ ) is equivalent to minimising $\langle U\rangle$ , since these both amount to making $U(z)$ as flat as possible away from the boundary layer (§ 2.5). In other words, provided that the above assumptions are valid, minimising mean flow KE is equivalent to minimising $R\propto \langle U\rangle$ under the conditions of fixed $U_{b}^{\prime }$ , a minimum boundary thickness and profile concavity. In turn this is equivalent to maximising $D$ for fixed $R$ (§ 2.5).