2.1. Problem set-up

We begin with the compressible Navier–Stokes equations in an arbitrary coordinate system, written abstractly as

(2.1)\begin{equation} \frac{\partial q}{\partial t} = \mathcal{N}(q). \end{equation}

Equation (2.1) contains mass, momentum and energy equations, and the state vector $q(x,y,z,t)$ contains an appropriate set of variables, e.g. velocity components and two thermodynamic variables such as density and pressure. Incompressible Navier–Stokes equations can also be easily accommodated within a global resolvent analysis. For the one-way methodology developed in § 3, the lack of a time derivative within the incompressible continuity equation changes the mathematical nature of the equations. Instead, incompressible flows can be addressed within the one-way framework by selecting a small Mach number, e.g. $0.01$, which does not lead to stiffness issues in the spatial march as it would for time stepping.

Applying the Reynolds decomposition

(2.2)\begin{equation} q(x,y,z,t) = \bar{q}(x,y,z) + q^{\prime}(x,y,z,t), \end{equation}

and moving terms that are linear and nonlinear in the fluctuation $q^{\prime }$ to the left- and right-hand sides of the equations, respectively, leads to an equation of the form

(2.3)\begin{equation} \frac{\partial q^{\prime}}{\partial t} - \mathcal{A} q^{\prime} = f. \end{equation}

The left-hand-side of (2.3) is the linearized Navier–Stokes equations, while the right-hand-side vector $f(x,y,z,t)$ contains all remaining nonlinear terms, which within the context of resolvent analysis are interpreted as an external forcing on the linearized equations. The linear operator $\mathcal {A}(x,y,z)$ is the Jacobian of $\mathcal {N}$ evaluated at the mean flow $\bar {q}$. We also introduce an observable

(2.4)\begin{equation} y^{\prime} = \mathcal{C} q^{\prime}, \end{equation}

that is extracted from the state vector by the output matrix $\mathcal {C}(x,y,z)$.

2.2. Global resolvent analysis

In practice, computing resolvent modes requires numerical discretization of (2.3) in all inhomogeneous spatial directions (if the base flow is homogenous in a coordinate direction, the solution can be decomposed into Fourier modes in that direction to reduce the number of dimensions in which the linearized equations must be discretized). After discretization and incorporation of appropriate boundary conditions, (2.3) may be written as

(2.5a)\begin{gather} \frac{\partial \boldsymbol{q}_{g}^{\prime} }{\partial t} - \boldsymbol{\mathsf{A}}_g \boldsymbol{q}^{\prime} = \boldsymbol{\mathsf{B}}_g\boldsymbol{f}_{g}, \end{gather}

(2.5b)\begin{gather}\boldsymbol{y}_{g}^{\prime} = \boldsymbol{\mathsf{C}}_g \boldsymbol{q}_{g}^\prime, \end{gather}

where $\boldsymbol {q}_{g}^\prime$, $\boldsymbol{\mathsf{A}}_g$ and $\boldsymbol{\mathsf{C}}_g$ are the globally discretized state vector, linear operator and output matrix, respectively. We have also introduced an input matrix $\boldsymbol{\mathsf{B}}_g$ that can be used to tailor the properties of the forcing $\boldsymbol {f}_{g}$, e.g. to restrict it to certain parts of the domain. The ‘$g$’ subscripts are used to distinguish these globally discretized variables from analogous semi-discretized variables defined later in the context of OWNS.

The relationship between the nonlinear forcing term and the observable can be expressed in the frequency domain as

(2.6)\begin{equation} \hat{\boldsymbol{y}}_{g} = \boldsymbol{\mathsf{C}}_g \boldsymbol{\mathsf{R}}_{g} \boldsymbol{\mathsf{B}}_g \hat{\boldsymbol{f}}_{g}, \end{equation}

where $\hat {\boldsymbol {y}}_{g}$ and $\hat {\boldsymbol {f}}_{g}$ are the Fourier transforms of $\boldsymbol {y}_{g}^{\prime }$ and $\boldsymbol {f}_{g}$, respectively, and

(2.7)\begin{equation} \boldsymbol{\mathsf{R}}_{g} = ( -{\rm i} \omega \boldsymbol{\mathsf{I}} - \boldsymbol{\mathsf{A}}_{g} )^{{-}1}, \end{equation}

is the global resolvent operator.

Resolvent analysis seeks the forcing and response pairs that produce the largest gain. To make this concept precise, we must define a global inner product

(2.8)\begin{equation} \langle \boldsymbol{a},\boldsymbol{b} \rangle_g = \langle \boldsymbol{a},\boldsymbol{\mathsf{W}}_e \boldsymbol{b} \rangle = \boldsymbol{a}^H \boldsymbol{\mathsf{W}}_{xyz} \boldsymbol{\mathsf{W}}_e \boldsymbol{b} = \boldsymbol{a}^H \boldsymbol{\mathsf{W}}_g \boldsymbol{b}, \end{equation}

where $H$ represents the Hermitian transpose and $\boldsymbol{\mathsf{W}}_g$ is a positive-definite weight matrix. Here $\boldsymbol{\mathsf{W}}_g$ is constructed as $\boldsymbol{\mathsf{W}}_g = \boldsymbol{\mathsf{W}}_{xyz} \boldsymbol{\mathsf{W}}_e$, where $\boldsymbol{\mathsf{W}}_e$ is chosen so that the output represents a physical quantity of interest (e.g. energy) and $\boldsymbol{\mathsf{W}}_{xyz}$ is a diagonal positive-definite matrix of quadrature weights so that the inner product represents, up to a discretization error, the volume-integrated quantity. Inner products without any subscripts involve only quadrature weights.

The gain between the forcing and response is defined by the global Rayleigh quotient

(2.9)\begin{equation} \sigma^2_g(\omega) = \frac{ \langle \hat{\boldsymbol{y}}_{g},\hat{\boldsymbol{y}}_{g} \rangle_g }{ \langle\, \hat{\boldsymbol{f}}_{g},\hat{\boldsymbol{f}}_{g} \rangle_g } = \frac{ \hat{\boldsymbol{f}}_{g}^H \boldsymbol{\mathsf{B}}_g^H \boldsymbol{\mathsf{R}}_g^H \boldsymbol{\mathsf{C}}_g^H \boldsymbol{\mathsf{W}}_{g} \boldsymbol{\mathsf{C}}_g\boldsymbol{\mathsf{R}}_g\boldsymbol{\mathsf{B}}_g\hat{\boldsymbol{f}}_{g} }{ \hat{\boldsymbol{f}}_{g}^H \boldsymbol{\mathsf{W}}_{g} \hat{\boldsymbol{f}}_{g} }. \end{equation}

The forcing and response that maximize the gain are then sought, and solutions to this standard problem can be obtained via SVD. Owing to the weight matrix, one can either work with a generalized SVD or transform to the standard one by defining $\hat {\boldsymbol {f}}_{gW}=\boldsymbol{\mathsf{W}}_{g}^{1/2}\hat {\boldsymbol {f}}_{g}$ and maximizing

(2.10)\begin{equation} \sigma^2_g(\omega) = \frac{ \hat{\boldsymbol{f}}_{gW}^H \boldsymbol{\mathsf{R}}_{gW}^H \boldsymbol{\mathsf{R}}_{gW} \hat{\boldsymbol{f}}_{gW} }{ \hat{\boldsymbol{f}}_{gW}^H \hat{\boldsymbol{f}}_{gW} }, \end{equation}

where $\boldsymbol{\mathsf{R}}_{gW} = \boldsymbol{\mathsf{W}}_{g}^{1/2} \boldsymbol{\mathsf{C}}_g\boldsymbol{\mathsf{R}}_g\boldsymbol{\mathsf{B}}_g \boldsymbol{\mathsf{W}}_{g}^{-1/2}$ is a weighted form of the resolvent operator Towne et al. (Reference Towne, Schmidt and Colonius2018). Optimal gains and forcings are obtained by computing the eigenvalue decomposition of $\boldsymbol{\mathsf{R}}_{gW}^H \boldsymbol{\mathsf{R}}_{gW}$ or, equivalently, the SVD

(2.11)\begin{equation} \boldsymbol{\mathsf{R}}_{gW} = \boldsymbol{\mathsf{U}}_{gW} \boldsymbol{\varSigma}_{g} \boldsymbol{\mathsf{V}}_{gW}^{H}. \end{equation}

The singular values, which appear within the diagonal positive-semi-definite matrix $\boldsymbol {\varSigma }_{g}$, give the square root of the optimal gains between the input and output modes contained in the columns of the matrices $\boldsymbol{\mathsf{V}}_{g} = \boldsymbol{\mathsf{W}}_{g}^{-1/2} \boldsymbol{\mathsf{V}}_{gW}$ and $\boldsymbol{\mathsf{U}}_{g} = \boldsymbol{\mathsf{W}}_{g}^{-1/2} \boldsymbol{\mathsf{U}}_{gW}$, respectively.

While the explicit construction of $\boldsymbol{\mathsf{R}}_{g}$ can usually be avoided (Jeun et al. Reference Jeun, Nichols and Jovanović2016; Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018; Ribeiro et al. Reference Ribeiro, Yeh and Taira2020), it is necessary to compute the action of the resolvent operator on an arbitrary forcing vector, i.e. to evaluate (2.6). Typically, a direct LU decomposition is performed to factorize $(-{\rm i} \omega \boldsymbol{\mathsf{I}} + \boldsymbol{\mathsf{A}}_{g} )$, and this step constitutes the bulk of the computational cost of resolvent analysis. In § 3 we will show how the slow variation of the mean flow present in many flows can be leveraged to efficiently and accurately approximate the action of the resolvent operator on a forcing vector using a spatial marching method. Then in § 4 we will show how this capability can be used to computer optimal forcings and responses, i.e. the singular modes of the approximate resolvent operator.

3.1. Identifying downstream- and upstream-travelling waves

The first critical step in developing a well-posed one-way equation is to identify parts of the solution that transfer energy in the positive and negative streamwise directions, which we call downstream-travelling and upstream-travelling, respectively. To this end, we rewrite (2.3) as

(3.1)\begin{equation} \frac{\partial q^{\prime}}{\partial t} + {\mathsf{A}} \frac{\partial q^{\prime}}{\partial x} + {\mathsf{B}} q^{\prime} + {\mathsf{C}} \frac{\partial q^{\prime}}{\partial x} + {\mathsf{D}} \frac{\partial^2 q^{\prime}}{\partial x^2} = f. \end{equation}

Here, $x \in \mathbb {R}$ is the slowly varying direction along which we will apply spatial marching, and $y$ and $z$ are additional, transverse spatial dimensions. In (3.1) we have isolated $x$-derivative terms arising from the convective and pressure terms in the Navier–Stokes equations (${\mathsf{A}}({\partial }/{\partial x})$) and from streamwise viscous terms (${\mathsf{C}}({\partial }/{\partial x})$, ${\mathsf{D}} ({\partial ^2 }/{\partial x^2})$); the linear operator ${\mathsf{B}}$ contains all other terms in the linearized Navier–Stokes equations. While typically unimportant, the ${\mathsf{C}}({\partial }/{\partial x})$ term can arise due to compressibility or from non-Cartesian coordinate systems.

