Hydrodynamic stability is an important part of fluid-mechanics that has a relevant role in understanding how an unstable flow can evolve into a turbulent state of motion with chaotic three-dimensional vorticity fields and a broad spectrum of small temporal and spatial scales. The essential problems of hydrodynamic stability were recognised and formulated in 19th century, notably by Helmholtz, Kelvin, Rayleigh and Reynolds.

Conventional linear stability assumes a normal representation of the perturbation fields that can be represented as independent wave packets, meaning that the system is self-adjoint. The main aim of the global stability analysis is to evaluate the amplitude of the eigenmodes as time grows and tends to infinity. However, in most industrial applications, it is also interesting to study the behaviour at intermediate states that might affects significantly the functionality and performance of a device. The study of the transient evolution of the perturbations is seen to be strictly related to the non-normality of the linearised Navier-Stokes equations, therefore the normality assumptiong of the system is no longer assumed. The eigenmodes of a non-normal system do not evolve independently and their interaction is responsible for a non-negligible transient growth of the energy. Conventional stability analysis generally does not capture this behaviour, therefore other techniques should be used.

Conventional linear stability assumes a normal representation of the perturbation fields that can be represented as independent wave packets, meaning that the system is self-adjoint. The main aim of the global stability analysis is to evaluate the amplitude of the eigenmodes as time grows and tends to infinity. However, in most industrial applications, it is also interesting to study the behaviour at intermediate states that might affects significantly the functionality and performance of a device. The study of the transient evolution of the perturbations is seen to be strictly related to the non-normality of the linearised Navier-Stokes equations, therefore the normality assumption of the system is no longer assumed. The eigenmodes of a non-normal system do not evolve independently and their interaction is responsible for a non-negligible transient growth of the energy. Conventional stability analysis generally does not capture this behaviour, therefore other techniques should be used.

A popular approach to study the hydrodynamic stability of flows consists in performing a direct numerical simulation of the linearised Navier-Stokes equations using iterative methods for computing the solution of the associated eigenproblem. However, since linearly stable flows could show a transient increment of energy, it is necessary to extend this analysis considering the combined effect of the direct and adjoint evolution operators. This phenomenon has noteworthy importance in several engineering applications and it is known as transient growth.

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\subsubsection{Direct stability analysis}

The equations that describe the evolution of an infinitesimal disturbance in the flow can be derived decomposing the solution into a basic state $(\mathbf{U}, p)$ and a perturbed state $\mathbf{U}+\varepsilon\mathbf{u'}$ with $\varepsilon\ll1$ that both satisfy the Navier-Stokes equations. Substituting into the Navier Stokes equations and considering that the quadratic terms $\mathbf{u'}\cdot\nabla\mathbf{u'}$can be neglected, we obtain the linearised Navier-Stokes equations:

The equations that describe the evolution of an infinitesimal disturbance in the

flow can be derived decomposing the solution into a basic state $(\mathbf{U},

p)$ and a perturbed state $\mathbf{U}+\varepsilon\mathbf{u'}$ with

$\varepsilon\ll1$ that both satisfy the Navier-Stokes equations. Substituting

into the Navier-Stokes equations and considering that the quadratic terms $\mathbf{u'}\cdot\nabla\mathbf{u'}$can be neglected, we obtain the linearised Navier-Stokes equations:

\begin{subequations}

\begin{equation}

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@@ -518,11 +522,14 @@ Then we obtain the associated eigenproblem:

The dominant eigenvalue determines the behaviour of the flow. If it the real part is positive there exists exponentially growing solutions, conversely if every single eigenvalues has negative real part then the flow is linearly stable. If the real part of the eigenvalue is zero, it is present a bifurcation point.

The dominant eigenvalue determines the behaviour of the flow. If the real part

is positive then there exist exponentially growing solutions. Conversely, if all

the eigenvalues have negative real part then the flow is linearly stable. If the real part of the eigenvalue is zero, it is a bifurcation point.

\subsubsection{Adjoint Stability Analysis}

The adjoint of a linear operator is one of the most important concept in functional analysis and has an it has important role to tackle the transition to turbulence. Let us write the linearised Navier-Stokes equation in a compact form:

The adjoint of a linear operator is one of the most important concepts in

functional analysis and it plays an important role in understanding transition to turbulence. Let us write the linearised Navier-Stokes equation in a compact form: