2.1. Eulerian to Lagrangian transformation

Consider a real Euclidean vector space $\mathsf { {E}}$ of dimension $d$, with the inner product $\langle \boldsymbol {x}, \boldsymbol {x} \rangle > 0$ for non-zero $\boldsymbol {x}$, and real norm $\| \boldsymbol {x} \|=\sqrt {\langle \boldsymbol {x}, \boldsymbol {x} \rangle }$. Here, for convenience, we consider $\mathsf { {E}}$ as a $d$-dimensional point space with a coordinate system and a frame of reference, on which the Euclidean space with $\mathbb {R}^{d=3}$ can be realized by considering an orthonormal basis. A flow in a suitable closed domain $\mathsf { {D}} \subseteq \mathsf { {E}} = \mathbb {R}^3$ may be represented in terms of a vector field $\boldsymbol {u}$ through the mapping (the mathematical terminology on the Eulerian and Lagrangian flow descriptions, to some degree, follows Talpaert (Reference Talpaert2002))

(2.1)\begin{equation} \boldsymbol{u}:\mathsf{\textit{D}}\times [0,\mathsf{\textit{T}}] \rightarrow \mathbb{R}^3:(\boldsymbol{x},t)\mapsto \boldsymbol{u}(\boldsymbol{x},t)\quad \text{with } \boldsymbol{x}\in \mathsf{\textit{D}}, \end{equation}

where $t$ is an instant from the total time $\mathsf { {T}}\subset \mathbb {R}$. In the Eulerian description, all physical quantities (scalar, vector or tensor) are expressed at each instant and at every fixed spatial location with respect to the frame of reference. Thus, the fixed spatial coordinates $x_i$ of vector $\boldsymbol {x}$ and time $t$ constitute the Eulerian coordinates with respect to a fixed (Eulerian) frame of reference of $\mathsf { {E}}$. The Eulerian description refers to flow fields at an instant $t$ mapping on another time $t+\textrm {d} t$, where $\textrm {d} t$ is the time differential.

The Lagrangian description of the flow, on the other hand, identifies a flow state at an instant with respect to a time-dependent frame of reference. The flow field, $\boldsymbol {\mathcal {U}}$, over a closed domain $\mathcal {D} \subseteq \mathsf { {E}} = \mathbb {R}^3$ and time interval $[0,\mathcal {T}] \subset \mathbb {R}$ may be mapped as

(2.2)\begin{equation} \boldsymbol{\mathcal{U}}:\mathcal{D}\times [0,\mathcal{T}] \rightarrow \mathbb{R}^3:(\boldsymbol{\chi},\tau)\mapsto \boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\tau)\quad \text{with } \boldsymbol{\chi}\in \mathcal{D}, \tau\in [0,\mathcal{T}]. \end{equation}

The flow evolves from a reference state and maps on a deformed geometrical configuration. Thus, an initial reference configuration $\varOmega _0\in \mathcal {D}$ at $\tau =\tau _0$ and a current configuration $\varOmega \in \textit {SDiff}(\mathcal {D})$ at $\tau$ may be defined, where $\textit {SDiff}(\mathcal {D})$ is an orientation and measure-preserving diffeomorphism of $\mathcal {D}$. Mathematically, the flow map can be expressed as

(2.3a,b)\begin{equation} \left.\begin{array}{c} \mathcal{M}:\mathcal{D}\times [0,\mathcal{T}] \rightarrow \textit{SDiff}(\mathcal{D})\subseteq \mathsf{\textit{E}}=\mathbb{R}^3: (\boldsymbol{\chi},\tau) \mapsto \mathcal{M}(\boldsymbol{\chi},\tau)=(\boldsymbol{x},t)\\ \text{and} \qquad \mathcal{M}(\boldsymbol{\chi}_0,\tau_0)=\text{ identity map.} \end{array}\right\} \end{equation}

The triple components $\chi _i$ of vector $\boldsymbol {\chi }$ and time $\tau$ comprise the Lagrangian coordinates, which can be explicitly expressed using the Eulerian frame of reference as

(2.4)\begin{equation} (x_i,t) = \mathcal{M}^i(\boldsymbol{\chi},\tau) = \mathcal{M}^i(\chi_1,\chi_2,\chi_3,\tau). \end{equation}

The Lagrangian flow mapping from an initial configuration $\varOmega _0$ to a current configuration $\varOmega$ must meet the regularity conditions of the transformation, mainly that it be injective and $\mathcal {M}$ be a bijection. The inverse $\mathcal {M}^{-1}$ exists due to the regularity conditions, and, by considering the existence of the inverse at any instant $\tau$, we can state

(2.5)\begin{equation} (\boldsymbol{\chi},\tau) = \mathcal{M}^{{-}1}(\boldsymbol{x},t) \quad\Longleftrightarrow\quad (\boldsymbol{x},t)=\mathcal{M}(\boldsymbol{\chi},\tau). \end{equation}

Consequently, the Jacobian matrix $\boldsymbol {\mathcal {J}}=\partial (\boldsymbol {\chi },\tau )/\partial (\boldsymbol {x},t)$ is invertible, which plays an important role in the domain deformations. The vector fields of the Lagrangian and Eulerian frames of reference are related as

(2.6)\begin{equation} \boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\tau)=\boldsymbol{u}(\mathcal{M} (\boldsymbol{\chi},\tau)) \quad\Longleftrightarrow\quad \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{\mathcal{U}} (\mathcal{M}^{{-}1}(\boldsymbol{x},t)), \end{equation}

which also applies to each physical quantity of the flow.

The total or material derivative of a quantity, e.g. flow velocity vector, in the Lagrangian frame of reference is simply its partial derivative with respect to time $\tau$, written as

(2.7)\begin{equation} \frac{\textrm{D} \boldsymbol{\mathcal{U}}}{\textrm{D}\tau}=\left.\frac{\partial \boldsymbol{\mathcal{U}}}{\partial \tau}\right|_{\boldsymbol{\chi}}. \end{equation}

On the other hand, the Eulerian frame of reference accounts for the local and convective rates of change of a quantity. The total derivative from (2.6) is then

(2.8)\begin{align} \frac{\textrm{D}\boldsymbol{u}}{\textrm{D}t}&=\left.\frac{\partial \boldsymbol{u}}{\partial t} \right|_{\boldsymbol{x}} + \left.\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}}\boldsymbol{\cdot} \frac{\partial \boldsymbol{x}}{\partial t} \right|_{\boldsymbol{\chi}} \end{align}

(2.9)\begin{align} &=\underbrace{\frac{\partial \boldsymbol{u}}{\partial t}}_{\text{local rate of range}}+ \underbrace{ (\boldsymbol{u}_{\boldsymbol{\chi}}\boldsymbol{\cdot}\boldsymbol{\nabla}) \boldsymbol{u}}_{\text{convective rate of change}}. \end{align}

