2.1. The LNS equations

We consider the linearised incompressible Navier–Stokes equations,

(2.1) \begin{align} \frac {\partial \boldsymbol{u'}}{\partial t} + \boldsymbol{U} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{u'} + \boldsymbol{u'} \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{U} &= -\boldsymbol{\nabla }p' + \frac {\mathrm{1}}{\mathrm{R}}\varDelta \boldsymbol{u'} + \boldsymbol{d} ,\end{align}

(2.2) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} &= 0 ,\end{align}

where $\boldsymbol {u} = [u,v,w]^T$ , $p$ is pressure, $'$ denotes fluctuation about a base flow, $\boldsymbol {d} = [d_x,d_y,d_z]^T$ is an external force and $\varDelta := {\nabla} ^2$ is the Laplacian. The considered geometry is channel flow, where the streamwise, spanwise and wall-normal directions are denoted by $x$ , $y$ and $z$ , respectively. The streamwise, spanwise and wall-normal velocities are denoted as $u$ , $v$ and $w$ . The inclusion of a forcing term ( $\boldsymbol {d}$ in (2.1)), or external excitation, is consistent with the approach for linear systems in other works (Jovanović & Bamieh Reference Jovanović and Bamieh2005; Illingworth Reference Illingworth2020). The channel walls are at $z = \pm h$ and no-slip boundary conditions, $\boldsymbol{u}(\pm h) = 0$ and ${\partial w}/{\partial z}(\pm h) = 0$ , are applied at the two walls. The Reynolds number $R = u_0h/\nu$ is based on the channel half-height $h$ , kinematic viscosity $\nu$ and characteristic velocity $u_0$ . The base flow profile is denoted by $\boldsymbol{U}$ , which is either a laminar profile or a time-averaged mean flow profile. For laminar flows, the base flow is unidirectional and given by $\boldsymbol{U} = (U(z),0,0)$ , where $U(z) = 1 - z^2$ for Poiseuille flow and $U(z) = z$ for Couette flow. In the laminar cases considered, $u_0$ is the maximum velocity across the channel height and the flow is linearised about the laminar flow profile. For turbulent cases, $u_0$ is the friction velocity $u_\tau = \sqrt {\tau _w / \rho }$ , where $\tau _w$ is the wall shear stress and $\rho$ is density. Additional turbulent profile details are provided in § 2.3.

It is convenient to rewrite (2.1) and (2.2) as an input–output system containing the wall-normal velocity $w$ and the wall-normal vorticity $\eta$ (Schmid & Henningson Reference Schmid and Henningson2001). A Fourier transform in the streamwise and spanwise directions lets us rewrite the equations as

(2.3) \begin{align} \frac {\partial \tilde {w}}{\partial t} &= \frac {1}{\mathcal{D}^2 - k^2}\Bigg[\Bigg[(-i k_x U)\bigl(\mathcal{D}^2 - k^2\bigr) +ik_x U^{\prime \prime } + \frac {1}{R}\bigl(\mathcal{D}^2 - k^2\bigr)^2\Bigg]\tilde {w} \nonumber\\ &\quad\vphantom {\mathcal{D}^2 - k^2} - ik_x \mathcal{D}d_x - ik_y\mathcal{D}d_y - k^2 d_z\Bigg] , \end{align}

(2.4) \begin{align} \frac {\partial \tilde {\eta }}{\partial t} &= \Bigg[- i k_x U +\frac {1}{R} \bigl(\mathcal{D}^2 - k^2\bigr)\Bigg] \tilde {\eta } +i k_y U^{\prime } \tilde {w} + ik_xd_x - ik_yd_y ,\end{align}

where $k_x$ and $k_y$ are the streamwise and spanwise wavenumbers, $\mathcal{D} = {\partial }/{\partial z}$ , $k^2 = k_x^2 + k_y^2$ and a tilde denotes variables expressed in the transformed domain. The same notation is used for variables transformed in space only as well as those transformed in both space and time. These equations are put into the general state-space form,

(2.5) \begin{align} \boldsymbol{x}(t) &= A\boldsymbol{x}(t) +B\boldsymbol{u}(t), \nonumber \\ \boldsymbol{y}(t) &= C\boldsymbol{x}(t), \end{align}

as

(2.6) \begin{align} \begin{bmatrix} \frac {\partial \tilde {w}}{\partial t} \\[9pt] \frac {\partial \tilde {\eta }}{\partial t} \end{bmatrix} = \underbrace {\begin{bmatrix} \varDelta ^{-1} \mathcal{L}_{\textit{OS}} & 0 \\[5pt] -ik_yU^{\prime } & \mathcal{L}_{Sq} \end{bmatrix}}_{A} \begin{bmatrix} \tilde {w} \\ \tilde {\eta } \end{bmatrix} &+ \underbrace {\begin{bmatrix} -ik_x\varDelta ^{-1}\mathcal{D} & -ik_y\varDelta ^{-1}\mathcal{D} & -k^2\varDelta ^{-1} \\[3pt] ik_y & -ik_x & 0 \end{bmatrix}}_{B} \tilde {\boldsymbol{d}}, \end{align}

(2.7) \begin{align} \begin{bmatrix} \tilde {u} \\[3pt] \tilde {v} \\[3pt] \tilde {w} \end{bmatrix} =&\underbrace {\frac {1}{k^2} \begin{bmatrix} i k_x \mathcal{D} & -i k_y \\[5pt] i k_y \mathcal{D} & i k_x \\[5pt] k^2 & 0 \end{bmatrix}}_{C} \begin{bmatrix} \tilde {w} \\ \tilde {\eta } \end{bmatrix} ,\end{align}

where

(2.8) \begin{align} \mathcal{L}_{\textit{OS}} &= -i k_x U\varDelta + ik_x U^{\prime \prime } + \frac {1}{R}\varDelta ^2 ,\end{align}

(2.9) \begin{align} \mathcal{L}_{Sq} &= -i k_x U + \frac {1}{R} \varDelta \end{align}

and $\varDelta =(\mathcal{D}^2 - k^2)$ is the Laplacian in Fourier space.

The state-space model above is used to examine the response of the velocity fluctuations to external forcing. This model, hereon referred to as the full LNS system, has been evaluated in prior studies under stochastic (Jovanović & Bamieh Reference Jovanović and Bamieh2005) and harmonic (Schmid Reference Schmid2007; Hwang & Cossu Reference Hwang and Cossu2010a ) forcing conditions. When evaluated under harmonic forcing conditions, the frequency response of the velocity field due to the forcing vector can be measured. In order to do so, we Laplace transform the state-space system (2.6)–(2.7) and set the Laplace variable $s = i\omega$ . The result is the resolvent operator, $H$ , which takes the form of

(2.10) \begin{align} H \left ( k_x,k_y,\omega ,R \right ) = C \left ( k_x,k_y \right ) \left ( i\omega I - A \left ( k_x,k_y,R \right ) \right ) ^{-1}B \left ( k_x,k_y \right ) ,\end{align}

where the matrices $A$ , $B$ and $C$ are defined by our state-space system (2.6)–(2.7). For the full LNS system, the resolvent is written as

(2.11) \begin{align} H_{\textit{LNS}}(k_x,k_y,\omega ,R) &= \frac {1}{k^2} \begin{bmatrix} ik_x\mathcal D & -ik_y \\[5pt] ik_y\mathcal D & ik_x \\[5pt] k^2 & 0 \end{bmatrix} \left ( i\omega I - \begin{bmatrix} \varDelta ^{-1}\mathcal L_{\textit{OS}} & 0 \\[5pt] -ik_yU' & \mathcal L_{Sq} \end{bmatrix} \right )^{-1}\nonumber\\&\quad\times \begin{bmatrix} -ik_x\varDelta ^{-1}\mathcal D & -ik_y\varDelta ^{-1}\mathcal D & -k^2\varDelta ^{-1} \\[5pt] ik_y & -ik_x & 0 \end{bmatrix} \!.\end{align}

Here we note several items: (i) $H ( k_x,k_y,\omega ,R )$ is a four-dimensional operator-valued function in the wall-normal direction; (ii) the input to the state-space system is the external forcing field and the output is the velocity field; (iii) the state-space model can be modified to examine the effects of individual forcing components on individual velocity components by altering the $B$ or $C$ matrices; (iv) while $H$ is dependent on $k_x$ , $k_y$ , $\omega$ and $R$ , this study is primarily concerned with the effect of $k_x$ and $k_y$ at a given Reynolds number and ‘worst-case’ frequency. Here, ‘worst-case’ frequency refers to the frequency producing the maximum amplification. Before the simplified model is derived (§ 2.4), we first consider options for evaluation of our three-variable ( $k_x,k_y,\omega$ ) resolvent operator at a fixed Reynolds number.

