3.1 Input–output properties

We now define the three types of input–output properties that we study within the dissipativity framework and discuss their relation to common notions in the literature.

The first property that we study is the maximum energy growth due to initial perturbation velocities that are square integrable over the spatial domain for the nonlinear Navier–Stokes equation (2.4). In the context of linear systems, this property corresponds to maximum transient growth (Gustavsson Reference Gustavsson1991a ; Butler & Farrell Reference Butler and Farrell1992; Reddy & Henningson Reference Reddy and Henningson1993). For a vector $\boldsymbol{l}$ , we denote by $\boldsymbol{l}^{\prime }$ the transpose of $\boldsymbol{l}$ .

Definition 3.1 (Energy growth).

For any $\boldsymbol{u}$ satisfying (2.4) with $\boldsymbol{d}\equiv 0$ and initial flow perturbations that are square integrable over the spatial domain, i.e.  $\Vert \boldsymbol{u}_{0}\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}=(\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{u}_{0}^{\prime }(\unicode[STIX]{x1D703})\boldsymbol{u}_{0}(\unicode[STIX]{x1D703})\,\text{d}\unicode[STIX]{x1D703})^{1/2}<\infty$ , if there exists a constant $\unicode[STIX]{x1D6FE}>0$ such that

(3.5) $$\begin{eqnarray}\Vert \boldsymbol{u}\Vert _{{\mathcal{L}}_{[0,\infty ),\unicode[STIX]{x1D6FA}}^{2}}\leqslant \unicode[STIX]{x1D6FE}\Vert \boldsymbol{u}(0,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}},\end{eqnarray}$$

where $\Vert \boldsymbol{u}\Vert _{{\mathcal{L}}_{[0,T),\unicode[STIX]{x1D6FA}}^{2}}=(\int _{0}^{T}\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{u}^{\prime }(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D703})\boldsymbol{u}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D703})\,\text{d}\unicode[STIX]{x1D70F}\,\text{d}\unicode[STIX]{x1D703})^{1/2}$ , then we say that the flow perturbations have bounded energy growth with bound $\unicode[STIX]{x1D6FE}$ .

The next property of interest is related to amplifications from square integrable body forces or disturbances (see Jovanović (Reference Jovanović2004, chap. 9) for analogous results pertaining to a linearized model of channel flows). The square integrable forcings are of special interest, because they can be interpreted as finite-energy forcings. We refer to this class of amplifications as worst-case disturbance amplification.

Definition 3.2 (Worst-case disturbance amplification).

If there exist $\unicode[STIX]{x1D702}_{i}>0$ , $i\in I$ such that

(3.6) $$\begin{eqnarray}\Vert \boldsymbol{u}\Vert _{{\mathcal{L}}_{[0,\infty ),\unicode[STIX]{x1D6FA}}^{2}}^{2}\leqslant \mathop{\sum }_{i\in I}\unicode[STIX]{x1D702}_{i}^{2}\Vert d_{i}\Vert _{{\mathcal{L}}_{[0,\infty ),\unicode[STIX]{x1D6FA}}^{2}}^{2},\end{eqnarray}$$

for all $\boldsymbol{u}$ satisfying (2.4) subject to zero initial perturbations $\boldsymbol{u}(0,x)\equiv 0,\forall x\in \unicode[STIX]{x1D6FA}$ and square integrable disturbances over both spatial domain and time, i.e.  $\Vert d_{i}\Vert _{{\mathcal{L}}_{[0,\infty ),\unicode[STIX]{x1D6FA}}^{2}}^{2}<\infty$ , $i\in I$ , then we say that the flow has bounded worst-case disturbance amplification with bounds $\unicode[STIX]{x1D702}_{i},i\in I$ .

The above property is equivalent to the induced ${\mathcal{L}}^{2}$ -norm in control theory (Van der Schaft Reference Van der Schaft2017). In other words, each $\unicode[STIX]{x1D702}_{i}$ upper-bounds the peak amplification of perturbation velocities from the forcing in the direction $i$ , $d_{i}$ , when the forcings in other directions are set to zero, i.e.  $d_{j}=0$ , $j\in I$ , $i\neq j$ . That is,

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{i}\geqslant \sup _{\Vert d_{i}\Vert _{{\mathcal{L}}_{[0,\infty ),\unicode[STIX]{x1D6FA}}^{2}}\neq 0}\frac{\Vert \boldsymbol{u}\Vert _{{\mathcal{L}}_{[0,\infty ),\unicode[STIX]{x1D6FA}}^{2}}}{\Vert d_{i}\Vert _{{\mathcal{L}}_{[0,\infty ),\unicode[STIX]{x1D6FA}}^{2}}}.\end{eqnarray}$$

Note that the above relation provides an upper bound for the amplification of disturbances of any magnitude. For nonlinear systems, such as the flow dynamics studied here, the amplification may depend on the magnitude of the disturbance $\Vert \boldsymbol{d}\Vert _{{\mathcal{L}}^{2}}$ in a nonlinear fashion (Ahmadi, Valmorbida & Papachristodoulou Reference Ahmadi, Valmorbida and Papachristodoulou2016, Example I). In this case, any value of $\unicode[STIX]{x1D702}$ satisfying (3.6) provides an upper bound to the worst-case disturbance amplification. Therefore, in order to obtain tight bounds to the worst-case amplification, we are interested in the minimum values of $\unicode[STIX]{x1D702}_{i}$ , $i\in I$ for which (3.6) holds.

