The Orr–Sommerfeld equation, in
fluid dynamics
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'' (the study of air and other gases in motion) an ...
, is an
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
equation describing the linear two-dimensional modes of disturbance to a
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
parallel flow. The solution to the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for
hydrodynamic stability
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 insta ...
are.
The equation is named after
William McFadden Orr
250px
William McFadden Orr, FRS (2 May 1866 – 14 August 1934) was a British and Irish mathematician.
He was born in Comber, County Down and educated at Methodist College Belfast and Queen's College, Belfast under John Purser, before en ...
and
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
, who derived it at the beginning of the 20th century.
Formulation
The equation is derived by solving a
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
ized version of the Navier–Stokes equation for the perturbation velocity field
:
,
where
is the unperturbed or basic flow. The perturbation velocity has the
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
-like solution
(real part understood). Using this knowledge, and the
streamfunction representation for the flow, the following dimensional form of the Orr–Sommerfeld equation is obtained:
:
,
where
is the dynamic
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
of the fluid,
is its
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
, and
is the potential or stream function. In the case of zero viscosity (
), the equation reduces to
Rayleigh's equation. The equation can be written in non-dimensional form by measuring velocities according to a scale set by some characteristic velocity
, and by measuring lengths according to channel depth
. Then the equation takes the form
:
,
where
:
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 domi ...
of the base flow. The relevant boundary conditions are the
no-slip boundary conditions at the channel top and bottom
and
,
:
at
and
in the case where
is the potential function.
Or:
:
at
and
in the case where
is the stream function.
The eigenvalue parameter of the problem is
and the eigenvector is
. If the imaginary part of the wave speed
is positive, then the base flow is unstable, and the small perturbation introduced to the system is amplified in time.
Solutions
For all but the simplest of velocity profiles
, numerical or asymptotic methods are required to calculate solutions. Some typical flow profiles are discussed below. In general, the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of the equation is discrete and infinite for a bounded flow, while for unbounded flows (such as
boundary-layer
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 condi ...
flow), the spectrum contains both continuous and discrete parts.
For plane
Poiseuille flow
The poiseuille (symbol Pl) has been proposed as a derived SI unit of dynamic viscosity, named after the French physicist Jean Léonard Marie Poiseuille (1797–1869).
In practice the unit has never been widely accepted and most international s ...
, it has been shown that the flow is unstable (i.e. one or more eigenvalues
has a positive imaginary part) for some
when
and the neutrally stable mode at
having
,
. To see the stability properties of the system, it is customary to plot a dispersion curve, that is, a plot of the growth rate
as a function of the wavenumber
.
The first figure shows the spectrum of the Orr–Sommerfeld equation at the critical values listed above. This is a plot of the eigenvalues (in the form
) in the complex plane. The rightmost eigenvalue is the most unstable one. At the critical values of Reynolds number and wavenumber, the rightmost eigenvalue is exactly zero. For higher (lower) values of Reynolds number, the rightmost eigenvalue shifts into the positive (negative) half of the complex plane. Then, a fuller picture of the stability properties is given by a plot exhibiting the functional dependence of this eigenvalue; this is shown in the second figure.
On the other hand, the spectrum of eigenvalues for
Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow. ...
indicates stability, at all Reynolds numbers. However, in experiments, Couette flow is found to be unstable to small, but ''finite,'' perturbations for which the linear theory, and the Orr–Sommerfeld equation do not apply. It has been argued that the non-normality of the eigenvalue problem associated with Couette (and indeed, Poiseuille) flow might explain that observed instability. That is, the eigenfunctions of the Orr–Sommerfeld operator are complete but non-orthogonal. Then, the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the disturbance contains contributions from all eigenfunctions of the Orr–Sommerfeld equation. Even if the energy associated with each eigenvalue considered separately is decaying exponentially in time (as predicted by the Orr–Sommerfeld analysis for the Couette flow), the cross terms arising from the non-orthogonality of the eigenvalues can increase transiently. Thus, the total energy increases transiently (before tending asymptotically to zero). The argument is that if the magnitude of this transient growth is sufficiently large, it destabilizes the laminar flow, however this argument has not been universally accepted.
A nonlinear theory explaining transition, has also been proposed. Although that theory does include linear transient growth, the focus is on 3D nonlinear processes that are strongly suspected to underlie transition to turbulence in shear flows. The theory has led to the construction of so-called complete 3D steady states, traveling waves and time-periodic solutions of the Navier-Stokes equations that capture many of the key features of transition and coherent structures observed in the near wall region of turbulent shear flows. Even though "solution" usually implies the existence of an analytical result, it is common practice in fluid mechanics to refer to numerical results as "solutions" - regardless of whether the approximated solutions satisfy the Navier-Stokes equations in a mathematically satisfactory way or not. It is postulated that transition to turbulence involves the dynamic state of the fluid evolving from one solution to the next. The theory is thus predicated upon the actual existence of such solutions (many of which have yet to be observed in a physical experimental setup). This relaxation on the requirement of exact solutions allows a great deal of flexibility, since exact solutions are extremely difficult to obtain (contrary to numerical solutions), at the expense of rigor and (possibly) correctness. Thus, even though not as rigorous as previous approaches to transition, it has gained immense popularity.
An extension of the Orr–Sommerfeld equation to the flow in porous media has been recently suggested.
Mathematical methods for free-surface flows
For Couette flow, it is possible to make mathematical progress in the solution of the Orr–Sommerfeld equation. In this section, a demonstration of this method is given for the case of free-surface flow, that is, when the upper lid of the channel is replaced by a free surface. Note first of all that it is necessary to modify upper boundary conditions to take account of the free surface. In non-dimensional form, these conditions now read
at
,
,
at
.
The first free-surface condition is the statement of continuity of tangential stress, while the second condition relates the normal stress to the surface tension. Here
:
are the
Froude and
Weber number
The Weber number (We) is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces. It is named ...
s respectively.
For Couette flow
, the four
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
solutions to the non-dimensional Orr–Sommerfeld equation are,
:
,
:
:
where
is the
Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solutio ...
of the first kind. Substitution of the
superposition solution
into the four boundary conditions gives four equations in the four unknown constants
. For the equations to have a non-trivial solution, the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
condition
must be satisfied. This is a single equation in the unknown ''c'', which can be solved numerically or by
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
methods. It can be shown that for a range of wavenumbers
and for sufficiently large Reynolds numbers, the growth rate
is positive.
See also
*
Gravitational comet-asteroid forcing events
*
Gravity wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
*
Lee waves
In meteorology, lee waves are atmospheric stationary waves. The most common form is mountain waves, which are atmospheric internal gravity waves. These were discovered in 1933 by two German glider pilots, Hans Deutschmann and Wolf Hirth, above ...
*
Rogue wave
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
References
Further reading
*
*
*
{{DEFAULTSORT:Orr-Sommerfeld equation
Fluid dynamics
Equations of fluid dynamics
de:Lineare_Stabilitätstheorie#Orr-Sommerfeld-Gleichung
Fluid dynamic instabilities