The Eulerian convective flow velocity $\boldsymbol {u}_{\boldsymbol {\chi }}$ is the mapping of the vector field from the fixed spatial coordinates $\boldsymbol {x}$ to $\boldsymbol {x}+\textrm {d}\kern0.06em \boldsymbol {x}$, where $\textrm {d}\kern0.06em \boldsymbol {x}$ is the differential of space. The flow velocity vector fields can be expressed in terms of the total derivative of the space vector fields in the Eulerian and Lagrangian approaches, respectively, as

(2.10a,b)\begin{equation} \underbrace{\boldsymbol{u}(\boldsymbol{x},t)=\frac{\textrm{D} \boldsymbol{x}}{\textrm{D}t}}_{\text{Eulerian}} \quad \text{and}\quad \underbrace{\boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\tau) =\left.\frac{\partial \boldsymbol{\chi}}{\partial \tau}\right|_{\boldsymbol{\chi}}}_{\text{Lagrangian}}. \end{equation}

Note that the velocity field in the Lagrangian frame of reference is always a function of time for a non-uniform flow.

2.2. Lagrangian proper orthogonal decomposition (LPOD)

The POD method is based on forming a two-point correlation tensor leading to an eigenvalue problem (Lumley Reference Lumley1967). The procedure yields an expansion in terms of orthogonal real basis functions or modes, which are coherent flow structures with associated modal energies. The technique may be applied in the space or spectral domains (Lumley Reference Lumley1967; Moin & Moser Reference Moin and Moser1989; Citriniti & George Reference Citriniti and George2000; Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018), each with its own advantages. Various mathematical properties of POD, such as the optimal modal energy representation and spatiotemporal modal dynamics (Lumley Reference Lumley1970; Aubry Reference Aubry1991; Aubry et al. Reference Aubry, Guyonnet and Lima1991), are instrumental in the popularity of the technique. The most popular method is that of Sirovich (Reference Sirovich1987), which uses snapshots gathered from successive flow instants to form an equivalent two-point correlation tensor, adhering to the conventional Eulerian frame of reference. Here we will consider the equivalent Lagrangian approach; for concreteness, we develop the spatial form of POD, with the understanding that the correspondence to the spectral form is straightforward.

Let us consider a real matrix $\boldsymbol { {\mathsf {X}}}\in \mathbb {R}^{m\times n}$ that comprises Lagrangian flow fields in discrete form, where $m$ and $n$ are the space and time dimensions, respectively. The Lagrangian flow-field matrix $\boldsymbol { {\mathsf {X}}}$ can be redefined accounting for the weight tensor as $\boldsymbol { {\mathsf {Y}}}=\boldsymbol {{w}}^\textrm {T} \boldsymbol { {\mathsf {X}}} \in \mathbb {R}^{m\times n}$. For a given Lagrangian flow field, the objective of POD is to distil out functions $\boldsymbol {\varPhi }_l \in \mathbb {R}^m$, such that

(2.11)\begin{equation} \lambda_l = \arg \max \left\{ \frac{\boldsymbol{\varPhi}_l^\textrm{T} \boldsymbol{{\mathsf{Y}}} \boldsymbol{{\mathsf{Y}}}^\textrm{T} \boldsymbol{\varPhi}_l} {\boldsymbol{\varPhi}_l^\textrm{T}\boldsymbol{\varPhi}_l}\right\}. \end{equation}

The LPOD spatial modes $\boldsymbol {\varPhi }_l$ are the eigenfunctions of the eigenvalue problem,

(2.12)\begin{equation} \boldsymbol{{\mathsf{Y}}}\boldsymbol{{\mathsf{Y}}}^\textrm{T} \boldsymbol{\varPhi}_l = \lambda_l \boldsymbol{\varPhi}_l, \end{equation}

where the matrix $\boldsymbol { {\mathsf {Y}}}\boldsymbol { {\mathsf {Y}}}^\textrm {T}$ is symmetric positive semidefinite, ensuring a set of orthonormal eigenvectors and corresponding eigenvalues: $\{ \boldsymbol {\varPhi }_l, \lambda _l\} _{l\in \{1,\ldots,m\}}$ ordered as $\lambda _l \geq \lambda _{l+1} \geq 0$. The matrix of these eigenvectors $\boldsymbol {\varPhi } \in \mathbb {R}^{m\times m}$ with $\boldsymbol {\varPhi }^\textrm {T}\boldsymbol {\varPhi } = \boldsymbol { {\mathsf {I}}}$ forms a complete orthonormal basis of $\mathbb {R}^m$. Here $\boldsymbol { {\mathsf {I}}}\in \mathbb {R}^{m\times m}$ is an identity matrix. The Lagrangian flow fields can be expressed as

(2.13)\begin{equation} \boldsymbol{{\mathsf{X}}} = \boldsymbol{\varPhi} \boldsymbol{\varLambda}^{{1}/{2}}\boldsymbol{\varPsi}^\textrm{T}, \end{equation}

where $\boldsymbol {\varLambda }=\text {diag}\{\lambda _1,\ldots,\lambda _m\}$ and $\boldsymbol {\varPsi }\in \mathbb {R}^{n\times m}$ are the LPOD temporal coefficients which are associated with the LPOD spatial modes $\boldsymbol {\varPhi }$. The LPOD time coefficient can be obtained as

(2.14)\begin{equation} \boldsymbol{\varPsi} = \boldsymbol{{\mathsf{Y}}}^\textrm{T}\boldsymbol{{w}}^{{-}1} \boldsymbol{\varPhi} \boldsymbol{\varLambda}^{-{1}/{2}}. \end{equation}

The LPOD temporal coefficients matrix is also orthonormal, i.e. $\boldsymbol {\varPsi }^\textrm {T}\boldsymbol {\varPsi }=\boldsymbol { {\mathsf {I}}}$, forming another set of basis functions. Thus, the eigenvalue problem of (2.12) can be alternatively stated as

(2.15)\begin{equation} \boldsymbol{{\mathsf{Y}}}^\textrm{T}\boldsymbol{{\mathsf{Y}}}\boldsymbol{\varPsi} = \boldsymbol{\varPsi} \tilde{\boldsymbol{\varLambda}} \quad\text{with}\quad \boldsymbol{\varPhi}=\boldsymbol{{w}}^{-\textrm{T}} \boldsymbol{{\mathsf{Y}}} \boldsymbol{\varPsi} \tilde{\boldsymbol{\varLambda}}^{-{1}/{2}} = \boldsymbol{{\mathsf{X}}}\boldsymbol{\varPsi} \tilde{\boldsymbol{\varLambda}}^{-{1}/{2}}, \end{equation}

where $\tilde {\boldsymbol {\varLambda }}=\text {diag}\{\lambda _1,\ldots, \lambda _n\}$. Typically, the LPOD procedure via (2.15) is much more efficient compared to (2.12) due to the fewer degrees of freedom that arise in the time discretization as opposed to the spatial discretization, i.e. $n \ll m$, which is the key aspect underlying the method of snapshots (Sirovich Reference Sirovich1987).

