Recommended reading: Lautrup (Chapter 24) & Acheson (Chapter 9)
Recommended video: Flow instabilities https://youtu.be/yutbmcO5g2o by National Com- mittee for Fluid Mechanics Films
In 1883 Osborne Reynolds, an eminent figure in the early days of fluid dynamics, published a paper An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels [1] in which he described his milestone experiment on the stability of flow in a cylindrical tube.
His experimental setup, sketched in Fig. 1(left) consisted of tube through which water was pumped at a constant speed with a streak of coloured dye in the middle. Depending on the flow speed, Reynolds described three regimes presented in Fig.1(right). In his own words:
(1) When the velocities were sufficiently low, the streak of colour extended in a beautiful straight line through the tube (a).
(2) If the water in the tank had not quite settled to rest, at sufficiently low velocities, the streak would shift about the tube, but there was no appearance of sinuosity.
(3) As the velocity was increased by small stages, at some point in the tube, always at a considerable distance (...) from intake, the colour band would all at once mix up with the surrounding water, and fill the rest of the tube with a mass of coloured water(b). Any increase in the velocity caused the point of breakdown to approach the intake, but with no velocities tried did it reach this. On viewing the tube by the light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls, showing eddies(c).
Reynolds quantified the results using the now well-known Reynolds number Re = U L
ν (1)
as a distinguishing factor. In his experiment, U was the typical velocity inside the tube, L was the diameter, and ν was the kinematic viscosity of water. Reynolds described the transition but stated clearly that there was not a critical value of the Reynolds number below which the flow was stable or above which an instability would set in. He remarked instead that
(...) the critical velocity was very sensitive to the disturbance in the water before entering the tubes (...) This at once suggested the idea that the condition might be
Figure 1. Sketches from Reynolds’ original paper from 1883. Left: the experimental setup. Right:
drawings of the flow of dye in the experiment.
one of instability for disturbances of a certain magnitude and stability for smaller disturbances.
By taking great care of the alignment of the setup, Reynolds was able to push the laminar- turbulent flow transition to Re as high as 13000, with further experiments shifting this limit even to 90000. At the same time, when no great care is taken, the transition may occur already at Re 2000. Wonderful photographs of the experiments made on the same experimental setup can be found in Van Dyke’s album.
We shall keep in mind the conclusion that hydrodynamic instability might set in only when a certain magnitude of a disturbance is achieved, and the system would be stable to perturbations below this limit. However, it is natural in stability analysis to first consider the effect of infinites- imal disturbances, which allows us to neglect terms that are products of small quantities and we can equip ourselves with linear stability theory, a powerful tool for the analysis. Firstly, let us examine two instabilities occurring on a fluid-fluid interface.
INTERNAL WAVES AND STABILITY OF INTERFACES
Waves or disturbances at fluid interfaces may easily be found in nature. Examples involve layers of immiscible fluids (e.g. water and oil) or oceanic water layers (with differences in tempreature, salinity etc.). It is therefore vital to know whether an interface will be stable or whether disturbances would grow, leading to the disruption of subsequent layers and forced
water). Even if the density difference is small, we would expect surface disturbances to grow in such a system. These can be examined by looking at the evolution of waves propagating at the interface.
We will be using the standard theory of waves for irrotational flows, where the velocity field is encoded in the velocity potential Ψ such that
u = ∇Ψ. (2)
The standard treatment is described in detail inLautrup, Chapter 24or inHydrodynamics and Elasticity lectures. If you feel you need to review the basic wave formalism, please have a look there.