To obtain a one-way equation, we wish to work in terms of a system including only first $x$-derivatives (Towne & Colonius Reference Towne and Colonius2015). This can be accomplished in one of several ways. The viscous terms can be parabolized by redefining the state vector to include both $q^{\prime }$ and ${\partial q^{\prime }}/{\partial x}$ and writing (3.1) as an expanded system (of twice the original size) using this new state variable (Towne Reference Towne2016; Harris & Hack Reference Harris and Hack2020), analogous to the standard approach for solving quadratic eigenvalue problems (Tisseur & Meerbergen Reference Tisseur and Meerbergen2001). Alternatively, the streamwise viscous terms can be moved to the right-hand-side of (3.1) and treated as a forcing term, which is later evaluated using the solution at the previous step in the spatial march (Kamal et al. Reference Kamal, Rigas, Lakebrink and Colonius2020). Finally, following the standard boundary-layer approximation, the streamwise viscous terms can simply be neglected. We have found this simplification to be sufficient for all flows, including both free-shear and wall-bounded flows, to which OWNS has been applied to date. Accordingly, we neglect ${\mathsf{C}}({\partial }/{\partial x})$ and ${\mathsf{D}} ({\partial ^2 }/{\partial x^2})$ in what follows.

Next, we discretize (3.1) in the transverse directions using a collocation method (such as finite differences) with $N_{c}$ collocation points. The semi-discrete approximation of (3.1) can then be written as

(3.2)\begin{equation} \frac{\partial \boldsymbol{q}^{\prime}}{\partial t} + \boldsymbol{\mathsf{A}}(x) \frac{\partial \boldsymbol{q}^{\prime}}{\partial x} + \boldsymbol{\mathsf{B}} (x)\boldsymbol{q}^{\prime} = \boldsymbol{f}(x,t), \end{equation}

where $\boldsymbol {q}^{\prime }(x,t), \boldsymbol {f}(x,t) \in \mathbb {C}^{N}$ and $\boldsymbol{\mathsf{A}}, \boldsymbol{\mathsf{B}} \in \mathbb {C}^{N \times N}$ are semi-discrete analogues of $q$, $f$, ${\mathsf{A}}$ and ${\mathsf{B}}$, respectively. The entries of $\boldsymbol{\mathsf{A}}$ consist of the values of ${\mathsf{A}}$ at the collocation points, while the matrix $\boldsymbol{\mathsf{B}}$ contains discrete approximations of transverse derivatives contained within ${\mathsf{B}}$ as well as modifications required to enforce the desired transverse boundary conditions. The total size of the semi-discrete system is $N = N_{q}N_{c}$, where $N_{q}$ is the number of state variables in the Navier–Stokes equations (e.g. $N_{q} = 5$ for three-dimensional problems).

Equation (3.2) is a one-dimensional strongly hyperbolic system since $\boldsymbol{\mathsf{A}}$ is diagonalizable and has real eigenvalues (Kreiss & Lorenz Reference Kreiss and Lorenz2004). That is, there exists a transformation $\boldsymbol{\mathsf{T}}(x)$ such that

(3.3)\begin{equation} \boldsymbol{\mathsf{T}}\boldsymbol{\mathsf{A}}\boldsymbol{\mathsf{T}}^{{-}1} = \tilde{\boldsymbol{\mathsf{A}}} = \left[ \begin{array}{@{}ccc@{}} \tilde{\boldsymbol{\mathsf{A}}}_{+{+}} & \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \tilde{\boldsymbol{\mathsf{A}}}_{-{-}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} & \tilde{\boldsymbol{\mathsf{A}}}_{00} \end{array} \right], \end{equation}

where $\tilde {\boldsymbol{\mathsf{A}}}$ is a diagonal matrix and the diagonal entries of the submatrices $\tilde {\boldsymbol{\mathsf{A}}}_{++} \in \mathbb {R}^{N_{+} \times N_{+}} > 0$, $\tilde {\boldsymbol{\mathsf{A}}}_{--} \in \mathbb {R}^{N_{-} \times N_{-}} < 0$ and $\tilde {\boldsymbol{\mathsf{A}}}_{00} \in \mathbb {R}^{N_{0} \times N_{0}} = 0$ contain the positive, negative and zero eigenvalues of $\boldsymbol{\mathsf{A}}$, respectively. Here, $N_{+}$, $N_{-}$ and $N_{0}$ denote the number of positive, negative and zero eigenvalues, respectively, and $N = N_{+} + N_{-} + N_{0}$. The transformation $\boldsymbol{\mathsf{T}}$ is known analytically since it is the discretization of the matrix that diagonalizes ${\mathsf{A}}$.

To derive a one-way equation, it is convenient to work in terms of the characteristic variables of (3.2), which are defined in terms of the transformation $\boldsymbol{\mathsf{T}}(x)$,

(3.4)\begin{equation} \boldsymbol{\phi}(x,t) = \boldsymbol{\mathsf{T}}(x)\boldsymbol{q}^{\prime}(x,t). \end{equation}

These characteristic variables can be split into three components associated with the positive, negative and zero blocks of $\tilde {\boldsymbol{\mathsf{A}}}$,

(3.5)\begin{equation} \boldsymbol{\phi} = \left\{\begin{array}{@{}c@{}} \boldsymbol{\phi}_{+} \\ \boldsymbol{\phi}_{-} \\ \boldsymbol{\phi}_{0} \end{array}\right\}, \end{equation}

with $\boldsymbol {\phi }_{+} \in \mathbb {R}^{N_{+}}$, $\boldsymbol {\phi }_{-} \in \mathbb {R}^{N_{-}}$ and $\boldsymbol {\phi }_{0} \in \mathbb {R}^{N_{0}}$. For later use, we also define

(3.6)\begin{equation} \boldsymbol{\phi}_{{\pm}} = \left\{\begin{array}{@{}c@{}} \boldsymbol{\phi}_{+} \\ \boldsymbol{\phi}_{-} \end{array}\right\}, \end{equation}

which contains only the characteristic variables associated with the non-zero block of $\tilde {\boldsymbol{\mathsf{A}}}$,

(3.7)\begin{equation} \tilde{\boldsymbol{\mathsf{A}}}_{{\pm}{\pm}} = \left[ \begin{array}{@{}cc@{}} \tilde{\boldsymbol{\mathsf{A}}}_{+{+}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \tilde{\boldsymbol{\mathsf{A}}}_{-{-}} \end{array} \right]. \end{equation}

To be clear, the $\pm$ subscript here and throughout the paper does not indicate that we are choosing either the plus or minus characteristic, as in the typical usage of that symbol, but rather both component, as exemplified in (3.6) and (3.7).

In terms of the characteristic variables, (3.2) becomes

(3.8)\begin{equation} \frac{\partial \boldsymbol{\phi}}{\partial t} + \tilde{\boldsymbol{\mathsf{A}}} (x)\frac{\partial \boldsymbol{\phi}}{\partial x} + \tilde{\boldsymbol{\mathsf{B}}} (x)\boldsymbol{\phi} = {\boldsymbol{f}}_{\phi}(x,t), \end{equation}

where $\tilde {\boldsymbol{\mathsf{B}}} = \boldsymbol{\mathsf{T}}\boldsymbol{\mathsf{B}}\boldsymbol{\mathsf{T}}^{-1} + \tilde {\boldsymbol{\mathsf{A}}}\boldsymbol{\mathsf{T}} ({{\rm d}\boldsymbol{\mathsf{T}}^{-1}}/{{\rm d} x})$ and $\boldsymbol {f}_{\phi } = \boldsymbol{\mathsf{T}} \boldsymbol {f}$.

We ultimately wish to obtain the response to a forcing in the frequency domain. However, we proceed by applying a Laplace transform in time, rather than a Fourier transform, to (3.8), giving

(3.9)\begin{equation} s \boldsymbol{\hat{\phi}} + \tilde{\boldsymbol{\mathsf{A}}}(x) \frac{{\rm d} \boldsymbol{\hat{\phi}}}{{\rm d} x} + \tilde{\boldsymbol{\mathsf{B}}} (x)\boldsymbol{\hat{\phi}} = \boldsymbol{\hat{f}}_{\phi}(x,s), \end{equation}

where $\boldsymbol {\hat {\phi }}(x,s)$ is the Laplace transform of $\boldsymbol {\phi }(x,t)$ and $s = \eta - {\rm i} \omega$ ($\eta, \omega \in \mathbb {R}$) is the Laplace dual of $t$. We will ultimately take $\eta = 0$ and set $\omega$ to a particular value to obtain the response to a forcing at that frequency, but keeping the possibility of non-zero $\eta$ will help us distinguish between upstream- and downstream-travelling solutions of (3.8).

Up to this point, we have made no approximation (aside from discarding a small subset of viscous terms), and the action of the resolvent operator on a forcing vector could be computed by discretizing (3.9) in $x$, applying boundary conditions at the beginning and end of the $x$ domain, solving for $\boldsymbol {\hat {\phi }}$, and inverting the transformation (3.4) to obtain $\boldsymbol {\hat {q}}$. However, this involves solving a large system of equations, as discussed in § 2, which constitutes a large fraction of the cost of resolvent analysis. Instead, we will obtain an approximate solution of (3.9) via spatial integration. Directly integrating (3.9) is ill-posed (Li & Malik Reference Li and Malik1997; Towne & Colonius Reference Towne and Colonius2015; Towne et al. Reference Towne, Rigas and Colonius2019), leading to exponential divergence of the solution. To circumvent this, we will derive a well-posed one-way equation that can be stably integrated.

To proceed, we isolate the $x$-derivatives within (3.9), giving

(3.10a)\begin{gather} \tilde{\boldsymbol{\mathsf{A}}}_{{\pm}{\pm}} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}}{{\rm d} x} = \boldsymbol{\mathsf{L}}_{{\pm}{\pm}} \boldsymbol{\hat{\phi}}_{{\pm}} + \boldsymbol{\mathsf{L}}_{\pm0} \boldsymbol{\hat{\phi}}_{0} + \boldsymbol{\hat{f}}_{\phi,\pm}, \end{gather}

(3.10b)\begin{gather}\boldsymbol{0} = \boldsymbol{\mathsf{L}}_{0\pm} \boldsymbol{\hat{\phi}}_{{\pm}} + \boldsymbol{\mathsf{L}}_{00} \boldsymbol{\hat{\phi}}_{0} + \boldsymbol{\hat{f}}_{\phi,0} , \end{gather}

with

(3.11)\begin{equation} \boldsymbol{\mathsf{L}}(x,s) = \left[ \begin{array}{@{}cc@{}} {\boldsymbol{\mathsf{L}}}_{{\pm}{\pm}} & {\boldsymbol{\mathsf{L}}}_{\pm0} \\ {\boldsymbol{\mathsf{L}}}_{0\pm} & {\boldsymbol{\mathsf{L}}}_{00} \end{array} \right] ={-} (s\boldsymbol{\mathsf{I}} + {\boldsymbol{\mathsf{B}}}(x) ). \end{equation}

Here and throughout the paper, the subscripts of a matrix indicate its size, e.g. ${\boldsymbol{\mathsf{L}}}_{0\pm } \in \mathbb {C}^{N_0 \times (N_{+}+N_{-})}$.