2.3. Lagrangian dynamic mode decomposition (LDMD)

The popularity of DMD has grown recently as a complementary approach to POD. DMD extracts coherent features based on the Koopman operator and may be applied to snapshots, which usually represent progress in time, though spatially evolving features can also be extracted if desired (Rowley et al. Reference Rowley, Mezić, Bagheri, Schlatter and Henningson2009; Schmid Reference Schmid2010). When the snapshots represent time progression, the DMD modes represent spatially coherent structures evolving in time with unique frequencies and growth/decay rates. For a linearized flow about a steady state (in Eulerian frame of reference), the DMD modes are equivalent to global stability modes (Schmid Reference Schmid2010). As noted earlier, the Lagrangian formulation is inherently unsteady for non-uniform flows; thus the Lagrangian DMD may be performed directly on a (non-uniform Eulerian) steady base flow, leading to modal information pertinent to the stability of the base flow.

Typically, DMD derives a mapping between suitably constructed sequences of flow states. It then solves for the basis functions (eigenvectors) of a reduced-order representation of the mapping. The equivalent in a Lagrangian frame of reference may be developed as follows. We consider $\boldsymbol { {\mathsf {X}}}$ and $\boldsymbol { {\mathsf {Y}}}$ as tensors whose elements are the Lagrangian flow fields, e.g. the velocity vector $\boldsymbol {\mathcal {U}}(\boldsymbol {\chi },\tau )$, where time $\tau \in [0,\mathcal {T}]$, such that $\tau =\{\tau _1, \tau _2,\ldots,\tau _{n} \}$ for $\boldsymbol { {\mathsf {X}}}$ and $\tau =\{\tau _2, \tau _3,\ldots,\tau _{n+1} \}$ for $\boldsymbol { {\mathsf {Y}}}$. Similarly, the Lagrangian space coordinate vector $\boldsymbol {\chi }$ is considered to be discrete of size $m$, thus $\boldsymbol { {\mathsf {X}}}, \boldsymbol { {\mathsf {Y}}}\in \mathbb {R}^{m\times n}$. The aim of the DMD procedure is to find $\boldsymbol { {\mathsf {A}}} \in \mathbb {R}^{m\times m}$ such that

(2.16)\begin{equation} \boldsymbol{{\mathsf{A}}}\boldsymbol{{\mathsf{X}}} = \boldsymbol{{\mathsf{Y}}} \quad\text{or} \quad \boldsymbol{{\mathsf{A}}}= \boldsymbol{{\mathsf{Y}}}\boldsymbol{{\mathsf{X}}}^+, \end{equation}

where $\boldsymbol { {\mathsf {X}}}^+$ is the Moore–Penrose pseudoinverse of $\boldsymbol { {\mathsf {X}}}$. As in the traditional Eulerian approach, in practice $m \gg n$, which complicates the use of (2.16). A low-order representation of $\boldsymbol { {\mathsf {A}}}$ is sought through the compact singular value decomposition of $\boldsymbol { {\mathsf {X}}}$:

(2.17)\begin{equation} \boldsymbol{{\mathsf{X}}} = \boldsymbol{{\mathsf{U}}}\boldsymbol{\varSigma} \boldsymbol{{\mathsf{V}}}^\textrm{T}. \end{equation}

This leads to an approximate representation of $\boldsymbol { {\mathsf {A}}}$ as

(2.18)\begin{equation} \tilde{\boldsymbol{A}} = \boldsymbol{{\mathsf{U}}}^\textrm{T} \boldsymbol{{\mathsf{A}}}\boldsymbol{{\mathsf{U}}} = \boldsymbol{{\mathsf{U}}}^\textrm{T} \boldsymbol{{\mathsf{Y}}}\boldsymbol{{\mathsf{V}}}\boldsymbol{\varSigma}^{{-}1} \in \mathbb{R}^{n \times n}, \end{equation}

where $\boldsymbol { {\mathsf {U}}}\in \mathbb {R}^{m \times n}$ and $\boldsymbol { {\mathsf {V}}}\in \mathbb {R}^{n\times n}$ are orthogonal matrices, while $\boldsymbol {\varSigma }$ is a diagonal matrix of size $n\times n$ with non-zero real singular values. Lastly, the Lagrangian DMD modes, $\boldsymbol {\phi }_l \in \mathbb {C}^m$, are obtained by

(2.19)\begin{equation} \boldsymbol{\phi}_l = \boldsymbol{{\mathsf{U}}}\boldsymbol{v}_l, \end{equation}

where the $l$th eigenvector $\boldsymbol {v}_l \in \mathbb {C}^n$ is a solution of the eigenvalue problem

(2.20)\begin{equation} \tilde{\boldsymbol{A}}\boldsymbol{v}_l=\kappa_l \boldsymbol{v}_l, \end{equation}

with the corresponding eigenvalue $\kappa _l \in \mathbb {C}$. The growth rate and angular frequency of the LDMD mode are $\ln |\kappa _l|/\delta \tau$ and $\arg (\kappa _l)/\delta \tau$, respectively, where $\delta \tau$ is the Lagrangian uniform time discretization.

The LDMD formulation naturally connects to the Lagrangian flow map, which may comprise fixed points, periodic orbits, stable and unstable manifolds, and chaotic attractors (Ottino Reference Ottino1989; Shadden et al. Reference Shadden, Lekien and Marsden2005; Wiggins Reference Wiggins2005; Lekien, Shadden & Marsden Reference Lekien, Shadden and Marsden2007; Haller Reference Haller2015). Indeed, the LDMD matrix $\tilde {\boldsymbol {A}}$, which is an approximation for $\boldsymbol { {\mathsf {A}}}$, seeks properties of the Lagrangian flow map $\mathcal {M}(\boldsymbol {\chi },\tau )$ (of (2.3a,b)) in terms of the Lagrangian flow fields $\boldsymbol { {\mathsf {X}}}$ (see figure 1). These properties include eigenvalues, eigenvectors, energy amplification and resonance behaviour (Schmid Reference Schmid2010), which reveal the dynamic characteristics of the process that is governing the flow map. The Lagrangian flow fields at any time instant $\tau$ can be expressed as

(2.21)\begin{equation} \boldsymbol{{\mathsf{X}}}_\tau =\boldsymbol{\phi}\exp( \boldsymbol{\kappa}\tau ) \boldsymbol{a} \quad\text{with}\quad \boldsymbol{a}=\boldsymbol{\phi}^+ \boldsymbol{{\mathsf{X}}}_{\tau_1}. \end{equation}

Here $\boldsymbol {\phi }=\{\boldsymbol {\phi }_l\}_{l\in \{1,\ldots,n\}}\in \mathbb {C}^{m\times n}$ is a complex set of LDMD modes, while $\boldsymbol {\kappa }=\text {diag}\{\kappa _1,\ldots,\kappa _n\}$ are the eigenvalues (also Ritz values). The initial conditions $\boldsymbol {a}\in \mathbb {C}^{n}$ are obtained by means of the pseudoinverse $\boldsymbol {\phi }^+$ and the identity map $\boldsymbol { {\mathsf {X}}}_{\tau _1}=\mathcal {M}(\boldsymbol {\chi }_0,\tau _0)$.