Boundary conditions We consider two fluid layers of infinite depth, separated by an interface located at z = h(x, y, t). Let the lower fluid have the desnity ρ1 and the upper fluid ρ2. The velocity potentials are Ψ1 and Ψ2, respectively. The kinematic boundary condition at the interface states that the interfacial velocity in the vertical direction must equal that of both fluids, so
u1z = u2z = ∂h
∂t (3)
The dynamic boundary condition is then
p1+ ∆p = p2 for z = h, (4)
where ∆p denotes a contribution from surface tension
∆p = −α ∂2
∂x2 + ∂2
∂y2
h, (5)
and the pressure can be written as
pi = p0− ρigz + ρi
∂Ψi
∂t , i ∈ {1, 2}, (6)
for each fluid.
Dispersion relation Suppose the interface disturbance takes the form of a plane wave h = a cos(kx − ωt). Since both fluid layers are deep, we require the velocity field to vanish at z → ±∞, and thus the velocity potentials are of the form
Ψ1= A1ekzsin(kx − ωt), Ψ2 = A2e−kzsin(kx − ωt). (7) For k|h| 1, the boundary conditions yield
kA1 = −kA2 = aω, and ρ1(ga − ωA1) + αk2a = ρ2(ga − ωA2), (8)
from which we find
A1 = −A2 = aω k = a
ω
g(ρ1− ρ2) + αk2 ρ1+ ρ2
. (9)
Solving the last equality for ω, we find the dispersion law for deep-water internal waves to be
ω = s
gk(ρ1− ρ2) + αk3 ρ1+ ρ2
. (10)
If the upper density is much smaller than the lower (think of air and water) so that ρ2 ρ1, these waves become ordinary capillary gravity waves with ω =p
gk + αk3, which additionally reduce to deep-water gravity waves (with ω = √
gk) when surface tension can be neglected (αk2 g). When the densities are nearly equal, ρ2 . ρ1, internal waves have a much lower frequencies (and velocities) than those at a free interface. The length scale Rc when surface tension contribution becomes comparable to the gravitational contribution, called the capillary length, is thus defined as
Rc=
r α
|ρ1− ρ2|g. (11)
We note that Rc diverges when the densities become equal. In this limit gravity plays no role and surface waves are of purely capillary nature, with ω =pαk3/2ρ1.
Example Consider a brackish surface layer above a saline layer with 4% higher density. The capillary wavelength for internal waves becomes λc= 2πRc= 8.6 cm. At this wavelength, the wave has a period of τ = 1.2 s and a phase velocity c = 7.2 cm s−1.
Check out the Videos:
(a) by Megan Davies Wykes and Stuart Dalziel at University of Cambridge [2] https://youtu.
be/NI85oC-3mJ0
(b) RT instability in a foam filmhttps://youtu.be/UVC0y3DIzNY (c) Magnetically induced RT instability
https://www.jove.com/video/55088/magnetically-induced-rotating-rayleigh-taylor-instability
Examples of the Rayleigh-Taylor instability are presented in Fig. 2.
Rayleigh-Taylor instability
When the heavier fluid lies below the lighter one, the dispersion relation in Eq. (10) is purely real. However, if such a containes is rapidly turned upside down and the lighter fluid is located
Figure 2. Examples of the Rayleigh-Taylor instability in various contexts across length and time scales. (Top left) A two-layer density-stratified liquid can be spun-up into solid body rotation and subsequently induced into Rayleigh-Taylor instability by applying a gradient magnetic field. Source:
Ref. [3]. (Top right) Hydrodynamics simulation of a single "finger" of the Rayleigh–Taylor instability.
Note the formation of Kelvin–Helmholtz instabilities, in the second and later snapshots shown, as well as the formation of a "mushroom cap" at a later stage in the third and fourth frame in the sequence.
Source: Shengtai Li, Hui Li, Parallel AMR Code for Compressible MHD or HD Equations, Los Alamos.
