2.1. Linear frequency response

In this section we consider the frequency response associated with the LNS system. The derivations in this section will be used as the foundation for the more complex nonlinear analysis, which we employ to obtain a stability criterion that is based on perturbation magnitude. Here, we consider a forcing term $\boldsymbol{d}$ , which is harmonic in time and in the $x$ and $z$ directions, i.e. $\boldsymbol{d}(t,x,y,z) = \overline {\boldsymbol{d}}(y)\exp [{i(k_x x + k_z z + \omega t)}]$ , where $\omega \in \mathbb{R}$ is the temporal frequency and $k_x,k_z\in \mathbb{R}$ are wavenumbers of the streamwise and spanwise flow directions, respectively, and $\overline {\boldsymbol{d}}(y)$ is some function of $y$ , following the formulation in Jovanović & Bamieh (Reference Jovanović and Bamieh2005). In our study we do not consider any specific form of $\overline {\boldsymbol{d}}(y)$ , instead the forcing component is viewed as the feedback in the interconnected system in figure 1 to account for the effect of the nonlinear advection term of the full NS system as will be detailed in § 2.3.

The frequency response operator that maps the harmonic forcing term applied to the state-space system in (2.2) to the resulting perturbation velocity field is given by (Zhou et al. Reference Zhou, Doyle and Glover1995)

(2.8) \begin{equation} \begin{aligned} \mathscr{H}\,(y;k_x,k_z,\omega )=[\mathscr{C}(k_x,k_z)[i\omega \mathbf{I} - \mathscr{A}(k_x,k_z)]^{-1}\mathscr{B}(k_x,k_z)](y). \end{aligned} \end{equation}

Here, $ \mathbf{I} $ is the identity matrix. As can be seen in (2.4) and (2.5), the operators $\mathscr{B}$ and $\mathscr{C}$ have three components each corresponding to different inputs and outputs, respectively.

The frequency response operator is used extensively in the input–output analysis of Jovanović & Bamieh (Reference Jovanović and Bamieh2005). Herein, we define an alternative operator, as proposed by Liu & Gayme (Reference Liu and Gayme2021), and used in several structured input–output works (such as Mushtaq et al. Reference Mushtaq, Bhattacharjee, Seiler and Hemati2023; Shuai et al. Reference Shuai, Liu and Gayme2023):

(2.9) \begin{equation} \mathscr{H}_{\,\boldsymbol{\nabla }}(y;k_x,k_z,\omega ) \equiv \text{diag} (\boldsymbol{\nabla },\boldsymbol{\nabla },\boldsymbol{\nabla }) \mathscr{H}\,(y;k_x,k_z,\omega ). \end{equation}

This frequency response operator of the system $\mathscr{H}_{\,\boldsymbol{\nabla }}$ relates the forcing term to the resulting gradients of velocity perturbations. This operator will be useful for the derivation of a perturbation-based stability criterion that can incorporate nonlinear interactions via a structured input–output formulation, as will be shown in § 2.3.

2.2. Choice of norms

The selection of the norm plays a critical role in the analysis of input–output behaviours in flow systems. Norms serve as the mathematical infrastructure for quantifying the magnitude of the response to a specific input and provide a basis for defining and computing the amplification between an input and its corresponding output. It is important to note that different choices of norms are associated with the computation of different quantities. For example, the $\mathscr{H}_2$ norm is specifically used to represent perturbation kinetic energy, as referenced in Jovanović & Bamieh (Reference Jovanović and Bamieh2005). In our research, we utilise the $\mathscr{H}_{\infty }$ norm instead of the $\mathscr{H}_2$ norm, which is related to the amplification of one specific mode (which is the most amplified) in the flow, as shown next.

Given the operator $\mathscr{H}\,(y;k_x,k_z,\omega )$ after discretisation in $y$ , we define the $\mathscr{H}_\infty$ system norm as (Zhou et al. Reference Zhou, Doyle and Glover1995)

(2.10) \begin{equation} \|{\mathscr{H}}\,\|_{\infty }(k_x,k_z) \equiv \sup _{\omega \in \mathbb{R}}{\overline {\sigma }[ \mathscr{H}\,(k_x,k_z,\omega )]}, \end{equation}

and similarly,

(2.11) \begin{equation} \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }(k_x,k_z) \equiv \sup _{\omega \in \mathbb{R}}{\overline {\sigma }[\mathscr{H}_{\,\boldsymbol{\nabla }}(k_x,k_z,\omega )]}. \end{equation}

Here, $\overline {\sigma }[ \boldsymbol{\cdot }]$ represents the largest singular value under the standard Euclidean inner product and $\sup _{\omega \in \mathbb{R}}{}$ represents the supremum operation over all temporal frequencies. Hence, the $\mathscr{H}_\infty$ norm does not quantify the energy amplification of the whole field, but rather the most amplified mode of the system due to harmonic temporal forcing.

2.3. Nonlinear frequency response

In this section we use structured input–output analysis to model the nonlinear response associated with the full NS system. This modelling of nonlinear interactions will be used in our stability analysis. We consider the full NS equations for velocity and pressure perturbations as we seek to incorporate nonlinear behaviours in our analysis. The NSE without a forcing term are as follows:

(2.12) \begin{equation} \begin{matrix} \dfrac {\partial \boldsymbol{u}}{\partial t} = -\boldsymbol{\bar {u}} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u} - \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\bar {u}} - \boldsymbol{\nabla }p + \dfrac {1}{\textit{Re}} \Delta \boldsymbol{u} - \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}, \\ \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} = 0. \end{matrix} \end{equation}

Equation (2.12) is very similar to (2.1), with the only difference being that the forcing term is swapped by the nonlinear advection term $\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}$ . Thus, we can model nonlinear interactions as a forcing input to the LNS system. We model the nonlinear advection term in the NSE using the block diagram representation in figure 1. Here, $\boldsymbol{u}_{\varXi }$ is the gain applied to the vectorised velocity gradient $\begin{bmatrix} {\nabla} ^T u & {\nabla} ^T v & {\nabla} ^T w \end{bmatrix}^T$ that retains the componentwise structure of the advection term $\boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}$ (Liu & Gayme Reference Liu and Gayme2021). To interpret the feedback interconnection in figure 1 physically, we view the nonlinear advection term as an internal forcing mechanism acting on the LNS dynamics. In a standard ‘unstructured’ analysis, the feedback block ${\boldsymbol {u}}_{\!\varXi}$ is treated as a generic operator bounded only by its magnitude. Thus, unstructured analysis can be thought of as a worst case model of the nonlinearity, as there are no limitations on the individual terms in ${\boldsymbol {u}}_{\!\varXi}$ . By contrast, the ‘structured’ feedback connection explicitly imposes constraints on ${\boldsymbol {u}}_{\!\varXi}$ , such that the value of some of the terms in it is fixed to be zero. Structure constraints are imposed to limit feedback pathways for the goal of replicating the structure of the nonlinearity in the NSE (Liu & Gayme Reference Liu and Gayme2021). As will be shown in § 2.4, the unstructured analysis results in more conservative, lower bounds on velocity perturbations.

Figure 1. Interconnection loop block diagram for the nonlinear interactions. Adapted from Liu & Gayme (Reference Liu and Gayme2021, figure 3).

The exact expression for ${\boldsymbol {u}}_{\!\varXi}$ that captures the nature of the nonlinear term in the NSE is (see Liu & Gayme Reference Liu and Gayme2021, equation (2.9))

(2.13) \begin{equation} {{\boldsymbol {u}}}_\varXi \equiv \begin{bmatrix} -{\boldsymbol {u}}^T & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & -{\boldsymbol {u}}^T & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & -{\boldsymbol {u}}^T \end{bmatrix}\!. \end{equation}

For the rest of this work, we consider a discretised version of the operator $\boldsymbol{u}_\varXi$ , which we denote as $\boldsymbol{U_\varXi }$ . The discretisation in the wall-normal direction causes $\boldsymbol{U}_\varXi$ to be a constant-valued matrix. The exact expression for $\boldsymbol{U}_\varXi$ can be found in Appendix A. To incorporate this nonlinearity, we calculate SSVs of the system. The SSVs are a generalisation of the concept of singular values for the case of a transfer function or a frequency response operator that is feedback interconnected to an uncertain matrix, which has a predetermined structure (Zhou et al. Reference Zhou, Doyle and Glover1995). In detail, similarly to the unstructured case, we calculate SSVs for the frequency response operator $\mathscr{H}_{\,\boldsymbol{\nabla }}$ by solving the following minimisation problem (Packard & Doyle Reference Packard and Doyle1993; Zhou et al. Reference Zhou, Doyle and Glover1995):

(2.14) \begin{equation} \mu _{\boldsymbol{\Delta }}[\mathscr{H}_{\,\boldsymbol{\nabla }}(k_x,k_z,\omega )] = \frac {1}{{\textit{min}}\{ \overline {\sigma } ({\boldsymbol {U}}_{\!\varXi} ) : {\boldsymbol {U}}_{\!\varXi} \in \boldsymbol{\Delta }, \text{det}[I - \mathscr{H}_{\,\boldsymbol{\nabla }}(k_x,k_z,\omega ) {\boldsymbol {U}}_{\!\varXi} ] = 0 \}}. \end{equation}

In (2.14), $\overline {\sigma }(\boldsymbol{\cdot })$ represents the largest singular value, $\text{det}[\boldsymbol{\cdot }]$ is the determinant of a matrix and ${\boldsymbol{\Delta}}$ is a set of matrices that includes all the matrices that have a certain structure. Solving the SSV problem under the same (or similar) structure as ${{\boldsymbol {U}}}_\varXi$ allows modelling the ‘worst case’ nonlinear response to external velocity perturbations. In this framework, the matrix ${{\boldsymbol {U}}}_\varXi$ is not known a priori. Rather, it is computed by solving (2.14) such that the smallest structured uncertainty (in terms of the largest singular value) that will cause instability of the system. While in the current study ${{\boldsymbol {U}}}_\varXi$ models the nonlinear interactions, it is treated in the SSV framework as modelling uncertainty. Thus, we refer to ${{\boldsymbol {U}}}_\varXi$ for the rest of this work as the structured uncertainty. A detailed discussion about uncertainty structures is provided in § 2.5.