Figure 1.

Schematic representation of a fluid element in the Lagrangian frame of reference, where the deforming flow trajectories lead to Lyapunov exponents and a data matrix for LMA over a finite time.

2.4. Lyapunov exponents and Lagrangian modal analysis ansatz

The Lagrangian flow map $\mathcal {M}(\boldsymbol {\chi },\tau )$ of (2.3a,b) represents a dynamical system, evolving from an initial state, i.e. from the identity map $\mathcal {M}(\boldsymbol {\chi }_0,\tau _0)$. Lyapunov exponents characterize the rate of separation between two points on the manifold $\textit {SDiff}(\mathcal {D})$, with divergence between the points being constrained to the linear approximation; in addition, the Lyapunov exponent spectrum is analogous to the eigenvalue spectrum of the linearized stability equations at steady state (Goldhirsch, Sulem & Orszag Reference Goldhirsch, Sulem and Orszag1987; Vastano & Moser Reference Vastano and Moser1991). For a $d$-dimensional state space, there are $d$ number of Lyapunov exponents; however, among these, the largest is significant in determining the system behaviour. If $\delta \boldsymbol {\chi }_0$ and $\delta \boldsymbol {\chi }_\tau$ are the separations between any two points at an initial time $\tau _0$ and a later time $\tau$, respectively, then the maximum Lyapunov exponent is given by

(2.22)\begin{equation} \lambda^{LE} = \lim_{\tau \rightarrow \infty} \lim_{|\delta \boldsymbol{\chi}_0| \rightarrow 0} \frac{1}{\tau} \ln \frac{|\delta \boldsymbol{\chi}_\tau|}{|\delta \boldsymbol{\chi}_0|}, \end{equation}

where the limits $\tau \rightarrow \infty$ and $|\delta \boldsymbol {\chi }_0| \rightarrow 0$ ensure time asymptotic and linear considerations, respectively. The Lyapunov exponents provide insights into a vector space that is tangent to the state space. The Jacobian matrix $\boldsymbol {\mathcal {J}}$ governs the evolution of the small separation $\delta \boldsymbol {\chi }_0$ as

(2.23)\begin{equation} \delta \boldsymbol{\chi}_\tau = \exp\left(\int_0^\tau\boldsymbol{\mathcal{J}}(\tau')\,\textrm{d} \tau'\right) \delta \boldsymbol{\chi}_0. \end{equation}

A matrix $\boldsymbol {\mathcal {X}}$, defined as (Oseledets Reference Oseledets1968)

(2.24)\begin{equation} \boldsymbol{\mathcal{X}} = \lim_{\tau \rightarrow \infty} \frac{1}{\tau} \ln \sqrt{\delta \boldsymbol{\chi}_\tau \delta \boldsymbol{\chi}_\tau^\textrm{T}}, \end{equation}

provides the Lyapunov exponent spectrum in terms of its eigenvalues, giving the average exponential growth rates of the separation at time $\tau$. Furthermore, in the time limit $\tau \rightarrow \infty$, the Lyapunov spectrum offers a global measure of the strange attractor of the dynamical system (Yoden & Nomura Reference Yoden and Nomura1993).

Alternatively, Lyapunov exponents may be estimated locally (in the limit $\tau \to 0$) or for a finite time (for $\tau \in [0,\mathcal {T}]$) in order to investigate the local dynamics of the system (Goldhirsch et al. Reference Goldhirsch, Sulem and Orszag1987; Thiffeault & Boozer Reference Thiffeault and Boozer2001; Nolan, Serra & Ross Reference Nolan, Serra and Ross2020). The FTLEs for $\tau \in [0,\mathcal {T}]$ are estimated as

(2.25)\begin{equation} \lambda^{LE}_d = \frac{1}{\mathcal{T}} \ln{\sqrt{\lambda_d(\boldsymbol{\mathcal{C}})}}, \end{equation}

where $\lambda _d(\boldsymbol {\mathcal {C}})$ denotes the $d$th eigenvalue of the right Cauchy–Green strain tensor,

(2.26)\begin{equation} \boldsymbol{\mathcal{C}} = \boldsymbol{\mathcal{J}}^\textrm{T} \boldsymbol{\mathcal{J}} = \boldsymbol{\nabla} \boldsymbol{\mathcal{M}}^\textrm{T} \boldsymbol{\nabla} \boldsymbol{\mathcal{M}}. \end{equation}

The maximum Lyapunov exponent is expressed in terms of the maximum eigenvalue of $\boldsymbol {\mathcal {C}}$ as

(2.27)\begin{equation} \lambda^{LE} = \frac{1}{\mathcal{T}} \ln{\sqrt{\lambda_{max}(\boldsymbol{\mathcal{C}})}}. \end{equation}

Here, $\lambda _{max}(\boldsymbol {\mathcal {C}})$ identifies flow regions with high shear, which are further illuminated by the maximum FTLE field due to the logarithmic definition (Haller Reference Haller2002).