(Bottom left) RT instability in an elastic-plastic material. Hellman’s Real Mayonnaise was poured into a Plexiglass container. Different wave-like perturbations were formed on the mayonnaise and the sample was then accelerated on a rotating wheel experiment. The growth of the material was tracked using a high-speed camera (500 fps). An image processing algorithm, was then applied to compute various parameters associated with the instability. Source: Ref. [4]. (Bottom right) A mosaic image, one of the largest ever taken by NASA’s Hubble Space Telescope of the Crab Nebula, a six-light-year-wide expanding remnant of a star’s supernova explosion. The orange filaments are the tattered remains of the star and consist mostly of hydrogen. The rapidly spinning neutron star embedded in the center of the nebula is the dynamo powering the nebula’s eerie interior bluish glow. The blue light comes from electrons whirling at nearly the speed of light around magnetic field lines from the neutron star. The RT instability structure is evident, in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from the supernova explosion 1000 years ago [5]. Source: Wikipedia.
below the heavier one, i.e. ρ2 > ρ1, we expect a possible instability which we explore below.
The dispersion relation can now be written as
ω = r
gkρ1− ρ2 ρ1+ ρ2
k(k2R2c− 1). (12)
Note that in this case the argument of the square root will be negative for kRc < 1, so for λ > λc= 2πRc. Then ω becomes imaginary and the sinusoidal form of the wave is replaced by an exponential e|ω|t growing fast as time proceeds. This exponential growth is a signature of an instability, called the Rayleigh-Taylor instability. In a large fluid container, such as the ocean, there is always room for long-wavelength perturbation and thus the situation described above will always inevitably be unstable and can only be maintained for a short time. However, if a finite-sized container is considered, this puts a limitation on the maximal wavelength. A box of side length L puts a limitation on the wavelength by imposing a no-slip boundary condition of vx= 0 for x = 0 and x = L. Thus for vx∼ sin kx cos ωt the only permissible wave numbers are k = nπ/L with n being an integer. So the largest wavelength is found for n = 1 is λ = 2π/k = 2L, so the unstable mode cannot occur as long as
L < λc
2 = πRc= π
r α
|ρ1− ρ2|g. (13)
If you invert a container with horizontal size smaller than half the capillary wavelength, the heavier liquid will remain stably on top of the lighter!
Example or a home experiment Air against water has as we have seen before a capillary wavelength of λc= 1.7 cm, so we require that L < 0.85 cm. Try it yourself with a glass tube of, for example, 5 mm diameter. It works!
Kelvin-Helmholtz instability
Another possibly unstable situation may arise in the case when layers of fluid are sliding past each other. For inviscid fluids we know this situation to be possible, particularly if we disregrd the possible viscous boundary layers forming to soften the velocity discontinuity across the interface. However, wavy disturbances of the surface may ’get in the way’ of the sliding layers and get dragged along the interface, thereby disturbing the surface even further when the slip velocity is high enough.
Suppose now the upper layer is moving at a velocity U with respect to the lower layer. Taking into account the slope of the interface, ∂h/∂x, the horizontal flow in the upper layer will add U ∂h/∂x to the vertical velocity. Thus the kinematic boundary condition needs to be replaced
Figure 3. Plot of U/Uc as a function of λ/λc. For air-water interface we have λc = 1.7 cm. For a given velocity, the range of unstable wavelengths may be read off from the graph.