From a practical perspective, stability of a base flow is often not very meaningful, because small disturbances may cause an unstable behaviour. We instead require a notion of stability that relates disturbances to perturbation velocities. In particular, since the definition of the worst-case disturbance amplification requires the forcings to be square integrable, it excludes persistent forcings, e.g. constant and sinusoidal forcings, that are defined for all time. To include these classes of forcings in a nonlinear context, we employ the concept of ISS (Sontag Reference Sontag, Nistri and Stefani2008) to study the class of upper-bounded forcings. (In the fluids literature, the ensemble average energy density or the ${\mathcal{H}}^{2}$ -norm has been used to study amplifications from Gaussian stochastic forcings to the linearized flow dynamics (Farrell & Ioannou Reference Farrell and Ioannou1993; Jovanović & Bamieh Reference Jovanović and Bamieh2005). The ${\mathcal{H}}^{2}$ -norm is equivalent to the root-mean-square value of the linearized flow response to white noise forcings. However, extension of ${\mathcal{H}}^{2}$ analysis to the nonlinear Navier–Stokes equations is an open problem.) We refer to this extended notion of stability as stability to persistent disturbances. Prominent among the features of this property are that forcings that are bounded, eventually small, integrally small or convergent lead to perturbation velocities with the respective property. Furthermore, this property quantifies in what manner initial perturbation velocities affect transient behaviour. Flows with this property do not have unstable behaviour for persistent (non-vanishing) forcings.

To characterize this property, let us introduce a few comparison functions. Let ${\mathcal{K}}$ denote the class of non-negative functions that are strictly increasing and zero for zero argument, and ${\mathcal{K}}_{\infty }$ denote the class of functions that, in addition, become unbounded as their argument goes to infinity. For a vector $\boldsymbol{l}$ , denote by $|\boldsymbol{l}|$ its absolute value given by $\sqrt{\boldsymbol{l}^{\prime }\boldsymbol{ l}}$ .

Definition 3.3 (Stability to persistent disturbances).

For all $\boldsymbol{u}$ satisfying (2.4) with upper-bounded disturbances, i.e.  $\Vert \boldsymbol{d}\Vert _{{\mathcal{L}}_{[0,\infty )}^{\infty }}=\sup _{\unicode[STIX]{x1D70F}\in [0,\infty )}|\boldsymbol{d}(\unicode[STIX]{x1D70F})|<\infty$ , if there exist some scalar $\unicode[STIX]{x1D713}>0$ , functions $\unicode[STIX]{x1D6FD},\tilde{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D712}\in {\mathcal{K}}_{\infty }$ and $\unicode[STIX]{x1D70E}\in {\mathcal{K}}$ , such that

(3.8) $$\begin{eqnarray}\Vert \boldsymbol{u}(t,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}\leqslant \unicode[STIX]{x1D6FD}(\text{e}^{-\unicode[STIX]{x1D713}t}\unicode[STIX]{x1D712}(\Vert \boldsymbol{u}(0,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}))+\tilde{\unicode[STIX]{x1D6FD}}\left(\sup _{\unicode[STIX]{x1D70F}\in [0,t)}\left(\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D70E}(|\boldsymbol{d}(\unicode[STIX]{x1D70F},x)|)\,\text{d}\unicode[STIX]{x1D6FA}\right)\right),\end{eqnarray}$$

for all $t>0$ , then we call the flow stable to persistent disturbances.

Property (3.8) implies convergence to the base flow $(\boldsymbol{U},P)$ in the ${\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}$ -norm (the norm corresponding to the space of square integrable functions over the spatial domain) when the disturbances are not present ( $\boldsymbol{d}\equiv 0$ ). Indeed, the $\unicode[STIX]{x1D6FD}(\text{e}^{-\unicode[STIX]{x1D713}t}\unicode[STIX]{x1D712}(\Vert \boldsymbol{u}(0,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}))$ term dominates for small $t$ , and this serves to quantify the magnitude of the transient growth as a function of the size of the initial state $\Vert \boldsymbol{u}(0,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}$ .

Moreover, as $t\rightarrow \infty$ , we obtain

(3.9) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\Vert \boldsymbol{u}(t,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}\leqslant \tilde{\unicode[STIX]{x1D6FD}}\left(\int _{\unicode[STIX]{x1D6FA}}\Vert \unicode[STIX]{x1D70E}(|\boldsymbol{d}(\cdot ,x)|)\Vert _{{\mathcal{L}}_{[0,\infty )}^{\infty }}\,\text{d}\unicode[STIX]{x1D6FA}\right)\leqslant \tilde{\unicode[STIX]{x1D6FD}}\left(\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D70E}(\Vert \boldsymbol{d}(\cdot ,x)\Vert _{{\mathcal{L}}_{[0,\infty )}^{\infty }})\,\text{d}\unicode[STIX]{x1D6FA}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6FD}\in {\mathcal{K}}$ . Since $\Vert \boldsymbol{d}(\cdot ,x)\Vert _{{\mathcal{L}}_{[0,\infty )}^{\infty }}$ is bounded by assumption, $\unicode[STIX]{x1D70E}\in {\mathcal{K}}$ (continuous and bounded), and the integral is over a bounded domain $\unicode[STIX]{x1D6FA}$ , the quantity on the right-hand side of (3.9) is also bounded. Hence, as long as the external excitations or body forces $\boldsymbol{d}$ are upper-bounded, the perturbation velocities $\boldsymbol{u}$ are bounded in the ${\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}$ -norm, meaning that they remain square integrable over the flow geometry.