To establish the relation between the Lyapunov exponents and LMA, we express the Lagrangian velocity of (2.10a,b) as

(2.28)\begin{equation} \boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\tau) = \frac{\partial \mathcal{M}(\boldsymbol{\chi},\tau)}{\partial \tau} = \frac{\partial \mathcal{M}(\boldsymbol{\chi},\tau)}{\partial \boldsymbol{\chi}_0} \frac{\partial \boldsymbol{\chi}_0}{\partial \tau} = \boldsymbol{\nabla} \boldsymbol{\mathcal{M}}\,\boldsymbol{\mathcal{U}}(\boldsymbol{\chi}_0,\tau). \end{equation}

Reconsider the real matrix $\boldsymbol { {\mathsf {X}}}\in \mathbb {R}^{m\times n}$ utilized in the formulation of LPOD and LDMD. Here $\boldsymbol { {\mathsf {X}}}$ comprises the Lagrangian flow fields of the absolute velocity $\|\boldsymbol {\mathcal {U}}(\boldsymbol {\chi },\tau )\|$. For a time instant $\tau$ with $n=1$, we can write

(2.29)\begin{align} \text{diag} \{ \boldsymbol{{\mathsf{X}}}\boldsymbol{{\mathsf{X}}}^\textrm{T} \} &= \|\boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\tau)\|^2 = \boldsymbol{\mathcal{U}}^\textrm{T}(\boldsymbol{\chi},\tau)\, \boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\tau) \end{align}

(2.30)\begin{align} &= \boldsymbol{\mathcal{U}}^\textrm{T}(\boldsymbol{\chi}_0,\tau) \{ \boldsymbol{\nabla} \boldsymbol{\mathcal{M}}^\textrm{T}\boldsymbol{\nabla} \boldsymbol{\mathcal{M}}\} \,\boldsymbol{\mathcal{U}}(\boldsymbol{\chi}_0,\tau) \end{align}

(2.31)\begin{align} &=\boldsymbol{\mathcal{U}}^\textrm{T}(\boldsymbol{\chi}_0,\tau)\, \boldsymbol{\mathcal{C}}\,\boldsymbol{\mathcal{U}}(\boldsymbol{\chi}_0,\tau), \end{align}

where $\boldsymbol {\mathcal {C}} \in \mathbb {R}^{d\times d}$ is the right Cauchy–Green strain tensor of (2.26). The alignment between $\boldsymbol {\mathcal {U}}(\boldsymbol {\chi }_0,\tau )$ and the eigenvectors of $\boldsymbol {\mathcal {C}}$ manifests in the value of $\text {diag}\{\boldsymbol { {\mathsf {X}}}\boldsymbol { {\mathsf {X}}}^\textrm {T}\}$. For a finite time $\tau \in [0,\mathcal {T}]$, we can rewrite (2.31) as

(2.32)\begin{align} \text{diag} \{ (\boldsymbol{{\mathsf{X}}}\boldsymbol{{\mathsf{X}}}^\textrm{T})_d \} &= \| \boldsymbol{\mathcal{U}}_d(\boldsymbol{\chi}_0,\tau) \|^2 \lambda_d(\boldsymbol{\mathcal{C}}) \end{align}

(2.33)\begin{align} &= \| \boldsymbol{\mathcal{U}}_d(\boldsymbol{\chi}_0,\tau) \|^2 \exp(2\mathcal{T}\lambda_d^{LE}). \end{align}

The (maximum) FTLE relates to the maximum of $\text {diag}\{ (\boldsymbol { {\mathsf {X}}}\boldsymbol { {\mathsf {X}}}^\textrm {T})_{d} \}$ for a specific argument $d$, which corresponds to the alignment of $\boldsymbol {\mathcal {U}}(\boldsymbol {\chi }_0,\tau )$ and the eigenvector of $\boldsymbol {\mathcal {C}}$ with the largest eigenvalue $\lambda _{max}(\boldsymbol {\mathcal {C}})$, as

(2.34)\begin{equation} \lambda^{LE} =\frac{1}{\mathcal{T}}\ln\sqrt{\text{diag} \left\{\max_d\{( \boldsymbol{{\mathsf{X}}}\boldsymbol{{\mathsf{X}}}^\textrm{T})_d\} \right\}}. \end{equation}

The FTLE field represents the local maxima of

(2.35)\begin{align} \text{diag} \left\{\max_d\{( \boldsymbol{{\mathsf{X}}}\boldsymbol{{\mathsf{X}}}^\textrm{T})_d\} \right\} &=\text{diag} \left\{\max_d\{(\boldsymbol{\varPhi}\boldsymbol{\varLambda} \boldsymbol{\varPhi}^\textrm{T})_d\} \right\} \end{align}

(2.36)\begin{align} &=\text{diag}\left\{\max_d\left\{\left(\sum_{l=1}^m \boldsymbol{\varPhi}_l{\lambda}_l \boldsymbol{\varPhi}_l^\textrm{T}\right)_d\right\}\right\}, \end{align}

whereas LPOD provides the global eigenfunctions $\boldsymbol {\varPhi }_l$ and associated energies ordered as $\lambda _l\geq \lambda _{l+1} \geq 0$. In addition to the real symmetry and positive semidefiniteness, the autocorrelation tensor $\boldsymbol { {\mathsf {X}}}\boldsymbol { {\mathsf {X}}}^\textrm {T}$ is also diagonally dominant, which is a consequence of the rearrangement inequality (Hardy, Littlewood & Pólya Reference Hardy, Littlewood and Pólya1952, chap. X). Thus, the maximum Lyapunov exponent field closely relates to the first eigenmode (LPOD mode with maximum $\lambda _l$) of the autocorrelation tensor of the velocity magnitude.

A key feature of the FTLE field is that it is objective, i.e. independent of the observer's frame of reference. This is due to the functional dependence of Lyapunov exponents on the invariants of the right Cauchy–Green strain tensor, which satisfy the principle of material frame independence (Truesdell & Noll Reference Truesdell and Noll2004). In general, the objectivity in terms of Euclidean measures is ensured for an observer transformation from ($\boldsymbol {\chi },\tau$) to ($\boldsymbol {\chi }^\ast,\tau ^\ast$) as

(2.37a,b)\begin{equation} \boldsymbol{\chi}^\ast\,{=}\, \boldsymbol{{\mathsf{Q}}}\boldsymbol{\chi}+\boldsymbol{c}, \quad \tau^\ast\,{=}\,\tau+\mathsf{\textit{b}}, \end{equation}

where $\mathsf { {b}}$ is an arbitrary constant, $\boldsymbol {c}$ is a time-dependent vector and $\boldsymbol { {\mathsf {Q}}}$ is a time-dependent proper orthogonal tensor. The scalar, vector and tensor fields are objective if, respectively,

(2.38a–c)\begin{equation} \beta^\ast\,{=}\,\beta, \quad \boldsymbol{b}^\ast\,{=}\, \boldsymbol{{\mathsf{Q}}}\boldsymbol{b} \quad\text{and}\quad \boldsymbol{{\mathsf{B}}}^\ast\,{=}\, \boldsymbol{{\mathsf{Q}}}\boldsymbol{{\mathsf{B}}}\boldsymbol{{\mathsf{Q}}}^\textrm{T}. \end{equation}