We quantify amplification with nonlinear interactions that are modelled by a structured uncertainty ${\boldsymbol {U}}_{\!\varXi} \in \boldsymbol{\Delta }$ using the supremum over all temporal frequencies, i.e.

(2.15) \begin{equation} \|{\mathscr{H}\,\|_{\boldsymbol{\nabla }}}_{\mu _{\boldsymbol{\Delta }}}(k_x,k_z) \equiv \sup _{\omega \in \mathbb{R}}{\mu _{\boldsymbol{\Delta }}[ \mathscr{H}_{\,\boldsymbol{\nabla }}(k_x,k_z,\omega )]}, \end{equation}

this is the ‘worst case’ amplification of perturbation gradients under the uncertainty structure $\boldsymbol{\Delta}$ . This amplification is not an amplification of the energy of the whole flow field, but rather the gain applied to the most amplified mode. Thus, we note that $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}$ is a generalisation of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }$ for the case of interconnection of discretised $\mathscr{H}_{\,\boldsymbol{\nabla }}$ with a structured uncertainty ${\boldsymbol {U}}_{\!\varXi}$ . Additionally, we use the notation $\|{\boldsymbol{\cdot }}\|_{\mu }$ for consistency with previous works (Packard & Doyle Reference Packard and Doyle1993; Liu & Gayme Reference Liu and Gayme2021), despite this operator not being a proper norm since it does not satisfy the triangle inequality.

2.4. Small gain theorem formulation of stability

In this section we use the small gain theorem to develop an understanding of the stability of flow systems represented by the NSE. This theorem is used to define the stability of structured feedback interconnections in the structured input–output formulation, providing stability margins for wall-bounded shear flows (Liu & Gayme Reference Liu and Gayme2021; Mushtaq et al. Reference Mushtaq, Bhattacharjee, Seiler and Hemati2023, Reference Mushtaq, Bhattacharjee, Seiler and Hemati2024; Shuai et al. Reference Shuai, Liu and Gayme2023). Herein, we utilise the small gain theorem to establish a threshold on velocity disturbance magnitude that triggers flow instability. The interconnection in figure 1 is structured in a way compatible with the small gain theorem (see Zhou et al. Reference Zhou, Doyle and Glover1995, Chapter 11), which for a fixed-in-time $\boldsymbol{U}_\varXi$ is phrased in the following manner.

Theorem 1. Suppose $\mathscr{H}_{\,\boldsymbol{\nabla }}(k_x,k_z)$ is stable for a given wavenumber pair $(k_x,k_z)$ and let $0 \lt \gamma \lt \infty$ . The following is a sufficient condition for the feedback loop in figure 1 to be stable:

(2.16) \begin{equation} \|{\boldsymbol{U}_\varXi }\|_\infty \lt 1/\gamma \end{equation}

if and only if

(2.17) \begin{equation} \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}(k_x,k_z) \leqslant \gamma . \end{equation}

Here, $1/\gamma$ constitutes a finite bound on $\|{\boldsymbol{U}_\varXi }\|_\infty$ and $\gamma$ on $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}$ . Using singular value decomposition, it can be shown that (see Appendix A) $\|{{\boldsymbol{U}}_\varXi }\|_\infty$ is an approximation of the quantity $\max _{y\in \mathcal{Y}}\|{\boldsymbol{u}(y)}\|_2$ under the discretisation in the wall-normal direction. Here, $\mathcal{Y}$ is the relevant domain in the wall-normal direction corresponding to the flow system (for channel flows $\mathcal{Y}=[-1,1]$ , for boundary layer flows $\mathcal{Y}=[0,\infty )$ ); $\max _{y\in \mathcal{Y}}\|{\boldsymbol{u}(y)}\|_2$ denotes the largest velocity perturbation magnitude that is present in the flow. Combining equations (2.16) and (2.17) leads to the following condition for system stability:

(2.18) \begin{equation} \|{\boldsymbol{U}_\varXi }\|_\infty \lt \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}^{-1}(k_x,k_z). \end{equation}

Therefore, (2.18) provides a bound of value $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}^{-1}(k_x,k_z)$ on the magnitude of perturbations that are allowed in the flow to maintain stability. If the perturbation exceeds this bound, stability is no longer guaranteed and a transition to turbulence may occur. This formulation is capable of capturing the non-modal instability, i.e. transient growth mechanisms and flow instability triggered via bypass routes in the case of the presence of finite-size disturbances surpassing the threshold set by the small gain theorem criterion. Alternatively, for a given magnitude of external velocity perturbations, (2.18) can be used to bound the gain applied by the flow system, quantified by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}(k_x,k_z)$ . This interpretation relates disturbance size and the magnitude of non-modal growth allowed in the flow. Following (2.18), the wavenumber pair that determines the stability threshold bound for a given Reynolds number can be defined as

(2.19) \begin{equation} \left[\left(k^*_x,k^*_z\right)_{\mu _{\boldsymbol{\Delta }}}\right]({\textit{Re}})\equiv \arg \min \left\{ \left[\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}^{-1}(k_x,k_z)\right]({\textit{Re}}) \right\}\!. \end{equation}

Here, we denote this mode as the most dominant mode for a particular Reynolds number that arises from our structured analysis. We note that in (2.17) in Theorem1, the norm-like quantity $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}$ can be replaced by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }$ , representing a version of the small gain theorem in which the structure of the uncertainty is not considered (see Zhou et al. Reference Zhou, Doyle and Glover1995, Chapter 9). This operation yields the following relation:

(2.20) \begin{equation} \|{\boldsymbol{U}_\varXi }\|_\infty \lt {\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|^{-1}_{\infty }}(k_x,k_z). \end{equation}

The use of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }$ instead of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}$ is equivalent to replacing the set $\Delta$ with the set of all complex-valued matrices of compatible size (Packard & Doyle Reference Packard and Doyle1993; Zhou et al. Reference Zhou, Doyle and Glover1995; Liu & Gayme Reference Liu and Gayme2021), meaning that no particular feedback structure is considered. As denoted earlier, $\|{\boldsymbol{U}_\varXi }\|_\infty$ corresponds to the largest velocity perturbation magnitude (i.e. corresponds to $\max _{y\in \mathcal{Y}}\|{\boldsymbol{u}(y)}\|_2$ ) that is found in the flow. The inequality in (2.20) provides a stability criterion, which determines the maximal perturbation value allowed in the flow to guarantee the flow system stability.

Equivalently to the structured case, the most dominant mode for a given Reynolds number for the unstructured case can be defined as

(2.21) \begin{equation} \left[\left(k^*_x,k^*_z\right)_{\infty }\right]({\textit{Re}})\equiv \arg \min \left\{ \left[\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}(k_x,k_z)\right]({\textit{Re}}) \right\}\!. \end{equation}

In general, we note that the following inequality always holds for any complex-valued matrix $\boldsymbol{M}$ and some set of matrices $\boldsymbol{\Delta}$ (Zhou et al. Reference Zhou, Doyle and Glover1995):

(2.22) \begin{equation} \forall {\boldsymbol{\Delta },\boldsymbol{M}}: \mu _{\boldsymbol{\Delta }}(\boldsymbol{M}) \leqslant \overline {\sigma }(\boldsymbol{M}). \end{equation}

By replacing $\boldsymbol{M}$ with $\mathscr{H}_{\,\boldsymbol{\nabla }}$ and taking the supremum of both terms in (2.22), we obtain

(2.23) \begin{equation} \forall {\boldsymbol{\Delta }}: \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }(k_x,k_z). \end{equation}

From (2.23) it follows that

(2.24) \begin{equation} \forall {\boldsymbol{\Delta }}: \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}^{-1}(k_x,k_z) \geqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}(k_x,k_z). \end{equation}

Both $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ are bounds on the magnitude of the maximal velocity perturbation via structured and unstructured approaches, respectively. Our approach provides a threshold on disturbance magnitude to trigger instability. In both (structured and unstructured) cases, the wavenumber domain is the same, allowing for the same disturbances to be considered in both cases. Here, the structured case is a subset of an unstructured case. This, in turn, implies that there are more amplification routes/scenarios for the unstructured case than for structured cases that consider only a subset of such scenarios, limited by the imposed structure on the gain block. The structured approach takes into account the structure of ${\boldsymbol {U}}_{\!\varXi}$ to model the physics of the nonlinear interactions that are an integral part of the realistic transitional flow physics. Therefore, we expect $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}^{-1}$ to be a more representative criterion for stability, while $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ will be overly strict, and can predict lower bounds on velocity perturbation magnitude than what the actual flow allows.

2.5. Structures of uncertainty

In this section we discuss several possible structures for the structured uncertainty ${\boldsymbol {U}}_{\!\varXi}$ , that are used for approximating the nonlinear term in the NSE and obtaining the best approximation to $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }}}$ . These structures arise after a suitable discretisation in the wall-normal direction is applied to the system. The uncertainty structure that approximates the product of ${\boldsymbol {u}}_{\!\varXi}$ and the vector of velocity perturbation gradient reproducing the structure of the advection term in the Navier–Stokes equation is

(2.25) \begin{equation} \boldsymbol{\Delta }_{\boldsymbol{u}} = \left \{\begin{bmatrix} \Delta _u & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \Delta _u & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & \Delta _u \end{bmatrix}\! , \Delta _u = [\text{diag}(u_{\xi }), \text{diag}(v_{\xi }), \text{diag}(w_{\xi })]: u_{\xi },v_{\xi },w_{\xi } \in \mathbb{C}^{N_y \times 1} \right \}\!, \end{equation}

where $u_{\xi },v_{\xi },w_{\xi }$ can have complex entries because they represent Fourier-transformed velocity components; $N_y$ is the number of points that are used in the discretisation of the wall-normal direction. Equation (2.25) represents the structure that arises after a discretisation of (2.13). This structure ensures that the feedback interconnection loop in figure 1 accurately recreates the structure of the nonlinear advection term. Any additional terms that are not included in the set $\boldsymbol{\Delta }_{\boldsymbol{u}}$ may be regarded as fictitious nonlinear interactions that are not part of the term $\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u}$ and, thus, do not represent physical flow behaviours.