Equation (3.10) is a differential-algebraic equation (DAE) due to the zero left-hand-side of (3.10b), which occurs because of the zero eigenvalues of $\boldsymbol{\mathsf{A}}$ contained in the zero matrix $\tilde {\boldsymbol{\mathsf{A}}}_{00}$. These zero eigenvalues correspond to points in the base flow where the streamwise velocity is zero or exactly sonic (Towne & Colonius Reference Towne and Colonius2015). The algebraic conditions in (3.10b) function as a constraint on the allowable form of $\boldsymbol {\hat {\phi }}$. Assuming that ${\boldsymbol{\mathsf{L}}}_{00}$ is invertible, then (3.10) is a DAE of index 1 and the zero characteristic variable $\boldsymbol {\hat {\phi }}_{0}$ is slaved to positive and negative characteristic variables as

(3.12)\begin{equation} \boldsymbol{\hat{\phi}}_{0} ={-} \boldsymbol{\mathsf{L}}_{00}^{{-}1} ( \boldsymbol{\mathsf{L}}_{0\pm} \boldsymbol{\hat{\phi}}_{{\pm}} + \boldsymbol{\hat{f}}_{\phi,0} ). \end{equation}

To obtain a one-way equation, it is convenient to reduce (3.10) to an ordinary differential equation (ODE) for $\boldsymbol {\hat {\phi }}_{\pm }$, which is accomplished by using (3.12) to eliminate $\boldsymbol {\hat {\phi }}_{0}$ from (3.10a), leading to

(3.13)\begin{equation} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}}{{\rm d} x} = \boldsymbol{\mathsf{M}}(x, s) \boldsymbol{\hat{\phi}}_{{\pm}} + \boldsymbol{\hat{h}}(x,s), \end{equation}

with

(3.14)\begin{equation} \boldsymbol{\mathsf{M}} = \boldsymbol{\mathsf{A}}_{{\pm}{\pm}}^{{-}1} ( \boldsymbol{\mathsf{L}}_{{\pm}{\pm}} - \boldsymbol{\mathsf{L}}_{\pm0} \boldsymbol{\mathsf{L}}_{00}^{{-}1} \boldsymbol{\mathsf{L}}_{0\pm}) \end{equation}

and

(3.15)\begin{equation} \boldsymbol{\hat{h}} = \tilde{\boldsymbol{\mathsf{A}}}_{{\pm}{\pm}}^{{-}1} (\,\boldsymbol{\hat{f}}_{\phi,\pm} - \boldsymbol{\mathsf{L}}_{\pm0} \boldsymbol{\mathsf{L}}_{00}^{{-}1} \boldsymbol{\hat{f}}_{\phi,0} ). \end{equation}

While these expressions formally contain $\boldsymbol{\mathsf{L}}_{00}^{-1}$, this inverse, and its potential detrimental effect on the sparsity of $\boldsymbol{\mathsf{M}}$, is in practice avoided by reversing the contraction of the system (from (3.10) to (3.13)) in the final set of one-way equations (see § 3.5 and Appendix C).

If $N_0 = 0$, which is the case, e.g. in subsonic free-shear flows, then all matrices containing a zero index vanish, $\boldsymbol {\hat {\phi }} = \boldsymbol {\hat {\phi }}_{\pm }$, and (3.13) and (3.15) reduce to the simpler forms

(3.16)\begin{equation} \boldsymbol{\mathsf{M}} ={-}\tilde{\boldsymbol{\mathsf{A}}}^{{-}1}(s\boldsymbol{\mathsf{I}} + \tilde{\boldsymbol{\mathsf{B}}} ) \end{equation}

and

(3.17)\begin{equation} \boldsymbol{\hat{h}} = \tilde{\boldsymbol{\mathsf{A}}}^{{-}1} \boldsymbol{\hat{f}}_{\phi} = \tilde{\boldsymbol{\mathsf{A}}}^{{-}1} \boldsymbol{\mathsf{T}}\boldsymbol{\hat{f}}. \end{equation}

Since (3.2) is hyperbolic, the solution $\boldsymbol {\hat {\phi }}_{\pm }$ of (3.13) consists of a summation of downstream- and upstream-travelling modes, i.e. waves that transfer energy in the positive and negative $x$ direction, respectively (waves that do not propagate were eliminated by the contraction of the system that lead to $\boldsymbol{\mathsf{M}}$). These downstream- and upstream-travelling components of the solution of (3.13) can be identified based on the eigenvalues and eigenvectors of $\boldsymbol{\mathsf{M}}$. Consider the eigen-expansion of the solution

(3.18)\begin{equation} \boldsymbol{\hat{\phi}}_{{\pm}}(x,s) = \sum_{k = 1}^{N} \boldsymbol{v}_{k}(x,s) \boldsymbol{\psi}_{k}(x,s), \end{equation}

where each $\boldsymbol {v}_{k}$ is an eigenvector of $\boldsymbol{\mathsf{M}}$ with an associated eigenvalue $i \alpha _{k}$, and $\boldsymbol {\psi }_{k}$ is an expansion coefficient defining the contribution of mode $k$ to the solution. The well-posedness theory of Kreiss (Reference Kreiss1970), which can be thought of as an extension to $x$-dependent systems of Briggs’ criteria (Briggs Reference Briggs1964), provides a means to distinguish downstream- and upstream-travelling components of the solution: the mode associated with the eigenvalue $\alpha _{k}(x,s)$ is downstream travelling at $x = x_{0}$ if

(3.19)\begin{equation} \lim_{\eta \to +\infty} \textit{Im}[\alpha_{k}(x_{0},s)] ={+}\infty, \end{equation}

and upstream travelling if

(3.20)\begin{equation} \lim_{\eta \to +\infty} \textit{Im}[\alpha_{k}(x_{0},s)] ={-}\infty. \end{equation}

The eigenvalue decomposition of $\boldsymbol{\mathsf{M}}$ can be written as

(3.21)\begin{equation} \boldsymbol{\mathsf{M}} = [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{D}}_{+{+}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{D}}_{-{-}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right], \end{equation}

where the columns of $\boldsymbol{\mathsf{V}}_{+} \in \mathbb {C}^{N \times N_{+}}$ and $\boldsymbol{\mathsf{V}}_{-}\in \mathbb {C}^{N \times N_{-}}$ and the rows of $\boldsymbol{\mathsf{U}}_{+}\in \mathbb {C}^{N_{+} \times N}$ and $\boldsymbol{\mathsf{U}}_{-}\in \mathbb {C}^{N_{-} \times N}$ contain the left and right eigenvectors associated with the downstream- and upstream-travelling eigenvalues of $\boldsymbol{\mathsf{M}}$, respectively, which are contained in the diagonal matrices $\boldsymbol{\mathsf{D}}_{++} \in \mathbb {C}^{N_{+} \times N_{+}}$ and $\boldsymbol{\mathsf{D}}_{--}\in \mathbb {C}^{N_{-} \times N_{-}}$. The eigenvectors are normalized such that

(3.22)\begin{equation} [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right] = \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right] [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] = \boldsymbol{\mathsf{I}}. \end{equation}

Using this block matrix notation, (3.18) can be written as

(3.23)\begin{equation} \boldsymbol{\hat{\phi}}_{{\pm}} = \boldsymbol{\mathsf{V}} \boldsymbol{\psi} = \boldsymbol{\mathsf{V}}_{+} \boldsymbol{\psi}_{+} + \boldsymbol{\mathsf{V}}_{-} \boldsymbol{\psi}_{-}, \end{equation}

where

(3.24)\begin{equation} \boldsymbol{\psi} = \left\{\begin{array}{@{}c@{}} \boldsymbol{\psi}_{+} \\ \boldsymbol{\psi}_{-} \end{array}\right\}, \end{equation}

and $\boldsymbol {\psi }_{+}$ and $\boldsymbol {\psi }_{-}$ are vectors of expansion coefficients for the downstream- and upstream-travelling modes, respectively. Therefore, the downstream-travelling part of the solution is

(3.25)\begin{equation} \boldsymbol{\hat{\phi}}_{{\pm}}' = \boldsymbol{\mathsf{V}}_{+} \boldsymbol{\psi}_{+}, \end{equation}

and the upstream-travelling part is

(3.26)\begin{equation} \boldsymbol{\hat{\phi}}_{{\pm}}'' = \boldsymbol{\mathsf{V}}_{-} \boldsymbol{\psi}_{-}. \end{equation}

3.2. Exact projection operator

We define a projection operator

(3.27)\begin{equation} \boldsymbol{\mathsf{P}} = [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{I}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right], \end{equation}

that exactly splits the solution $\boldsymbol {\hat {\phi }}_{\pm }$ into downstream- and upstream-travelling components at each $x$. That is,

(3.28a)\begin{gather} \boldsymbol{\hat{\phi}}_{{\pm}}' = \boldsymbol{\mathsf{P}} \boldsymbol{\hat{\phi}}_{{\pm}}, \end{gather}

(3.28b)\begin{gather}\boldsymbol{\hat{\phi}}_{{\pm}}'' = (\boldsymbol{\mathsf{I}} - \boldsymbol{\mathsf{P}}) \boldsymbol{\hat{\phi}}_{{\pm}}. \end{gather}

Equation (3.28a) follows from (3.23) and (3.25),

(3.29a)\begin{align} \boldsymbol{\mathsf{P}} \boldsymbol{\hat{\phi}}_{{\pm}} &= [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{I}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right] [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}c@{}} \boldsymbol{\psi}_{+} \\ \boldsymbol{\psi}_{-} \end{array} \right] \end{align}

(3.29b)\begin{align} &= \boldsymbol{\mathsf{V}}_{+} \boldsymbol{\psi}_{+} \end{align}

(3.29c)\begin{align} &= \boldsymbol{\hat{\phi}}_{{\pm}}', \end{align}

and (3.23) and (3.26) can be similarly manipulated to verify (3.28b). Using (3.27) and (3.22), it is straightforward to show that $\boldsymbol{\mathsf{P}}$ is a projection operator, i.e. that $\boldsymbol{\mathsf{P}} \boldsymbol{\mathsf{P}} = \boldsymbol{\mathsf{P}}$.

3.3. One-way equation

We now obtain a one-way equation for the downstream-travelling component of the solution $\boldsymbol {\hat {\phi }}_{\pm }'$ by applying the projection $\boldsymbol{\mathsf{P}}$ to (3.13). This gives

(3.30)\begin{equation} \boldsymbol{\mathsf{P}} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}}{{\rm d} x} = \boldsymbol{\mathsf{P}} \boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}} + \boldsymbol{\mathsf{P}} \boldsymbol{\hat{h}}. \end{equation}

To obtain an evolution equation for $\boldsymbol {\hat {\phi }}_{\pm }'$, we must move $\boldsymbol{\mathsf{P}}$ inside the derivative. Using the chain rule, we have

(3.31)\begin{equation} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}'}{{\rm d} x} = \frac{{\rm d} \boldsymbol{\mathsf{P}} \boldsymbol{\hat{\phi}}_{{\pm}}}{{\rm d} x} = \boldsymbol{\mathsf{P}} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}}{{\rm d} x} + \frac{{\rm d} \boldsymbol{\mathsf{P}} }{{\rm d} x} \boldsymbol{\hat{\phi}}_{{\pm}}. \end{equation}

Using the fact that $\boldsymbol{\mathsf{P}}$ and $\boldsymbol{\mathsf{M}}$ commute (since they have the same eigenvectors by (3.21) and (3.27)), the first term on the right-hand-side of (3.30) can also be written in terms of $\boldsymbol {\hat {\phi }}_{\pm }'$,

(3.32)\begin{equation} \boldsymbol{\mathsf{P}}\boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}} = \boldsymbol{\mathsf{P}} \boldsymbol{\mathsf{P}}\boldsymbol{\mathsf{M}} \boldsymbol{\hat{\phi}}_{{\pm}} =\boldsymbol{\mathsf{P}}\boldsymbol{\mathsf{M}} \boldsymbol{\mathsf{P}} \boldsymbol{\hat{\phi}}_{{\pm}} = \boldsymbol{\mathsf{P}}\boldsymbol{\mathsf{M}} \boldsymbol{\hat{\phi}}_{{\pm}}'. \end{equation}

Therefore, using (3.31) and (3.32), (3.30) can be written as

(3.33)\begin{equation} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}'}{{\rm d} x} = \boldsymbol{\mathsf{P}} ( \boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}}' + \boldsymbol{\hat{h}}) + \frac{{\rm d} \boldsymbol{\mathsf{P}} }{{\rm d} x} \boldsymbol{\hat{\phi}}_{{\pm}}. \end{equation}

Following the same steps, but with $\boldsymbol{\mathsf{I}}-\boldsymbol{\mathsf{P}}$ replacing $\boldsymbol{\mathsf{P}}$, we similarly obtain