Let us now consider the orthonormal basis $\boldsymbol {\varPhi }=\{\boldsymbol {\varPhi }_l\}_{l \in \{ 1,\ldots,m \}}$, and a second orthonormal basis $\boldsymbol {\varPhi }^\ast =\{\boldsymbol { {\mathsf {Q}}}\boldsymbol {\varPhi }_l\}_{l \in \{1,\ldots,m \}}$. For a frame-independent vector $\boldsymbol {b}$ in the basis $\boldsymbol {\varPhi }$, an equivalent $\boldsymbol {b}^\ast$ in the basis $\boldsymbol {\varPhi }^\ast$ is

(2.39)\begin{equation} \boldsymbol{b}^*_l=\boldsymbol{b}^{{\ast} \textrm{T}} \boldsymbol{{\mathsf{Q}}} \boldsymbol{\varPhi}_l=\boldsymbol{b} ^\textrm{T}\boldsymbol{{\mathsf{Q}}}^\textrm{T} \boldsymbol{{\mathsf{Q}}}\boldsymbol{\varPhi}_l=\boldsymbol{b}^\textrm{T} \boldsymbol{\varPhi}_l=\boldsymbol{b}_l, \end{equation}

i.e. the components of $\boldsymbol {b}^\ast$ in basis $\boldsymbol {\varPhi }^*$ and $\boldsymbol {b}$ in basis $\boldsymbol {\varPhi }$ are identical. Similarly, an objective tensor $\boldsymbol { {\mathsf {B}}}$ in the basis $\boldsymbol {\varPhi }$ can be expressed as $\boldsymbol { {\mathsf {B}}}_{kl}=\boldsymbol {\varPhi }_k^\textrm {T} \boldsymbol { {\mathsf {B}}}\boldsymbol {\varPhi }_l$, while the components of a second tensor $\boldsymbol { {\mathsf {B}}}^\ast$ in the basis $\boldsymbol {\varPhi }^\ast$ are

(2.40)\begin{equation} \boldsymbol{{\mathsf{B}}}_{kl}^\ast\,{=}\, ( \boldsymbol{{\mathsf{Q}}}\boldsymbol{\varPhi}_k) ^\textrm{T} \boldsymbol{{\mathsf{B}}}^\ast(\boldsymbol{{\mathsf{Q}}}\boldsymbol{\varPhi}_l)= \boldsymbol{\varPhi}_k ^\textrm{T}\boldsymbol{{\mathsf{Q}}}^\textrm{T}\boldsymbol{{\mathsf{Q}}} \boldsymbol{{\mathsf{B}}}\boldsymbol{{\mathsf{Q}}}^\textrm{T} \boldsymbol{{\mathsf{Q}}}\boldsymbol{\varPhi}_l = \boldsymbol{\varPhi}_k ^\textrm{T} \boldsymbol{{\mathsf{B}}}\boldsymbol{\varPhi}_l= \boldsymbol{{\mathsf{B}}}_{kl}, \end{equation}

leading to exactly the same tensor. Thus, the objectivity of the flow fields, including scalar, vector and tensor fields, is preserved under LMA, ensuring the principle of material frame independence.

The FTLE and Lagrangian DMD relate through the well-documented connections between the POD and DMD in the literature (Schmid et al. Reference Schmid, Meyer and Pust2009; Schmid Reference Schmid2010). As noted before, the POD optimally extracts the most energetic coherent flow structures, whereas the DMD focuses on the coherent structures with unique frequency and growth/decay rate. In the LMA ansatz, the dominant LPOD modes are the coherent flow structures that comprise maximum stretching of the flow fields, while the LDMD modes are the coherent flow structures that evolve at unique frequencies. The relation between the FTLE and Lagrangian POD/DMD modes is illustrated in § 4.4 by considering the simple mathematical model comprising the double-gyre pattern.

3.1. Flow governing equations and numerical methods

The flow fields are governed by the full compressible Navier–Stokes equations, which are solved in non-dimensional form using curvilinear coordinates:

(3.1)\begin{equation} \frac{\partial}{\partial\tau}\left(\frac{\boldsymbol{S}}{J}\right) + \frac{\partial \boldsymbol{F}}{\partial\xi_1} + \frac{\partial \boldsymbol{G}}{\partial\xi_2} + \frac{\partial \boldsymbol{H}}{\partial\xi_3} = \frac{1}{Re}\left[ \frac{\partial \hat{\boldsymbol{F}}}{\partial\xi_1} + \frac{\partial \hat{\boldsymbol{G}}}{\partial\xi_2} + \frac{\partial \hat{\boldsymbol{H}}}{\partial\xi_3} \right], \end{equation}

where $\boldsymbol {S}=[\rho, \rho \boldsymbol {u}, \rho E ]^\textrm {T}$ is the conserved solution vector. The flow variables are non-dimensionalized by their reference ($\infty$) values, except for pressure, which is normalized by using the reference density and reference velocity. Here, $\boldsymbol {u}$ is the velocity vector, while $\rho$ and $E$ are the density and internal energy, respectively. The non-dimensionalized flow variables are defined by

(3.2a–f)\begin{align} \rho = \frac{\rho^\ast}{\rho_\infty^\ast}, \quad \boldsymbol{u} = \frac{\boldsymbol{u}^\ast}{u_\infty^\ast}, \quad p = \frac{p^\ast}{\rho^\ast_\infty u^{{\ast} 2}_\infty},\quad T=\frac{T^\ast}{T^\ast_\infty},\quad \boldsymbol{x}=\frac{\boldsymbol{x}^\ast}{L_{ref}^\ast} \quad \text{and}\quad t=\frac{t^\ast u^\ast_\infty}{L_{ref}^\ast}, \end{align}

where $L_{ref}^\ast$ is a dimensional reference length and the asterisk denotes a dimensional quantity. The non-dimensional Reynolds and Mach numbers are then

(3.3a,b)\begin{equation} Re \equiv \frac{\rho^\ast_\infty u^\ast_\infty L_{ref}^\ast}{\mu_\infty^\ast}\quad \text{and} \quad M_\infty \equiv \frac{u^\ast_\infty}{\sqrt{\gamma p^\ast_\infty/\rho^\ast_\infty}}. \end{equation}

The Jacobian, denoted by $J$, of the Cartesian-to-curvilinear coordinate transformation $(\boldsymbol {x},t)\rightarrow (\boldsymbol {\xi },\tau )$ is given by $J={\partial (\boldsymbol {\xi },\tau )}/{\partial (\boldsymbol {x},t)}$. The inviscid and viscous fluxes, for instance $\boldsymbol {F}$ and $\hat {\boldsymbol {F}}$, respectively, are given as