The structure of the uncertainty should ideally be similar to $\boldsymbol{\Delta }_{\boldsymbol{u}}$ for an accurate model of the physics of nonlinear interactions. The solution for uncertainties in $\boldsymbol{\Delta }_{\boldsymbol{u}}$ involves the solution of the optimisation problem in (2.14) with a very complex set of constraints that are required for preserving the structure in (2.25). Such a solution has not yet been found, as far as we know. Alternatively, approximating this structure of the uncertainties is considered, which we detail next. In this work we use two different sets of matrices that are simpler in structure and are meant to approximate the uncertainty structure from (2.14). These sets are as follows.

1. (i) Set $\boldsymbol{\Delta }_{r}$ : the set of block-diagonal matrices with three repeating full blocks with compatible sizes as was used in the works of Mushtaq et al. (Reference Mushtaq, Bhattacharjee, Seiler and Hemati2023, Reference Mushtaq, Bhattacharjee, Seiler and Hemati2024),

   (2.26) \begin{equation} \boldsymbol{\Delta }_{r} = \left \{ \begin{bmatrix} \Delta _r & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \Delta _r & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & \Delta _r \end{bmatrix} : \Delta _r \in \mathbb{C}^{N_y \times 3N_y}\right \}\!. \end{equation}
   Here, $\Delta _r$ can be any matrix of compatible dimensions ( $N_y \times 3N_y$ ).

2. (ii) Set $\boldsymbol{\Delta }_{\textit{nr}}$ : the set of block-diagonal matrices with three full, independent non-repeating blocks with compatible sizes, as was used in the works of Liu & Gayme (Reference Liu and Gayme2021), Shuai et al. (Reference Shuai, Liu and Gayme2023):

   (2.27) \begin{equation} \boldsymbol{\Delta }_{\textit{nr}} = \left \{\begin{bmatrix} \Delta _1 & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \Delta _2 & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & \Delta _3 \end{bmatrix} : \Delta _1,\Delta _2,\Delta _3 \in \mathbb{C}^{N_y \times 3N_y}\right \}\!. \end{equation}
   The difference here between $\boldsymbol{\Delta }_{\textit{nr}}$ and $\boldsymbol{\Delta }_{r}$ lies in the fact that $\boldsymbol{\Delta }_{\textit{nr}}$ does not include the constraint of equal blocks, as $\Delta _1$ , $\Delta _2$ and $\Delta _3$ can be different matrices.

Visualisation of the uncertainty matrices that correspond to the structures $\boldsymbol{\Delta }_{\boldsymbol{u}}$ , $\boldsymbol{\Delta }_{r}$ and $\boldsymbol{\Delta }_{\textit{nr}}$ are illustrated in figure 2. We note that solving the minimisation problem for $\boldsymbol{\Delta }_{r}$ is computationally more demanding than for $\boldsymbol{\Delta }_{\textit{nr}}$ . While $\boldsymbol{\Delta }_{\textit{nr}}$ imposes a single constraint on the uncertainty structure to be a set of block-diagonal matrices, $\boldsymbol{\Delta }_{r}$ imposes an additional constraint by requiring these blocks to be equal.

Figure 2. Illustration of uncertainty matrices with relevant structures: (a) uncertainty structure that is based on the feedback interconnection as defined in (2.25), (b) repeated block structure as defined in (2.26), (c) non-repeated block structure as defined in (2.27). Visualisations of the uncertainty matrices are shown for (a) ${\boldsymbol {U}}_{\!\varXi} \in \boldsymbol{\Delta }_{\boldsymbol{u}}$ , (b) ${\boldsymbol {U}}_{\!\varXi} \in \boldsymbol{\Delta }_{r}$ , (c) ${\boldsymbol {U}}_{\!\varXi} \in \boldsymbol{\Delta }_{\textit{nr}}$ .

The definitions in (2.25), (2.26) and (2.27) yield

(2.28) \begin{equation} \boldsymbol{\Delta }_{\boldsymbol{u}} \subset \boldsymbol{\Delta }_{r} \subset \boldsymbol{\Delta }_{\textit{nr}}\subset \mathbb{C}^{3N_y \times 9N_y}. \end{equation}

These relations between sets are manifested in the values of the SSVs for each set. Based on the monotonicity of the SSV with respect to set inclusion (see Scherer Reference Scherer2001, § 4.10), solving the minimisation problem (2.14) over these sets of matrices that satisfy the relations in (2.28), yields

(2.29) \begin{equation} \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{u}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }(k_x,k_z). \end{equation}

As we will show next, this hierarchical relationship allows us to explain the difference in the predictions of the most dominant mode by unstructured and structured input–output methods. Moreover, the norm-like property of the smallest set $\boldsymbol{\Delta }_{\boldsymbol{u}}$ is $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{u}}}(k_x,k_z)$ , which inverse ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{u}}}^{-1}$ ) would provide the least tight and most accurate bound to perturbation magnitude to keep the system stable. In particular, the inequality in (2.20) can serve as a generalised stability criterion, which is more physically justified to study pathways to transition, in contrast to classical stability theory, in which the mathematical formulation is restricted to infinitesimally small perturbations. However, as mentioned previously, the solution to such a set requires the solution of the optimisation problem in (2.14), which is elaborate and has not yet been found. The best next approximation of this set would be to use $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , which is the most representing bound on velocity perturbation magnitude that we can compute, while $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ provide more restricting bounds, respectively.

4.1. Couette base flow

In this section we present the results of our analysis for the Couette base flow profile of the form $U=y$ . In figure 3 we show the most dominant flow structures in terms of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}(k_x,k_z)$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}(k_x,k_z)$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}(k_x,k_z)$ for both of the Reynolds numbers ${\textit{Re}}=358$ (figure 3 a) and ${\textit{Re}}=8000$ (figure 3 b). The Reynolds number ${\textit{Re}}=358$ was chosen to be the same as used in Liu & Gayme (Reference Liu and Gayme2021) and Mushtaq et al. (Reference Mushtaq, Bhattacharjee, Seiler and Hemati2024), to validate our results with these works. The value ${\textit{Re}}=8000$ was selected arbitrarily after the post-critical value observed in experiments. We chose a very high value to demonstrate that the stability characteristics do not change drastically when changing the Reynolds number for Couette flow. Over these contour maps, we overlay the modes from table 1. The most dominant mode is marked by an X symbol. If the most dominant mode is not denoted in table 1, it is coloured in cyan. We note that the relationships between the three types of amplification satisfy the relation in (2.29) for both Reynolds numbers, i.e. $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }(k_x,k_z)$ . Here, $\|{\mathscr{H}\,\|_{\boldsymbol{\nabla }}}^{-1}_{\infty }$ reaches values lower than $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|^{-1}_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|^{-1}_{\mu _{\boldsymbol{\Delta }_{r}}}$ , respectively, as discussed in § 2.5.

Figure 3. Contour plots in logarithmic scale of (from left to right): $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ at (a) ${\textit{Re}}=358$ and (b) ${\textit{Re}}=8000$ for Couette flow. The most dominant mode is marked by an X (cyan colour indicates that the mode’s $(k_x,k_z)$ value is not denoted in the table), whereas the other modes from table 1 are marked by coloured circles (magenta – DLR mode, black – TS mode, red – SPS mode, blue – DHR mode). Results are shown for (a) ${\textit{Re}}=358$ and (b) ${\textit{Re}}=8000$ .

The use of a structured approach for both cases (repeated and non-repeated approaches) leads to an increased stability threshold. The dominant structures that arise in the unstructured case are streamwise elongated structures, whereas oblique modes become the most dominant structures in structured cases, which is more clearly evident in the non-repeated blocks case. The latter result agrees better with DNS and nonlinear optimal perturbation (NLOP)-based predictions than with traditional linear input–output methods (Prigent et al. Reference Prigent, Grégoire, Chaté and Dauchot2003; Duguet et al. Reference Duguet, Schlatter and Henningson2010b ). The contour map of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ shows a most dominant oblique wave mode of $(k_x,k_z) \approx (0.1,1)$ , which overpowers the streamwise elongated structures that are also evident on the same colour map by an order of magnitude. It was stated in Liu & Gayme (Reference Liu and Gayme2021), ‘this result is consistent with findings of Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998) showing that oblique waves require less perturbation energy to trigger turbulence in plane Couette flow than streamwise vortices.’ As we will show next, the observed dominance of oblique waves is affected by the type of uncertainty that is considered and the Reynolds number. In the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , where repeated uncertainty is considered, the lowest stability threshold region that is evident in $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ spreads over a wide range of modes in the bottom left quadrant of the domain, corresponding to a wide range of dominant oblique modes. This result is consistent with the results of Mushtaq et al. (Reference Mushtaq, Bhattacharjee, Seiler and Hemati2024) for ${\textit{Re}}=358$ .