2.2. Transfer function norms

The energy in the velocity fields can be quantified by defining a suitable norm for the resolvent operator. The norm provides a mathematical measure of the energy contained in the velocity field, integrated over the wall-normal direction. This reduces the dimensions of our problem, allowing for analysis of the resolvent as a function of streamwise wavenumber, spanwise wavenumber and frequency. The treatment of the frequency changes with the choice of norm. The two norm (Farrell & Ioannou Reference Farrell and Ioannou1993; Bamieh & Dahleh Reference Bamieh and Dahleh2001; Jovanović & Bamieh Reference Jovanović and Bamieh2005; Hwang & Cossu Reference Hwang and Cossu2010a ), which is an average over all frequencies, and the infinity norm (Zhou, Doyle & Glover Reference Zhou, Doyle and Glover1996; Schmid Reference Schmid2007; Illingworth Reference Illingworth2020), which examines only the ‘worst-case’ frequency, are commonly used to measure the energy amplification of the resolvent operator, $H$ , in shear flows.

The infinity norm is defined as

(2.12) \begin{align} \Vert H \Vert _\infty = \underset {\omega }{\text{max}} \; \sigma _1 \left ( i \omega \right ), \end{align}

where $\sigma _1$ is the maximum singular value of $H(i \omega )$ at a given $\omega$ . The infinity norm measures the maximum gain over all frequencies and forcing directions; this represents the ‘worst-case’ response to deterministic inputs. The infinity norm is particularly useful in this study due to its sub-multiplicative property;

(2.13) \begin{align} \Vert AB\Vert _{\infty }\leqslant \Vert A \Vert _{\infty }\boldsymbol{\cdot }\Vert B \Vert _{\infty }. \end{align}

The sub-multiplicative property provides a way to compare the combined amplification of systems, e.g. Orr–Sommerfeld and Squire systems, to their maximum potential amplification; where the maximum amplification is found by multiplying the optimal responses of each individual component.

On the other hand, the two norm represents an average gain over all forcing frequencies and directions. The two norm is defined as

(2.14) \begin{align} \Vert H \Vert _{2}^2 = \frac {1}{2\pi }\int _{{-\infty }}^{{\infty }} {\text{Trace}[H^*(i\omega )H(i\omega )]} \: {\rm d}{\omega }. \end{align}

The two norm is used in studies that examine stochastic forcing to determine the ensemble-average energy density of the statistically steady state (Farrell & Ioannou Reference Farrell and Ioannou1993). The two norm integrates contributions from all frequencies, helping reveal features present across a broad range of frequencies within the velocity response. The two norm can also be obtained without integration over $\omega$ by solving the Lyapunov equations

(2.15) \begin{align} A\big(k_x,k_y\big) X\big(k_x,k_y\big) + X\big(k_x,k_y\big) A^*\big(k_x,k_y\big) = -B\big(k_x,k_y\big) B^*\big(k_x,k_y\big) ,\end{align}

(2.16) \begin{align} A^*\big(k_x,k_y\big) Y\big(k_x,k_y\big) + Y\big(k_x,k_y\big) A\big(k_x,k_y\big) = -C^*\big(k_x,k_y\big) C\big(k_x,k_y\big) ,\end{align}

where a $(\boldsymbol{\cdot })^*$ denotes the adjoint of the operator, and $X$ and $Y$ are referred to as the controllability and observability Gramians, respectively. The two norm is then calculated by solving the following equation:

(2.17) \begin{align} \Vert H \Vert _{2}^2 = \text{trace}\big(C\big(k_x,k_y\big)X\big(k_x,k_y\big)C^*\big(k_x,k_y\big)\big) .\end{align}

When considering amplification of the resolvent operator, the wavenumber pairs at which the maximum energy is observed are insensitive to the choice of norm. In § 3 we choose to use the infinity norm due to its sub-multiplicative property (2.13). This property allows for examination of the individual Orr–Sommerfeld and Squire systems, as well as the full LNS system (models derived in § 2.4). We will see that this analysis helps visualise how the amplification in the full system arises from the combined effects of the individual Orr–Sommerfeld and Squire components. In § 5 we examine the leading modal structures by using an eigenvalue decomposition. Unlike amplification, where different norms may yield different numerical values of the gain but typically preserve the qualitative trends, the structures obtained are more sensitive to the choice of norm. If the infinity norm is used, it can provide a limited view when evaluating leading modal structures as only one frequency is considered. For this reason, a two-norm-like approach is used to consider all frequencies by taking the eigenvalue decomposition of the response covariance matrix ( $CXC^*$ ). In summary, the infinity norm is used for evaluation of amplifications to make use of the sub-multiplicative property; for modal analysis of structures, all frequencies are considered in the analysis.

2.3. Turbulent Poiseuille flow details

The turbulent Poiseuille flow case requires introduction of a mean turbulent profile via an eddy-viscosity model into our state-space system. In order to evaluate our linear model about a turbulent mean profile, we introduce a Cess (Cess Reference Cess1958) eddy-viscosity model (Reynolds & Hussain Reference Reynolds and Hussain1972). The same base equation (2.1) is used with two main differences. The first is a change in the definition of the forcing terms to be equal to the remaining nonlinear terms, so that $\boldsymbol{d} = -(\boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol{u} + (\overline {\boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u}}$ . From this viewpoint, we are now evaluating the linear response to an unknown nonlinear forcing. The second change is that we replace the laminar base flow with the time-averaged mean flow profile $\boldsymbol{U}$ . The nonlinear effects are realised in both the forcing term and in setting the mean profile. The mean velocity profile is $\boldsymbol{U} = (U(z),0,0)$ , where $U(z)$ is obtained by integrating $R_{\tau }(1-z)\nu / \nu _{\tau }(z)$ . The eddy viscosity is defined as

(2.18) \begin{align} \nu _{\tau }(z) = \frac {\nu }{2}\left ( 1+\frac {\kappa R^2_{\tau }}{9}\big(2z - z^2\big)^2\big(3-4z+z^2\big) \left [ 1 - \exp \left ( \frac {-R_{\tau }}{A} \right ) \right ]^2 \right )^{\tfrac {1}{2}} + \frac {\nu }{2}. \end{align}

Here $\kappa = 0.426$ and $A = 25.4$ (del Álamo & Jiménez Reference del Álamo and Jiménez2006; Pujals et al. Reference Pujals, García-Villalba, Cossu and Depardon2009; Hwang & Cossu Reference Hwang and Cossu2010b ) where these values are optimised for $R_{\tau }=2003$ . This approximation has proven to be effective at other Reynolds numbers (Symon et al. Reference Symon, Madhusudanan, Illingworth and Marusic2023). By including the eddy viscosity, the equations for the Orr–Sommerfeld and Squire operators become

(2.19) \begin{align} \mathcal{L}_{\textit{OS}} &= -i k_x U\varDelta + ik_x U^{\prime \prime } + \nu _{\tau }\varDelta ^2 + 2 \nu _{\tau }^{\prime } \mathcal{D} \varDelta + \nu _{\tau }^{\prime \prime } \big(\mathcal{D}^2 + k^2\big), \end{align}

(2.20) \begin{align} \mathcal{L}_{Sq} &= -i k_x U + \nu _{\tau }\varDelta + 2 \nu _{\tau }^{\prime }\mathcal{D} .\end{align}

The introduction of the eddy viscosity changes the meaning of the forcing term, $\boldsymbol{d}$ , to be an unknown nonlinear forcing as opposed to an external forcing. This change has no effect on the lift-up mechanism in the simplified model.