(3.34)\begin{equation} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}''}{{\rm d} x} = ( \boldsymbol{\mathsf{I}}-\boldsymbol{\mathsf{P}} ) ( \boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}}'' + \boldsymbol{\hat{h}}) - \frac{{\rm d} \boldsymbol{\mathsf{P}} }{{\rm d} x} \boldsymbol{\hat{\phi}}_{{\pm}}. \end{equation}

By neglecting $({{\rm d} \boldsymbol{\mathsf{P}}}/{{\rm d} x}) \boldsymbol {\hat {\phi }}_{\pm }$ in (3.33) and (3.34), we arrive at one-way equations for the downstream- and upstream-travelling components of the solution,

(3.35)\begin{gather} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}'}{{\rm d} x} = \boldsymbol{\mathsf{P}} ( \boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}}' + \boldsymbol{\hat{h}}), \end{gather}

(3.36)\begin{gather}\frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}''}{{\rm d} x} = ( \boldsymbol{\mathsf{I}}-\boldsymbol{\mathsf{P}} ) (\boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}}'' + \boldsymbol{\hat{h}}). \end{gather}

When $\boldsymbol{\mathsf{M}}$ is $x$-independent, ${{\rm d} \boldsymbol{\mathsf{P}} }/{{\rm d} x} = \boldsymbol {0}$ and (3.35) and (3.36) exactly describe the evolution of downstream- and upstream-travelling waves, respectively. When $\boldsymbol{\mathsf{M}}$ is $x$-dependent, ${{\rm d} \boldsymbol{\mathsf{P}} }/{{\rm d} x} \neq \boldsymbol {0}$ and (3.35) and (3.36) are approximate. Insight into the nature of the approximation can be gained by solving both (3.33) and (3.34) for $({{\rm d} \boldsymbol{\mathsf{P}}}/{{\rm d} x}) \boldsymbol {\hat {\phi }}_{\pm }$ and equating the expressions, giving

(3.37)\begin{equation} \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}'}{{\rm d} x} - \boldsymbol{\mathsf{P}} ( \boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}}' + \boldsymbol{\hat{h}}) ={-}\left( \frac{{\rm d} \boldsymbol{\hat{\phi}}_{{\pm}}''}{{\rm d} x} - ( \boldsymbol{\mathsf{I}}-\boldsymbol{\mathsf{P}} ) ( \boldsymbol{\mathsf{M}}\boldsymbol{\hat{\phi}}_{{\pm}}'' + \boldsymbol{\hat{h}}) \right) = \frac{{\rm d} \boldsymbol{\mathsf{P}}}{{\rm d} x} \boldsymbol{\hat{\phi}}_{{\pm}}. \end{equation}

Comparing (3.35) and (3.37) reveals that neglecting $({{\rm d} \boldsymbol{\mathsf{P}}}/{{\rm d} x}) \boldsymbol {\hat {\phi }}_{\pm }$ is equivalent to setting $\boldsymbol {\phi }'' = 0$ when calculating $\boldsymbol {\phi }'$. In other words, the one-way equation (3.35) neglects the influence of the upstream-travelling waves on the evolution of the downstream-travelling waves. In the same way, comparing (3.36) and (3.37) reveals that neglecting $({{\rm d} \boldsymbol{\mathsf{P}}}/{{\rm d} x}) \boldsymbol {\hat {\phi }}_{\pm }$ is equivalent to setting $\boldsymbol {\phi }' = 0$ when calculating $\boldsymbol {\phi }''$; the one-way equation (3.36) neglects the influence of the downstream-travelling waves on the evolution of the upstream-travelling waves. As discussed in detail by Towne & Colonius (Reference Towne and Colonius2015), it is reasonable to neglect the influence of upstream-travelling waves on the downstream-travelling waves (and vice versa) when $\boldsymbol{\mathsf{M}}$ is slowly varying in $x$. Since $\boldsymbol{\mathsf{M}}$ inherits its $x$-dependence from the mean flow $\bar {q}$, the one-way equation will yield an accurate approximation of the downstream- or upstream-travelling response to the forcing for slowly varying flows, i.e. for flows in which the gradient of the mean flow is much smaller in the streamwise direction than in the cross-stream directions.

Next, we will show that (3.35) and (3.36) are well posed as one-way equations. This amounts to showing that their eigenvalues correspond to downstream- and upstream-travelling modes, respectively. Focusing first on (3.35), the relevant operator is

(3.38a)\begin{align} \boldsymbol{\mathsf{P}}\boldsymbol{\mathsf{M}} &= [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{I}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right] [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{D}}_{+{+}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{D}}_{-{-}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right] \end{align}

(3.38b)\begin{align} &= [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{D}}_{+{+}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right]. \end{align}

Compared with the original elliptic operator $\boldsymbol{\mathsf{M}}$, the eigenvectors and downstream-travelling eigenvalues of the one-way operator $\boldsymbol{\mathsf{P}}\boldsymbol{\mathsf{M}}$ are unchanged, but the upstream-travelling eigenvalues have been eliminated. Therefore, (3.35) is well posed as a one-way equation and can be solved by integrating the equations in the positive $x$ direction.

The relevant operator for the well posedness of (3.36) is

(3.39a)\begin{align} (\boldsymbol{\mathsf{I}}-\boldsymbol{\mathsf{P}}) \boldsymbol{\mathsf{M}} &= [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{I}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right] [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{D}}_{+{+}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{D}}_{-{-}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right] \end{align}

(3.39b)\begin{align} &= [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{D}}_{-{-}} \end{array} \right] \left[ \begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right]. \end{align}

The downstream-travelling eigenvalues have been eliminated, so (3.36) is well posed as a one-way equation and can be solved by integrating the equations in the negative $x$ direction.

This projection-based paradigm for obtaining one-way equations and the projection operator defined in (3.27) were originally derived by Towne (Reference Towne2016) and was recently rederived by Harris & Hack (Reference Harris and Hack2020). While it is well posed, as we have shown, its computational efficiency is problematic; the eigen-decomposition of $\boldsymbol{\mathsf{M}}$ is required at every $x$ to construct the exact projection $\boldsymbol{\mathsf{P}}$, resulting in an intolerably high computational cost for large $N$ due to the nominal $O(N^{3})$ scaling of the number of operations required to solve each eigenvalue problem. To obtain a practically useful one-way equation, we construct in the next section an approximation of $\boldsymbol{\mathsf{P}}$ that can be efficiently computed.

3.4. Approximate projection operator

The following set of recursion equations approximate the action of $\boldsymbol{\mathsf{P}}$ on an arbitrary vector $\boldsymbol {\hat {\phi }}_{\pm }$:

(3.40a)\begin{gather} \boldsymbol{\hat{\phi}}^{{-}N_{\beta}}_{+} = 0, \end{gather}

(3.40b)\begin{gather}(\boldsymbol{\mathsf{M}} - i \beta_{-}^{j} \boldsymbol{\mathsf{I}} ) \boldsymbol{\hat{\phi}}_{{\pm}}^{{-}j} - (\boldsymbol{\mathsf{M}} - i \beta_{+}^{j} \boldsymbol{\mathsf{I}} ) \boldsymbol{\hat{\phi}}_{{\pm}}^{{-}j-1} = 0 \quad j = 1,\ldots,N_{\beta}-1 , \end{gather}

(3.40c)\begin{gather}(\boldsymbol{\mathsf{M}} - i \beta_{-}^{0} \boldsymbol{\mathsf{I}} ) \boldsymbol{\hat{\phi}}_{{\pm}}^{0} - (\boldsymbol{\mathsf{M}} - i \beta_{+}^{0} \boldsymbol{\mathsf{I}} ) \boldsymbol{\hat{\phi}}_{{\pm}}^{{-}1} = (\boldsymbol{\mathsf{M}} - i \beta_{-}^{0} \boldsymbol{\mathsf{I}} ) \boldsymbol{\hat{\phi}}_{{\pm}}, \end{gather}

(3.40d)\begin{gather}(\boldsymbol{\mathsf{M}} - i \beta_{+}^{j} \boldsymbol{\mathsf{I}} ) \boldsymbol{\hat{\phi}}_{{\pm}}^{j} - (\boldsymbol{\mathsf{M}} - i \beta_{-}^{j} \boldsymbol{\mathsf{I}} ) \boldsymbol{\hat{\phi}}_{{\pm}}^{j+1} = 0 \quad j = 0,\ldots,N_{\beta}-1, \end{gather}

(3.40e)\begin{gather}\boldsymbol{\hat{\phi}}^{N_{\beta}}_{-} = 0. \end{gather}

Here, we have introduced a set of auxiliary variables $\{ \boldsymbol {\hat {\phi }}_{\pm }^{j}: j = -N_{\beta },\ldots,N_{\beta } \}$ and a set of complex scalar recursion parameters $\{ \beta ^{j}_{+}, \beta ^{j}_{-}: j = 0,\ldots,N_{\beta }-1 \}$. Here $N_{\beta }$ is the order of the approximate projection.

In Appendix A we show that the zero-indexed variable is the approximate projection of $\boldsymbol {\hat {\phi }}_{\pm }$, i.e. that

(3.41)\begin{equation} \boldsymbol{\hat{\phi}}_{{\pm}}^{0} = \boldsymbol{\mathsf{P}}_{N_{\beta}} \boldsymbol{\hat{\phi}}_{{\pm}} \approx \boldsymbol{\hat{\phi}}_{{\pm}}'. \end{equation}

The operator

(3.42)\begin{equation} \boldsymbol{\mathsf{P}}_{N_{\beta}} = [ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{V}}_{+} & \boldsymbol{\mathsf{V}}_{-} \end{array} ] \left[\begin{array}{@{}cc@{}} \boldsymbol{\mathsf{I}} & -\boldsymbol{\mathsf{R}}_{+{-}} \\ -\boldsymbol{\mathsf{R}}_{-{+}} & \boldsymbol{\mathsf{I}} \end{array}\right]^{{-}1} \left[ \begin{array}{@{}cc@{}} \boldsymbol{\mathsf{I}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \end{array} \right] \left[\begin{array}{@{}cc@{}} \boldsymbol{\mathsf{I}} & -\boldsymbol{\mathsf{R}}_{+{-}} \\ -\boldsymbol{\mathsf{R}}_{-{+}} & \boldsymbol{\mathsf{I}} \end{array}\right] \left[\begin{array}{@{}c@{}} \boldsymbol{\mathsf{U}}_{+} \\ \boldsymbol{\mathsf{U}}_{-} \end{array} \right], \end{equation}

is the approximation of the exact projection $\boldsymbol{\mathsf{P}}$ that is implicitly defined by the recursions (3.40). It is easy to verify that $\boldsymbol{\mathsf{P}}_{N_{\beta }} \boldsymbol{\mathsf{P}}_{N_{\beta }} = \boldsymbol{\mathsf{P}}_{N_{\beta }}$, so $\boldsymbol{\mathsf{P}}_{N_{\beta }}$ is itself a projection. Furthermore, comparing (3.27) and (3.42) we see that $\boldsymbol{\mathsf{P}}_{N_{\beta }} \to \boldsymbol{\mathsf{P}}$ as $\boldsymbol{\mathsf{R}}_{+-},\boldsymbol{\mathsf{R}}_{+-} \to \boldsymbol{\mathsf{0}}$. Therefore, the approximation converges if every entry of $\boldsymbol{\mathsf{R}}_{+-}$ and $\boldsymbol{\mathsf{R}}_{+-}$ converges toward zero as the order of the approximation increases. In Appendix A we show that the $(n,m)$ entry of $\boldsymbol{\mathsf{R}}_{+-}$ and the $(m,n)$ entry of $\boldsymbol{\mathsf{R}}_{-+}$ are, respectively,

(3.43a)\begin{gather} (\boldsymbol{\mathsf{R}}_{+{-}})_{nm} = \frac{\mathcal{F}(\alpha_{+,n})}{\mathcal{F}(\alpha_{-,m})} (\boldsymbol{w}_{+{-}})_{nm}, \end{gather}

(3.43b)\begin{gather}(\boldsymbol{\mathsf{R}}_{-{+}})_{mn} = \frac{\mathcal{F}(\alpha_{+,n})}{\mathcal{F}(\alpha_{-,m})} (\boldsymbol{w}_{-{+}})_{mn}, \end{gather}

where

(3.44)\begin{equation} \mathcal{F}(\alpha) = \prod_{j=0}^{N_{\beta}-1} \frac{\alpha-\beta^{j}_{+}}{\alpha-\beta^{j}_{-}}, \end{equation}

$\alpha _{+,n}$ is the $n$th downstream-travelling eigenvalue, $\alpha _{-,m}$ is the $m$th upstream-travelling eigenvalue, and $(\boldsymbol {w}_{+-})_{nm}$ and $(\boldsymbol {w}_{-+})_{mn}$ are scalar weights that do not depend on the recursion parameters. Since the weights are fixed, the recursion parameters must be chosen such that $\mathcal {F}(\alpha _{+,n})/\mathcal {F}(\alpha _{-,m})$ goes to zero for all $m,n$.