We note that while the relationship in (2.29) shows that $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ provides a closer bound than $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ to the accurate value $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\mu }}}^{-1}$ on velocity perturbations from a stability viewpoint, there is no guarantee that it represents more accurately the contour plots in the wavenumber domain. Approximations of the structure of the nonlinear term in the NSE by either non-repeating or repeating blocks still may include additional feedback pathways that are not physically associated with nonlinear flow physics, but to a significantly smaller extent than in the unstructured case. Therefore, considering two structured cases (repeated and non-repeated blocks) is important in our analysis, where common features may be associated with flow physics, whereas the differences require further investigation. One such common feature is the shift of dominant modes to lower $k_x$ values with Reynolds number for unstructured and both structured cases, which means this trend is associated with a linear mechanism. This trend agrees with results obtained from stability analysis of optimal disturbance growth (Butler & Farrell Reference Butler and Farrell1992; Schmid et al. Reference Schmid, Henningson and Jankowski2002), where it is shown that for Couette flow, the maximum growth is achieved for $k_x Re \sim \text{C}$ , where $C$ is some constant value, which means that for a given non-zero $k_z$ , an increase in the Reynolds number results in a decrease of $k_x$ , which corresponds to oblique and $\varLambda$ -shaped structures that transform to streaks at higher Reynolds numbers (Hwang & Cossu Reference Hwang and Cossu2010b ; Dokoza & Oberlack Reference Dokoza and Oberlack2023). Observing $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (with non-repeated uncertainty) for both Reynolds numbers, we see that there is a change of the most dominant mode with a Reynolds number increase, where the DLR mode (indicated by a magenta coloured circle) of $(k_x,k_z)=(0.196,0.628)$ for ${\textit{Re}}=358$ in figure 3(a) has a larger stability threshold associated with it than the DHR mode (indicated by blue coloured circle) of $(k_x,k_z)=(0.01, 0.13)$ for ${\textit{Re}}=8000$ in figure 3(b). Additionally, in both plots, we added a horizontal dashed black line overlaid on the contour maps of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}$ . This horizontal line represents the border between two regions of relatively unstable modes in the contour maps of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ for both Reynolds numbers, Re = 358 and Re = 8000, respectively. This line migrates towards a lower $k_x$ value with the Reynolds number increase, due to the shift of dominant modes to lower $k_x$ values. Thus, it is evident that there is a set of oblique modes with this streamwise wavenumber (of the dashed line) and spanwise wavenumber values of $k_z\lt 1$ that will exhibit increased stability compared with neighbouring modes.

In the results for the higher Reynolds number, lower stability threshold values are observed for all of the quantities $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}(k_x,k_z)$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}(k_x,k_z)$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}(k_x,k_z)$ . In figure 3(a) the smallest perturbations that cause instability have a magnitude of $10^{-3.6}$ for the unstructured case. In figure 3(b), significantly lower values appear, with the smallest perturbations, which can cause instability, being of magnitude $10^{-5.5}$ for the unstructured case.

Comparing only two Reynolds numbers is insufficient to reveal the interplay between the dominant structures that are evident in these contour maps with Reynolds number variation. Therefore, next, we focus our analysis on key preselected modes of interest from these contour maps and monitor their behaviour for a wide range of Reynolds numbers. In figure 4 we show the evolution of the preselected modes that are listed in table 1 as a function of Reynolds number in the range ${\textit{Re}} \in [100,10\,000]$ . The trends from each plot are accompanied by insets of zoomed-in views for certain limited Reynolds number ranges at different locations, revealing the interesting interplay between the mode curves for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ . In particular, the following insights are gained from these plots.

Figure 4. Evolution curves of imposed thresholds due to flow structures associated with preselected modes of interest denoted in table 1 (magenta solid - DLR mode, black dotted - TS mode, red dashed - SPS mode, blue dashed-dotted - DHR mode) for Couette base flow as a function of the Reynolds number in terms of (a) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , (b) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and (c) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ .

First, for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ (figure 4 a), we observe that the SPS mode (red curve) remains the most dominant, i.e. providing the lowest threshold, through the entire range of Reynolds numbers. A higher threshold is shown for the DLR mode (magenta curve) that overpowers the DHR mode (blue curve) for Reynolds numbers up to ${\textit{Re}}=750$ , and vice versa afterward. The SPS modes are extensively reported in the literature to be the dominant structures for a variety of base flows at different Reynolds numbers when using linear input–output analysis (Bamieh & Dahleh Reference Bamieh and Dahleh2001; Jovanovic Reference Jovanovic2004; Jovanović & Bamieh Reference Jovanović and Bamieh2004, Reference Jovanović and Bamieh2005; Schmid Reference Schmid2007). Thus, it is not surprising to us that the dominance of the SPS mode for a wide range of Reynolds numbers in the linear analysis represented by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }$ is in agreement with previous works. In contrast, this mode is not always dominant if structured analysis is considered. In particular, in the non-repeated case, for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (figure 4 b), the threshold that is associated to the SPS mode is always higher than the DLR mode (magenta) and the DHR mode (blue) thresholds. The DLR and DHR modes are most dominant for the entire Reynolds number range, where the oblique DLR mode (magenta curve) sets a lower threshold on disturbance to make the flow unstable than the DHR mode (blue curve) for Reynolds number up to ${\textit{Re}}\approx 3800$ (denoted by a red vertical dashed line), and vice versa afterward. In the repeated case, for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ (figure 4 c), we observe first that the oblique DLR mode (magenta curve) provides the highest threshold (out of the four modes that were tested) for the entire range of Reynolds numbers that were considered. Second, the oblique DHR mode (blue curve) is the most dominant mode for ${\textit{Re}}\lt 900$ and ${\textit{Re}}\gt 3500$ , whereas in the range $900\lt Re\lt 3500$ , the SPS mode (red curve) imposes a slightly lower threshold on perturbation magnitudes for instability to occur than the DHR mode. The blue curve trend is shown to be decreasing, but with a subtle kink at ${\textit{Re}}\approx 1900$ where the DHR mode drops close to the DLR mode value and then grows again. This kink can be explained by inspecting the contour maps of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{r}}}}^{-1}$ . In figures 3(a) and 3(b) we mark dashed horizontal lines representing modes that are stable for larger perturbations, dividing two distinct regions of relative instability for both Reynolds numbers, ${\textit{Re}}=358$ and ${\textit{Re}}=8000$ , respectively. This horizontal line is located at $k_x\approx 0.05$ for ${\textit{Re}}=358$ , i.e. above the $k_x$ value of the DHR mode (blue circle) – which is defined by the wavenumber pair $(k_x,k_z)=(0.01, 0.13)$ – whereas it shifts towards $k_x\approx 0.0025$ , below the $k_x$ value of the DHR mode at ${\textit{Re}}=8000$ . This means that when this line migrates towards lower streamwise wavenumbers with a Reynolds number increase, it intersects the $k_x=0.01$ value at ${\textit{Re}}\approx 1900$ where this line matches the same $k_x$ value of the DHR mode.

Second, the TS mode (black curve) imposed threshold is the highest in the unstructured case for all Reynolds numbers, whereas for both $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ (figure 4 c) and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (figure 4 b), the TS mode imposed threshold is higher than the SPS mode threshold (red curve) for almost the entire range of Reynolds numbers that were considered, besides a limited range between $200\lt Re\lt 700$ . When using the structured analysis, the most dominant mode changes with the Reynolds number, demonstrating the importance of looking at a range of Reynolds numbers as opposed to a single one to understand the full physical behaviour and stability characteristics of the flow system at hand.

Third, we note that for all Reynolds numbers and modes tested, the curve trends of the most dominant modes agree with the relationship in (2.29) that holds for all Reynolds numbers as expected. In particular, one can observe that the graphs of all of the chosen modes for both of the structured quantities, $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , remain less spread out than in the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ through the whole range of Reynolds numbers, which is mostly apparent for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ . This observation correlates with the number of constraints imposed on the amplification routes that yield the hierarchical relation of subsets in (2.28). Here, all of the modes impose similar thresholds on the magnitude of perturbations that can be supported by the flow without losing stability. The stability margin represented by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ is significantly more strict (with the lowest threshold) than in the structured case with repeated block uncertainty, which is represented by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ . For example, at ${\textit{Re}}=1000$ , the system represented by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ can remain stable up to perturbations of a magnitude of $\|{\boldsymbol{u}}\| \sim O(10^{-4.5})$ , whereas for the system represented by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ , the perturbations magnitude increase to a magnitude of $\|{\boldsymbol{u}}\| \sim O(10^{-3.5})$ , and the system represented by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ can support perturbations up to a magnitude of $\|{\boldsymbol{u}}\| \sim O(10^{-3})$ . We consider here the results of previous works (Reddy et al. Reference Reddy, Schmid, Baggett and Henningson1998; Duguet et al. Reference Duguet, Brandt and Larsson2010a , Reference Duguet, Monokrousos, Brandt and Henningson2013), which used optimisation methods and DNS to predict the minimal perturbation kinetic energy that induces transition to turbulence. These works predict a minimal kinetic energy between $O(10^{-6})$ and $O(10^{-5})$ to cause transition at ${\textit{Re}}=1000$ . Kinetic energy is proportional to the square of velocity magnitude; therefore, it corresponds to the disturbance of $O(10^{-3})$ and $O(10^{-2.5})$ , where it is clear that the predictions by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ are overly strict whereas $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ produce stability thresholds that better agree with DNS results. Since the main cause for the large difference in stability thresholds between the unstructured and structured approaches is associated with the SPS mode, we conclude that the unstructured analysis exaggerates the effect of SPS compared with the real flow physics. A more thorough comparison between the predictions of our stability criterion and the results from numerical simulations and experimental works is provided in § 5.

Figure 5. Contour plots in logarithmic scale of (from left to right): $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu_{\boldsymbol{\Delta}_{r}}}^{-1}$ at (a) ${\textit{Re}}=690$ , (b) ${\textit{Re}}=5700$ and (c) ${\textit{Re}}=6000$ for plane Poiseuille flow. The most dominant mode is marked by an X (cyan colour indicates that the mode’s $(k_x,k_z)$ value is not denoted in the table), whereas the other modes from table 1 are marked by coloured circles (magenta - DLR mode, black - TS mode, red - SPS mode). Results are shown for (a) ${\textit{Re}}=690$ , (b) ${\textit{Re}}=5700$ , (c) ${\textit{Re}}=6000$ .

4.2. Plane Poiseuille base flow

In this section we present stability analysis results obtained using the plane Poiseuille base flow profile of the form $U=1-y^2$ for three-dimensional perturbations. In figure 5 we show the most dominant flow structures in terms of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}(k_x,k_z)$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}(k_x,k_z)$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}(k_x,k_z)$ for the Reynolds numbers ${\textit{Re}}=690$ (figure 5 a), ${\textit{Re}}=5700$ (figure 5 b) and ${\textit{Re}}=6000$ (figure 5 c). The Reynolds number ${\textit{Re}}=690$ is the same as used in Liu & Gayme (Reference Liu and Gayme2021) and Mushtaq et al. (Reference Mushtaq, Bhattacharjee, Seiler and Hemati2023) and is chosen to validate our results. The Reynolds numbers ${\textit{Re}}=5700$ and ${\textit{Re}}=6000$ are chosen as one is below and one is above the critical Reynolds number, ${\textit{Re}}_{c}=5772$ , predicted by LST for plane Poiseuille flow. Over these contour maps, we overlay the modes from table 1. The most dominant mode is marked by an X symbol. If the most dominant mode is not denoted in table 1, it is coloured in cyan. For the three Reynolds numbers that were tested, the relationship of the norm-like quantities in (2.29), i.e. $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }(k_x,k_z)$ , is satisfied as in the Couette case.