In fact, by input-to-state superposition theorem (Sontag Reference Sontag, Baillieul and Samad2013), we can show that stability to persistent disturbances is the conjunction of two properties, one of them concerned with asymptotic bounds on the perturbation velocities, in the sense of $\Vert \boldsymbol{u}(t,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}$ , as a function of the magnitude of the forcings, and the other one providing a transient term obtained when we ignore forcings (see figure 1).

Figure 1.

The stability to persistent disturbances property combines transient growth (overshoot) and asymptotic behaviour.

We now demonstrate how the problem of verifying the properties in Definitions 3.1–3.3 can be cast as verifying a set of dissipation inequalities. These results, which can be derived from Ahmadi et al. (Reference Ahmadi, Valmorbida and Papachristodoulou2016, Theorem 6), allow for the extension of well-known methods for stability, input–output and optimal perturbation analysis of linear systems to the full nonlinear Navier–Stokes equations.

We begin by studying energy growth.

Theorem 3.4. Consider the perturbation model (2.4) with $\boldsymbol{d}\equiv 0$ and let $\Vert \boldsymbol{u}(0,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}^{2}<\infty$ , i.e. the initial perturbations be square integrable over the spatial domain. If there exist a positive semi-definite storage functional $V(\boldsymbol{u})$ and a positive scalar $\unicode[STIX]{x1D6FE}$ such that

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle V(\boldsymbol{u})\leqslant \unicode[STIX]{x1D6FE}^{2}\Vert \boldsymbol{u}(t,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}^{2}, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}V(\boldsymbol{u}(t,x))}{\text{d}t}\leqslant -\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{u}^{\prime }(t,x)\boldsymbol{u}(t,x)\,\text{d}\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

then it has bounded energy growth as given by (3.5).

The next result pertains to worst-case energy amplification.

Theorem 3.5. Consider the perturbation model (2.4) and let $\Vert \boldsymbol{d}(t,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}^{2}<\infty$ , i.e. the disturbances be square integrable over the spatial domain. If there exist a positive semi-definite storage functional $V(\boldsymbol{u})$ and positive scalars $\{\unicode[STIX]{x1D702}_{i}\}_{i\in I}$ such that

(3.12) $$\begin{eqnarray}\displaystyle \frac{\text{d}V(\boldsymbol{u}(t,x))}{\text{d}t}\leqslant -\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{u}^{\prime }(t,x)\boldsymbol{u}(t,x)\,\text{d}\unicode[STIX]{x1D6FA}+\int _{\unicode[STIX]{x1D6FA}}\mathop{\sum }_{i\in I}\unicode[STIX]{x1D702}_{i}^{2}d_{i}^{2}(t,x)\,\text{d}\unicode[STIX]{x1D6FA}, & & \displaystyle\end{eqnarray}$$

then the perturbation velocities (2.4) have worst-case disturbance amplification upper bounds $\unicode[STIX]{x1D702}_{i}$ , $i\in I$ as in (3.6).

Lastly, we study amplifications to persistent forcings.

Theorem 3.6. Consider the perturbation model (2.4) and let $\Vert \boldsymbol{d}\Vert _{{\mathcal{L}}_{[0,\infty )}^{\infty }}<\infty$ , i.e. the disturbances have an upper bound. If there exist a positive semi-definite storage functional $V(\boldsymbol{u})$ , a positive scalar $\unicode[STIX]{x1D713}$ and functions $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2},\unicode[STIX]{x1D70E}\in {\mathcal{K}}_{\infty }$ such that

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}_{1}(\Vert \boldsymbol{u}(t,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}})\leqslant V(\boldsymbol{u})\leqslant \unicode[STIX]{x1D6FD}_{2}(\Vert \boldsymbol{u}(t,\cdot )\Vert _{{\mathcal{L}}_{\unicode[STIX]{x1D6FA}}^{2}}), & \displaystyle\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}V(\boldsymbol{u}(t,x))}{\text{d}t}\leqslant -\unicode[STIX]{x1D713}V(\boldsymbol{u}(t,x))+\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D70E}(|\boldsymbol{d}(t,x)|)\,\text{d}\unicode[STIX]{x1D6FA}, & \displaystyle\end{eqnarray}$$

then the perturbation velocities described by (2.4) are stable to persistent disturbances as given by (3.8) with $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D6FD}_{2}$ , $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{1}^{-1}\circ 2$ and $\tilde{\unicode[STIX]{x1D6FD}}=\unicode[STIX]{x1D6FD}_{1}^{-1}\circ 2/\unicode[STIX]{x1D713}$ , where $\circ$ implies function composition.

In the following, we derive classes of storage functionals $V(\boldsymbol{u})$ suitable for the analysis of perturbation dynamics (2.4) invariant in one of the three spatial coordinates. We consider two classes of flows, namely channel flows with perturbations that vary in two spatial dimensions and time discussed in § 3.2 and pipe flows invariant in the axial direction discussed in appendix C.