(3.4a,b)\begin{equation} \boldsymbol{F} = \frac{1}{J}\begin{bmatrix} \rho U_1\\ \rho u_1U_1+\dfrac{\partial \xi_1}{\partial x_1}p\\ \rho u_2U_1+\dfrac{\partial \xi_1}{\partial x_2}p\\ \rho u_3U_1+\dfrac{\partial \xi_1}{\partial x_3}p\\ (\rho E+p)U_1-\dfrac{\partial \xi_1}{\partial t}p \end{bmatrix},\quad \hat{\boldsymbol{F}} = \frac{1}{J}\begin{bmatrix} 0\\ \dfrac{\partial \xi_1}{\partial x_i}\sigma_{1i}\\ \dfrac{\partial \xi_1}{\partial x_i}\sigma_{2i}\\ \dfrac{\partial \xi_1}{\partial x_i}\sigma_{3i}\\ \dfrac{\partial \xi_1}{\partial x_i}(u^j\sigma_{ij}-\varTheta_i) \end{bmatrix}, \end{equation}

where $i$ and $j$ are summation indices. Here, $U_i$ is the contravariant velocity component, which is expressed by using a summation index $j$ as

(3.5)\begin{equation} U_i = \frac{\partial \xi_i}{\partial t}+\frac{\partial \xi_i}{\partial x_j}u_j. \end{equation}

The internal energy is given by

(3.6)\begin{equation} E = \frac{T}{\gamma (\gamma - 1) M^2_\infty} +\frac{1}{2}\| \boldsymbol{u} \|^2, \end{equation}

where $T$, $\gamma$ and $M_\infty$ are the temperature, the ratio of specific heats and the reference Mach number, respectively. The fluid is assumed to be a perfect gas, with pressure $p=\rho T/\gamma M^2_\infty$. The ratio of specific heats for air, $\gamma$, is assumed to be $1.4$. The components of the stress tensor and the heat flux vector are given by, respectively,

(3.7)\begin{equation} \sigma_{ij}= \mu \left( \frac{\partial \xi_k}{\partial x_j} \frac{\partial u_i}{\partial \xi_k} + \frac{\partial \xi_k}{\partial x_i}\frac{\partial u_j}{\partial \xi_k} - \frac{2}{3}\frac{\partial \xi_l}{\partial x_k}\frac{\partial u_k}{\partial \xi_l} \delta_{ij} \right) \end{equation}

and

(3.8)\begin{equation} \varTheta_i={-} \frac{\mu/Pr}{(\gamma-1)M_\infty^2} \frac{\partial \xi_j}{\partial x_i} \frac{\partial T}{\partial \xi_j}. \end{equation}

The Prandtl number is set to $Pr=0.72$. Here, $\mu$ denotes the dynamic viscosity of the fluid, while the bulk viscosity is $-2\mu /3$, assuming the Stokes’ hypothesis. The fluid viscosity change due to the temperature is modelled using Sutherland's law, given as

(3.9)\begin{equation} \mu=T^{3/2}\left(\frac{1+C_1}{T+C_1}\right), \end{equation}

where $C_1=0.37$ is the non-dimensionalized Sutherland's constant.

The second-order implicit time marching scheme of Beam & Warming (Reference Beam and Warming1978) is adopted, with two Newton-like sub-iterations to reduce factorization and explicit boundary condition application errors. Further details on the time scheme are provided in Visbal & Gordnier (Reference Visbal and Gordnier2004). The spatial derivatives are discretized using a sixth-order compact finite difference scheme with the central difference, ensuring no dissipation error. An eighth-order implicit low-pass Padé-type filtering, with $\alpha _f=0.4$, is used to provide dissipation at high spatial wavenumbers. Detailed validation studies may be found in Visbal & Gaitonde (Reference Visbal and Gaitonde1999), Gaitonde & Visbal (Reference Gaitonde and Visbal2000) and Visbal & Gaitonde (Reference Visbal and Gaitonde2002).

3.2. Two-dimensional lid-driven cavity

The first test case considers a compressible two-dimensional lid-driven cavity flow. The flow inside a lid-driven cavity exhibits relatively complex vortex dynamics with increasing Reynolds number, including the onset of Hopf bifurcation, making it one of the classical configurations for flow stability and transition (Ghia, Ghia & Shin Reference Ghia, Ghia and Shin1982; Shen Reference Shen1991; Ramanan & Homsy Reference Ramanan and Homsy1994). Although three-dimensionality and endwall effects are significant for the flow physics (Koseff et al. Reference Koseff, Street, Gresho, Upson, Humphrey and To1983; Sheu & Tsai Reference Sheu and Tsai2002; Albensoeder & Kuhlmann Reference Albensoeder and Kuhlmann2005; Lopez et al. Reference Lopez, Welfert, Wu and Yalim2017), high-fidelity two-dimensional numerical simulations continue to be canonical benchmarks (Bruneau & Saad Reference Bruneau and Saad2006), in situations such as the present. For generality, the effects of compressibility are retained by considering a Mach number of $M_\infty =0.5$, where the flow stability and dynamics of the lid-driven cavity have been discussed by Bergamo et al. (Reference Bergamo, Gennaro, Theofilis and Medeiros2015), Ohmichi & Suzuki (Reference Ohmichi and Suzuki2017) and Ranjan, Unnikrishnan & Gaitonde (Reference Ranjan, Unnikrishnan and Gaitonde2020). Simulations were performed at Reynolds numbers based on cavity size $L$ ranging from $Re_L=5000$ to $Re_L=15\,000$ at intervals of $2000$. The flow remains steady until $Re_L=9000$ but becomes unsteady at $Re_L=11\,000$, consistent with the critical Reynolds number $Re_c=10\,500$ at this Mach number Ohmichi & Suzuki (Reference Ohmichi and Suzuki2017). Details on the geometry and grid convergence are provided in Appendix B. For concreteness, the Lagrangian modal analysis is discussed for $Re_L=7000$ (steady) and $Re_L=15\,000$ (unsteady).

Figure 2(a) displays the steady pre-critical Reynolds number flow at $Re_L=7000$ in terms of the absolute flow velocity $\boldsymbol {u}$ and select flow streamlines. Several recirculation regions are apparent: in addition to the large central feature, three smaller regions are evident near the top-left, bottom-left and bottom-right corners of the cavity. On the other hand, at the post-critical Reynolds number of $Re_L=15\,000$, the flow is unsteady. The pattern of figure 2(b) shows the time-averaged absolute flow velocity $\bar {\boldsymbol {u}}$ at $Re_L=15\,000$; the mean streamlines in this case indicate additional flow recirculation patterns near the lower corners. Furthermore, the skin-friction coefficient (B3) estimated along the bottom wall (figure 15b) also indicates the regions of recirculation, corresponding to the flow pattern of figure 2(b).

Figure 2.

Flow recirculation patterns inside the lid-driven cavity at Mach number $M_\infty =0.5$ and pre- and post-critical Reynolds numbers. (a) Steady flow velocity $|\boldsymbol {u}|$ at $Re_L=7000$ and (b) time-averaged flow velocity $|\bar {\boldsymbol {u}}|$ at $Re_L=15\,000$. Streamlines display the flow recirculation patterns.