A geometric interpretation of $\mathcal {F}$ is helpful for choosing parameters that accomplish this objective. Notice that the magnitude of each term in the product defining $\mathcal {F}$ is less than one for regions of the complex $\alpha$ plane that are closer to $\beta _{+}^{j}$ than $\beta _{-}^{j}$ ($|\alpha - \beta _{+}^{j}| < |\alpha - \beta _{-}^{j}|$) and greater than one for regions that are closer to $\beta _{-}^{j}$ than $\beta _{+}^{j}$ ($|\alpha - \beta _{+}^{j}| > |\alpha - \beta _{-}^{j}|$). Therefore, $\mathcal {F}(\alpha _{+,n})$ is driven to zero by placing the $\beta _{+}^{j}$ parameters near the downstream-travelling eigenvalues in the complex plane, and $\mathcal {F}(\alpha _{-,m})$ is driven to infinity by placing the $\beta _{-}^{j}$ parameters near the upstream-travelling eigenvalues.

This is exactly the same requirement for convergence as derived for the OWNS-O method by Towne & Colonius (Reference Towne and Colonius2015). This has several important implications. First, as established by Towne & Colonius (Reference Towne and Colonius2015), if $\alpha _{+,n} \neq \alpha _{-,m}$ for all $m$,$n$, there always exist recursion parameters that make the approximation error arbitrarily small. Second, if the recursion parameters are well placed, the convergence of the approximation is exponential. Third, any recursion parameters derived for OWNS-O can be used without modification for OWNS-P. A general strategy for choosing recursion parameters was outlined by Towne & Colonius (Reference Towne and Colonius2015), and effective sets of recursion parameters have been developed for mixing layers (Towne & Colonius Reference Towne and Colonius2014), jets (Towne & Colonius Reference Towne and Colonius2013), subsonic boundary layers (Rigas et al. Reference Rigas, Colonius and Beyar2017a) and supersonic boundary layers (Kamal et al. Reference Kamal, Rigas, Lakebrink and Colonius2020, Reference Kamal, Rigas, Lakebrink and Colonius2021).

Finally, to be rigorous, we must show that (3.35) and (3.36) remain well posed as one-way equations when $\boldsymbol{\mathsf{P}}_{N_{\beta }}$ is used in place of $\boldsymbol{\mathsf{P}}$. This is confirmed in Appendix B.

3.5. Implementation

The recursion equations defining the approximate projection operator define a system of equations of the form

(3.45a)\begin{gather} \boldsymbol{\hat{\phi}}_{{\pm}}' = \boldsymbol{\mathsf{P}}_3 \boldsymbol{\hat{\phi}}^{{aux}}, \end{gather}

(3.45b)\begin{gather}\boldsymbol{\mathsf{P}}_2 \boldsymbol{\hat{\phi}}^{{aux}} = \boldsymbol{\mathsf{P}}_1 \boldsymbol{\hat{\phi}}_{{\pm}}, \end{gather}

where $\boldsymbol {\hat {\phi }}^{{aux}} \in \mathbb {C}^{N_{{aux}}}$ is a vector containing all of the auxiliary variables, the matrices $\boldsymbol{\mathsf{P}}_{1} \in \mathbb {C}^{N_{{aux}} \times N}$ and $\boldsymbol{\mathsf{P}}_{2} \in \mathbb {C}^{N_{{aux}} \times N_{{aux}}}$ are defined by the recursion equations (3.40), $\boldsymbol{\mathsf{P}}_{3} \in \mathbb {C}^{N \times N_{{aux}}}$ is a matrix that extracts the projected state from the auxiliary variables via (3.41) and $N_{{aux}} = 2 N N_{\beta } + N_{0}$. The structure of these matrices is exemplified in Appendix C.

From (3.45), we see that applying the projection operator to a vector $\boldsymbol {\hat {\phi }}_{\pm }$ to obtain the projected state $\boldsymbol {\hat {\phi }}_{\pm }'$ requires the solution of a linear system of size $N_{{aux}}$; the cost of this operation compared with those associated with a global solution strategy is discussed in the next section. We stress that in practice we never form the approximate projection operator $\boldsymbol{\mathsf{P}}_{N_{\beta }}$.

The approximate form of the one-way equation (3.35) can be expressed as a DAE input/output system

(3.46a)\begin{gather} \boldsymbol{\mathsf{A}}^\ddagger \frac{{\rm d} \boldsymbol{\hat{\phi}}^\ddagger}{{\rm d} x} = \boldsymbol{\mathsf{L}}^\ddagger \boldsymbol{\hat{\phi}}^\ddagger + \boldsymbol{\mathsf{B}}^{{\ddagger}} \boldsymbol{\hat{f}}_{\phi}, \end{gather}

(3.46b)\begin{gather}\boldsymbol{\hat{\phi}}' = \boldsymbol{\mathsf{C}}^\ddagger \boldsymbol{\hat{\phi}}^\ddagger. \end{gather}

The expanded state vector is

(3.47)\begin{equation} \hat{\boldsymbol{\phi}}^\ddagger{=} \begin{bmatrix} \boldsymbol{\hat{\phi}}_{{\pm}}' \\ \boldsymbol{\hat{\phi}}_{0}' \\ \boldsymbol{\hat{\phi}}^{{aux}} \end{bmatrix}, \end{equation}

and the operators in (3.46) are

(3.48)\begin{equation} \boldsymbol{\mathsf{A}}^\ddagger{=} \begin{bmatrix} \boldsymbol{\mathsf{I}} & & \\ & \boldsymbol{\mathsf{0}} & \\ & & \boldsymbol{\mathsf{0}} \end{bmatrix}, \quad \boldsymbol{\mathsf{B}}^\ddagger{=} \begin{bmatrix} \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{P}}_1\tilde{\boldsymbol{\mathsf{A}}}_{{\pm}{\pm}}^{{-}1} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{I}} \end{bmatrix}, \quad \boldsymbol{\mathsf{C}}^{{\ddagger}} = \begin{bmatrix} \boldsymbol{\mathsf{I}} & \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} \\ \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{I}} & \boldsymbol{\mathsf{0}} \end{bmatrix} \end{equation}

and

(3.49)\begin{equation} \boldsymbol{\mathsf{L}}^\ddagger{=} \begin{bmatrix} \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{0}} & \boldsymbol{\mathsf{P}}_3\\ \boldsymbol{\mathsf{P}}_1\tilde{\boldsymbol{\mathsf{A}}}_{{\pm}{\pm}}^{{-}1} \boldsymbol{\mathsf{L}}_{{\pm}{\pm}} & \boldsymbol{\mathsf{P}}_1\tilde{\boldsymbol{\mathsf{A}}}_{{\pm}{\pm}}^{{-}1} \boldsymbol{\mathsf{L}}_{\pm0} & -\boldsymbol{\mathsf{P}}_2 \\ \boldsymbol{\mathsf{L}}_{0\pm} & \boldsymbol{\mathsf{L}}_{00} & \boldsymbol{\mathsf{0}} \end{bmatrix}. \end{equation}

The input to the system is the forcing $\boldsymbol {\hat {f}}_{\phi } = \boldsymbol{\mathsf{T}} \boldsymbol {\hat {f}}$, while the output is $\boldsymbol {\hat {\phi }}'$, the downstream-travelling component of the characteristic variable, from which the OWNS-P approximation of the physical state variable can be obtained using (3.4) as $\boldsymbol {\hat {q}}' = \boldsymbol{\mathsf{T}}^{-1} \boldsymbol {\hat {\phi }}'$. The DAE (3.46) can be efficiently integrated in the positive $x$ direction given a specified value for $\boldsymbol {\hat {\phi }}_{\pm }'$ at the inlet of the domain; this value is physically a boundary condition for the global problem but functions as an initial condition for the spatial integration of the DAE.

An additional issue arises in the practical implementation of the method. Specifically, errors incurred during the numerical integration of (3.46) will not lie entirely in the downstream-travelling subspace. In other words, the numerical approximation of $\boldsymbol {\hat {\phi }}_{\pm }'$ collects an error that projects onto the zero eigenvalues of $\boldsymbol{\mathsf{P}}_{N_{\beta }} \boldsymbol{\mathsf{M}}$, which is then propagated along during the march. This causes an accumulation of error that contaminates the solution. Fortunately, there is an easy fix: apply the projection operator to the solution after each step in the march.

We emphasize that the OWNS-P approach differs significantly from PSE, the latter of which achieves a stable spatial march by numerically damping upstream-travelling waves, either by using an implicit axial discretization along with a restriction on the minimum step size (Li & Malik Reference Li and Malik1996) or by explicitly adding damping terms to the equations (Andersson, Henningson & Hanifi Reference Andersson, Henningson and Hanifi1998). The damping prevents the upstream-travelling waves from destabilizing the spatial march, but also introduces uncontrollable errors, to different degrees that depend on the complexity of the solution, into all of the downstream-travelling waves (Towne et al. Reference Towne, Rigas and Colonius2019).

3.6. Computational cost

The standard approach for obtaining the action of the resolvent operator on a forcing vector requires the solution of a linear system of equations of size $N_{q} N_{x} N_{c}$, where $N_{x}$ and $N_{c}$ are the number of discretization points in $x$ and in all transverse directions, respectively, and $N_{q}$ is the number of state variables, e.g. $N_{q} = 5$. In the following scaling estimates, we drop the dependence on $N_{q}$ since it is a constant for a given problem. Assuming the use of sparse discretization schemes and direct solution via multi-frontal LU decomposition, the CPU cost (FLOPS) of solving the linear system is found, empirically, to scale as $O(N_{x}^{a} N_{c}^{a})$, and the memory usage scales as $O(N_{x}^{b} N_{c}^{b})$, where the factors $1 < a \le 3$ and $1 < b \le 2$ depend on the sparsity and structure of the matrix and the sophistication of the algorithm employed (Duff, Erisman & Reid Reference Duff, Erisman and Reid2017). In our global computations for two-dimensional (2-D) base flows corresponding to turbulent jets presented in § 5, we observed $a \approx 1.6$ and $b \approx 1.2$, whereas in some preliminary computations for three-dimensional (3-D) base flows, we observed $a \approx 2$ and $b \approx 2$. We have found these values to be quite robust across multiple software packages and operating platforms. We have also implemented iterative solvers based on generalized minimal residual (GMRES) and biconjugate gradient stabilized (Bi-CG-Stab) methods, which decrease both $a$ and $b$ to near unity, but, without preconditioning, these were typically much slower as the number of iterations required grew large.