The use of structured approaches shows that the lowest threshold is governed by dominance of oblique modes rather than the streamwise elongated streaky structures in the unstructured case, similarly to the Couette base flow. Structured case results were found to better agree with the NLOP-based predictions study of Farano et al. (2015), in which it was shown that hairpin vortex structures (associated with oblique modes) can be the outcome of a nonlinear optimal growth process in plane Poiseuille flow, in a similar way as streaky structures can be the result of a linear optimal growth mechanism.

It was stated in Liu & Gayme (Reference Liu and Gayme2021), ‘This result is consistent with findings of Reddy et al., (Reddy et al. Reference Reddy, Schmid, Baggett and Henningson1998) showing that oblique waves require slightly less perturbation energy to trigger turbulence in plane Poiseuille flow than streamwise vortices.’ Similarly to the case of Couette base flow, herein we show that this conclusion is highly affected by the type of uncertainty that is considered and the Reynolds number. In the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ the smallest perturbation magnitudes are associated to a wide range of modes, manifested as a uniform region in the bottom left quadrant of the $k_x,k_z$ plane without any obvious dominant oblique mode. Moreover, observing $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (non-repeated uncertainty) for ${\textit{Re}}=5700$ (figure 5 b) and ${\textit{Re}}=6000$ (figure 5 c) reveals that as in the case of Couette base flow, the most dominant mode is highly dependent on the Reynolds number. While in figure 5(a) the most dominant mode is the DLR mode (indicated by a magenta coloured circle) of $(k_x,k_z)=(1.561,0.692)$ , in figure 5(b) the most dominant mode is the TS mode (indicated by a black-coloured circle) of $(k_x,k_z)=(1.02,10^{-2})$ . In figure 5(c), where the Reynolds number is increased further to ${\textit{Re}}=6000$ , the most dominant mode in the sense of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ shifts again to be an oblique mode located between the DLR mode and the TS mode on the $k_x,k_z$ domain.

Increasing the Reynolds number from the subcritical value ${\textit{Re}}=690$ to ${\textit{Re}}=5700$ (as shown in figure 5 b) leads to the dominance of the TS mode and slightly oblique TS modes (represented by a streamwise wavenumber of $k_x \approx 1$ and a spanwise wavenumber of $k_z \ll 1$ ), which impose a very low threshold for instability to occur for unstructured and both structured cases. Further increasing the Reynolds number to ${\textit{Re}}=6000$ (as shown in figure 5 c) causes a difference in the stability characteristics between the structured approaches ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ ) and the unstructured ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ ) approaches. While in the contour maps of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ , TS modes and slightly oblique modes govern the lowest threshold on perturbations up to a magnitude of $O(10^{-6})$ , in the contour map of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ the imposed threshold governed by the same modes drops to an infinitesimally small value, i.e. the flow will lose stability for perturbations of any size (even asymptotically small), indicated by the red rectangle in figure 5(c), which corresponds to $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}\rightarrow 0$ .

Next, we study the effect of Reynolds numbers on stability threshold over a wide range in plane Poiseuille flow. In figure 6 we show the evolution of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ for the chosen modes of interest as a function of the Reynolds number in the interval ${\textit{Re}} \in [100,10000]$ . The trends in each graph are accompanied by insets of zoomed-in views for a limited Reynolds number range at different locations, revealing the interesting interplay between the dominance of modes for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ . In particular, four insights are gained from these plots.

Figure 6. Evolution curves of imposed thresholds due to flow structures associated to preselected modes of interest denoted in table 1 (magenta solid - DLR mode, black dotted - TS mode, red dashed - SPS mode) for plane Poiseuille base flow as a function of the Reynolds number in terms of (a) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , (b) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and (c) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ .

First, for the unstructured case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , we observe that the SPS mode (red) is the most dominant up until very close to ${\textit{Re}}=5772$ , the critical Reynolds number that is predicted by LST (Orszag Reference Orszag1971; Schmid et al. Reference Schmid, Henningson and Jankowski2002). The TS mode (black) becomes most dominant, imposing the lowest threshold at proximity and after the critical Reynolds number. When using both of the structured approaches ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ ), the TS mode becomes the most dominant mode at a lower Reynolds number compared with the unstructured analysis. In the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ , the TS mode becomes the most dominant after ${\textit{Re}} \approx 4100$ (marked by a dashed red line in figure 6 b) and in the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , it becomes dominant at ${\textit{Re}} \approx 2175$ (marked by a dashed red line in figure 6 c). A recent study by Huang et al. (2024) showed that in the case of two-dimensional plane Poiseuille flow, TS waves are the optimal perturbation and can cause a sustained transition to turbulence for ${\textit{Re}} \geqslant 2400$ , in agreement with the results obtained for the case of repeated blocks. considering structured analysis methods. The non-repeated case ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ ) and repeated case ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ ) differ mainly in the stability threshold that are imposed by the DLR mode (magenta), which represents an oblique wave mode. The DLR mode imposes a lower stability threshold in the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ compared with $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ for the entire range of Reynolds numbers that were tested, similarly to the Couette base flow.

Second, the TS mode curve of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ (figure 6 a) exhibits an asymptotic behaviour where it approaches zero at ${\textit{Re}}=5772$ . The fact that $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ approaches zero for the critical mode of actuation and the critical Reynolds number predicted by LST is, of course, not a coincidence. Similarly to figure 12, the results of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1} \rightarrow 0$ correspond to the flow being unstable even for asymptotically small perturbations, and thus, results that match the predictions of LST are to be expected. It is important to note that even though close to the critical Reynolds number and beyond, a TS wave sets the lowest threshold for disturbance to trigger transition, it does not necessarily translate to transition being caused by a TS mode. Any mode or combination of modes that surpasses the corresponding stability threshold will destabilise the flow system, which consequently may trigger transition. For example, due to a transient growth scenario or the presence of finite-size disturbances in a noisy flow environment.

Third, in the graphs of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (figure 6 b) and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ (figure 6 c), the TS mode curve reaches a finite value minimum exactly at the critical Reynolds number value that is predicted by LST. After this minimum, the values of both $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ increase. This minimum has a finite value in contrast to the behaviour of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , where the TS mode threshold curve asymptotes towards zero. We explain this observation in more detail in § 5, where we show the associated stability diagrams for both unstructured and structured cases. At higher post-critical Reynolds numbers, the most dominant mode shifts to an oblique mode as the Reynolds number increases. Lastly, the curve trends of the dominant modes that arise in our analysis agree with the relationship in (2.29) for all Reynolds numbers.

4.3. Blasius base flow

Here, we analyse the effect of the Reynolds number on the stability and the dominant structures in Blasius flow. The Blasius base flow profile is obtained numerically by solving the Blasius equation $f''' + ff'' = 0$ (Schlichting & Gersten Reference Schlichting and Gersten2016), where $f^{\prime}$ is the normalised base flow velocity varying with a function of the similarity variable $\eta =y/\delta ^*$ and $\delta ^*$ is a displacement thickness. The boundary conditions used for the solution of the Blasius equation are $f(0)=f'(0)=0$ and $\lim _{\eta \to \infty } f'(\eta ) = 1$ . In figure 7 we show the most dominant flow structures in terms of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}(k_x,k_z)$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}(k_x,k_z)$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}(k_x,k_z)$ for the Reynolds numbers ${\textit{Re}}=400$ (figure 7 a), ${\textit{Re}}=500$ (figure 7 b) and ${\textit{Re}}=530$ (figure 7 c). The Reynolds numbers ${\textit{Re}}=500$ and ${\textit{Re}}=530$ are chosen as one is below and one is above the critical Reynolds number, ${\textit{Re}}_c=520$ , predicted by LST for Blasius flow (Schmid et al. Reference Schmid, Henningson and Jankowski2002). The Reynolds number ${\textit{Re}}=400$ was chosen to observe the stability characteristics of the flow at a subcritical Reynolds number, which is further away from the critical value. Over these contour maps, we overlay the modes from table 1. The most dominant mode is marked by an X symbol. If the most dominant mode is not denoted in table 1, it is coloured in cyan. For the three Reynolds numbers that were tested, the relationship between norm-like quantities in (2.29) ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}(k_x,k_z) \leqslant \|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }(k_x,k_z)$ ) is satisfied, similarly to the cases of Couette flow and plane Poiseuille flow. This in turn shows that the relation in (2.29) is invariant to the base flow geometry considered (at least for the cases we tested).

Figure 7. Contour plots in logarithmic scale of (from left to right): $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ at (a) ${\textit{Re}}=400$ , (b) ${\textit{Re}}=500$ and (c) ${\textit{Re}}=530$ for Blasius flow. The most dominant mode is marked by an X (cyan colour indicates that the mode’s $(k_x,k_z)$ value is not denoted in the table), whereas the other modes from table 1 are marked by coloured circles (magenta - DLR mode, black - TS mode, red - SPS mode). Results are shown for (a) ${\textit{Re}}=400$ , (b) ${\textit{Re}}=500$ , (c) ${\textit{Re}}=530$ .

The most dominant mode of actuation is highly dependent on the Reynolds number and the type of structured uncertainty considered (repeated versus non-repeated). In the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ , the most dominant mode at ${\textit{Re}}=400$ (figure 7 a) is the DLR mode, which is an oblique mode. However, increasing the Reynolds number causes the most dominant mode to change, as can be observed in figures 7(b) and 7(c). When using repeated uncertainty ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ ), the DLR mode and neighbouring modes have significantly lower stability thresholds.

For Reynolds numbers below ${\textit{Re}}_c$ , the quantities $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ all display similar characteristics of TS modes and slightly oblique TS modes (represented by a streamwise wavenumber of $k_x \approx 0.3$ and a spanwise wavenumber of $k_z \ll 1$ ) being the most dominant. At ${\textit{Re}}=400$ (displayed in figure 7 a), there is a band of TS and slightly oblique TS modes with relatively small stability thresholds for both the unstructured and structured approaches. When increasing the Reynolds number to ${\textit{Re}}=500$ (as shown in figure 7 b), the TS and slightly oblique TS modes become unstable for significantly smaller perturbations and are the most dominant for both $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ . In the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , these modes have slightly higher stability thresholds than the dominant streamwise elongated streak mode that imposes the lowest threshold. Further increasing the Reynolds number to ${\textit{Re}}=530$ (as shown in figure 7 c) causes a difference in the stability characteristics of TS and slightly oblique TS modes between the structured approaches ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ ) and the unstructured ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ ) approach. In the contour map of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , the region of slightly oblique TS modes becomes unstable even for asymptotically small perturbations, as in these regions the stability threshold approaches zero (marked as red regions in the contour map). In the structured cases, the most dominant mode of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ shifts to a larger value of $k_z$ , i.e. the mode becomes more oblique, similarly to the case of plane Poiseuille flow.

Next, we explore the stability thresholds variation over a wider range of Reynolds numbers. In figure 8 the evolution of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ for the chosen modes of interest as a function of the Reynolds number in the interval ${\textit{Re}} \in [100,1000]$ is shown. The trends in each graph are accompanied by insets of zoomed-in views for a certain Reynolds number range at different locations where intersections between different mode curves are observed for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ . In particular, these plots reveal the following observations.

Figure 8. Evolution curves of imposed thresholds due to flow structures associated to preselected modes of interest denoted in table 1 (magenta solid - DLR mode, black dotted - TS mode, red dashed - SPS mode) for Blasius base flow as a function of the Reynolds number in terms of (a) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , (b) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and (c) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ .

First, for the unstructured case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , we observe that the SPS mode (red) is the most dominant up to ${\textit{Re}}=520$ , the critical Reynolds number as predicted by LST (Schmid et al. Reference Schmid, Henningson and Jankowski2002). After this critical Reynolds number, the TS mode (black) becomes the most dominant mode. When using both structured analysis methods ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ ), the TS mode becomes the most dominant mode at lower-Reynolds-number values as we detail next. In the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ , the TS mode becomes the most dominant mode at ${\textit{Re}} \approx 430$ (marked by a dashed red line in figure 8 b) and in the case of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ , it becomes the most dominant as early as ${\textit{Re}} \approx 370$ (marked by a dashed red line in figure 8 c). The non-repeated case ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ ) and the repeated case ( $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ ) differ only in the magnitude of perturbations that cause instability of the DLR mode (magenta), which represents an oblique wave mode. The perturbation magnitude that causes instability of the DLR mode is slightly smaller for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ compared with $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ for all of the tested range of Reynolds numbers, similarly to the cases of Couette and plane Poiseuille flow, and in agreement with (2.29).

Second, in the graph of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ (figure 8 a) there is an asymptotic behaviour of the TS mode threshold curve that is approaching zero at ${\textit{Re}}=520$ . This asymptotic behaviour is explained by LST, which predicts that the chosen TS mode causes the Blasius base flow to become unstable for ${\textit{Re}} \gt 520$ (Schmid et al. Reference Schmid, Henningson and Jankowski2002) for asymptotically small perturbations. In the graphs of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (figure 8 b) and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ (figure 8 c), there is a finite amplitude minimum at the critical Reynolds number predicted by LST, reminiscent of the one observed for the case of plane Poiseuille flow. After this minimum, the value of the TS modes in both $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ increase. As in the case of plane Poiseuille, this minimum corresponds to the flow being extremely sensitive to perturbations and very close to the verge of instability. However, this minimum still reaches a finite value in contrast to the behaviour of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ (that approaches zero) due to the imposed structure of the nonlinear term in the Navier–Stokes equation. The existence of this minimum in the graph of the TS mode will be explained similarly to the case of plane Poiseuille flow in more detail in § 5, where we show the associated stability diagrams for both unstructured and structured cases.

Third, using structured analysis creates a much more complicated interplay between the different modes than in the unstructured analysis. In detail, for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (figure 8 b), the SPS mode is the most dominant for ${\textit{Re}} \lt 255$ and, for higher Reynolds numbers, the most dominant mode is the DLR mode, in the range $255 \lt Re \lt 430$ . The TS mode is most dominant for $430 \lt Re \lt 590$ and, for higher Reynolds numbers, the DLR mode is again the most dominant. For $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ (figure 8 c), the SPS mode is the most dominant for ${\textit{Re}} \lt 367$ . The TS mode is most dominant for $367 \lt Re \lt 770$ and, for higher Reynolds numbers, the DLR mode becomes the most dominant. The results for the repeated blocks analysis in figure 8(c) show that after ${\textit{Re}}=800$ the oblique DLR mode dominates the flow, setting a threshold value, which is the lowest but very close to the SPS mode curve for both structured cases. Support for such results may be found in the experimental study by Elofsson & Alfredsson (Reference Elofsson and Alfredsson2000), where it was noted that for the post-critical transition process in the Blasius boundary layer, the combined effect of non-modal growth of streaks and a second stage with exponential growth of oblique waves is responsible for the initiation of the final breakdown stage. The intricate relationship between the different modes that set the stability threshold, as depicted in figure 8, highlights the necessity of analysing a broad range of Reynolds numbers. Limiting the examination to a single Reynolds number would result in overlooking significant physical characteristics of the flow system in this case.

Lastly, we note that for all Reynolds numbers, the curve trends of the imposed thresholds that are set by the most dominant modes agree with the relationship we derived in (2.29). As explained in § 2.5, $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ is the best approximation available to us for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{u}}}^{-1}$ , which incorporates the accurate structure of the nonlinear term in the NSE. Thus, we expect $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ to be the bound on velocity perturbations, which is the closest to the accurate value $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{u}}}^{-1}$ from a stability viewpoint. As denoted in previous sections, there is no guarantee that it represents more accurately the contour plots in the wavenumber domain compared with the $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ case. Using the formulation of stability with the small gain theorem shows that instability can occur in Blasius flow at any Reynolds number. Specifically, instability can occur at smaller Reynolds numbers than the critical value predicted via LST, ${\textit{Re}}=520$ , given that perturbations of finite magnitude are present in the flow, surpassing the threshold provided by our stability criterion.

The results presented here can be used to explain experimental evidence of subcritical transition in flat plate boundary layers. In the work of Asai & Nishioka (Reference Asai and Nishioka1995), subcritical transition is triggered by energetic hairpin eddies. The boundary layer is found to be stable below ${\textit{Re}}_x=2.8\times 10^{4}$ , which is equivalent to ${\textit{Re}}=288$ (based on the displacement thickness) for a perturbation magnitude of $0.3$ % (as shown in Asai & Nishioka Reference Asai and Nishioka1995 (figure 3) at $x \approx 100$ mm that correspond to this Reynolds number) at the leading edge (for the case of weak forcing). This magnitude corresponds to a value of $10^{-2.5}$ , which is in relatively good agreement with our analysis for the structured two cases, where the imposed threshold by the most dominant mode is $O(10^{-3})$ , which is the SPS mode at this Reynolds number. Subcritical transition induced by finite-size perturbations is also observed in DNS studies, such as Cherubini et al. (Reference Cherubini, De Palma, Robinet and Bottaro2011) and Vavaliaris, Beneitez & Henningson (Reference Vavaliaris, Beneitez and Henningson2020), where it was found that a finite threshold value exists for ${\textit{Re}}=300$ and ${\textit{Re}}=275$ , respectively. In particular, the energy threshold for transition was computed in Vavaliaris et al. (Reference Vavaliaris, Beneitez and Henningson2020), where a value of $E_c=3.64\times 10^{-2}$ was obtained. This value refers to the combined energy of the velocity perturbation field inside the entire spatial domain, whereas in the current work, we focus only on the specific perturbation of the largest magnitude. For that reason, a comparison between the results can be performed by dividing the energy by the volume of the computational domain reported in Vavaliaris et al. (Reference Vavaliaris, Beneitez and Henningson2020) to obtain an energy density value of $1.2\times 10^{-8}$ . The square root of this value corresponds to the average magnitude of localised perturbations in the flow, which turns out to be $1.1\times 10^{-4}$ . While the value of the largest perturbation magnitude is not provided in their work, it is reasonable to assume that it is significantly larger than the average value computed here. Thus, a maximal perturbation magnitude of $O(10^{-3})$ , which was found to be the stability threshold using our methodology, seems reasonable.