3.2 A Lyapunov/storage functional for flows between parallel plates

In Cartesian coordinates, for a scalar function $v$ , $\unicode[STIX]{x1D735}v=\sum _{i}\unicode[STIX]{x2202}_{x_{i}}v\boldsymbol{e}_{i}$ and $\unicode[STIX]{x1D6FB}^{2}v=\sum _{i}\unicode[STIX]{x2202}_{x_{i}}^{2}v$ , where $\boldsymbol{e}_{i}$ is the unit vector in the direction $x_{i}$ . For a vector valued function $\boldsymbol{w}=\sum _{i}w_{i}\boldsymbol{e}_{i}$ , and the divergence $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{w}$ is given by $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{w}=\sum _{i}\unicode[STIX]{x2202}_{x_{i}}w_{i}$ . In the following, $\{x_{1},x_{2},x_{3}\}$ correspond to $\{x,y,z\}$ (the streamwise, wall-normal and spanwise directions) and $I=\{1,2,3\}$ . Additionally, we adopt Einstein’s multi-index notation over index $j$ , that is the sum over repeated indices $j$ , e.g.  $v_{j}\unicode[STIX]{x2202}_{x_{j}}u_{i}=\sum _{j}v_{j}\unicode[STIX]{x2202}_{x_{j}}u_{i}$ .

The perturbation model (2.4) can be rewritten as

(3.15) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x2202}_{t}u_{i}=\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}u_{i}-u_{j}\unicode[STIX]{x2202}_{x_{j}}u_{i}-U_{j}\unicode[STIX]{x2202}_{x_{j}}u_{i}-u_{j}\unicode[STIX]{x2202}_{x_{j}}U_{i}-\unicode[STIX]{x2202}_{x_{i}}p+d_{i},\\ \displaystyle 0=\unicode[STIX]{x2202}_{x_{j}}u_{j},\end{array}\right\}\end{eqnarray}$$

where $i,j\in I$ . To simplify the exposition, without loss of generality, we assume that the perturbations are invariant with respect to $x_{1}$ . Since $x_{i},i=1,2,3$ are arbitrary, this does not affect the formulation.

The next proposition states that, by choosing a suitable storage functional structure (weighted kinetic energy of the perturbation velocities), the time derivative of the storage functional turns out to be upper-bounded by a quadratic form in the velocity fields $\boldsymbol{u}$ and their spatial derivatives. This property paves the way for a convex-optimization-based method to check input–output properties. Convex optimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The convexity makes optimization easier than the general case since local minimum must be a global minimum, and first-order conditions are sufficient conditions for optimality (Boyd & Vandenberghe Reference Boyd and Vandenberghe2004). Convex optimization problems can be solved efficiently by interior-point methods (Nesterov & Nemirovskii Reference Nesterov and Nemirovskii1994). Convex optimization was used by Moarref et al. (Reference Moarref, Jovanovic, Tropp, Sharma and McKeon2014) to obtain a low-order decomposition of the Navier–Stokes equations based on resolvent modes.

Proposition 3.7. Consider the perturbation model (3.15) subject to periodic or no-slip boundary conditions $\boldsymbol{u}|_{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}=0$ . Assume the velocity perturbations in (3.15) are invariant with respect to $x_{1}$ . Let $I_{0}=\{2,3\}$ and

(3.16) $$\begin{eqnarray}V(\boldsymbol{u})=\frac{1}{2}\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{u}^{\prime }\unicode[STIX]{x1D64C}\boldsymbol{u}\,\text{d}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where $\unicode[STIX]{x1D64C}=\Big[\begin{smallmatrix}q_{1} & 0 & 0\\ 0 & q_{j} & 0\\ 0 & 0 & q_{i}\end{smallmatrix}\Big]>0$ , $q_{i}=q_{j}$ for $i\neq j$ , $i,j\in I_{0}$ , be a candidate storage functional. Then, the time derivative of (3.16) along the solutions to (3.15) satisfies

(3.17) $$\begin{eqnarray}\frac{\text{d}V(\boldsymbol{u})}{\text{d}t}\leqslant -\mathop{\sum }_{i\in I}q_{i}\int _{\unicode[STIX]{x1D6FA}}\left(\frac{C}{Re}u_{i}^{2}+U_{j}u_{i}\unicode[STIX]{x2202}_{x_{j}}u_{i}+u_{j}u_{i}\unicode[STIX]{x2202}_{x_{j}}U_{i}-u_{i}d_{i}\right)\,\text{d}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where $C$ is a positive constant that only depends on the domain $\unicode[STIX]{x1D6FA}$ .

The proof of this proposition is given in appendix A.

Note that a special case of (3.16) was used in Joseph & Hung (Reference Joseph and Hung1971) to study the stability of viscous flows (subject to streamwise-constant perturbations) in pipes and between rotating cylinders. The authors referred to this structure as the two energy function. In the formulation presented in this paper, assuming invariant perturbations in the $x_{1}$ -direction, and since the inequalities hold if scaled by a positive scalar, we can represent the two energy function as

(3.18) $$\begin{eqnarray}V(\boldsymbol{u})=\frac{1}{2}\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{u}^{\prime }\left[\begin{array}{@{}ccc@{}}q & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]\boldsymbol{u}\,\text{d}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where $q$ is a positive scalar. The ‘optimal’ value for this constant was then calculated analytically for the pipe Poiseuille and the Taylor–Couette flow by Joseph & Hung (Reference Joseph and Hung1971).