2.4. A hypothesis for simplifying the linear system via a lift-up mechanism argument

In order to form our hypothesis and simplified model, we first present several methods of amplification in the LNS equations. Two methods of energy amplification in the resolvent are due to the non-normal amplification associated with the coupling term (i.e. the lift-up mechanism Ellingsen & Palm Reference Ellingsen and Palm1975; Landahl Reference Landahl1990; Jiménez Reference Jiménez2018) and the critical layer mechanism (McKeon et al. Reference McKeon, Sharma and Jacobi2013). A third mechanism is another non-normal amplification mechanism, the Orr mechanism (Orr Reference Orr1907). The lift-up mechanism is driven by the wall-normal shear and results in the generation of streamwise streaks (Brandt Reference Brandt2014). This mechanism can amplify an initial disturbance, where energy amplification in the time domain scales with the square of the Reynolds number (Reddy & Henningson Reference Reddy and Henningson1993). The critical layer mechanism leads to resonant interactions even when the mean shear is small (e.g. ${\rm d}U/{\rm d}z \approx 0$ ), and becomes increasingly dominant at high Reynolds numbers (McKeon & Sharma Reference McKeon and Sharma2010). This mechanism is present at non-streamwise-constant wavenumbers when the local velocity is equal to the local wave speed ( $u(z) = \omega / k_x$ ). The Orr mechanism amplifies upstream-tilted wall-normal velocity structures; this amplification occurs as the shear tilts the structures from an upstream-tilted orientation towards a vertical orientation (Jiménez Reference Jiménez2018). While the Orr mechanism can be identified directly from the linearised equations, Orr-like transient growth events have also been observed in fully turbulent flows and linked to momentum transfer and energy amplification (Encinar & Jiménez Reference Encinar and Jiménez2020; Lozano-Durán et al. Reference Lozano-Durán, Constantinou, Nikolaidis and Karp2021). The net effect of amplifying wall-normal velocity through this mechanism is the creation of streaks by the lift-up mechanism.

We now use an argument based on the lift-up mechanism to simplify the governing equations (2.6)–(2.7). While there are multiple ways to amplify a disturbance through linear mechanisms, the presence of the lift-up mechanism stands out; it naturally follows amplification from the Orr mechanism and we observe the streamwise streaks created by it in turbulent flows. If the lift-up mechanism is important for amplification in the linearised equations then a simplified model that retains this mechanism is an interesting way to explore the underlying dynamics. The purpose of the simplified model is that amplification mechanisms can be viewed in individual linear operators. In other words, we can examine how the Orr–Sommerfeld operator (or Squire operator) amplifies streaks, oblique waves and TS waves.

The pathways in which the lift-up mechanism, the critical layer mechanism and the Orr mechanism act can be visualised by representing the LNS equations in a block diagram. The block diagram for the LNS system is depicted in figure 1(a), where we can trace individual pathways from the three forcing components to the three velocity components.

Figure 1.

(a) Block diagram of the full LNS system, where individual forcing terms are related to individual velocity terms. (b) Block diagram of the simplified model, where individual transfer functions are linked in series, and pathways that do not contain the lift-up mechanism have been removed. (a) Full LNS model. (b) Simplified model.

Figure 1. Long description

The image contains two block diagrams. The first diagram, labeled (a), represents the full LNS system. It shows how individual forcing terms are related to individual velocity terms. The diagram includes various components such as 0, −ikx, iky, dx, dy, dz, k2, Δ, HOS(iω), −ikyU’, Hsq(iω), I, and several summation points. The second diagram, labeled (b), represents a simplified model of the LNS system. It shows individual transfer functions linked in series, with pathways that do not contain the lift-up mechanism removed. This diagram includes components such as dx, dy, dz, −ikxΔ−1D, −ikyΔ−1D, −k2Δ−1, HOS(iω), −ikyU’, Hsq(iω), −iky/k2, ikx/k2, and 0. The diagrams illustrate the relationships and flow between these components in the context of the LNS system.

The critical layer mechanism and Orr mechanism are both encoded in the Orr–Sommerfeld equation; the critical layer arising as a resolution to a singularity in the Orr–Sommerfeld equation at a critical point where the perturbation and wave speeds are equal (Maslowe Reference Maslowe1986), and the Orr mechanism in the Laplacian acting on wall-normal velocity (Jiménez Reference Jiménez2018). The lift-up mechanism requires the interaction of the Orr–Sommerfeld and Squire operators via the wall-normal shear term. The only pathway that contains the lift-up mechanism is the solid pathway in the centre of figure 1(a). We simplify the model by retaining only the centre pathway that contains the lift-up mechanism. The simplified model is shown in figure 1(b).

The model depicted in figure 1(b) is hereon referred to as the simplified model. While the simplified model is useful in enabling analysis of individual operators, we must also address two shortcomings. The first is that the simplified model only outputs streamwise and spanwise velocity components, and therefore, the model will have a lower kinetic energy under conditions where wall-normal velocity is significant. The second is that the Orr–Sommerfeld operator becomes unstable after a critical Reynolds number. For these reasons, we need to identify conditions where wall-normal velocity is not significant (as a function of $k_x$ and $k_y$ ) and give consideration to stable and unstable Reynolds number regimes.

2.4.1. Deriving the simplified model

A natural consequence of the simplified model, as shown in figure 1(b), is that energy is contained only in the $\tilde {u}$ and $\tilde {v}$ components. This allows for scalar relationships between velocity and vorticity to be derived in Fourier space. We first consider an equation for the wall-normal vorticity as forced by the wall-normal velocity, i.e.

(2.21) \begin{align} \tilde {\eta } = \left (i\omega I - \mathcal{L}_{Sq} \right )^{-1} \left (-ik_yU'\right ) \left ( i\omega I - \varDelta ^{-1} \mathcal{L}_{\textit{OS}} \right )^{-1} B_w \tilde {\boldsymbol{d}} ,\end{align}

where $B_w = \begin{bmatrix}-ik_x\varDelta ^{-1}\mathcal{D} & -ik_y\varDelta ^{-1}\mathcal{D} & -k^2\varDelta ^{-1} \end{bmatrix}$ .

The streamwise and spanwise velocity can then be expressed as

(2.22) \begin{align} \tilde {u} &= \frac {-ik_y}{k^2} \tilde {\eta },\\[-12pt]\nonumber \end{align}

(2.23) \begin{align} \tilde {v} &= \frac {ik_x}{k^2} \tilde {\eta }.\end{align}

These relationships hold if the contribution of the wall-normal velocity as an output can be neglected. In other words, if the flow field is primarily composed of a streamwise and spanwise velocity response (for a given $k_x$ , $k_y$ ), we can ignore the small wall-normal velocity contribution to the flow field. To identify the parameter space in which the model will perform the best, the infinity norm for the wall-normal velocity component, $\Vert H_{w}\Vert _\infty$ , is compared with the infinity norm for all three velocity components, $\Vert H \Vert _\infty$ , in figure 2 for laminar Poiseuille flow.

