**Background **

Welcome back. Today’s post will be the first part of a multi-part series about hydrodynamic stability, which is the study of how fluid flows transition from laminar states to turbulent states. Although “hydro” is in its name, this topic covers all types of fluids, such as water, air, honey, and liquid nitrogen. Predicting when a fluid will become unstable is important for several reasons. One of the most important is that the transition to turbulence can be detrimental to aircraft performance. Indeed, turbulent boundary layers on wings produce more skin friction drag than laminar boundary layers. This additional drag increases the amount of fuel needed to fly, which in turn increases pollution and ticket prices. Another motivation for studying hydrodynamic stability is that the transition to turbulence can actually be desirable. An example is the flow around a bluff body, such as a golf ball. When the flow is laminar, the boundary layer around the body separates quickly. This creates a large wake region with low pressure. The difference between the pressure in the wake and the pressure at the front of the body causes pressure drag to be high. If the bluff body is a golf ball traveling through the air, this drag can slow it down and impede performance. However, if the flow is tripped by a rough surface, the boundary layer becomes turbulent. This turbulent boundary layer brings more kinetic energy toward the ball’s surface, which encourages the flow to stay attached. When this happens, the wake region is reduced in size and pressure drag decreases.

Regardless of whether turbulence is desirable or undesirable, hydrodynamic stability is a valuable tool that can determine how turbulence develops. Ultimately, these insights can lead to active and passive control strategies that can encourage or hinder the growth of disturbances. Our exploration of hydrodynamic stability will begin by considering what instability even means. Next we will derive the general, 3-dimensional, linearized perturbation equations that govern the growth of very small disturbances near a steady base state. Our focus will then turn to 2-dimensional parallel shear flows and the celebrated Orr-Sommerfeld equation. This discussion will lead to deeper explorations of non-normality and pseudospectra. Finally, we will put everything together by deriving the fundamental equations of resolvent analysis and exploring how resolvent analysis fits into the broader topic of hydrodynamic stability. Throughout this discussion, our focus will be on the temporal stability of disturbances that are distributed in space. Although there is a rich theory behind spatial stability, this topic will be saved for a future post.

**What is Instability? **

In fluid dynamics, instability is often framed in terms of energy, and an organized base flow **U**(**x**,t) is considered (temporally) unstable to disturbances if the kinetic energy of a disturbance increases in time. If we wish to frame this more formally, we can begin by considering a disturbance velocity field **u**(**x**,t) that depends on the spatial position **x** and the time t. The flow is assumed to be restricted to a spatial region 𝛀. In this case, disturbance energy at the initial time t = 0 is

E(0) = ½ ∫_{ }_{𝛀} |**u**(**x**,0)|^{2} d**x**,

and the disturbance energy at a later time t is

E(t) = ½ ∫_{ }_{𝛀} |**u**(**x**,t)|^{2} d**x**.

We will then say that a base flow **U**(**x**,t) is stable to perturbations **u**(**x**,t) if

E(t)/E(0) → 0 as t → ∞.

Whenever this condition is *not* satisfied, the flow is unstable. If there is an energy threshold δ > 0 such that **U** is stable if E(0) < δ but unstable if E(0) > δ, then **U** is conditionally stable. If the energy threshold δ is infinite, then the flow is globally stable. If δ = 0, then the flow is linearly unstable.

Another concept related to the definition of stability is that of *monotonic* *stability*. If a flow is monotonically stable, then dE/dt < 0 for all t, and the disturbance energy is always decreasing. Non-monotonically stable flows are important because they have finite-time, transient energy growth, which can exceed the initial disturbance energy E(0) by several orders of magnitude. This short-term energy growth can cause turbulence to appear in shear-dominated flows, such as boundary layers, jets, and wakes. Although a linear eigenvalue analysis might predict disturbances to decay at very large times, the transient growth at short-time scales might be so large that the flow becomes turbulent anyway. More on this will be explored next week. Until then, please take care.

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