5.1. Couette base flow

In figure 9 we show an example of the stability diagram in the ${\textit{Re}} - k_x$ domain, showing only a single isocontour for a threshold value equal to $10^{-2.43}=3.7\times 10^{-3}$ for $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ at $k_z=0$ . The $(Re,k_x)$ plane is divided into two regions: the stable region is found to the left of the isocontour given the presence of perturbations of magnitude below $3.7\times 10^{-3}$ , while the unstable region is found to its right, i.e. at higher Reynolds numbers. In our case, the critical Reynolds number for this perturbation magnitude is defined as the smallest Reynolds number for which the perpendicular line (denoted by the red dashed line in the figure) is tangent to the contour of the threshold $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}=3.7\times 10^{-3}$ . In this case, the critical Reynolds number turns out to be ${\textit{Re}}_c=325$ for perturbations that are $0.37\,\%$ of the characteristic base flow velocity (in the Couette case, it is half the velocity difference between the walls). This critical Reynolds number matches the experimental results of Dauchot & Daviaud (Reference Dauchot and Daviaud1995) and simulations by Kreiss, Lundbladh & Henningson (Reference Kreiss, Lundbladh and Henningson1994). In Dauchot & Daviaud (Reference Dauchot and Daviaud1995), it was noted that very large perturbations are introduced via transverse jet flow injected into the laminar flow, where the information on the exact magnitude was not available. In Kreiss et al. (Reference Kreiss, Lundbladh and Henningson1994), the relation $A_c\sim Re^{-1}$ is established for the critical perturbation amplitude in Couette flow. Considering the Reynolds number ${\textit{Re}}=325$ , this relation yields a critical value of ${\textit{Re}}^{-1}= 3.1\times 10^{-3}$ (upper bound on the disturbance by their analysis), which is similar to the value we predict using $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and also for the structured case, which yields the same results as shown in figure 10. We note that in Kreiss et al. (Reference Kreiss, Lundbladh and Henningson1994) there is a large prefactor of $\sqrt {2000}$ that multiplies ${\textit{Re}}^{-1}$ . As noted by Kreiss et al. (Reference Kreiss, Lundbladh and Henningson1994), they compute an upper bound for the perturbation magnitude and not necessarily the exact value. In Kreiss et al. (Reference Kreiss, Lundbladh and Henningson1994), only one initial condition is considered (two counter-rotating vortices added to Stokes’ noise), and thus, their result can be significantly higher than the bound for the least stable initial condition. This highlights a significant advantage of our analysis, as we do not require exhaustive simulations of many different initial conditions to obtain the optimal bound on the perturbation magnitude.

Figure 9. Contour plot of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}=10^{-2.43}$ and stability interpretation based on (2.20) at $k_z=0$ .

The analysis presented here can be extended to any range of threshold values, as shown in figure 10, where we show stability diagrams in the ${\textit{Re}}-k_x$ domain at $k_z=0$ , showing several isocontours for different threshold values for both cases: $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ . The stability diagrams in both structured and unstructured cases display very similar behaviour, showing that the structure of the interconnected nonlinearity does not affect the stability characteristics of Couette flow for $k_z=0$ , i.e. if only two-dimensional TS modes are considered. Both subfigures (panels a and b) show that as the Reynolds number increases, the flow becomes more sensitive to smaller velocity perturbations. We observe that Couette flow is more sensitive to perturbations characterised by smaller $k_x$ values, especially at higher Reynolds numbers.

Stability analysis via our small gain theorem approach implies that there exists a perturbation of a finite value that could cause the flow to become unstable. With the increase in Reynolds number, the magnitude of perturbation that will not trigger transitioning to turbulence becomes smaller, and thus, there is a higher probability for transition to occur. In contrast to LST, which predicts that Couette flow is asymptotically stable for any Reynolds number (Schmid Reference Schmid2007), our small gain theorem approach provides mathematical proof that Couette flow can become unstable for a finite disturbance magnitude that exceeds a particular bound such as $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ or $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ for any Reynolds number, which agrees with experimental observations, such as Tillmark & Alfredsson (Reference Tillmark and Alfredsson1992), Dauchot & Daviaud (Reference Dauchot and Daviaud1995), and simulation results (Barkley & Tuckerman Reference Barkley and Tuckerman2005; Dou & Khoo Reference Dou and Khoo2012) showing transition at ${\textit{Re}} = 320-370$ . In particular, the vertical dashed red lines in figures 10(a) and 10(b) correspond to ${\textit{Re}}=325$ , the critical Reynolds number found experimentally by Dauchot & Daviaud (Reference Dauchot and Daviaud1995), and the vertical dashed–dotted red lines correspond to ${\textit{Re}}=360$ , the critical Reynolds number found experimentally by Tillmark & Alfredsson (Reference Tillmark and Alfredsson1992). In their study, the authors introduced controlled finite-size disturbances of short duration that were made by a solenoid activated, 1 mm diameter fluid jet. However, no information is available on the magnitude of the disturbance. According to our stability criterion, the stability bound depends on the geometric shape of the disturbance (represented by its wavenumber pair $k_x,k_z$ ), its temporal frequency $\omega$ and its magnitude. Thus, differences in the magnitude of the introduced disturbances, in geometric shape and in the time period between disturbances can all cause the discrepancy between the critical Reynolds number predictions of Tillmark & Alfredsson (Reference Tillmark and Alfredsson1992) and Dauchot & Daviaud (Reference Dauchot and Daviaud1995).

Figure 10. Stability diagrams for Couette flow showing contour plots on the ${\textit{Re}} - k_x$ domain for $k_z=0$ of imposed thresholds of (a) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and (b) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ .

Next, to validate our analysis for a wide range of Reynolds numbers with results from the literature, we compute the critical perturbation energy that causes instability. In figure 11(a) we conduct a comparison between the critical perturbation energy from our analysis versus DNS data from three studies (Reddy et al. Reference Reddy, Schmid, Baggett and Henningson1998; Duguet et al. Reference Duguet, Brandt and Larsson2010a , Reference Duguet, Monokrousos, Brandt and Henningson2013). Figure 11(a) shows that the predictions for the critical Reynolds number and the critical energy for instability are similar to the prediction made by Duguet et al. (Reference Duguet, Monokrousos, Brandt and Henningson2013). While both Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998) and Duguet et al. (Reference Duguet, Brandt and Larsson2010a ) consider a specific oblique wave scenario, by employing a nonlinear optimisation method, in Duguet et al. (Reference Duguet, Monokrousos, Brandt and Henningson2013) a more general transition scenario was considered, leading to lower critical energy predictions. The linear trend (in logarithmic scale) in figure 11(a) corresponds to a relationship of the form $E_c\propto Re^{-\gamma }$ . Using our methodology, a value of $\gamma \approx 2$ is predicted, similar to the results of Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998) and Duguet et al. (Reference Duguet, Brandt and Larsson2010a ). However, a value of $\gamma \approx 2.7$ was reported by Duguet et al. (Reference Duguet, Monokrousos, Brandt and Henningson2013), which better agrees with our results. It is possible that considering modes with a non-zero spanwise wavenumber can lead to even better agreement. Lastly, in figure 11(b) we show that the obtained relationship $E_c\propto Re^{-2}$ is sustained for a wide range of Reynolds numbers, by increasing the domain to ${\textit{Re}} \in [100,10000]$ . We note that we only display the results using $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , as both the unstructured and structured methods yielded the same results, similarly to the results in figure 10.

Figure 11. Critical perturbation energy versus critical Reynolds number for Couette flow. (a) Our analysis versus results from literature: magenta solid - our results; black dashed-dotted - results from Duguet et al. (Reference Duguet, Brandt and Larsson2010a ); red dotted - results from Duguet et al. (Reference Duguet, Monokrousos, Brandt and Henningson2013), blue dashed - results from Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998). (b) Our analysis for a wider range of Reynolds numbers, ${\textit{Re}} \in [100,10000]$ .

5.2. Plane Poiseuille base flow

In figure 12 we show stability diagrams for plane Poiseuille flow in the ${\textit{Re}}-k_x$ domain at $k_z=0$ , showing several isocontours for different threshold values for both cases: $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ (figure 12 a) and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ (figure 12 b). The dashed red contour in figures 12(a) and 12(b) corresponds to the neutral curve for Poiseuille base flow computed using eigenvalue analysis (LST), which is shown for comparison with our results.

Figure 12. Stability diagrams for plane Poiseuille flow showing contour plots on the ${\textit{Re}} - k_x$ domain for $k_z=0$ of imposed thresholds of (a) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and (b) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ . The vertical dotted line corresponds to ${\textit{Re}}=1000$ and the dashed-dotted line corresponds to ${\textit{Re}}=5772$ . The red dashed contour is the neutral stability contour computed using LST.

In figure 12(a), as the Reynolds number increases, the threshold on disturbance that is required to make the flow unstable decreases similarly to the case of Couette flow. However, in the case of plane Poiseuille flow, as the threshold decreases, the resulting contours approach the neutral curve obtained by LST. When $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}(k_x,k_z) \rightarrow 0$ , infinitesimal velocity perturbations will cause the system to become unstable. This is how our small gain formulation of stability relates to LST, according to which, the region within the neutral curve is where the modes are infinitely amplified. The predictions of LST, including critical Reynolds numbers, are valid only for the case of asymptotically small perturbations. Herein, the small gain theorem formulation provides an extension of LST and allows us to determine stability based on the magnitude of the perturbations in the flow for any Reynolds number. Using our small gain theorem approach to study transition states shows that instability can occur in plane Poiseuille flow at lower Reynolds numbers than the critical value ${\textit{Re}}=5772$ (denoted by vertical dashed–dotted red lines in the figures) that is predicted via LST and our use of the bounds $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ . The exact Reynolds number associated with the loss of stability depends on the magnitude of velocity perturbations that exist in the flow field. The flow will become unstable at earlier Reynolds number if the velocity perturbation exceeds the bound of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{u}}}^{-1}$ , which can be approximated by $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ , $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ or $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ . Thus, experimental observations of subcritical transition in plane Poiseuille flow, such as in Nishioka & Asai (Reference Nishioka and Asai1985) and Sano & Tamai (Reference Sano and Tamai2016), can be explained by our novel stability criterion. The perturbation magnitude that corresponds to a critical Reynolds number of ${\textit{Re}}=1000$ (denoted by the vertical dotted red lines in the figures) is approximately $10^{-2.9}=1.1\times 10^{-3}$ (for both structured and unstructured cases), meaning that perturbations that are $0.11\,\%$ of the centreline velocity are needed to be present in the flow for instability to occur for this Reynolds number (Carlson, Widnall & Peeters Reference Carlson, Widnall and Peeters1982; Nishioka & Asai Reference Nishioka and Asai1985). In both studies, finite-size disturbances are introduced to the laminar flow to induce instability, though we could not determine their magnitudes from these studies.

