These animations illustrates a variety of hydrodynamic
instabilities. All numerical simulations are 2-D. Although many
hydrodynamic instabilities are initially two-dimensional, which is what makes
these simulations relevent to the real world, they eventually become
three dimensional and turbulent. This would occur well before the end
of these simulations.
Main animations page
Kelvin-Helmholtz and Holmboe Instability
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Kelvin Helmholtz instability (2.2MB
gif file) An easy way to generate Kelvin Helmholtz
instability in the laboratory is with a tilting tank. Filled
with a continuous two-layer stratified water the tank is
initial tilted. After the fluid has come to rest the tank is
rapidly brought to the horizontal creating an initial
stratification with a sloping pycnocline. The baroclinic
adjustment creates a sheared flow across the pycnocline. If
the initial pycnocline slope is large enough the shear is
strong enough to destabilize the flow. This process is
illustrated in this numerical simulation which also illustrates
the merger of KH billows. In this simulation the tank
is 6 m long and 30 cm deep (-3 < x < 3 and -0.3 < z < 0) . The animation shows
a section in the central portion of the tank
(-1 < x < 1 and -0.2 < z < -0.1).
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Kelvin Helmholtz instability (2.2MB
gif file) A zoom in of the previous simulation, showing
a smaller part of the tank (-0.4 < x < 0.4 and -0.2 < z <
-0.1). This provides a closer look at the vortex pairing
that occurs.
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Spatial Instability in an internal solitary wave (1.5MB
gif file) This shows the results of a 2D-simulations of
a shear instability in an internal solitary wave. The
stratification consists of two sharp pycnoclines embedded in
a broader pycnocline based on observations on the Oregon
Shelf. Wave is propagating to the left through the fluid and
the simulation is done in a reference frame moving with the
wave. A short pulse perturbation is added in front of the
wave. It is advected in the wave where it goes unstable in
the upper sharp pycnocline. The minimum Richardson number in
the pycnocline is 0.09 at the centre of the wave in the
upper sharp pycnocline.
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Eigenvalues for Narrow pycnocline: Case 1(0.5MB
gif file) This shows the eigenvalues predicted by
linear temporal instability for an inviscid, parallel,
stratified shear layer. The background flow is U(z)
= Umax*tanh(z) and the background density is rho(z) =
1-0.001*tanh(5z). The vertical domain is -10 <= z <= 10 with solid
boundaries. The shear layer is five times thicker than
the stratified layer and the Richardson number has its maximum
value at the centre of the shear layer. The red and green curves are
the zero contours of the real and imaginary parts of the
eigenfunction at the top boundary (it is integrated up from
the lower boundary). The eigenvalues are found where the red
and green curves intersect. The black curve is the circle
cr^2+ci^2=Umax^2 of Howard's semi-circle theorem. Unstable
modes (and the stable, decaying complex conjugates) must lie
inside this circle. The results shown here are for horizonal
wave number k=0.2. The eigenvalues are symmetric about the
imaginary and the real axis. Only values to the right of the
imaginary axis are shown.
When Umax=0 there is no shear flow. The
eigenvalues are real and they correspond to the different
internal gravity wave modes. The resolution (201x201 grid)
can only resolve a finite number of the countable infinity
of eigenmodes. As Umax increases they mode 1 wave (largest
propagation speed cr) slowly decreases while the others
disappear (perhaps a resolution problem). A closed red curve
appears inside the black circle (e.g., at Umax =
0.0925). These modes have cr = U(z) at some value of z and
hence would have a critical layer. At about Umax = 0.1725
the gravest mode crosses into the black circle and a closed
green curve appears. At something like Umax = 0.19 (poorly
resolved) the red curve intersects this green curve and
there are now a pair of eigenvalues off the real axis. This
is Holmboe instability. The
move away from the real axis and approach the imaginary axis
as Umax continues to increase. The imaginary axis is reached
at Umax just above 0.37 at which point they bifurcate and
split into two modes, moving up and down the imaginary axis
(recall the eigenvalues are symmetric about the imaginary
axis). Now the instability is a Kelvin-Helmholtz
instability. The mode moving down the imaginary axis reaches
the origin at Umax = 0.4475 and then disappears.
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Eigenvalues for stratified shear layer: Case 2(0.5MB
gif file) This shows the eigenvalues predicted by
linear temporal instability for an inviscid, parallel,
stratified shear layer. Same as above but now the shear
layer and pycnocline have the same thickness: the background
flow is U(z) = Umax*tanh(z) and the background density is rho(z) =
1-0.001*tanh(z). The vertical domain is -10 <= z <= 10 with solid
boundaries. The Richardson number now has a minimum at the
centre of the shear layer. In this case the eigenvalues stay
on the real axis until an eigenvalue reaches the origin at
Umax slightly above 0.2425 after which a bifurcation occurs
and a pair of eigenmodes move up and down the imaginary
axis. The flow is now unstable with stationary
eigenmodes. This is Kelvin-Helmoltz instability.
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Narrow pycnocline: Case 1(3.2MB
gif file) This shows the results of a 2D-simulations of
a stratified shear instability. The background flow is U(z)
= 0.38*tanh(z). The background density is rho(z) =
1-0.001*tanh(5z). The shear layer is five times thicker than
the stratified layer and the Richardson number has a maximum
value of 0.34 at the centre of the shear layer. In this
similation the flow is initialized with a density
perturbation with wave number 0.2, modulated by a
sech(x/100.0) envelope. Domain is -20 <= z <= 0 with solid
boundaries. Linear temporal stability theory predicts a pair
of unstable eigenmode with phase speeds c1 = 0.121i and c2 =
0.121i. Because there is a pair of unstable waves
propagating slowly in opposite directions relative to the centre of
the shear layer weak characteristics of a Holmboe Instability.