Indeed, considering streamwise-constant perturbations provides a degree of freedom in the Lyapunov/storage functional, which can be used to study flow properties at higher Reynolds numbers. When the perturbations are spatially varying in all three directions, the Lyapunov/storage functional converts to the energy functional, which has been used to study the stability of viscous flows (Serrin Reference Serrin1959). However, since for three-dimensional perturbations there is no degree of freedom in the structure of the Lyapunov/storage functional, the flow properties can only be studied for low Reynolds numbers.

Note that in (3.17) the Poincaré constant, $C$ , appears. There are several estimates for the optimal Poincaré constant. The optimal constant (Payne & Weinberger Reference Payne and Weinberger1960) we use in this paper is

(3.19) $$\begin{eqnarray}C(\unicode[STIX]{x1D6FA})=\frac{\unicode[STIX]{x03C0}^{2}}{D(\unicode[STIX]{x1D6FA})},\end{eqnarray}$$

where $D(\unicode[STIX]{x1D6FA})$ is the diameter of the domain $\unicode[STIX]{x1D6FA}$ (the longest distance between any two point in the domain $\unicode[STIX]{x1D6FA}$ ).

Proposition 3.7 allows us to provide an algorithmic method for input–output analysis of fluid flows based on convex optimization. These convex optimization problems are in terms of linear matrix inequalities and polynomial matrix inequalities. This formulation is delineated in more detail in the next section.

5.1 Plane Couette flow

We consider the flow of viscous fluid between two parallel plates, where the gap between the plates is much smaller than the length of the plates as illustrated in figure 2. The base flow is given by $\boldsymbol{U}=(y,0,0)^{\prime }=y\boldsymbol{e}_{x}$ and $P=P_{0}$ .

Figure 2.

Schematic of the plane Couette flow geometry.

We consider no-slip boundary conditions $\boldsymbol{u}|_{y=-1}^{1}=0$ in the wall-normal direction and $\boldsymbol{u}(t,y,z)=\boldsymbol{u}(t,y,z+L)$ in the spanwise direction. The Poincaré constant is then given by $C=\unicode[STIX]{x03C0}^{2}/\sqrt{L^{2}+2^{2}}$ .

A storage functional of the form (3.16) can be used to compute bounds on all three quantities of interest (i.e. energy growth, worst-case amplification and stability to persistent forcings):

(5.1) $$\begin{eqnarray}V(u)=\int _{0}^{L}\int _{-1}^{1}\left[\begin{array}{@{}c@{}}u_{x}\\ u_{y}\\ u_{z}\end{array}\right]^{\prime }\left[\begin{array}{@{}ccc@{}}q_{x} & 0 & 0\\ 0 & q_{y} & 0\\ 0 & 0 & q_{z}\end{array}\right]\left[\begin{array}{@{}c@{}}u_{x}\\ u_{y}\\ u_{z}\end{array}\right]\,\text{d}y\,\text{d}z,\end{eqnarray}$$

with $q_{y}=q_{z}$ .

We assume that spatial invariance in the $x$ -direction, and thus ( $m=x,j=y,i=z$ ). The $\unicode[STIX]{x1D648}$ matrix in (4.1) is then given by

(5.2) $$\begin{eqnarray}\unicode[STIX]{x1D648}=\left[\begin{array}{@{}ccc@{}}\displaystyle \frac{q_{x}C}{Re} & \displaystyle \frac{q_{x}}{2} & 0\\ \displaystyle \frac{q_{x}}{2} & \displaystyle \frac{q_{y}C}{Re} & 0\\ 0 & 0 & \displaystyle \frac{q_{y}C}{Re}\end{array}\right].\end{eqnarray}$$

Given a channel width $L=\unicode[STIX]{x03C0}$ we can compute the input–output energy growth by solving the optimization problem (4.5) with $\unicode[STIX]{x1D648}$ given by (5.2). The resulting energy growth $\unicode[STIX]{x1D6FE}^{2}$ as a function of Reynolds number is plotted in figure 3. For small Reynolds numbers $\unicode[STIX]{x1D6FE}^{2}\sim O(Re)$ , whereas for larger Reynolds numbers $\unicode[STIX]{x1D6FE}^{2}\sim O(Re^{3})$ . Therefore, it can be inferred that $\unicode[STIX]{x1D6FE}^{2}\sim c_{0}Re+c_{1}Re^{3}$ with $c_{0},c_{1}>0$ . This scaling is consistent with the results obtained by Bobba, Bamieh & Doyle (Reference Bobba, Bamieh and Doyle2002), where the maximum energy growth of streamwise-constant (nonlinear) plane Couette flow was calculated analytically based on a streamwise-constant $(2\text{-}\text{D}/3\text{-}\text{C})$ model. It is worth noting that we calculated the scalings by interpolation of the upper bounds computed from the convex optimization problems in terms of the Reynolds numbers. In the cases for which a polynomial fit could not be found, we measured the slope of the line in the amplification versus Reynolds number plot (figure 3).