Figure 2.

Plot of ${\Vert H_{w}\Vert _\infty }/{\Vert H \Vert _\infty }$ using the infinity norm at $R$ = 2000 for Poiseuille flow. Oval region (labelled I) depicts a wavenumber space around $k_x \sim O(1)$ that can contain TS-wave amplification in Poiseuille flow.

Figure 2. Long description

A heat map with logarithmic axes labeled kx and ky represents wavenumber space for Poiseuille flow analysis. The color scale ranges from light blue to dark blue, indicating varying levels of intensity. An oval region labeled I highlights a specific area of interest where TS-wave amplification can occur. The heat map shows a gradient of intensity, with darker colors representing higher values and lighter colors representing lower values. The axes range from 10^−4 to 10^2 for both kx and ky, capturing a wide spectrum of wavenumbers. The heat map is used to analyze the linear amplification of disturbances in Poiseuille flow, particularly focusing on transient growth mechanisms.

In figure 2, for wavenumber pairs in the region bounded by $1\times 10^{-4} \lt k_x \lt 1$ and $0.01 \lt k_y \lt 50$ , we see that the wall-normal velocity component contributes $\lt$ 30 % of the total energy. The majority of wavenumber pairs in this same region have a wall-normal velocity contribution below 1 % (white). Therefore, in the considered wavenumber space (§ 2.5), the contribution of the wall-normal velocity to the total energy is negligible when compared with the streamwise and spanwise contribution. There is one exception for the horizontal band around $k_x \sim O(1)$ , a region containing TS waves (see region labelled I in figure 2). Here, the assumption of negligible wall-normal velocity is no longer valid and model discrepancies are expected. These discrepancies are highlighted in § 3 and characterised further in § 3.2.

The equations for the simplified model contain the individual contributions of the Orr–Sommerfeld operator and separately the Squire operator. When placed in series, they produce a system that is equivalent to figure 1(b). Here, the wall-normal velocity serves as the input to the Squire system, and the wall-normal vorticity is the only input to the velocity components. Now that the model is simplified, it is translated back into a state-space system for calculations. The modified equations are shown below in state-space format, i.e.

(2.24) \begin{align} \frac {\partial \tilde {w}}{\partial t} = \varDelta ^{-1} \mathcal{L}_{\textit{OS}} \tilde {w} &+ \begin{bmatrix} -ik_x\varDelta ^{-1}\mathcal{D} & -ik_y\varDelta ^{-1}\mathcal{D} & -k^2\varDelta ^{-1} \end{bmatrix} \tilde {\boldsymbol{d}}, \\[-12pt]\nonumber \end{align}

(2.25) \begin{align} &\quad \quad \tilde {w} = I \tilde {w}, \\[-12pt]\nonumber \end{align}

(2.26) \begin{align} &\frac {\partial \tilde {\eta }}{\partial t} = \mathcal{L}_{Sq} \tilde {\eta } + -ik_yU^{\prime } \tilde {w}, \\[-12pt]\nonumber \end{align}

(2.27) \begin{align} &\, \begin{bmatrix} \tilde {u} \\ \tilde {v} \end{bmatrix} = \frac {1}{k^2} \begin{bmatrix} -i k_y \\ i k_x \end{bmatrix} \lbrack \tilde {\eta } \rbrack ,\end{align}

so that the overall system can be written as

(2.28) \begin{align} \begin{bmatrix} \frac {\partial \tilde {w}}{\partial t} \\[9pt] \frac {\partial \tilde {\eta }}{\partial t} \end{bmatrix} = \begin{bmatrix} \varDelta ^{-1} \mathcal{L}_{\textit{OS}} & 0 \\[5pt] -ik_yU^{\prime } & \mathcal{L}_{Sq} \end{bmatrix} \begin{bmatrix} \tilde {w} \\ \tilde {\eta } \end{bmatrix} &+ \begin{bmatrix} -ik_x\varDelta ^{-1}\mathcal{D} & -ik_y\varDelta ^{-1}\mathcal{D} & -k^2\varDelta ^{-1} \\[5pt] 0 & 0 & 0 \end{bmatrix} \tilde {\boldsymbol{d}},\\[-12pt]\nonumber \end{align}

(2.29) \begin{align} \begin{bmatrix} \tilde {u} \\[3pt] \tilde {v} \end{bmatrix} &= \frac {1}{k^2} \begin{bmatrix} 0 & -i k_y \\[5pt] 0 & i k_x \\ \end{bmatrix} \begin{bmatrix} \tilde {w} \\[3pt] \tilde {\eta } \end{bmatrix}.\end{align}

Equations (2.24)–(2.25) represent a simplified Orr–Sommerfeld system, (2.26)–(2.27) represent a simplified Squire system and (2.28)–(2.29) represent the simplified model where the two subsystems are connected in series. Equations (2.24)–(2.29) are the mathematical equivalent of the block diagram in figure 1(b).

Forming the simplified model allows us to individually express the linear operators using the resolvent. The resolvent forms of the operators are

(2.30) \begin{align} H_{\textit{OS}}(k_x,k_y,\omega ,R) = \left ( i\omega I - \varDelta ^{-1}\mathcal L_{\textit{OS}} \right )^{-1} \end{align}

and

(2.31) \begin{align} H_{Sq}(k_x,k_y,\omega ,R) = \left ( i\omega I - \mathcal L_{Sq} \right )^{-1}. \end{align}

For the simple model, the resolvent then becomes

(2.32) \begin{align} \begin{aligned} H_{\textit{simple}}(k_x,k_y,\omega ,R) &= \frac {1}{k^2} \begin{bmatrix} 0 & -ik_y \\[3pt] 0 & ik_x \end{bmatrix} \left ( i\omega I - \begin{bmatrix} \varDelta ^{-1}\mathcal L_{\textit{OS}} & 0 \\[3pt] -ik_yU' & \mathcal L_{Sq} \end{bmatrix} \right )^{-1} \\ &\quad\times \begin{bmatrix} -ik_x\varDelta ^{-1}\mathcal D & -ik_y\varDelta ^{-1}\mathcal D & -k^2\varDelta ^{-1} \\[3pt] 0 & 0 & 0 \end{bmatrix} .\end{aligned} \end{align}

These operators can then be evaluated to determine their response to input forcings. For example, the input–output response of the Orr–Sommerfeld subsystem to the forcing $\tilde {\boldsymbol d}$ is given by

(2.33) \begin{align} \tilde {w} = H_{\textit{OS}}(k_x,k_y,\omega ,R) \begin{bmatrix} -ik_x\varDelta ^{-1}\mathcal D & -ik_y\varDelta ^{-1}\mathcal D & -k^2\varDelta ^{-1} \end{bmatrix} \tilde {\boldsymbol d}. \end{align}

Similarly, the response of the Squire subsystem to $\tilde {w}$ (including the coupling term $-ik_y U'$ ) is evaluated as

(2.34) \begin{align} \tilde {\eta } = H_{Sq}(k_x,k_y,\omega ,R) \left ( -ik_y U' \right ) \tilde {w}. \end{align}

This relationship can then be used with (2.22)–(2.23) to obtain $\tilde {u}$ and $\tilde {v}$ .

