**1. Introduction**

The transition from laminar to turbulent flow is extremely difficult and has eluded engineers, physicists, and mathematicians for more than a century, despite massive efforts. Given that transition may be observed in many engineering flows and has a significant impact on the aerodynamics and heat transfer properties of those flow systems, it has attracted a great deal of attention and research. Transition is so complicated because it can take many different paths depending on flow configuration and geometry, and the presence of many different flow disturbances, such as wall roughness or obstructions, free-stream turbulence, acoustic noise, pressure gradient, surface heating or cooling, suction or blowing of fluid from the wall, and so on, greatly influences the transition process.

It is commonly known that Osborne Reynolds was the first person to conduct systematic experimental studies of pipe flow transition in the late 19th century. William McFadden Orr and Arnold Sommerfeld pioneered the study of transition. They separately created a mechanism for explaining the start of turbulence, which was dubbed the Orr-Sommerfeld approach in their honor (more commonly known as linear stability theory). Active theoretical research of transition began in the early 20th century, following the Orr-Sommerfeld paradigm.

Numerical studies of transition, which included calculating a simplified, linearized version of the Navier–Stokes equations numerically, began long before computers were invented. Nonetheless, effective numerical studies of transition employing large-eddy simulation (LES) and direct numerical simulation (DNS) did not begin to emerge until the late 1980s, with significant development in the last two decades.

Transition has traditionally been divided into three broad types for wallbounded flows.