Figure 3.

Upper bounds on the maximum energy growth for plane Couette flow in terms of Reynolds numbers.

We next compute the worst-case amplification using the matrix inequality (4.3), which for this particular flow is given by

(5.3)

with $\unicode[STIX]{x1D648}$ as in (5.2).

Figure 4.

Upper bounds on the worst-case amplification for perturbation velocities of plane Couette flow for different Reynolds numbers.

The resulting upper bounds on the worst-case amplification for $2\text{-}\text{D}/3\text{-}\text{C}$ plane Couette flow obtained by minimizing $\unicode[STIX]{x1D702}_{x}^{2}$ , $\unicode[STIX]{x1D702}_{y}^{2}$ and $\unicode[STIX]{x1D702}_{z}^{2}$ are given in figure 4. Since the flow is stable for all Reynolds numbers, the worst-case amplifications are increasing monotonically with Reynolds number. The results in figure 4 imply $\unicode[STIX]{x1D702}_{x}^{2}\sim a_{0}Re^{2}+a_{1}Re^{3}$ , $\unicode[STIX]{x1D702}_{y}^{2}\sim b_{0}Re^{2}+b_{1}Re^{4}$ and $\unicode[STIX]{x1D702}_{z}^{2}\sim c_{0}Re^{2}+c_{1}Re^{4}$ with $a_{0},a_{1},b_{0},b_{1},c_{0},c_{1}>0$ . This implies that worst-case amplifications in all three components of disturbances grow with a $Re^{2}$ ratio for very low Reynolds numbers. For Reynolds numbers approximately greater than $1$ , the streamwise disturbances are amplified at a rate proportional to $Re^{3}$ ; whereas, the wall-normal and spanwise disturbance component amplification scales as  $Re^{4}$ .

The upper bounds depicted in figure 4 can be compared with corollary F.2 (see appendix F), wherein it was demonstrated that $\unicode[STIX]{x1D702}_{x}^{2}=f_{0}Re^{2}$ , $\unicode[STIX]{x1D702}_{y}^{2}=g_{0}Re^{2}+g_{1}Re^{4}$ and $\unicode[STIX]{x1D702}_{z}^{2}=h_{0}Re^{2}+h_{1}Re^{4}$ for the linearized plane Couette flow with constants $f_{0},g_{0},g_{1},h_{0},h_{1}>0$ by solving Lyapunov equations.

Finally, we compute the stability to persistent forcings (ISS) property using inequality (4.4) from corollary 4.1. Applying this to the $2\text{-}\text{D}/3\text{-}\text{C}$ Couette flow of interest leads to

(5.4)

with $\unicode[STIX]{x1D648}$ given in (5.2) and $\unicode[STIX]{x1D652}=\Bigl[\begin{smallmatrix}q_{x}\unicode[STIX]{x1D713} & 0 & 0\\ 0 & q_{y}\unicode[STIX]{x1D713} & 0\\ 0 & 0 & q_{y}\unicode[STIX]{x1D713}\end{smallmatrix}\Bigr]$ . We fix $\unicode[STIX]{x1D713}=10^{-4}$ and the channel width to $L=2\unicode[STIX]{x03C0}$ . In this case, we obtain $Re_{ISS}=316$ . The quantity $Re_{ISS}=316$ is interestingly close to the empirical transition Reynolds number $Re\approx 350$ obtained experimentally by Tillmark & Alfredsson (Reference Tillmark and Alfredsson1992). In this sense, it turns out that $Re_{ISS}$ is surprisingly close to the Reynolds number above which transition occurs. Understanding this connection is an interesting direction for future work.

Figure 5.

The perturbation flow structures with maximum amplification from persistent forcings at $Re=316$ for plane Couette flow.

In order to understand the above result on stability to persistent disturbances, we carried out numerical experiments to investigate the flow structures that receive maximum amplification from persistent disturbances. The experiments were undertaken for the linearized Navier–Stokes equation through the Orr–Sommerfeld equations. Appendix E discusses the details of these numerical experiments. Notice that these results are based on solving linear matrix inequalities that ensure stability to persistent forcings for the ODE obtained through spatial discretization of the Orr–Sommerfeld equations. Our simulations were carried out using a $50\times 50$ grid on the wavenumber space $k_{x}-k_{z}$ ( $k_{x},k_{z}\in [0,150]$ ) and the linear matrix inequalities were computed at each point in the grid. Then, the wavenumbers corresponding to the maximum amplification are selected (especially, we are interested to find $k_{x}$ corresponding to maximum amplification, as this is the streamwise direction) and the corresponding flow structure is simulated. As expected the maximum amplification corresponds to the streamwise-constant case $k_{x}=0$ . Figure 5 illustrates the flow structures that receive maximum amplification at $Re=316$ .

It is also worth mentioning that certificates for stability to persistent disturbances of the linearized Navier–Stokes equation, as discussed in appendix E, could be constructed for all Reynolds numbers, which is in contrast to the nonlinear case studied in this example. This illustrates that stability to persistent disturbances is a fundamentally nonlinear phenomenon.

Figure 6.

Schematic of the plane Poiseuille flow geometry.