2.4.2. Pathways affected by Orr–Sommerfeld

For laminar Poiseuille flow there is a scenario where the Orr–Sommerfeld operator can become unstable; this occurs when the Reynolds number exceeds the stability limit. When $R \gt 5772$ , the Orr–Sommerfeld operator becomes unstable (Orszag Reference Orszag1971) and any pathway that includes this operator can experience exponential growth. In figure 1(a) there are three pathways from the forcing inputs $(\tilde {d}_x,\tilde {d}_y,\tilde {d}_z)$ to the velocity outputs $(\tilde {u},\tilde {v},\tilde {w})$ . The simplified model includes one of these paths (solid pathway); however, the Orr–Sommerfeld operator is also included in the dash-dot pathway in figure 1(a), and is therefore of interest as well. To address this, we form an additional model that captures this pathway.

Figure 3.

Block diagram of the Orr–Sommerfeld system, where only the dash-dot path from figure 1(a) is retained.

Figure 3. Long description

The block diagram of the OrrSommerfeld system features three input variables labeled d_tilde_x, d_tilde_y, and d_tilde_z. Each input variable is processed through a series of operations: d_tilde_x through −i k_x Δ inverse D, d_tilde_y through −i k_y Δ inverse D, and d_tilde_z through −k squared Δ inverse. The outputs of these operations are summed together and fed into a block labeled H_O S (i ω). The output of this block, labeled w_tilde, is then split into three paths. Each path processes w_tilde through different operations: i k_y D over k squared, i k_x D over k squared, and an identity operation labeled I. These operations produce the final outputs labeled u_tilde, v_tilde, and w_tilde respectively.

The block diagram in figure 3 follows the pathway from the external forcing through the Orr–Sommerfeld operator and directly to $\tilde {u}$ , $\tilde {v}$ and $\tilde {w}$ (as opposed to passing through the Squire operator). This block diagram is exactly the dash-dot pathway from figure 1(a). We refer to this model as the Orr–Sommerfeld model. While we do not examine cases where $R \gt 5772$ , analysis of the Orr–Sommerfeld model can still provide insights into which structures are amplified in this subsystem. The Orr–Sommerfeld model is analysed in § 5.3, where the structural analysis provides a clear picture of the dynamics of the dash-dot pathway in terms of the leading modal shape, and gives insight into the modes that become unstable in laminar Poiseuille flow.

2.5. Numerical parameters and structure details

The governing equations are discretised using Chebyshev polynomials with $N=75$ (laminar base flow) and $N=151$ (turbulent base flow) collocation points in the wall-normal direction. Convergence of the results has been checked by doubling the number of Chebyshev collocation points and ensuring that there was a negligible change in the results. In the wall-parallel plane we discretise using $k_{x_{{min}}}$ = $10^{-4}$ , $k_{x_{{max}}}$ = 3.02, $k_{y_{{min}}}$ = $10^{-2}$ and $k_{y_{{max}}}$ = 15.84 with 50 Fourier modes in the streamwise and spanwise directions.

In this paper we frequently reference well-known structures such as streamwise streaks, oblique waves and TS waves. While these structures are commonly referred to in the literature, there is no universal definition of the wavenumbers that correspond to these features. For example, while an oblique wave can be referred to as any structure with a non-zero streamwise and spanwise wavenumber, the oblique waves that receive the most attention are $(k_x = 1, k_y=1)$ in linear studies (Jovanović & Bamieh Reference Jovanović and Bamieh2005; Schmid Reference Schmid2007) or oblique waves between the angles of $20^{\circ }$ and $32^{\circ }$ that play a role in laminar–turbulent bands (Prigent et al. Reference Prigent, Grégoire, Chaté and Dauchot2003; Kanazawa Reference Kanazawa2018; Liu & Gayme Reference Liu and Gayme2021). To address this, in this paper we use the definitions provided in table 1 when referring to streaks, oblique waves and TS waves. Additionally, figure 4 denotes the generalised locations of structures as a function of $k_x$ and $k_y$ .

Table 1.

Summary of structures and their corresponding wavenumbers $k_x$ and $k_y$ .

Figure 4.

Cartoon showing the generalised structure locations for streaks, oblique waves (shown here with $k_x \approx 1$ , $k_y \approx \pm 1$ superimposed) and TS waves. Note that while all wavenumber pairs where $k_x$ and $k_y$ are non-zero are technically oblique, this study focuses mainly on the most amplified pair at $k_x \approx 1$ , $k_y \approx 1$ . Three-dimensional call-outs depict physical structures observed in a DNS. Streaks and oblique waves are shown using streamwise velocity contours and TS waves are shown using wall-normal velocity contours (positive, red; negative, blue).

Figure 4. Long description

A line graph illustrates the generalised structure locations for streaks, oblique waves, and TS waves in the context of NavierStokes equations. The x-axis represents the wavenumber k_y, ranging from 10^−2 to 10^1, while the y-axis represents the wavenumber k_x, ranging from 10^−4 to 10^0. The graph highlights three key regions: TS waves approximately at (1, 0), oblique waves approximately at (1, 1), and streaks approximately at (0, 1.66). Three-dimensional call-outs depict physical structures observed in direct numerical simulations (DNS). Streaks and oblique waves are shown using streamwise velocity contours, while TS waves are shown using wall-normal velocity contours, with positive values in red and negative values in blue. All values are approximated.

The locations in figure 4 represent the highly amplified wavenumber pairs previously identified in linear studies (Jovanović & Bamieh Reference Jovanović and Bamieh2005; Schmid Reference Schmid2007). An analysis examining amplification from a single forcing component into a single velocity component identified several amplification mechanisms that are responsible for the creation of (i) streamwise streaks with peak values at $k_x \approx 0$ , $k_y \approx 1$ ; (ii) oblique waves with peak values at $k_x \approx 1$ , $k_y \approx 1$ ; (iii) TS waves with peak values at $k_x \approx 1$ , $k_y \approx 0$ (Jovanović & Bamieh Reference Jovanović and Bamieh2005). The observations were made by examining all nine possible combinations of input–output pairs. For example, the ‘mechanism that transforms low-amplitude streamwise vortices ( $v$ - $w$ -input) into large-energy streaks ( $u$ -output)’ (Schmid Reference Schmid2007) is seen in plots showing the influence of spanwise or wall-normal forcing on the streamwise velocity. In these studies it is observed that there are multiple instances of oblique waves that are amplified, particularly along a ridge where $k_x = k_y$ . Figure 4 uses the peak amplification values from all nine plots (using present data for contours) to highlight the general structure locations identified by Jovanović (Reference Jovanović and Bamieh2005). The call-outs in the figure show velocity contours that depict physical space representations of the structures.

3.1. Linear amplification in different shear flow profiles

Figure 5 shows energy amplification, using the infinity norm, for different shear flow profiles and both models. Plots contain the combined effects from all forcing directions, $\tilde {d}_x,\tilde {d}_y,\tilde {d}_z$ , and velocity components, $\tilde {u},\tilde {v},\tilde {w}$ . The rows contain the full LNS system (panels a,b,c) and the simplified model (panels d,e, f); columns contain different flow profiles.

Figure 5.

Plots of $\log _{10}({\Vert H\Vert _\infty })$ for (a,d) laminar Couette flow, (b,e) laminar Poiseuille flow for $R = 2000$ , and (c, f) turbulent Poiseuille flow at $R_{\tau } = 2000$ using the full LNS model (a,b,c) and the simplified model (d,e, f), where all forcing input directions ( $\tilde {d_x},\tilde {d_y},\tilde {d_z}$ ) and all velocity components ( $\tilde {u},\tilde {v},\tilde {w}$ ) are considered. Oval and rectangular regions (labelled I–VI) are used to highlight disagreement between the LNS model and the simple model.

Figure 5. Long description

The image contains six contour plots arranged in two rows and three columns. Each plot represents fluid flow analysis under different conditions and models. The top row shows results for laminar Couette flow, laminar Poiseuille flow, and turbulent Poiseuille flow using the full LNS model. The bottom row shows similar conditions using a simplified model. Each plot has axes labeled with kx and ky, representing wavenumbers in the streamwise and spanwise directions, respectively. The color scale indicates the magnitude of the response, with darker blues representing higher values. Oval and rectangular regions labeled I, II, III, IV, V, and VI highlight areas of disagreement between the LNS model and the simplified model.

From previous studies, it is known that streamwise streaks are a dominant feature in all the shear flow profiles we consider (Kitoh & Umeki Reference Kitoh and Umeki2008; Lee et al. Reference Lee, Lee, Choi and Sung2014). Additionally, oblique waves are observed as a path to transition in both Poiseuille and Couette flow due to three-dimensional interactions (Orszag & Kells Reference Orszag and Kells1980). On the other hand, TS waves are stable in Couette flow (Davey Reference Davey1973), making their instability of primary interest in only our Poiseuille flow cases. At a glance, these trends remain true for the full LNS model in figure 5 (streaks at $k_x \approx 0$ , $k_y \approx 1.66$ ; oblique waves at $k_x \approx 1$ , $k_y \approx 1$ ; TS waves at $k_x \approx 1$ , $k_y \approx 0$ ). However, a more detailed analysis of each shear flow profile reveals differences in the full LNS and simplified models.

The infinity norm of the resolvent as a function of ( $k_x,k_y$ ) for Couette flow is shown in figure 5(a,d). Couette flow contains persistent large-scale streamwise streaks, both in numerical simulations (Lee & Moser Reference Lee and Moser2018; Tao, Eckhardt & Xiong Reference Tao, Eckhardt and Xiong2018; Illingworth Reference Illingworth2020) and in experiments (Tillmark & Alfredsson Reference Tillmark and Alfredsson1998; Kitoh & Umeki Reference Kitoh and Umeki2008). The dominance of these structures increases with increasing Reynolds number, where they can contribute up to 50 % of the Reynolds shear stresses (Pirozzoli, Bernardini & Orlandi Reference Pirozzoli, Bernardini and Orlandi2014). Examining laminar Couette flow, with its single dominant mechanism, therefore presents the simplest case. Here, a successful model needs to capture amplification of streamwise streaks only. In figure 5(a,d), both the full LNS and simplified models show amplification in the streamwise streak region near $k_x \approx 0$ , $k_y \approx 1.66$ . The good agreement between the two models indicates that the forcing required to generate streamwise streaks has been retained in the simplified set of equations. While streamwise streaks are the most persistent feature in Couette flow, other less amplified structures still exist. For example, the amplification of oblique waves around $k_x \approx 1$ , $k_y \approx 1$ is still present, however, these structures are not as amplified as the streaks. The agreement in this region is good when comparing the two models. Tollmien–Schlichting waves are also observed at a lower amplification (compared with streaks), where in figure 5(a) a small, light blue, horizontal region exists in the contours around $k_x \approx 1$ , $k_y \approx 0$ . For the simplified model (figure 5 d) however, this same feature does not exist. Compare regions labelled I and II in figure 5(a,d).

The infinity norm of the resolvent for the laminar Poiseuille flow case is shown in figure 5(b,e). In the full LNS model (figure 5 b), there are three regions where amplification occurs. The first region corresponds to streamwise streaks for which $k_x$ $\approx$ 0, $k_y$ $\approx$ 1.66. The second region corresponds to TS waves for which $k_x$ $\approx$ 1, $k_y$ $\approx$ 0. The third is at the intersection of these regions, $k_x$ $\approx$ 1, $k_y$ $\approx$ 1, where the most amplified oblique waves are present. Figure 5(e) shows the laminar Poiseuille case for the simplified model. There is excellent agreement in the region corresponding to streamwise streaks and oblique waves. However, in the region corresponding to TS waves, the model shows poor agreement. Compare regions labelled III and IV in figure 5(b,e). It is clear the simplified model contains the necessary components for streamwise streaks and oblique waves, but not TS waves.

The infinity norm of the resolvent for the turbulent Poiseuille flow case is shown in figure 5(c, f). Here, we use the modified Orr–Sommerfeld and Squire equations from § 2.3. In terms of streaks, oblique waves and TS waves, the results for the simplified model and full model are in good agreement. The results for the streamwise streak region in both the simplified and full model are nearly identical in both magnitude and shape. The observed differences are similar to the laminar cases. The first notable difference is that the TS-wave region ( $k_x \approx 1$ , $k_y \approx 0$ ) disappears in the simplified model. This comparison indicates that the simplified model lacks the necessary forcing components to reproduce TS waves. The second notable difference is the region around $k_x \le 0.1, k_y \le 0.5$ where amplification is diminished in the simple model for the turbulent Poiseuille flow case (e.g. compare regions labelled V and VI in figure 5 c,f). In the full LNS model, the behaviour in region V arises from the inclusion of the eddy-viscosity term in the linearised equations. When the turbulent mean profile is used without an eddy viscosity, the amplification becomes qualitatively similar to that of the laminar Poiseuille case (as verified by additional computations, not shown). This demonstrates that the observed change in behaviour is primarily due to the eddy-viscosity modification of the linear dynamics, rather than the turbulent mean profile itself. We also note that the low wavenumbers (region V) do not become more amplified in the presence of an eddy viscosity; rather, the peak amplification in the streamwise streak region is reduced (e.g. $\log _{10} (\Vert H \Vert _{\infty } ) \approx 4$ in the laminar case and $\log _{10} (\Vert H \Vert _{\infty } ) \approx 1.4$ in the turbulent case), which makes the low-wavenumber response appear relatively more prominent. The drop in amplification observed in the simplified model is instead related to the removal of the forcing terms ( $\tilde {\boldsymbol{d}}$ ) acting on the Squire equation when deriving the simple model. This mechanism is discussed further in § 3.2.3.

3.2. Model validation

An analysis of individual input–output responses for the LNS system identified several amplification mechanisms responsible for creating streaks, oblique waves and TS waves (Jovanović & Bamieh Reference Jovanović and Bamieh2005). To address the agreements and disagreements in the model, we briefly consider the resolvent operator broken down into individual forcing components ( $\tilde {d}_x,\tilde {d}_y,\tilde {d}_z$ ) and individual velocity responses ( $\tilde {u},\tilde {v},\tilde {w}$ ). By conducting a componentwise comparison, where single-input, single-output pairs are examined, we expand on the comparison performed in figure 5 in two ways: (i) highlight specifically which dominant mechanisms are captured by the simplified model for individual forcing and response pairs (e.g. streamwise streaks via $\tilde {d}_y$ to $\tilde {u}$ ); (ii) identify the impact of the neglected paths in the models for individual forcing ( $\tilde {d}_x,\tilde {d}_y,\tilde {d}_z$ ) and velocity ( $\tilde {u},\tilde {v},\tilde {w}$ ) pairs. The componentwise breakdown is shown in figures 6–8 for the full LNS model (figure 1 a) in the top row and the simplified model (figure 1 b) in the bottom row. The velocity outputs in the streamwise, spanwise and wall-normal directions are shown as a function of $k_x$ and $k_y$ for each forcing input $\tilde {d}_x, \tilde {d}_y, \tilde {d}_z$ (columns). The notation $H_{rs}$ denotes the componentwise transfer function from the forcing component $s \in \{\tilde {d}_x,\tilde {d}_y,\tilde {d}_z\}$ to velocity response $r \in \{\tilde {u},\tilde {v},\tilde {w}\}$ . We examine the laminar Poiseuille flow case, as it contains amplified regions for streaks, oblique waves and TS waves. Conclusions made about the simplified model’s capability for this case can be extended to laminar Couette flow and turbulent Poiseuille flow, where the same discrepancies are less pronounced.

