# Tollmien–Schlichting wave

Last updated

In fluid dynamics, a Tollmien–Schlichting wave (often abbreviated T-S wave) is a streamwise unstable wave which arises in a bounded shear flow (such as boundary layer and channel flow). It is one of the more common methods by which a laminar bounded shear flow transitions to turbulence. The waves are initiated when some disturbance (sound, for example) interacts with leading edge roughness in a process known as receptivity. These waves are slowly amplified as they move downstream until they may eventually grow large enough that nonlinearities take over and the flow transitions to turbulence.

## Contents

These waves, originally discovered by Ludwig Prandtl, were further studied by two of his former students, Walter Tollmien and Hermann Schlichting after whom the phenomenon is named.

Also, the T-S wave is defined as the most unstable eigen-mode of Orr–Sommerfeld equations. [1]

## Physical mechanism

In order for a boundary layer to be absolutely unstable (have an inviscid instability), it must satisfy Rayleigh's criterion; namely ${\displaystyle D^{2}U=0}$ where ${\displaystyle D}$ represents the y-derivative and ${\displaystyle U}$ is the free stream velocity profile. In other words, the velocity profile must have an inflection point to be unstable.

It is clear that in a typical boundary layer with a zero pressure gradient, the flow will be unconditionally stable; however, we know from experience this is not the case and the flow does transition. It is clear, then, that viscosity must be an important factor in the instability. It can be shown using energy methods that

${\displaystyle {\frac {DE}{Dt}}=-\int _{V}u'v'\left({\frac {dU}{dy}}\right)-{\frac {1}{R}}\int _{V}\left(\nabla {\vec {v}}'\right)^{2}}$

The rightmost term is a viscous dissipation term and is stabilizing. The left term, however, is the Reynolds stress term and is the primary production method for instability growth. In an inviscid flow, the ${\displaystyle u'}$ and ${\displaystyle v'}$ terms are orthogonal, so the term is zero, as one would expect. However, with the addition of viscosity, the two components are no longer orthogonal and the term becomes nonzero. In this regard, viscosity is destabilizing and is the reason for the formation of T-S waves.

## Transition phenomena

### Initial disturbance

In a laminar boundary layer, if the initial disturbance spectrum is nearly infinitesimal and random (with no discrete frequency peaks), the initial instability will occur as two-dimensional Tollmien–Schlichting waves, travelling in the mean flow direction if compressibility is not important. However, three-dimensionality soon appears as the Tollmien–Schlichting waves rather quickly begin to show variations. There are known to be many paths from Tollmien–Schlichting waves to turbulence, and many of them are explained by the non-linear theories of flow instability.

### Final transition

A shear layer develops viscous instability and forms Tollmien–Schlichting waves which grow, while still laminar, into finite amplitude (1 to 2 percent of the freestream velocity) three-dimensional fluctuations in velocity and pressure to develop three-dimensional unstable waves and hairpin eddies. From then on, the process is more a breakdown than a growth. The longitudinally stretched vortices begin a cascading breakdown into smaller units, until the relevant frequencies and wave numbers are approaching randomness. Then in this diffusively fluctuating state, intense local changes occur at random times and locations in the shear layer near the wall. At the locally intense fluctuations, turbulent 'spots' are formed that burst forth in the form of growing and spreading spots — the result of which is a fully turbulent state downstream.

#### The simple harmonic transverse sound of Tollmien–Schlichting (T-S) waves

Tollmien (1931) [2] and Schlichting (1929) [3] theorized that viscosity-induced grabbing and releasing of laminae created long-crested simple harmonic (SH) oscillations (vibrations) along a smooth flat boundary, at a flow rate approaching the onset of turbulence. These T-S waves would gradually increase in amplitude until they broke up into the vortices, noise and high resistance that characterize turbulent flow. Contemporary wind tunnels failed to show T-S waves.