5.2 Plane Poiseuille flow

Similar to the plane Couette flow, we consider the flow of viscous fluid between two parallel plates, where the gap between the plates is much smaller than the length of the plates. Here, the plates are stationary and the flow is induced by a pressure gradient in the flow direction. The flow geometry is depicted in figure 6.

The domain $\unicode[STIX]{x1D6FA}$ is defined as $\unicode[STIX]{x1D6FA}=\{(y,z)\mid -1<y<1,0<z<L\}$ . The base flow is given by $\boldsymbol{U}=U_{m}(y)\boldsymbol{e}_{x}=(1-y^{2})\boldsymbol{e}_{x}$ and $P=1-(4x/Re)$ . We consider no-slip boundary conditions $\boldsymbol{u}|_{y=-1}^{1}=0$ and $\boldsymbol{u}(t,y,z)=\boldsymbol{u}(t,y,z+L)$ . The Poincaré constant is then given by $C=\unicode[STIX]{x03C0}^{2}/\sqrt{L^{2}+2^{2}}$ .

We can again study the the input–output properties of the flow using the storage functional (5.1). For this flow, the flow perturbations are assumed invariant in the streamwise direction $x$ , i.e. ( $m=x,j=y,i=z$ ).

We compute the upper bounds on maximum energy growth for this flow by solving the optimization problem (4.5) with

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D648}(y)=\left[\begin{array}{@{}ccc@{}}\displaystyle \frac{q_{x}C}{Re} & -yq_{x} & 0\\ -yq_{x} & \displaystyle \frac{q_{y}C}{Re} & 0\\ 0 & 0 & \displaystyle \frac{q_{y}C}{Re}\end{array}\right].\end{eqnarray}$$

The results are illustrated in figure 7, which shows that the maximum energy amplification is described by $\unicode[STIX]{x1D6FE}^{2}\sim b_{0}Re+b_{1}Re^{2}$ , with $b_{0},b_{1}>0$ . This result is consistent with transient growth calculations of Reddy & Henningson (Reference Reddy and Henningson1993), in which the authors showed that the transient growth of the linearized plane Poiseuille flow model behaves like $O(Re^{2})$ for large Reynolds numbers.

For worst-case amplification analysis, we use inequality (4.3) which for this flow is given by the following matrix inequality:

(5.6)

with $\unicode[STIX]{x1D648}$ as in (5.5). By minimizing $\unicode[STIX]{x1D702}_{x}^{2}$ , $\unicode[STIX]{x1D702}_{y}^{2}$ and $\unicode[STIX]{x1D702}_{z}^{2}$ , we compute upper bounds on the worst-case amplification for the plane Poiseuille flow that are depicted in figure 8. From figure 8, it can be inferred that $\unicode[STIX]{x1D702}_{x}^{2}\sim a_{0}Re^{2}+a_{1}Re^{3}$ , $\unicode[STIX]{x1D702}_{y}^{2}\sim b_{0}Re^{2.2}+b_{1}Re^{4}$ and $\unicode[STIX]{x1D702}_{z}^{2}\sim c_{0}Re^{2}+c_{1}Re^{4}$ with $a_{0},a_{1},b_{0},b_{1},c_{0},c_{1}>0$ . From this result, we can infer that the worst-case amplifications in all three components of disturbances grow with a $Re^{2}$ ratio for low Reynolds numbers. For Reynolds numbers approximately greater than ${\approx}5$ , the streamwise disturbances are amplified proportional to $Re^{3}$ ; whereas, the wall-normal and spanwise disturbance components show $Re^{4}$ amplification.

Figure 7.

Upper bounds on the maximum energy growth for plane Poiseuille flow in terms of Reynolds numbers.

Figure 8.

Upper bounds on the worst-case amplification of plane Poiseuille flow for different Reynolds numbers.

We again evaluate stability to persistent disturbances using inequality (4.4) from corollary 4.1 flow, i.e.

(5.7)

with $\unicode[STIX]{x1D648}$ given in (5.5) and $\unicode[STIX]{x1D652}=\left[\!\begin{smallmatrix}q_{x}\unicode[STIX]{x1D713} & 0 & 0\\ 0 & q_{y}\unicode[STIX]{x1D713} & 0\\ 0 & 0 & q_{y}\unicode[STIX]{x1D713}\end{smallmatrix}\!\right]$ . We fix $\unicode[STIX]{x1D713}=10^{-4}$ and $L=2\unicode[STIX]{x03C0}$ . In this case, we obtain $Re_{ISS}=1855$ . The quantity $Re_{ISS}=1855$ can be compared with the empirical Reynolds number at the onset of turbulence $Re\approx 2000$ as discussed by Grossmann (Reference Grossmann2000). Once again, we observe that $Re_{ISS}$ provides a bound that is consistent with the Reynolds number at which transition to turbulence occurs in channel flows, an observation that leads to some interesting future research questions that are beyond the scope of the current work.

Figure 9.

The perturbation flow structures with maximum amplification to persistent disturbances at $Re=1855$ for plane Poiseuille flow.

Figure 10.

Schematic of the Hagen–Poiseuille flow geometry.

Analogous to the plane Couette flow, we undertook numerical experiments to find the flow structures subject to maximum amplification from persistent forcings. Again, we found that the maximum amplification corresponds to the streamwise-constant case $k_{x}=0$ . Figure 9 illustrates the flow structures that receive maximum amplification from persistent forcings at $Re=1855$ .