3.2.1. Amplification and structures in wavenumber space

Figure 6(b,c) shows that the largest amplification occurs in the streamwise component near $k_x \approx 0$ , $k_y \approx 1.66$ when forced by the spanwise or wall-normal forcing component. This amplification corresponds to the creation of streamwise vortices and streaks. Figures 6(a–c) and 7(a–c) also have notable amplification near $k_x \approx 1$ , $k_y \approx 1$ . Amplification in this region corresponds to oblique waves. For oblique waves, the response to linear amplification appears in all three velocity components, but is largest in the streamwise and spanwise velocities along the ridge $k_x \approx k_y$ . Figure 6(a) has a significant velocity response in the streamwise direction around $k_x \approx 1$ , $k_y \approx 0$ . The amplification in these spanwise-constant wavenumbers corresponds to TS waves.

Figure 6.

Plots of $\log _{10}(\Vert H_{us} \Vert _\infty )$ for laminar Poiseuille flow for $R$ = 2000 using the full LNS model (a,b,c) and simplified model (d,e, f), where $s$ is the forcing input directions ( $\tilde {d}_x,\tilde {d}_y,\tilde {d}_z$ ). Oval regions (labelled I–II) are used to highlight disagreement between the LNS model and the simple model.

Figure 6. Long description

The image contains six contour plots arranged in a 2x3 grid, comparing the full linearized Navier-Stokes (LNS) model and a simplified model for laminar Poiseuille flow at a Reynolds number of 2000. The x-axis represents the wavenumber in the y-direction (k_y) and the y-axis represents the wavenumber in the x-direction (k_x) on a logarithmic scale. Each plot shows the logarithm of the infinity norm of different components of the LNS operator. Plots (a), (b), and (c) correspond to the full LNS model, while plots (d), (e), and (f) correspond to the simplified model. The color scale indicates the magnitude of the infinity norm, with darker blue representing higher values. Oval regions labeled III in plots (a) and (d) highlight areas where the full LNS model and the simplified model disagree. The plots illustrate how the simplified model deviates from the full LNS model in certain wavenumber ranges, emphasizing the importance of the full model for accurate analysis. All values are approximated.

Figure 7.

Plots of $\log _{10}(\Vert H_{vs}\Vert _\infty )$ for laminar Poiseuille flow for $R$ = 2000 using the full LNS model (a,b,c) and simplified model (d,e, f), where $s$ is the forcing input directions ( $\tilde {d}_x,\tilde {d}_y,\tilde {d}_z$ ). Rectangular regions (labelled III–IV) are used to highlight disagreement between the LNS model and the simple model.

Figure 7. Long description

Six contour plots compare the full LNS model and a simplified model for laminar Poiseuille flow at a Reynolds number of 2000. The plots are organized into two rows: the top row (a, b, c) represents the full LNS model, while the bottom row (d, e, f) represents the simplified model. Each plot shows the logarithm of the infinity norm of different components of the flow. The x-axis and y-axis of each plot are labeled with kx and ky, respectively, ranging from 10^−4 to 10^1. Rectangular regions labeled III and IV highlight areas where the models disagree. The color scale on the right of each plot indicates the magnitude of the values, with darker blues representing higher values.

3.2.2. Agreement between the full LNS model and the simplified model

The agreement in the peak magnitude and the shape for the streak and oblique wave regions is excellent when comparing the full and simplified models. Panels (b,c,e, f) in figure 6 have their largest values at $k_x \approx 0$ , $k_y \approx 1.66$ , corresponding to streamwise streaks. In both models, the peak amplification is $\log _{10} (\Vert H_{uy}\Vert _{\infty } ) \approx 4$ and $\log _{10} (\Vert H_{uz}\Vert _{\infty } ) \approx 4$ . The most amplified oblique waves are seen in figures 6(a,d) and 7(a,b,d,e). The largest amplification observed for oblique waves is $\log _{10} (\Vert H_{vy}\Vert _{\infty } ) \approx 2.3$ near $k_x \approx 1$ , $k_y \approx 1$ and $k_x \approx 0.01$ , $k_y \approx 0.01$ . The simplified model contains both of these features despite only including the streamwise and spanwise velocity components (the simplified model does not output a wall-normal velocity component; see figure 1 b). Therefore, the amplification of streaks and oblique waves is primarily contained in the streamwise and spanwise velocity components.

Figure 8.

Plots of $\log _{10}(\Vert H_{ws}\Vert _\infty )$ for Poiseuille flow for $R$ = 2000 using the full LNS model and simplified model, where $s$ is the forcing input directions ( $\tilde {d}_x,\tilde {d}_y,\tilde {d}_z$ ). Here, the wall-normal velocity component is the same for both systems, but is recovered from the simplified model using (2.25)–(2.26).

Figure 8. Long description

Three contour plots compare the logarithmic values of different matrix norms for Poiseuille flow at a Reynolds number of 2000 using full and simplified models. The x-axis represents the variable k_y, and the y-axis represents the variable k_x. The first plot (a) shows the logarithmic value of the infinity norm of the matrix H_wx, the second plot (b) shows the logarithmic value of the infinity norm of the matrix H_wy, and the third plot (c) shows the logarithmic value of the infinity norm of the matrix H_wz. Each plot uses a color gradient to indicate the magnitude of the logarithmic values, with darker blues representing higher values. The contour lines provide additional detail on the gradient of these values across the k_x and k_y variables. The plots reveal differences in the full LNS and simplified models, highlighting variations in the stability and behavior of streamwise streaks, oblique waves, and TS waves.

3.2.4. Influence of the wall-normal velocity component

Figure 8 shows the wall-normal velocity component as a function of ( $k_x,k_y$ ) in response to all three forcing components ( $\tilde {d}_x,\tilde {d}_y,\tilde {d}_z$ ). The wall-normal component is the same in both the full and simplified systems. However, the contribution of wall-normal velocity is different in each model. In the LNS model it serves as a forcing to the Squire system and also directly contributes to the output; whereas in the simplified model it serves only as an input forcing to the Squire system (figure 1 a,b). Figure 8 shows large amplification in the regions of wavenumber space that corresponds to TS waves, oblique waves and streamwise streaks. Therefore, the wall-normal velocity plays an important role in all of these features.

It is worth noting that although the wall-normal velocity $\tilde {w}$ is computed internally within the simplified model, via the Orr–Sommerfeld operator, it is not retained as an output of the simplified model by design. We know from figure 2 that omitting as an output $\tilde {w}$ will reduce the accuracy of the model for certain wavenumber pairs. However, this truncation is what enables the decomposition of the linear dynamics into two subsystems in series and isolates amplification pathways associated with the lift-up mechanism. Later, in § 5.3 we explore a separate transfer function that makes clear the impact of $\tilde {w}$ as an output.

Figure 9.

Plots of $\log _{10}(\Vert H_{\textit{OS}}\Vert _\infty )$ (a), $\log _{10}(\Vert H_{Sq-u}\Vert _\infty )$ (b) and $\log _{10}(\Vert H_{u}\Vert _\infty )$ (c) in Poiseuille flow for $R = 2000$ (logarithmic scaling) using the simplified model. All plots shown are a result of forcing in all directions ( $\tilde {d_x},\tilde {d_y},\tilde {d_z}$ ).