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Narrow pycnocline: Case 2(2.6MB
gif file) Same as preceeding case but with U(z) =
0.35*tanh(z). The Richardson number now has a maximum
value of 0.40 at the centre of the shear layer. Linear
temporal stability theory predicts a pair
of unstable eigenmode with phase speeds c1 = 0.051+0.077i and c2 =
-0.051+0.077i. Because there is a pair of unstable waves
propagating slowly in opposite directions relative to the centre of
the shear layer the instability has weak characteristics of
a Holmboe Instability.
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Narrow pycnocline: Case 3 (4.1MB
gif file) Same as preceeding case but with U(z) =
0.22*tanh(z). The
Richardson number has a maximum value of 1.01 at the centre of the
shear layer. Linear stability theory predicts a pair of
unstable eigenmodes with phase speed c1
= 0.130+0.010i and c2 = -0.130+0.010i. In this case the
instability more clearly illustrates the Holmboe
Instability. Because of the small growth of the perturbation
the animation zooms into the region -15 <= z <= -5.
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Narrow pycnocline: Case 4 (3.4MB
gif file) Same as preceeding case but with U(z) =
0.21*tanh(z). The Richardson
number has a maximum value of 1.11 at the centre of the
shear layer. Linear stability theory predicts a pair of
unstable eigenmodes with phase speed c1
= 0.141+0.004i and c2 = -0.141+0.004i. In this case the
instability more clearly illustrates the Holmboe
Instability. Because of the small growth of the perturbation
the animation zooms into the region -15 <= z <= -5.
Convective Instability in two layer fluid
- This shows the results of a 2D-simulations of
convective instability. The initial stratification has dense
fluid over light fluid with sinusoidal perturbations to the
pycnocline modulated by a sech squared envelope. Upper and
lower layer densities are 1010 and 990 kg/m^3. Kinematic viscosity
0.0002 m^2/s. Diffusivity 0.00001 m^2/s. Tank is 1m by
0.3m. No-slip boundary conditions used along top, bottom and
sides.
Large (10MB gif file)
- Same as preceeding case but initial
perturbation is a single sech squared depression. (
Long (10MB gif file) )
(
Short (6MB gif file) )
- Same as preceeding case but viscosity and
diffusivity both decreased by a factor of 10.
(
Long(12MB gif file) )
(
Short (6MB gif file) )
(
Small (2MB gif file) )
- Viscosity and diffusivity reduced even
further to 1.0e-6 m^2/s and 1.0e-7 m^2/6.
(
Long (15MB gif file) )
(
Short (6MB gif file) )
Parametric Subharmonic Instability
- Parametric subharmonic instability (PSI)
involves the transfer of energy to frequencies
of approximately half the frequency of the primary
wave. This numerical simulation was initialized with a
linear internal standing wave in a closed
basin. Perturbations are generated by the nonlinear terms.
Ultimately a perturbation of half the frequency of the
initial wave grows. From 2D-simulation.(
(9MB gif file) ) (
Small (5MB gif file) )
Multi-Layer Flows: Dispersion Curves
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Two Layer Gravity-Capillary Waves (0.6MB
gif file) This is an animation of the dispersion curves (frequency
vs wave number) for 2-D waves on the air-water interface as
the wind speed varies. Water is at rest. Both layers are
infinitely deep. For a critical value of the air speed
(just above 6.5 m/s) a gap appears in the dispersion curves
in which the flow is unstable. Long wave (small k) are
stabilized by gravity and short waves (large k) are
stabilized by surface tension.
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Two Layer Gravity-Capillary Waves (0.6MB
gif file) As above but now in a reference frame in
which the water is moving with velocity 0.1 m/s to the right.
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Three-Layer flow (1.06MB
gif file) This is an animation of the dispersion curves
for the three layer flow considered by Adam & Craik
(1979). See text by Craik as well. The upper, middle and
lower layers have densities of
1015, 1020 and 1026 kg/m^3 respectively. Surface tension of
0.074 N/m acts on the upper interface. The surface tension
on the lower interface is taken to be zero. The upper and
lower layers are infinitely deep while the middle layer is
16 cm thick. The lower and middle layers are at rest. The
dispersion curves (frequency vs horizontal wave number) vary
as U1, the current in the upper layer, is increased from
zero. When U1 = 0 an exchange of identities
between lower branches of the gravity waves on the
lower interface and the gravity-capillary waves on the upper
interface can be seen near k=15 1/m. A similar exhange of
identities occurs between the upper branches. As U1
increases the value of k at which the exchange occurs moves
off to the right and the two curves get closer and closer
together. Then never intersect. The dispersion curves
corresponding to the gravity-capillary waves on the upper
interface both move up. The lower branch eventually crosses
the sigma-axis and approaches the upper branch of the
gravity wave curve. These two curves coalesce and gaps
appear in which the flow is unstable. After the lower
gravity-capillary wave branch passes through the upper
gravity wave branch the two gravity-capillary wave branches
approach each other (with the upper one now moving down)
until they too coalesce forming and new unstable wave number band.
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Three-Layer flow (1.06MB
gif file) This is the same as the above but with
dispersion curves for gravity-capillary waves on the upper
interface (green curve) and for gravity waves on the lower
interface (blue curve) included for comparison. These
additional dispersion curves are those for two layer flows,
both layers infinitely deep.