In 1943, Schubauer and Skramstad (S and S) [4] created a wind tunnel that went to extremes to damp mechanical vibrations and sounds that might affect the airflow studies along a smooth flat plate. Using a vertical array of evenly spaced hot wire anemometers in the boundary layer (BL) airflow, they substantiated the existence of T-S oscillations by showing SH velocity fluctuations in the BL laminae. The T-S waves gradually increased in amplitude until a few random spikes of in-phase amplitude appeared, triggering focal vortices (turbulent spots), with noise. A further increase in flow rate resulted suddenly in many vortices, aerodynamic noise and a great increase in resistance to flow. An oscillation of a mass in a fluid creates a sound wave; SH oscillations of a mass of fluid, flowing in that same fluid along a boundary, must result in SH sound, reflected off the boundary, transversely into the fluid.

S and S found foci of in-phase spiking amplitude in the T-S waves; these must create bursts of high amplitude sound, with high energy oscillation of fluid molecules transversely through the BL laminae. This has the potential to freeze laminar slip (laminar interlocking) in these spots, transferring the resistance to the boundary: this breaking at the boundary could rip out pieces of T-S long-crested waves which would tumble head-over-heels downstream in the boundary layer as the vortices of turbulent spots. With further increase in flow rate, there is an explosion into turbulence, with many random vortices and the noise of aerodynamic sound.

Schubauer and Skramstad overlooked the significance of the co-generation of transverse SH sound by the T-S waves in transition and turbulence. However, John Tyndall (1867) in his transition-to-turbulence flow studies using flames, [5] deduced that SH waves were created during transition by viscosity acting around the walls of a tube and these could be amplified by blending with similar SH sound waves (from a whistle), triggering turbulence at lower flow rates. Schubauer and Skramstad introduced SH sound into the boundary layer by creating SH fluttering vibrations of a BL ferromagnetic ribbon in their 1941 experiments, similarly triggering turbulence at lower flow rates.

Tyndall’s contribution towards explaining the mystery of transition to turbulence 150 years ago is beginning to gain recognition. [6]

## Related Research Articles

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross-currents perpendicular to the direction of flow, nor eddies or swirls of fluids. In laminar flow, the motion of the particles of the fluid is very orderly with particles close to a solid surface moving in straight lines parallel to that surface. Laminar flow is a flow regime characterized by high momentum diffusion and low momentum convection.

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition. The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.

Parasitic drag, also known as profile drag, is a type of aerodynamic drag that acts on any object when the object is moving through a fluid. Parasitic drag is a combination of form drag and skin friction drag. It affects all objects regardless of whether they are capable of generating lift.

The Kelvin–Helmholtz instability is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. Kelvin-Helmholtz instabilities are visible in the atmospheres of planets and moons, such as in cloud formations on Earth or the Red Spot on Jupiter, and the atmospheres of the Sun and other stars.

In the field of fluid dynamics the point at which the boundary layer changes from laminar to turbulent is called the transition point. Where and how this transition occurs depends on the Reynolds number, the pressure gradient, pressure fluctuations due to sound, surface vibration, the initial turbulence level of the flow, boundary layer suction, surface heat flows, and surface roughness. The effects of a boundary layer turned turbulent are an increase in drag due to skin friction. As speed increases, the upper surface transition point tends to move forward. As the angle of attack increases, the upper surface transition point also tends to move forward.

In fluid dynamics, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the Reynolds number Re, the flow is steady and purely azimuthal. This basic state is known as circular Couette flow, after Maurice Marie Alfred Couette, who used this experimental device as a means to measure viscosity. Sir Geoffrey Ingram Taylor investigated the stability of Couette flow in a ground-breaking paper. Taylor's paper became a cornerstone in the development of hydrodynamic stability theory and demonstrated that the no-slip condition, which was in dispute by the scientific community at the time, was the correct boundary condition for viscous flows at a solid boundary.

Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real life scenarios, including the flow of blood through the cardiovascular system, the airflow over an aircraft wing, the re-entry of space vehicles, besides others. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid behind the obstacle flows into the void creating a swirl of fluid on each edge of the obstacle, followed by a short reverse flow of fluid behind the obstacle flowing upstream, toward the back of the obstacle. This phenomenon is naturally observed behind large emergent rocks in swift-flowing rivers.