The main cost for OWNS-P is solving the system of (3.45) of size $N_{{aux}} = 2 N_{\beta } N + N_{0} = 2 N_{\beta } N_{q} N_{c} + N_{0}$ if the DAE (3.46) is explicitly integrated or a system of size $N^{\ddagger } = (2 N_{\beta }+1) N + N_0 = (2 N_{\beta }+1) N_{q} N_{c} + N_0$ if it is implicitly integrated. In both cases, the dominant factor in these size expressions is $N_{c} N_{\beta }$, and other terms are dropped. As the sparsity and structure of the OWNS matrices are analogous to those in the global solution, the FLOPS and memory of solving (3.45) scale as $O(N_{x} N_{c}^{a} N_{\beta }^{a})$ and $O(N_{c}^{b} N_{\beta }^{b})$. The additional factor of $N_{x}$ in the FLOPS scaling follows from the fact that an equation of this form must be solved at each step in the spatial march. Assuming $N_x > N_\beta$, OWNS-P represents a FLOPS and memory speedup by factors of $N_x^{a-1}/N_\beta ^{a}$ and $N_x^{b}/N_\beta ^{b}$, respectively.

Clearly OWNS-P will achieve significant savings in memory, and, for sufficiently large $N_x$, significant saving in FLOPS, compared with global methods. These reductions allow problems that would otherwise require high-performance computing resources to be solved on a laptop. The advantage grows for 3-D base flows where the factors $a$ and $b$ are larger.

Finally, comparing OWNS-P to OWNS-O, the set of recursion equations that must be solved within the OWNS-P method is roughly twice as large as those required for the OWNS-O method; the index of the auxiliary variables for an approximation of order $N_{\beta }$ span the range $[ -N_{\beta }, N_{\beta } ]$ and $[ 0, N_{\beta } ]$ for OWNS-P and OWNS-O, respectively. This is the price that must be paid to accommodate the inhomogeneous forcing term.

5.1. Governing equations

The flow dynamics are governed by the compressible Navier–Stokes equations for an ideal gas,

(5.1a)\begin{gather} \frac{D\rho}{D t} +\rho (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}) = 0, \end{gather}

(5.1b)\begin{gather}\rho \frac{D \boldsymbol{u}}{D t} + \boldsymbol{\nabla} p = \frac{1}{Re} \boldsymbol{\nabla} \boldsymbol{\cdot} \tau, \end{gather}

(5.1c)\begin{gather}\rho \gamma R \frac{DT}{D t} -(\gamma-1)\frac{Dp}{D t} = \frac{\gamma-1}{Pr Re}\boldsymbol{\nabla} \boldsymbol{\cdot} (k\boldsymbol{\nabla} T ) + \frac{\gamma-1}{Re} (\tau:\boldsymbol{\nabla} \boldsymbol{u}), \end{gather}

(5.1d)\begin{gather}\tau = 2\mu S +\kappa (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u})I, \end{gather}

(5.1e)\begin{gather}S = \frac{1}{2}\left(\boldsymbol{\nabla} \boldsymbol{u} + (\boldsymbol{\nabla} \boldsymbol{u})^T - \frac{2}{3} (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u})I\right), \end{gather}

(5.1f)\begin{gather}p = \rho RT, \end{gather}

(5.1g)\begin{gather}Re = \frac{\rho_\infty c_\infty L}{\mu_\infty}, \end{gather}

(5.1h)\begin{gather}Pr = \frac{\mu_\infty{c_p}_\infty }{k_\infty}, \end{gather}

where $\rho$, $\boldsymbol {u}$, $p$, $\gamma$, $R$, $T$, $k$, $\kappa$, $c$, $\mu$, $c_p$, $Re$ and $Pr$ are the density, velocity vector, pressure, specific heat ratio, gas constant, temperature, thermal conductivity, bulk viscosity, speed of sound, dynamic viscosity, isobaric specific heat, Reynolds number and Prandtl number, respectively (Kamal et al. Reference Kamal, Rigas, Lakebrink and Colonius2020). All relevant quantities have been non-dimensionalized by the dimensional ambient quantities $c_\infty$, $\rho _\infty$, $k_\infty$, ${c_p}_\infty$ and $\mu _\infty$, and a problem dependent length scale $L$.

We define the state vector as $q = (\rho, \boldsymbol {u}, T)^T$ and apply the Reynolds decomposition (2.2) to obtain the linearized equations (2.3). For some of the calculations, e.g. the acoustics problem, we employ a simplified formulation of the compressible Navier–Stokes equations described by Towne (Reference Towne2016).

We assume the fluid to be an ideal gas with $c_v=c_v(T)$ and $c_p = c_p(T)$ and that fluid properties $k$, $\mu$, $\kappa$ and $\gamma$ depend solely on temperature. We denote any of the aforementioned fluid properties as $\varPhi$ and perform a Taylor series expansion about $\bar {T}$,

(5.2)\begin{equation} \varPhi(T) = \varPhi(\bar{T}) + \left.\frac{{\rm d} \varPhi}{{\rm d}T}\right|_{T=\bar{T}}T'+ \text{higher-order terms} . \end{equation}

The linearized fluid property perturbation is thus

(5.3)\begin{equation} \varPhi' = \left.\frac{{\rm d} \varPhi}{{\rm d}T}\right|_{T=\bar{T}}T'. \end{equation}

Unless otherwise stated, we will assume the fluid as calorically perfect air with $\gamma =1.4$ and $Pr=0.72$ with viscosity and thermal conductivity calculated using Sutherland's law

(5.4)\begin{equation} \mu^{\dagger}{=} \mu^{\dagger}_\infty \left(\frac{T^{\dagger}}{T^{\dagger}_\infty} \right)^{3/2} \frac{T^{\dagger}_\infty + S^{\dagger}}{T^{\dagger}{+} S^{\dagger}},\end{equation}

where $S^{\dagger} =110.4$ K and $()^{\dagger}$ denotes dimensional quantities.

The acoustics problem and the flat-plate boundary layer flow are solved in Cartesian coordinates with $\boldsymbol {x} = (x,y,z)$ corresponding to the streamwise, wall-normal and spanwise directions, respectively, and $\boldsymbol {u}=(u, v, w)$. The jet flow is solved in cylindrical coordinates with $\boldsymbol {x} = (x,r,\theta )$ corresponding to the streamwise, radial and azimuthal directions, respectively, and $\boldsymbol {u}=(u_x, u_r, u_\theta )$. Due to the periodicity of the examined flows in the spanwise or azimuthal directions, the perturbation fields are decomposed as $\boldsymbol {q}' = \sum {\hat {\boldsymbol {q}}}(x,y) \exp (i \beta z - i \omega t)$ and $\boldsymbol {q}' = \sum {\hat {\boldsymbol {q}}}(x,r) \exp (i m \theta - i \omega t)$, where $\beta$ and $m$ are the spanwise and azimuthal wavenumbers, respectively.

All inner products, e.g. $\langle \boldsymbol {a},\boldsymbol {b} \rangle _g$, are defined so as to induce the Chu energy norm (Chu Reference Chu1965)

(5.5)\begin{equation} E_{Chu} = \frac{1}{2}\int \int_\varOmega \frac{R \bar{T}}{ \bar{\rho}} \rho'^2 + \bar{\rho} |\boldsymbol{u'}|^2 + \frac{R \bar{\rho}}{\bar{T} (\gamma-1) }T'^2 \,{\rm d}x \,{\rm d}A \approx \int_\varOmega \boldsymbol{q}'^*\boldsymbol{\mathsf{W}}_{yz} \boldsymbol{\mathsf{W}}_e \boldsymbol{q}'\,{\rm d}x, \end{equation}

where $\boldsymbol{\mathsf{W}}_e$ is the diagonal Chu energy weight matrix defined as

(5.6)\begin{equation} \boldsymbol{\mathsf{W}}_e = \frac{1}{2}\begin{bmatrix} \dfrac{R \bar{T}}{ \bar{\rho}} & 0 & 0 & 0 & 0 \\ 0 & \bar{\rho} & 0 & 0 & 0 \\ 0 & 0 & \bar{\rho} & 0 & 0 \\ 0 & 0 & 0 & \bar{\rho} & 0 \\ 0 & 0 & 0 & 0 & \dfrac{R \bar{\rho}}{\bar{T}(\gamma-1) }\end{bmatrix}. \end{equation}

5.2. Global and OWNS solvers

The global and OWNS calculations are performed using the Caltech stability and transition analysis toolkit. The linearized equations are discretized in inhomogeneous transverse directions using fourth-order central finite differences with summation-by-parts boundary closure (Strand Reference Strand1994; Mattsson & Nordstróm Reference Mattsson and Nordstróm2004). Far-field radiation boundary conditions are enforced at free transverse boundaries using a super-grid damping layer (Appelo & Colonius Reference Appelo and Colonius2009) truncated by Thompson characteristic conditions (Thompson Reference Thompson1987). For global calculations, the streamwise direction is discretized using the same scheme and streamwise boundary conditions are implemented using inlet and outlet sponges in addition to the characteristic boundary conditions. For OWNS calculations, the numerical treatment of the $x$ direction is slightly different in each problem and is reported in the following subsections. Additional details of the code can be found in Kamal et al. (Reference Kamal, Rigas, Lakebrink and Colonius2021).

An important consideration in the calculation of the OWNS response is the placement of the recursion parameters. For all cases, we follow the general recipe presented by Towne & Colonius (Reference Towne and Colonius2015). An estimate for the eigenvalues of the downstream and upstream modes for each frequency and for each streamwise location can be obtained assuming locally parallel and uniform flow equal to the free-stream velocity. The effectiveness of the OWNS-P projection to accurately filter the upstream-travelling modes without modifying the downstream ones is demonstrated in figure 3, where the local (in $x$) spectra of the Navier–Stokes and OWNS-P equations are shown for the boundary layer problem discussed in § 5.5 for three values of $N_{\beta }$. The spectra were calculated by solving a generalized eigenvalue problem obtained from the unforced (homogeneous) form of (3.46) by assuming a locally parallel flow, for which $\partial _x \to i \alpha$ and the complex streamwise wavenumber $\alpha$ is the eigenvalue.

Figure 3. Placement of recursion parameters $\beta ^+$ and $\beta ^-$ for the $M=4.5$ flat-plate boundary layer flow. The local spectra of the Navier–Stokes and OWNS-P operators are shown. The $\beta ^+$ recursion parameters are placed near the downstream-travelling modes, whereas the $\beta ^-$ parameters are placed near the upstream acoustic modes. For a sufficiently large number $N_\beta$ of recursion parameters, the OWNS-P operator converges to the downstream-travelling modes, thus eliminating the ellipticity associated with upstream modes and enabling stable, convergent marching in the slowly varying $x$ direction.

5.3. Dipole forcing of a quiescent fluid

In this problem a 2-D dipole forcing is used to excite waves in an inviscid quiescent fluid. The right-hand-side forcing that is applied to the energy equation is shown in figure 4(a), and the proper forcing terms are applied to the other equations in order to produce a dipole response in the pressure field, which is shown in figure 4(d). This is an exact solution of the Euler equations.

Figure 4. Dipole test case: (a) forcing applied to the energy equation; (b) OWNS-P downstream-travelling pressure response; (c) OWNS-P upstream-travelling pressure response; (d) exact pressure response; (e) OWNS-P total pressure response recovered by summing the downstream- and upstream-travelling solutions; (f) error computed as the difference between the total OWNS-P and exact pressure responses, normalized by the maximum of the exact pressure.

We use the OWNS-P method to obtain an approximate solution. The computational domain extends ten wavelengths in both $x$ and $y$ and is discretized using $200$ equally spaced points in each direction. No incoming fluctuations are specified at the domain boundaries – the waves are excited exclusively by the inhomogeneous forcing terms.