3.3. Two-dimensional lid-driven cavity with mesh deformation

To construct a prototypical problem representing a flow with a deforming mesh, the bottom surface of the cavity is subjected to forced deformation, keeping the other flow conditions and simulation parameters the same. The deformation is governed by an analytical function expressed as

(3.10)\begin{equation} \left.\begin{array}{c} \chi_1 = x_1, \\ \chi_2(\tau)=a(1-x_2)x_1^2 (1-x_1)^2 \sin({\rm \pi} nx_1) \sin(2{\rm \pi} St_f t), \end{array}\right\} \end{equation}

where $a$ and $n$ are the deformation amplitude and mode number, respectively. The non-dimensional frequency of mesh deformation is $St_f$. The velocity of the mesh deformation, $\boldsymbol {\mathcal {U}}^\texttt{G}(\boldsymbol {\mathcal {\chi }},\tau )$, is then given as

(3.11)\begin{equation} \left.\begin{array}{c} \dfrac{\partial {\chi}_1}{\partial \tau} = 0, \\ \dfrac{\partial {\chi}_2}{\partial \tau} = a(1-x_2)x_1^2(1-x_1)^2\sin({\rm \pi} nx_1)\cos(2{\rm \pi} St_f t) 2{\rm \pi} St_f. \end{array}\right\} \end{equation}

The choice of parameters, $a=0.1$, $n=10$ and $St_f=1$, is based on obtaining a case that adequately tests the LMA development. The flow velocity in the Lagrangian (moving-mesh) frame of reference can be given by

(3.12)\begin{equation} \boldsymbol{\mathcal{U}}(\boldsymbol{\chi},\tau)= \boldsymbol{u}(\mathcal{M}(\boldsymbol{\chi},\tau))- \boldsymbol{\mathcal{U}}^\texttt{G}(\boldsymbol{\chi},\tau), \end{equation}

where $\mathcal {M}$ is the mapping of form (2.3a,b). The flow solver accounts for the Eulerian–Lagrangian effect by enforcing the geometric conservation law (Thomas & Lombard Reference Thomas and Lombard1979; Gordnier & Visbal Reference Gordnier and Visbal2002), which governs the spatial volume element under arbitrary mapping. The deformed computational domain ($\texttt{G}_3$) at an arbitrary instant is shown in the inset of figure 3(a). In addition, the figure displays the absolute velocity magnitude, $|\boldsymbol {\mathcal {U}}(\boldsymbol {\chi },\tau )|$, on the moving mesh at that instant.

Figure 3.

Lid-driven cavity flow at $M_\infty =0.5$ and $Re_L=15\,000$ with forced bottom surface/mesh deformation. (a) Instantaneous flow velocity magnitude with contours on a deformed domain. The inset shows a closer view of the deformed domain ($\boldsymbol {\chi }$). (b) Skin-friction coefficient on the bottom surface with/without deformation.

The lid-driven cavity with bottom surface deflection exhibits many of the main flow features of the baseline (no boundary motion) flow including the large central region and smaller recirculation regions near the no-slip walls. The domain deformation affects the motions of these flow features, of course, particularly the smaller recirculation regions near the bottom wall. In addition to the near-wall undulations on the velocity contours in figure 3(a), the entire flow is modified to some degree due to the surface deformation. Figure 3(b) displays the skin-friction coefficient (B3) on the deforming bottom wall, indicating a discernible increase at $\chi _1\approx 0.7$. The Lagrangian averaged flow field, estimated by accounting for the moving mesh, is largely similar to the Eulerian time-averaged flow field of figure 2(b), where the small differences can be attributed to the mild domain deformation.

3.4. Two-dimensional flow past a cylinder

The second flow considered is that past a circular cylinder, which is also a classical problem of engineering significance. The configuration highlights the fluid dynamics around bluff bodies, and encompasses many fundamental phenomena, including steady or unsteady separation, transition and wake vortex shedding (Williamson Reference Williamson1996), for all of which a large body of experimental and numerical data are available for validation. The problem is also a popular testbed for studies on flow stability, control, fluid–structure interaction, reduced-order modelling (Shinde et al. Reference Shinde, Longatte, Baj, Hoarau and Braza2016,  Reference Shinde, Longatte, Baj, Hoarau and Braza2019a) and compressibility effects (Canuto & Taira Reference Canuto and Taira2015).

In this configuration, the flow transitions from steady state to unsteady vortex shedding in distinct stages. As the Reynolds number based on cylinder diameter $D$ is increased, the initial unsteadiness is manifested for incompressible flow in $47 \lessapprox Re_D \lessapprox 178$ as periodic vortex shedding following a supercritical Hopf bifurcation at the critical Reynolds number (Sreenivasan, Strykowski & Olinger Reference Sreenivasan, Strykowski and Olinger1987; Noack & Eckelmann Reference Noack and Eckelmann1994). The initial two-dimensionality of the flow provides a suitable environment on which to demonstrate LMA. The onset of three-dimensionality at $Re_D\approx 178$, appears in the form of spanwise undulations (Behara & Mittal Reference Behara and Mittal2010).

Simulations are performed for a range of Reynolds numbers $20 \leq Re_D \leq 100$. Appendix C provides details on the geometry and grid convergence study. The flow fields are shown in figure 4 using the normalized streamwise velocity $u_1$ for $Re_D=40$ and $Re_D=100$. As noted earlier, the flow is steady at $Re_D=40$ (figure 4a). The corresponding pressure coefficient, defined as

(3.13)\begin{equation} C_p\equiv \frac{p-p_\infty}{\frac{1}{2}\rho_\infty u_\infty^2}, \end{equation}

is shown in figure 4(b) together with a favourable comparison with the DNS result of Canuto & Taira (Reference Canuto and Taira2015). For $Re_D=100$, on the other hand, a periodic vortex shedding is observed in the wake region of the cylinder (figure 4c). The time-averaged flow field for $Re_D=100$ is displayed in figure 4(d). The effect of increased Reynolds number is evident when compared to the $Re_D=40$ flow field of figure 4(a), particularly in the wake region, which becomes more compact for $Re_D=100$.

Figure 4.

Compressible flow past a cylinder, in terms of the streamwise velocity $u_1$, at Mach number $M_\infty =0.5$ and Reynolds numbers $Re_D=40$ and $Re_D=100$. (a) Steady flow at $Re_D=40$. (b) The pressure coefficient $C_p$ is compared with the DNS profile of Canuto & Taira (Reference Canuto and Taira2015) (dashed line.) (c) Unsteady instantaneous flow at $Re_D=100$. (d) Time-mean flow at $Re_D=100$.