In fluid dynamics, the law of the wall states that the average velocity of a turbulent flow at a certain point is proportional to the logarithm of the distance from that point to the "wall", or the boundary of the fluid region. This law of the wall was first published in 1930 by Hungarian-American mathematician, aerospace engineer, and physicist Theodore von Kármán. It is only technically applicable to parts of the flow that are close to the wall, though it is a good approximation for the entire velocity profile of natural streams.

This page describes some of the parameters used to characterize the thickness and shape of boundary layers formed by fluid flowing along a solid surface. The defining characteristic of boundary layer flow is that at the solid walls, the fluid's velocity is reduced to zero. The boundary layer refers to the thin transition layer between the wall and the bulk fluid flow. The boundary layer concept was originally developed by Ludwig Prandtl and is broadly classified into two types, bounded and unbounded. The differentiating property between bounded and unbounded boundary layers is whether the boundary layer is being substantially influenced by more than one wall. Each of the main types has a laminar, transitional, and turbulent sub-type. The two types of boundary layers use similar methods to describe the thickness and shape of the transition region with a couple of exceptions detailed in the Unbounded Boundary Layer Section. The characterizations detailed below consider steady flow but is easily extended to unsteady flow.

The term friction loss has a number of different meanings, depending on its context.

In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century. These foundations have given many useful tools to study hydrodynamic stability. These include Reynolds number, the Euler equations, and the Navier–Stokes equations. When studying flow stability it is useful to understand more simplistic systems, e.g. incompressible and inviscid fluids which can then be developed further onto more complex flows. Since the 1980s, more computational methods are being used to model and analyse the more complex flows.

In fluid dynamics, the process of a laminar flow becoming turbulent is known as laminar–turbulent transition. The main parameter characterizing transition is the Reynolds number.

In fluid mechanics, the Reynolds number is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation.

Turbulent flows are complex multi-scale and chaotic motions that need to be classified into more elementary components, referred to coherent turbulent structures. Such a structure must have temporal coherence, i.e. it must persist in its form for long enough periods that the methods of time-averaged statistics can be applied. Coherent structures are typically studied on very large scales, but can be broken down into more elementary structures with coherent properties of their own, such examples include hairpin vortices. Hairpins and coherent structures have been studied and noticed in data since the 1930s, and have been since cited in thousands of scientific papers and reviews.

Skin friction drag is a type of aerodynamic or hydrodynamic drag, which is resistant force exerted on an object moving in a fluid. Skin friction drag is caused by the viscosity of fluids and is developed from laminar drag to turbulent drag as a fluid moves on the surface of an object. Skin friction drag is generally expressed in terms of the Reynolds number, which is the ratio between inertial force and viscous force.

In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations. In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.

A bypass transition is a laminar–turbulent transition in a fluid flow over a surface. It occurs when a laminar boundary layer transitions to a turbulent one through some secondary instability mode, bypassing some of the pre-transitional events that typically occur in a natural laminar–turbulent transition.

## References

1. Schmid, Peter J., Henningson, Dan S., Stability and Transition in Shear Flows (https://www.springer.com/us/book/9780387989853), page 64.
2. Walter Tollmien (1931): Grenzschichttheorie, in: Handbuch der Experimentalphysik IV,1, Leipzig, S. 239–287.
3. Hermann Schlichting (1929) "Zur Enstehung der Turbulenz bei der Plattenströmung". Nachrichten der Gesellschaft der Wissenschaften – enshaften zu Göttingen, Mathematisch – Physikalische zu Göttingen, Mathematisch – Physikalische Klasse, 21-44.
4. G.B. Schubauer, H.K. Skramstad (1943) Laminar-boundary-layer-oscillations and transition on a flat plate. Advance Confidential Report. National Advisory Committee on Aeronautics, 1-70.
5. John Tyndall (1867) "On the action of sonorous vibrations on gaseous and liquid jets", Philosophical Magazine 33: 375-391.
6. Hamilton (2015) Simple Harmonics, pp. 2-4, Aylmer Express, Aylmer, Ontario