5.3 Hagen–Poiseuille flow

In this section we examine the input–output properties of Hagen–Poiseuille flow using an extension of the analysis framework that is detailed in appendix C. In particular, we consider the flow of viscous fluid driven by a pressure gradient, illustrated in figure 10. The domain $\unicode[STIX]{x1D6FA}$ is defined as $\unicode[STIX]{x1D6FA}=\{(r,\unicode[STIX]{x1D703})\mid 0<r<1,0<\unicode[STIX]{x1D703}<2\unicode[STIX]{x03C0}\}$ . The base flow is given by $\boldsymbol{U}=U_{m}(r)\boldsymbol{e}_{z}=(1-r^{2})\boldsymbol{e}_{z}$ and $P=1-4z/Re$ .

We again assume that flow is invariant in the streamwise direction $z$ , based on observations (e.g. Schmid & Henningson Reference Schmid and Henningson1994) that axially constant perturbations are subject to maximum background energy amplification in pipe flow. Then, the perturbation dynamics is given by (C 2) in appendix C with $U_{m}(r)=1-r^{2}$ . Moreover, we assume no-slip boundary conditions $\boldsymbol{u}|_{r=1}=0$ .

We consider the storage functional given in (C 3). Then, substituting $U_{m}$ and $F$ , we have

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D648}_{c}(r)=\left[\begin{array}{@{}ccc@{}}\displaystyle \frac{q_{z}C}{Re} & -rq_{z} & 0\\ -rq_{z} & \displaystyle \frac{q_{r}C}{Re} & 0\\ 0 & 0 & \displaystyle \frac{q_{\unicode[STIX]{x1D703}}C}{Re}\end{array}\right].\end{eqnarray}$$

Computation of the bounds on maximum energy growth in Hagen–Poiseuille flow as a function of disturbance direction and velocity component is an extension of previous analysis such as Jovanović & Bamieh (Reference Jovanović and Bamieh2005) that considered channel flows. We obtain these bounds by solving the optimization problem (4.5) with $\unicode[STIX]{x1D648}=\unicode[STIX]{x1D648}_{c}(r)$ as in (5.8). The results are illustrated in figure 11. The results imply that the maximum energy growth is described by $\unicode[STIX]{x1D6FE}^{2}\sim b_{0}Re+b_{1}Re^{2}$ , with $b_{0},b_{1}>0$ . This bound is in agreement with the calculations and numerical experiments of Schmid & Henningson (Reference Schmid and Henningson1994), where the authors investigated transient growth based on the linearized Navier–Stokes equations for pipe flow.

Figure 11.

Upper bounds on the maximum energy growth for Hagen–Poiseuille flow in terms of Reynolds numbers.

Considering $\unicode[STIX]{x1D648}_{c}(r)$ as in (5.8), inequality (C 14) becomes

(5.9)

Minimizing $\unicode[STIX]{x1D702}_{z}^{2}$ , $\unicode[STIX]{x1D702}_{r}^{2}$ and $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}^{2}$ subject to the above inequality provides upper bounds on the worst-case disturbance amplification, which are plotted in figure 12 as a function of Reynolds number. Figure 12 indicates that the perturbations are amplified as $\unicode[STIX]{x1D702}_{z}^{2}\sim a_{0}Re^{2}+a_{1}Re^{3}$ , $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D703}}^{2}\sim b_{0}Re^{2}+b_{1}Re^{4}$ and $\unicode[STIX]{x1D702}_{r}^{2}\sim c_{0}Re^{2}+c_{1}Re^{4}$ with $a_{0},a_{1},b_{0},b_{1},c_{0},c_{1}>0$ . Thus, similar to channel flows, for low Reynolds numbers, the worst-case amplifications from all three disturbance components scale as $Re^{2}$ . For Reynolds numbers greater than ${\approx}8$ , the amplifications from axial (flow direction) disturbances grow proportional to $Re^{3}$ ; whereas, the worst-case amplifications from azimuthal and radial disturbances increase as $Re^{4}$ .

Figure 12.

Upper bounds on the worst-case amplification of the Hagen–Poiseuille flow in terms of different Reynolds numbers.

We compute the bounds on stability to persistent forcings using the following polynomial matrix inequality:

(5.10)

where $\unicode[STIX]{x1D652}_{c}=\Big[\begin{smallmatrix}\unicode[STIX]{x1D713}q_{z} & 0 & 0\\ 0 & \unicode[STIX]{x1D713}q_{r} & 0\\ 0 & 0 & \unicode[STIX]{x1D713}q_{\unicode[STIX]{x1D703}}\end{smallmatrix}\Big]$ , which is analogous to inequality (C 15) in corollary C.2. The maximum Reynolds number for the system to remain stable to persistent forcings (i.e. remained input-to-state stable) was $Re_{ISS}=1614$ . Obtaining this bound required the use of degree-10 polynomials in $\unicode[STIX]{x1D70E}_{z}(r),\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D703}}(r)$ and $\unicode[STIX]{x1D70E}_{r}(r)$ . This result is again consistent with the lower bound on the Reynolds number for which transition to turbulence was observed empirically by Peixinho & Mullin (Reference Peixinho and Mullin2006), i.e.  $Re\approx 1800$ , and provides further support that this is a connection worth investigating.