The pressure field obtained by integrating the one-way equations from left to right is shown in figure 4(b). Clearly, the downstream-travelling waves are accurately captured. Similarly, the pressure field obtained by integrating from right to left is shown in figure 4(c). This time, the upstream-travelling waves are captured. Since the governing equations are $x$-independent in this problem, the full solution can be recovered by summing the downstream- and upstream-travelling solutions, which is shown in figure 4(e). The error in the full OWNS-P solution, relative to the exact solution, is shown in figure 4(f). The total $L_2$ error is $0.28\,\%$, and this error is concentrated at angles nearly perpendicular to the marching direction as measured from the position of the source. As shown by Towne & Colonius (Reference Towne and Colonius2015), these high angles converge with $N_{\beta }$ at a slower, but still exponential, rate than lower angles.

5.4. Supersonic turbulent jet

Next, we demonstrate our method using the example of a turbulent jet. Resolvent analysis has been applied to jets by numerous authors (Garnaud et al. Reference Garnaud, Lesshafft, Schmid and Huerre2013; Jeun et al. Reference Jeun, Nichols and Jovanović2016; Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018; Lesshafft et al. Reference Lesshafft, Semeraro, Jaunet, Cavalieri and Jordan2019), and resolvent modes have been shown to capture a rich set of physical phenomena. These include the Kelvin–Helmholtz instability, which leads to large-scale coherent wavepacket structures (Jordan & Colonius Reference Jordan and Colonius2013), the Orr mechanism (Tissot et al. Reference Tissot, Zhang, Lajús, Cavalieri and Jordan2017; Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018), the lift-up mechanism (Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019; Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020) and trapped acoustic waves within the jet core (Tam & Hu Reference Tam and Hu1989; Towne et al. Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017b). The slow spread of the jet makes OWNS applicable, and the diverse set of physics embedded within the resolvent operator make it a challenging test case.

We will make comparisons between a standard global solution of the linearized Navier–Stokes equation and those obtained using OWNS-P (and OWNS-O and PSE when applicable). The global solution comprises the result of applying the resolvent operator to a given forcing vector, giving a point of comparison for the approximation of this action provided by the OWNS-P method.

We consider the specific case of a jet with Mach number $M = U_{j} / c_{\infty } = 1.5$, Reynolds number $Re = \rho _{j} U_{j} D / \mu = 1760 000$ and temperature ratio $T_{j}/T_{\infty } = 1$, where the subscripts $j$ and $\infty$ denote conditions at the jet nozzle exit and in the far field, respectively. The mean flow about which the Navier–Stokes equations are linearized is obtained from a large-eddy simulation (LES) described by Brès et al. (Reference Brès, Ham, Nichols and Lele2017) and is shown in figure 5. Within the linearized equation, we use a turbulent Reynolds number of $Re_T = 1760$, three orders of magnitude less than the true Reynolds number. This choice is motivated by recent work showing that using an eddy-viscosity model or reduced effective Reynolds number improves both the near-field (Pickering et al. Reference Pickering, Rigas, Schmidt, Sipp and Colonius2021a) and far-field (Pickering et al. Reference Pickering, Towne, Jordan and Colonius2021b) predictions in free-shear flows.

Figure 5. Mean flow of the $M=1.5$ turbulent jet. Local Mach number is shown.

The linearized equations in cylindrical coordinates are discretized in the radial direction as described earlier, and boundary conditions at the polar axis are enforced using the approach of Mohseni, Colonius & Freund (Reference Mohseni, Colonius and Freund2002). The physical portion of the domain extends to $r/D = 6$ and is discretized using 150 grid points with a higher concentration around the shear layer at $r/D = 0.5$, while the damping layer contains an additional 50 points. While the LES from which the mean flow is obtained includes the nozzle within the computational domain, the resolvent calculations do not; previous publications have shown that excluding the nozzle does not hinder favourable comparisons with data (Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018; Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020). Since the mean flow is axisymmetric, the azimuthal direction is homogeneous and can be decomposed into a Fourier series. In what follows, we focus on the axisymmetric mode ($m=0$), which is often of foremost interest in the study of jet aeroacoustics (Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012; Chen & Towne Reference Chen and Towne2021).

The streamwise grid for each method extends from $x/D = 0.5$ to 30 with equidistant spacing of ${\rm \Delta} x = 0.05$ (with the exception of the PSE method, which determines its own step size) and an additional sponge region included in the global computation at three diameters upstream and downstream. The OWNS-O and OWNS-P equations are integrated in the streamwise direction using the second-order backward difference formula and a fifth-order Runge–Kutta Radau II scheme, respectively (Hairer & Wanner Reference Hairer and Wanner1971). Here $N_{\beta } = 13$ recursion parameters are used for estimating the OWNS operators. In the following subsections, we show results for two frequencies, $St = 0.26$ and 0.52, which are close to the frequency of maximum acoustic radiation (Jordan & Colonius Reference Jordan and Colonius2013) and maximum near-field resolvent gain (Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018), respectively.

5.4.1. Near-nozzle Kelvin–Helmholtz forcing

First, we compute the downstream response of the linearized equations to a disturbance near the nozzle exit, and compare the OWNS-P solution to those obtained using PSE, OWNS-O and a global computation. Specifically, the local Kelvin–Helmholtz eigenfunction is specified at $x/D = 0.5$ as an initial condition in the PSE and OWNS marches and as a boundary condition for the global method. The solution can be interpreted as the response to a localized forcing at the upstream boundary. In other words, we are computing the result of applying the resolvent operator, approximated by each method, to a forcing vector that is localized at the upstream boundary and which specifically excites the Kelvin–Helmholtz instability.

Results of this test case are shown in figure 6, which shows the pressure field computed by each method for $St = 0.26$ and 0.52. The OWNS-P solution closely matches the global solution for both frequencies, demonstrating that the OWNS-P method provides a good approximation of the action of the resolvent operator on the Kelvin–Helmholtz forcing vector. The OWNS-P solution also matches the OWNS-O solution, which has been previously validated for a similar turbulent jet (Towne & Colonius Reference Towne and Colonius2015). As shown by Sinha et al. (Reference Sinha, Rodriguez, Bres and Colonius2014) and others, PSE also provides a reasonable solution for supersonic jets forced by a Kelvin–Helmholtz mode. In this case PSE performs well because its underlying assumptions are valid; we have artificially excited a single instability mode by using the Kelvin–Helmholtz mode as an initial condition. While the acoustic radiation is slightly damped compared with the other three solutions, especially for $St = 0.26$, it is also reasonably captured because its wavelength is nearly the same as the Kelvin–Helmholtz mode for low-supersonic jets (Towne et al. Reference Towne, Rigas and Colonius2019). However, PSE cannot handle more complex forcings such as those considered next.

Figure 6. Response of the jet to the Kelvin–Helmholtz initial condition at $x/D = 0.5$ using global, PSE, OWNS-O and OWNS-P methods, top to bottom, respectively. Contours of pressure are shown in each panel.

5.4.2. Volumetric forcing using LES data

Second, we use both OWNS-P and the standard global approach to compute the response of the linearized equations to global forcing vectors extracted from LES data. The right-hand-side forcing term $\boldsymbol {f}$ in (2.3) is obtained by computing the two left-hand-side terms using the LES approximation of the nonlinear Navier–Stokes operator $\mathcal {N}$. The first term can be computed as

(5.7)\begin{equation} \frac{\partial q'}{\partial t} = \frac{\partial q}{\partial t} = \mathcal{N}(q), \end{equation}

since $\bar {q}$ does not depend on time (Towne Reference Towne2016). The second term can be approximated as

(5.8)\begin{equation} \mathcal{A}(\bar{q}) q' = \frac{\partial \mathcal{N}}{\partial q} (\bar{q})q' \approx \frac{\mathcal{N}(\bar{q}+\epsilon q') - \mathcal{N}(\bar{q})}{\epsilon} \end{equation}

for some $\epsilon \ll 1$ (de Pando, Sipp & Schmid Reference de Pando, Sipp and Schmid2012). We take $\epsilon = 10^{-7}$ and note that the results are nearly independent of $\epsilon$ over a range spanning at least four orders of magnitude. Details of the procedure can be found in Towne (Reference Towne2016).

After performing an azimuthal Fourier transform, an ensemble of realizations of $\boldsymbol {\hat {q}}$ and $\boldsymbol {\hat {f}}$ are obtained from the LES data by segmenting $\boldsymbol {q}^{\prime }$ and $\boldsymbol {f}$ into overlapping blocks, each containing $N_{f}$ snapshots of data. We use blocks of length $N_{f} = 256$ with 80 % overlap and employ the $w_{C_{4}^{\infty }}$ window proposed by Martini et al. (Reference Martini, Cavalieri, Jordan and Lesshafft2019) to minimize spectral leakage. This results in 189 realizations of the flow at discrete frequencies, separated by ${\rm \Delta} St = 0.02604$.

Each of these $\boldsymbol {\hat {q}}$, $\boldsymbol {\hat {f}}$ pairs approximately satisfy (2.6). That is, applying the resolvent operator to $\boldsymbol {\hat {f}}$ yields $\boldsymbol {\hat {q}}$. This relationship is approximate for the LES data for two reasons. First, by necessity, we used a finite length discrete Fourier transform in place of an infinite and continuous Fourier transform to obtain $\boldsymbol {\hat {q}}$ and $\boldsymbol {\hat {f}}$, leading to aliasing and spectral leakage. In particular, $\boldsymbol {\hat {q}}$ exhibits a relatively flat spectrum up to moderate frequencies, making it especially susceptible to aliasing (Towne, Brès & Lele Reference Towne, Brès and Lele2017a; Karban et al. Reference Karban, Martini, Jordan, Brès and Towne2022). Second, the forcing term $\boldsymbol {f}$ was defined implicitly using the LES operator as described above, whereas our resolvent operator is obtained by explicitly linearizing the Navier–Stokes equations and discretizing the linear equations using numerics different from those within the LES. Thus, we do not expect our global and OWNS-P approximation of the action of the resolvent operator on $\boldsymbol {\hat {f}}$ to exactly reproduce the corresponding $\boldsymbol {\hat {q}}$ obtained from the LES data. Nevertheless, the $\boldsymbol {\hat {f}}$ vectors obtained from the LES data provide a physically motivated forcing that can be used to compare the action of the resolvent operators obtained via the OWNS-P march and the standard global solution.

Figure 7 shows one realization of the forcing and response for $St = 0.26$ and 0.52. Specifically, we show the component of the forcing that is applied to the energy equation and the response of the pressure field; other components of the forcing and response are qualitatively similar (Towne Reference Towne2016). The forcing is rather incoherent and lacks readily visible large-scale structures. In contrast, the pressure field contains distinct wavepacket structures and an acoustic beam emanating from the near field to the far field. Despite the lack of structure within the forcing, its introduction as a volumetric forcing term to the global and OWNS-P operators produces a response that is quite similar to the LES pressure field. Some minor differences can be observed, e.g. both the global and OWNS-P solutions contain stronger acoustic waves downstream of the main beam near the top-right corner of the figure. These and other minor differences can be attributed to the factors described earlier.

Figure 7. Realizations of the Mach 1.5 jet, including an LES-derived forcing and pressure response realization (top two rows, respectively) and global and OWNS-P computed pressure realizations (bottom two rows, respectively) forced with the above LES forcing realization.

Of greater relevance to the current study, the OWNS-P response closely matches the global response. For both frequencies, the OWNS-P response accurately captures the relevant dynamics in the jet, including near-field structures such as Kelvin–Helmholtz wavepackets (similar to those observed in figure 3) and Orr-type wavepackets (farther downstream, e.g. $15< x/D,20$), as well as the angle and intensity of acoustic radiation to the far field. This substantial agreement between the global and OWNS-P responses indicates that the OWNS-P approximation of the action of the resolvent operator on this forcing vector accurately mimics that of the global resolvent operator.