Figure 9. Long description

Three contour plots compare logarithmic scaling in Poiseuille flow using a simplified model. Each plot represents different aspects of the flow dynamics. The x-axis and y-axis of each plot are labeled with kx and ky, respectively, and use a logarithmic scale. The color gradient indicates the magnitude of the values, with darker blues representing higher values. Plot (a) shows the Log10 of the HOS operator norm, plot (b) displays the Log10 of the H sub Sq-u operator norm, and plot (c) presents the Log10 of the H sub u operator norm. All plots are results of forcing in all directions. The contour lines within each plot illustrate the gradient and distribution of the values across the logarithmic scale.

Figure 10.

Plots of $\log _{10}(\Vert H_{\textit{OS}}\Vert _\infty )$ (a), $\log _{10}(\Vert H_{Sq-v}\Vert _\infty )$ (b) and $\log _{10}(\Vert H_{v}\Vert _\infty )$ (c) in Poiseuille flow for $R = 2000$ (logarithmic scaling) using the simplified model. All plots shown are a result of forcing in all directions ( $\tilde {d_x},\tilde {d_y},\tilde {d_z}$ ). Note panel (a) is the same as in figure 9.

Figure 10. Long description

Three contour plots compare logarithmic scaling of different flow analyses in Poiseuille flow. Each plot represents the result of forcing in all directions. Plot (a) shows the amplification of disturbances using the simplified model, similar to figure 9. Plot (b) illustrates the logarithmic scaling of another flow analysis, while plot (c) presents a different perspective on the same topic. The x-axis and y-axis of each plot are labeled with kx and ky, respectively, indicating wave numbers. The color gradient from light blue to dark blue represents the magnitude of the logarithmic scaling, with darker shades indicating higher values. The plots highlight how disturbances in fluid flow are amplified under different conditions, providing insights into turbulent mechanisms.

5.1. Streamwise streaks

Figure 11 contains the leading response mode for the streamwise velocity component, $u$ , projected onto the y–z plane for $k_x = 0, k_y = 1.66$ . This wavenumber pair corresponds to streamwise streaks. In § 3 the streamwise streak region contained the strongest amplifications for all three shear flow profiles.

Figure 11.

Leading response modes for the $u$ velocity component in the y–z plane. Panels (a,b,c) correspond to the LNS model, while panels (d,e, f) display the simplified model. Columns contain different shear flow profiles. The wavenumbers for all plots are $k_x = 0, k_y = 1.66$ .

Figure 11. Long description

A heat map displays the leading response modes for the velocity component in the yz plane. The map is divided into six panels, with panels (a), (b), and (c) corresponding to the LNS model, and panels (d), (e), and (f) displaying the simplified model. Each column represents different shear flow profiles: Laminar Couette, Laminar Poiseuille, and Turbulent Poiseuille. The color scale ranges from −0.4 to 0.4, with blue indicating lower values and red indicating higher values. The heat map shows distinct patterns of velocity distribution across the yz plane for each model and shear flow profile.

In figure 11 the rows correspond to the LNS model (panels a,b,c) and simplified model (panels d,e, f) while the columns contain the results for the different flow profiles. For this wavenumber pair, the simplified model is in excellent agreement with the LNS model for the first two cases. Agreement for the turbulent case is good, but there are exceptions in the shape and magnitude near the wall. This result confirms that streamwise streaks are a result of a favourable interaction between the Orr–Sommerfeld and Squire linear operators, as this mechanism is unaffected by the pathways that have been turned off in the simplified model.

5.2. Oblique waves

The leading modal responses for the streamwise velocity component, $u$ , are shown in figure 12 for the oblique wave case ( $k_x = 1, k_y = 1$ ). Based on results from § 3, good agreement is expected between the LNS and simplified models in this region, as the magnitude of energy amplification is similar between models.

Figure 12.

Leading response modes for the $u$ velocity component in the x–z plane. Panels (a,b,c) correspond to the LNS model, while panels (d,e, f) display the simplified model. Columns contain different shear flow profiles. The wavenumbers for all plots are $k_x = 1, k_y = 1$ .

Figure 12. Long description

A heat map displays the leading response modes for the velocity component in the xz plane. The heat map is divided into six panels, with panels (a), (b), and (c) corresponding to the linearized Navier-Stokes (LNS) model, while panels (d), (e), and (f) display the simplified model. Each column represents different shear flow profiles: Laminar Couette, Laminar Poiseuille, and Turbulent Poiseuille. The x-axis ranges from −4 to 3, and the z-axis ranges from −0.5 to 0.5. The color scale indicates the magnitude of the response, with red representing positive values up to 0.4 and blue representing negative values down to −0.4. The heat map shows distinct patterns of velocity response, with variations in intensity and distribution across different shear flow profiles and models.

All three shear flow profiles display structures tilted in the streamwise direction. In the case of laminar Couette flow, there is excellent agreement for the single tilted structure. In both the laminar Poiseuille and turbulent Poiseuille flow cases, there is anti-symmetry in the tilted structures about the channel half-height. While both Poiseuille flow cases are in good agreement when comparing the LNS and simplified model, there is slight disagreement in the profiles near the wall. These near-wall disagreements are more pronounced in the turbulent Poiseuille flow case.

5.3. Tollmien–Schlichting waves via the Orr–Sommerfeld model

The final structures we examine are for the wall-normal velocity response, $w$ , in TS waves at a wavenumber pair of $k_x = 1, k_y = 0$ . Based on results in § 3, it is evident that the simplified model in figure 1(b) is not capable of producing the amplifications for the TS-wave region due to the Squire operator (and the $w$ component is zero in the simplified model). Because of this, we instead compare the LNS model to the Orr–Sommerfeld model (figure 1 c) where results from figure 9(a) show significant response in the TS-wave region. Analysis of the Orr–Sommerfeld system gives attention to the amplification in stable TS waves and outlines what is stripped from the simplified model by not including the dash-dot pathway.

Figure 13.

Leading response modes for the $w$ velocity component in the x–z plane. Panels (a,b,c) correspond to the LNS model, while panels (d,e, f) display the Orr–Sommerfeld model (figure 3). Columns contain different shear flow profiles. The wavenumbers for all plots are $k_x = 1, k_y = 0$ .

Figure 13. Long description

A heat map displays the leading response modes for the velocity component in the xz plane. The map is divided into six panels, with the top row representing the LNS model and the bottom row representing the Orr-Sommerfeld model. Each column corresponds to different shear flow profiles: Laminar Couette, Laminar Poiseuille, and Turbulent Poiseuille. The color scale ranges from blue to red, indicating values from −0.2 to 0.2. The x-axis ranges from approximately −3 to 4, and the z-axis ranges from −0.5 to 0.5. The heat map shows symmetrical patterns with central concentration and gradients moving outward from the center.

There is excellent agreement between models in all three shear flow cases. The leading response modes in figure 13 show that the Orr–Sommerfeld system by itself is capable of reproducing structures that resemble TS-wave responses. In the simplified model, this response is attenuated by the Squire system; whereas in the LNS model and Orr–Sommerfeld model, we observe a nearly identical response. These results are in agreement with the idea from § 3 that TS waves are generated as a result of the Orr–Sommerfeld system alone.

This section highlights three key points: (i) in regions of wavenumber space where streaks are observed, the simplified model produces a similar response to the LNS model (figure 11), which means the lift-up mechanism is not significantly affected by the pathways that were removed; (ii) the simplified model is able to produce similar structures for oblique waves (figure 12); (iii) the Orr–Sommerfeld model (figure 3) can reproduce a TS-wave-like response using the Orr–Sommerfeld system alone (figure 13).