by
u1z = ∂h
∂t, u2z = ∂h
∂t + U∂h
∂x for z = h. (14)
Similarly, for the pressure we have p2= p0− ρ
gh + ∂Ψ2
∂t + U∂Ψ2
∂x
. (15)
Putting this all together, we find (check!) from the boundary conditions that
kA1 = aω, (16)
−kA2 = a(ω − kU ), (17)
ρ1(ga − ωA1) + αk2a = ρ2[ga − (ω − kU )A2], (18) which combine to a quadratic equation for the frequency
(ρ1+ ρ2)ω2− 2ρ2kU ω + ρ2k2U2= k[(ρ1− ρ2)g + αk2]. (19) For a given wave number k, we find the condition for the real roots to be
U2 < 1 ρ1
+ 1 ρ2
(ρ1− ρ2)g + αk2
k . (20)
If ρ1 > ρ2, the right-hand side has an absolute minimum for kRc= 1. For this particular k, the stability condition becomes
U < Uc= s
2gRc
ρ1 ρ2
−ρ2 ρ1
, (21)
which for air over water becomes Uc= 7.4 m s−1. The relation above may be recast as U
Uc
= s
1 2
λ λc
+λc λ
, (22)
and is plotted as a function of the scaled wavelength in Fig. 3. If U > Uc, there is a range of unstable weavelengths located symmetrically about λ = λc for which small disturbances will grow exponentially in time. This is the Kelvin-Helmholtz instability, which explains – at least qualitatively – how steady wind is able to generate waves from small disturbances. However, linear theory is unable to explain or predict how tiny waves amplified by the K-H instability grow into larger waves. For this, we need the full nonlinear theory. Nevertheless, some statistical aspects of sea waves can be explained using the linear approximation (for details, seeLautrup, Chaper 24.7).
Check out the Videos:
Amazing lab demonstration of the KH instability from DAMTP in Cambridge https://youtu.
be/UbAfvcaYr00
Vortex formation by KH instability in a free jethttps://youtu.be/ELaZ2x42dkU
Examples of the Kelvin-Helmholtz instability are presented in Fig. 4.
[1] O. Reynolds, Phil. Trans. Roy. Soc. London, 174, pp. 935-982 (1883).http://www.homepages.ucl.
ac.uk/~uceseug/Fluids3/Extra_Reading/Reynolds_1883.pdf.
[2] M. Davies-Wykes,http://www2.eng.cam.ac.uk/~msd38/gallery.html.
[3] M. M. Scase, K. A. Baldwin, and R. J. A. Hill,JoVE , e55088 (2017).
[4] R. Polavarapu, P. Roach, and A. Banerjee,Phys. Rev. E 99, 053104 (2019).
[5] J. J. Hester, Annual Review of Astronomy and Astrophysics 46, 127 (2008), https://doi.org/10.1146/annurev.astro.45.051806.110608.
Figure 4. Examples of the Kelvin-Helmholtz instability in various contexts across length and time scales. (Top left) KH instability on the surface of Saturn. Caption from NASA’s press release: This turbulent boundary between two latitudinal bands in Saturn’s atmosphere curls repeatedly along its edge in this Cassini image. This pattern is an example of a Kelvin-Helmholtz instability, which occurs when two fluids of different density flow past each other at different speeds. This type of phenomenon should be fairly common on the gas giant planets given their alternating jets and the different temperatures in their belts and zones. The image was taken with the Cassini spacecraft narrow angle camera on October 9, 2004, at a distance of 5.9 million km from Saturn through a filter sensitive to wavelengths of infrared light centered at 889 nm. The image scale is 69 km per pixel. Source: Wikipedia. (Top right) KH instability seen on a layer of clouds, sheared by wind in the layer of air above it. Source: Kim Elson, Pinterest. (Bottom left) Vortex formation occurs along the edge of a free axisymmetric jet. This is caused by Kelvin-Helmholtz instability from the shear force between the high velocity air jet and the surrounding stagnant air. Small vortices are created close to the nozzle and grow in size as they are carried along the flow until they are big enough to fully interrupt the jet. The vortices then collapse creating a fully turbulent flow. This visualization has been done by injecting smoke into the fluid (air).
The air is forced through a 100 mm diameter nozzle. A laser sheet makes it possible to visualize the flow in (almost) 2D. This is done for a Reynolds number of approx. 104. Source: Technical University of Denmark. (Bottom right) KH instability in the ocean. Temperature measured 530m deep, 15m off the bottom of the Great Meteor Seamount, North Atlantic Ocean, 2006. Horizontal scale (time): 300s;
Vertical scale: 13m; Colour scale: 12.75-13.25 degree Celsius. Source: Wikipedia.