We note that for both unstructured (figure 12 a) and structured (figure 12 b) cases, the stability threshold contours plot outside the neutral curve region are very similar, including the obtained infinitesimal thresholds on the curve of the neutral loop itself. The difference between the two is limited to the stability characteristics for threshold values that lie inside the neutral loop. While in figure 12(a) any perturbation that corresponds to a mode that is found inside the neutral is deemed unstable. However, in figure 12(b) there are contours that are found inside the neutral loop, corresponding to the fact that according to the structured analysis using $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ modes that are found inside the neutral loop can be stable given small enough perturbations. The interpretation of contours of the structured quantities $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{r}}}^{-1}$ or $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ inside the neutral curve should be treated with caution. The small gain theorem is valid only given that the nominal system $\mathscr{H}_{\,\boldsymbol{\nabla }}(k_x,k_z)$ is stable (as we denote in Theorem1). Linear stability theory predicts an exponentially growing mode within the neutral curve region, driven by linear interactions. Thus, our analysis holds only when the system within the neutral curve region is initially stable. Still, the uncertainty matrices in our analysis may provide a stabilising effect on the interconnected system represented by the loop block diagram in figure 1, allowing the flow to remain stable even at post-critical Reynolds numbers. This is in line with studies by Cossu & Brandt (Reference Cossu and Brandt2002), Fransson et al. (Reference Fransson, Brandt, Talamelli and Cossu2005) that have shown that finite-amplitude streaks stabilise TS instability. Furthermore, nonlinearity in the NSE has also been suggested to provide a stabilising effect, leading to the saturation of linear growth (Schmid et al. Reference Schmid, Henningson and Jankowski2002; McKeon Reference McKeon2017; Jin, Symon & Illingworth Reference Jin, Symon and Illingworth2021). The extent to which these effects are replicated using the current methodology with constant uncertainty matrices, and the implications for stability within the neutral curve, are subject to future study. Results from our analysis can be used to explain the experimental observations of the work by Nishioka et al. (Reference Nishioka1975), in which plane Poiseuille flow is found experimentally to remain stable up to a Reynolds number of ${\textit{Re}}=8000$ , which cannot be explained using LST. Our analysis suggests that the perturbation magnitude that leads to the loss of stability at a Reynolds number of ${\textit{Re}}=8000$ is $10^{-4.2}$ , which corresponds to a turbulence intensity of $0.006\,\%$ of the centreline velocity. This result matches the experimental set-up in Nishioka et al. (Reference Nishioka1975), where the turbulence intensity is reported to be ‘essentially less than $0.01\,\%$ ’.

Next, we validate our analysis for a wide range of Reynolds numbers with results from the literature by computing the critical perturbation energy. Figure 13(a) shows that the predictions of the critical Reynolds number and the critical energy for instability are found between the prediction by Parente et al. (Reference Parente, Robinet, De Palma and Cherubini2022) and the prediction by Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998). While in Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998), a specific oblique wave scenario is considered, Parente et al. (Reference Parente, Robinet, De Palma and Cherubini2022) employs a nonlinear variational optimisation method to consider a more general transition scenario, leading to lower critical energy predictions. As shown in figure 6, the TS mode is the least dominant for subcritical Reynolds numbers (according to both structured and unstructured methods). Thus, we expect that considering more general cases that include modes with non-zero spanwise wavenumbers will very likely (as discussed in § 4.2) lower the critical perturbation energy predicted by our methodology, making it closer to the results of Parente et al. (Reference Parente, Robinet, De Palma and Cherubini2022).

Figure 13. Critical perturbation energy versus critical Reynolds number for plane Poiseuille flow. (a) Our analysis versus results from literature: magenta solid - our results; black dotted - Reddy et al. (Reference Reddy, Schmid, Baggett and Henningson1998); red dashed-dotted - (Parente et al. Reference Parente, Robinet, De Palma and Cherubini2022) (minimal energy threshold using large domain). (b) Our analysis for the increased range ${\textit{Re}}\in [100,15\,000]$ .

The linear trend in figure 13(a) corresponds to a relationship of the form $E_c=O(Re^{-\gamma })$ . Similarly to the case of Couette flow, a value of $\gamma \approx 2$ is predicted using our methodology. However, this trend does not hold for all Reynolds numbers as we demonstrate in figure 13(b), which shows the relationship between the critical energy perturbation and critical Reynolds number for a wider range of Reynolds numbers, ${\textit{Re}} \in [100,15\,000]$ . Figure 13(b) shows that the linear trend (in logarithmic scale) does not continue for Reynolds numbers above $1700$ . Instead, there is a rapid decrease of the curve, that asymptotes towards zero at critical Reynolds number ${\textit{Re}}=5772$ for both $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ at ${\textit{Re}}=5772$ . This result reflects the imposed threshold by the TS mode in figure 6(b), where the perturbation energy that is required to cause instability approaches zero, matching the prediction of the critical Reynolds number by LST and our previous discussion on the results of figure 12. The difference between the structured and unstructured cases is restricted to post-critical Reynolds numbers (higher than ${\textit{Re}}=5772$ ). In detail, $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ predicts that any perturbation, even of infinitesimal energy, will cause a transition. However, the structured analysis using $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ predicts that while for ${\textit{Re}}=5772$ very small perturbations will cause instability, as the Reynolds number increases beyond this value the critical energy also increases. These results are consistent with the discussion regarding figure 12(b), in which thresholds on perturbation magnitude for stability exist inside the neutral curve.

Figure 14. Stability diagrams for Blasius flow showing contour plots on the ${\textit{Re}} - k_x$ domain for $k_z=0$ of imposed thresholds of (a) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and (b) $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ . The vertical dashed–dotted line corresponds to ${\textit{Re}}=520$ and the red-dashed contour is the neutral stability contour computed using LST.

5.3. Blasius base flow

Stability diagrams showing contour plots of threshold on disturbance magnitude using $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\infty }^{-1}$ and $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ for Blasius flow are shown in figures 14(a) and 14(b), respectively. The dashed red curve in both figures corresponds to the so-called ‘banana’ neutral curve for Blasius base flow, as predicted by LST, and the vertical dashed–dotted red lines correspond to ${\textit{Re}}=520$ , the critical Reynolds number predicted by LST for Blasius flow. These plots show that as the Reynolds number increases, the perturbation magnitude that is required to make the flow unstable decreases, similar to the cases of Couette and plane Poiseuille flows. Additionally, as the threshold magnitude decreases, the resulting threshold contours approach the neutral curve obtained via LST, as observed for the case of plane Poiseuille flow. This result shows once again that imposing an infinitesimally small threshold on velocity perturbations leads our analysis to converge to the results of LST, while extending them by allowing us to assess stability in the presence of finite-amplitude perturbations.

Both diagrams for unstructured and structured cases show a very similar topology of threshold isolines outside and on the neutral loop region. Whereas, inside the neutral loop of the structured case diagram (figure 14 b), there are finite-size threshold contours present. Imposing structure should integrate some aspects of nonlinearity in the computation of $\|{\mathscr{H}_{\,\boldsymbol{\nabla }}}\|_{\mu _{\boldsymbol{\Delta }_{\textit{nr}}}}^{-1}$ , causing the flow to become stable for small finite-sized perturbations inside the neutral loop. This is in contrast to the unstructured case, which predicts any mode inside the neutral loop to be unstable for any perturbation magnitude. However, the presence of a stability threshold isoline inside the neutral curve should be treated with caution, as explained in § 5.2 due to the inherent assumption of the small gain theorem that the interconnected system should be initially stable. It is shown to be feasible to enter the post-critical Reynolds number region, while keeping the flow stable via a nonlinear route to transition as reported in Cherubini et al. (Reference Cherubini, De Palma, Robinet and Bottaro2011, Reference Cherubini, De Palma, Robinet and Bottaro2012), where a finite perturbation magnitude is determined for causing transition of Blasius flow at a Reynolds number of ${\textit{Re}}=610$ . This result is in agreement with our results for the structured case, as illustrated in figure 14(b). In particular, in Cherubini et al. (Reference Cherubini, De Palma, Robinet and Bottaro2012) a critical energy value (of the entire velocity perturbation field) of $E_c=4.4\times 10^{-3}$ is obtained for this Reynolds number. As explained in the last paragraph of § 4.3, the average magnitude of perturbations in the flow can be computed given the energy, yielding an average magnitude of $3.3\times 10^{-4}=10^{-3.5}$ . This value is slightly higher than the value predicted in figure 14(b), which is approximately $10^{-4.05}$ . This difference can perhaps be attributed to approximating the structure of the nonlinear term in the NSE by non-repeated blocks, as we expect lower stability margin predictions using the exact structure according to (2.29), and also may be due to not accounting for non-parallel effects of the boundary layer in the current study.

Lastly, we now turn our attention to computing the critical energy perturbation for the transition of Blasius flow. Figure 15(a) displays the relationship between the critical perturbation energy and the critical Reynolds number on a logarithmic scale, defined similarly as in the cases of Couette and Poiseuille flows. The linear trend in logarithmic scale in figure 15(a) corresponds to a relationship of the form $E_c=O(Re^{-\gamma })$ . Similarly to the previous cases, a value of $\gamma \approx 2$ is predicted using our methodology. This linear trend holds up to about ${\textit{Re}}_c=320$ , after which there is a rapid drop in the critical perturbation energy, as shown in figure 15(b) that shows the same relationship between the critical perturbation energy and Reynolds number for a larger range of ${\textit{Re}} \in [0,1100]$ . Similar to the observed behaviour for plane Poiseuille flow, the $E_c$ curve asymptotes towards zero at ${\textit{Re}}=520$ for both structured and unstructured cases. The observed trend change in the curve after ${\textit{Re}}=300$ shown in figure 15(b) can be explained by a shift in the dominant flow structure that is associated with the least stable mode.

For ${\textit{Re}}\gt 520$ , the energy curve for structured methods departs from the unstructured case, exhibiting similar behaviour as the TS mode in figure 8(b), which we discussed in detail for the case of plane Poiseuille flow in § 5.2, and will not repeat it here again for brevity.

Details on validation of our stability criterion for Blasius flow with some results from experimental studies from literature are provided in the previous § 4.3 (last paragraph), where we discuss the general case for our analysis with non-zero $k_z$ , which is associated with an oblique transition scenario.

Figure 15. Critical perturbation energy versus critical Reynolds number for Blasius flow via our analysis: (a) for a limited range of ${\textit{Re}}$ up to 320, (b) for an increased range of Reynolds numbers up to 1100.