The ensemble of LES flow and forcing realizations can be used to compute the average error of the OWNS-P approximation over many potential forcing vectors. Figure 8 shows the power spectral density (PSD) of the LES, global and OWNS-P pressure fields for $St = 0.26$ and 0.52, i.e. the squared amplitude of the response averaged over the ensemble of realizations. In each case, the PSD is scaled by the maximum value of the PSD of the LES data. The PSD of the global and OWNS-P solutions accurately mimic that of the LES data. More importantly, the PSD computed from the ensemble of OWNS-P solutions closely matches the PSD computed from the global solutions. Both the envelope of the high-energy near-field region and intensity and directivity of the acoustic beam show good agreement for both frequencies.

Figure 8. Full-field pressure PSD results for LES (a,b), global (c,d) and OWNS-P (e,f).

A more quantitative assessment is provided in figure 9, which shows the PSD of the LES, global and OWNS-P pressure fields as a function of $x/D$ at $r/D = 0.5$ (the jet lip line) and $r/D = 6$ (the maximum radial position of the physical domain). The OWNS-P solution closely matches the global solution except in two low-energy regions where upstream-travelling waves are non-negligible. First, OWNS-P undershoots the global solution very close to the nozzle for $r/D=0.5$. Since the solution was initiated with a zero boundary condition, the downstream-travelling waves have not yet developed at the nozzle inlet, allowing the weak influence of upstream-travelling waves to be observed. Second, the global solution contains weak upstream-travelling acoustic waves that cannot be captured via one-way marching, resulting in an under prediction by OWNS-P along $r/D=6$ for $x/D<5$, well away from the main acoustic beam observed at higher values of $x/D$. Both of these minor discrepancies are expected given the one-way nature of OWNS.

Figure 9. Pressure PSD for LES (blue), global (orange) and OWNS-P (yellow) for (a) $St = 0.26$; (b) $St = 0.52$. Solid and dashed lines correspond to $r/D = 0.5$ and 6, respectively.

The PSD of the error between the global and OWNS-P solutions, normalized by the maximum value of the PSD of the global solution, is shown in figure 10. Comparing with figure 8, the average error is observed to be at least an order of magnitude smaller than the average solution in both the near-field hydrodynamic region and the acoustic field. Overall, these results indicate that the OWNS-P method provides an accurate approximation of the action of the resolvent operator for the ensemble of forcing realizations extracted from the LES data. Physically, the accuracy of the OWNS-P solution confirms the dominance of downstream-travelling waves in the forced jet.

Figure 10. Full-field error between global and OWNS-P pressure PSD.

5.4.3. Optimal forcing and response

Next, we show that the OWNS-P framework provides an accurate approximation of the global optimal forcing and response modes. The input forcing has been constrained to $0.5 < x/D < 30$, $R_{min} < r/D < R_{max}$. We use $R_{min}=0.0425$ to prevent the forcing from damaging the pole conditions (Mohseni et al. Reference Mohseni, Colonius and Freund2002) and $R_{max}$ is defined as the radial jet location where the velocity is greater than $5\,\%$ of the maximum jet velocity, i.e. the forcing is contained only within the jet. The output is defined in the full numerical domain, $0.5 < x/D < 30, 0 < r/D < 17$.

The real part of the optimal pressure forcing and response from the OWNS-P and global resolvent methods for $St=0.26$ and $0.52$ and $m=0$ are shown in figure 11. At these frequencies, the response is dominated by the amplification of disturbances due to the Kelvin–Helmholtz instability (Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018; Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020). The global and OWNS-P solutions are shown along the jet lip line $r/D=0.5$ in figure 12. Quantitative agreement is obtained, justifying the assumption of weak influence of the upstream-travelling modes during the OWNS-P parabolization procedure.

Figure 11. Optimal forcing (a–d) and response (e–h) for the $M=1.5$ turbulent jet. Comparison between global (a,b,e,f) and OWNS-P (c,d,g,h). Contours of the real pressure component at $St=0.26$ and $St=0.52$ are shown.

Figure 12. Optimal pressure response for the jet along $r/D = 0.5$ for (a) $St = 0.26$ and (b) $St = 0.52$. Solid and dashed lines show the global and OWNS-P solutions, respectively.

5.5. Mach 4.5 boundary layer

Finally, we consider a Mach 4.5 laminar flat-plate zero-pressure-gradient boundary layer. Optimal forcing and response modes are computed using the OWNS-P and global methods for several frequency and wavenumber combinations corresponding to different instability mechanisms. These cases, and the numerical parameters used for each, are summarized in table 1.

Table 1. Numerical parameters for the supersonic boundary layer cases.

All boundary layer quantities are non-dimensionalized using the free-stream velocity $U_{\infty }$ and the local compressible displacement thickness $\delta ^*(x)$ or the displacement thickness at the outlet of the domain $\delta ^*_0$. The Mach number is $M=U_{\infty } / c_{\infty }=4.5$. The outlet of the domain is at $Re_x^{out} = {\nu _\infty x}/{U_{\infty }} = 1.74\times 10^6$, or $Re_{\delta ^*_0}^{out}=11,216$. The domain inlet is located a small distance form the leading edge at $Re_x^{in} = 105$, corresponding to ${Re_{\delta ^*_0}^{in}=871}$, to avoid the singularity in the similarity solution that we use in this study. The domain size is similar to that of Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019), where in their calculations the outlet is at $Re_x^{out} = 1.75\times 10^6$ or $Re_{\delta ^*_0}^{out}=11 000$. However, in their calculations the flat-plate leading edge was also included in the computational domain, resulting in a weak shock at the leading edge and small discrepancies in the momentum thickness at the outlet compared with the similarity solution.

In order to properly resolve the instabilities near the wall and critical layer, grid stretching in the $y$ direction is employed clustering half of the points near the wall for $y/{\delta ^*_0}<0.9$ (Malik Reference Malik1990). No-slip and adiabatic boundary conditions ($\hat {u}=\hat {v}=\hat {w}=\partial \hat {T} / \partial y =0$) are enforced at the wall (Poinsot & Lele Reference Poinsot and Lele1992). Integration of the OWNS-P equations in $x$ is performed using a second-order backward differentiation formula. As indicated in table 1, fewer $x$ grid points were used for the global calculation due to the higher-order discretization and computational limitations due to its large memory requirements.

All calculations were performed with $N_{\beta }=15$, which gave a good approximation for the filtering of the upstream-travelling modes and accurate capturing of the downstream ones.

5.5.1. Base flow

The laminar base flow is obtained from a similarity solution of the compressible boundary layer equations. The Howarth–Dorodnitsyn transformation (Stewartson Reference Stewartson1964) was employed to reduce the governing equations to a set of ODEs, given in Appendix D. The similarity solution is shown in figure 13.

Figure 13. Base flow of the $M=4.5$ adiabatic flat-plate boundary layer obtained from a similarity transformation. Self-similar streamwise and temperature components (a). Local Mach number (b) with the dashed line corresponding to the displacement thickness.

5.5.2. Optimal gain

The resolvent modes for the adiabatic flat-plate boundary layer have been calculated with the optimal OWNS-P method for a range of frequencies and spanwise wavenumbers. In figure 14 the optimal input/output gains corresponding to three regions of locally maximum gain in the $\beta \unicode{x2013}\omega$ plane are shown. Their maxima correspond to the amplification of streaks ($\omega \to 0$), second Mack modes ($\beta =0$) and oblique first modes. For all three instability types, we observe excellent agreement of the optimal $\omega$ and/or $\beta$ when compared with the global resolvent calculations of Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019). For all the calculations presented here, the forcing has been restricted only to the momentum components $(f_u,f_v,f_w)$ to match the set-up of Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019).

Figure 14. Optimal input/output gain for three linear instability mechanisms for the $M=4.5$ flat-plate adiabatic boundary layer. Validation against normalized results from Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019) for a similar configuration.

5.5.3. Optimal forcing and response

For streaks, the optimal forcing consists of streamwise counter-rotating vortices that lift the streamwise base flow momentum. This is referred to as the lift-up mechanism and yields a response that contains primarily streaks of highly amplified streamwise velocity stretching in the streamwise direction. The dominant input ($f_v$) and output ($\hat {u}$) velocity modes from OWNS-P and global methods are shown in figure 15. Quantitative agreement is again obtained between the OWNS-P optimal modes and the global ones, but minor differences are observed near the inlet/outlet boundaries of the domain, where the sponges of the global method attenuate the response to avoid reflections. Based on our experience, the tuning of the sponges is a cumbersome procedure and problem specific, a step that is bypassed during the OWNS-P marching since upstream-travelling waves have been eliminated at each $x$ location during the parabolization procedure.

Figure 15. Streak (steady 3-D) optimal disturbances at $\omega =0.002$, $\beta =2.2$. Forcing and response amplitude components at $x/\delta ^*_0=35$ (a,c,e) and $x/\delta ^*_0=159$ (b,d,f). Circle symbols: Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019); triangle symbols: global; solid lines: OWNS-P.

In figure 15 we also plot the OWNS-P optimal forcing and response profiles for all the perturbation components at a fixed streamwise location near the inlet and outlet, respectively. The OWNS-P results are compared against our global calculations for the same configuration and the global calculations from Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019). Quantitative agreement is again obtained between OWNS-P and our global results, confirming the accuracy of the OWNS-P approximation. Some discrepancies are observed for the forcing away from the wall when our results (both global and OWNS-P) are compared against Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019). These discrepancies can be attributed to the differences in the computational domain (inclusion of leading edge and shock) and the different choice of the input norm between the two studies. The forcing amplitudes presented in Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019) likely correspond to the conservative form of the momentum equations, whereas we perform our analyses using primitive variables (Karban et al. Reference Karban, Bugeat, Martini, Towne, Cavalieri, Lesshafft, Agarwal, Jordan and Colonius2020). Essentially, our forcing exists in a different subspace, but we are optimizing the same quantity, i.e. the disturbance-energy amplification with respect to the Chu energy norm, which explains the agreement observed in the response profiles despite the differences in the forcing modes.

Contours of the optimal forcing and response for the oblique first mode are shown in figure 16 for the dominant components $f_w$ and $\hat {u}$ from the OWNS-P and global calculations. The optimal forcing field contains upstream-titled structures that are emblematic of the non-modal Orr mechanism. This generates an oblique wave response with relatively large streamwise velocity. Good agreement is obtained between the global and OWNS-P results for all the perturbation input and output components, although a small error of about $1\,\%$ in the wavelength can be observed.

Figure 16. Oblique first mode (unsteady 3-D) optimal disturbances at $\omega =0.32$, $\beta =1.2$. Forcing and response amplitude components at $x/\delta ^*_0=12$ (a,c,e) and $x/\delta ^*_0=159$ (b,d,f). Circle symbols: Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019); triangle symbols: global; solid lines: OWNS-P.

In figure 17 we compare the results for the second mode instability, following a similar procedure as in the two previous cases. We see the classical trapped acoustic waves between the wall and relative sonic line as well as thermodynamic amplification near the generalized inflection point in the response fields (two coexisting mechanisms). For such a response, the optimal forcing is localized near the generalized inflection point. Good agreement is also obtained between OWNS-P and our global results for this family of modes.

Figure 17. Planar second mode (unsteady 2-D) optimal disturbances at $\omega =2.5$, $\beta =0$. Forcing and response amplitude components at $x/\delta ^*_0=90$ (a,c,e) and $x/\delta ^*_0=148$ (b,d,f). Circle symbols: Bugeat et al. (Reference Bugeat, Chassaing, Robinet and Sagaut2019); triangle symbols: global; solid lines: OWNS-P.

The close agreement between the global and OWNS-P modes extends also to suboptimal modes. For example, figure 18 compares the global and OWNS-P streamwise velocity response for the first suboptimal modes, i.e. the mode with the second highest gain, for each of the three instability mechanisms.

Figure 18. Streamwise velocity response for the first suboptimal mode computed by the global and OWNS-P methods: (a,b) streaks; (c,d) oblique first mode; (e,f) planar second mode.