Aspects of Hydrodynamics and
Turbulence with Cryogens
“Perhaps the fundamental equation that describes the swirling
nebulae and the condensing, revolving, and exploding stars is
just a simple equation for the hydrodynamic behavior of nearly
pure hydrogen gas”.-- Richard Feynman, Nobel Laureate
J. Niemela
ICTP, Trieste
Turbulence is widespread, indeed almost the rule, in the flow of
fluids. It is a complex phenomenon, for which the development of
a satisfactory theoretical framework has been one of the greatest
unsolved challenges of classical physics.
The first scientific investigations of fluid turbulence are generally
attributed to Leonardo Da Vinci.
Turbulence is particularly useful because the equations of motion are known exactly and
can be simulated with precision. And so, even distant areas--perhaps even market
fluctuations [B.B. Mandelbrot, Scientific American 280, 50 (1999)]---may benefit from a
better understanding of it.
The complexity of the underlying equations, however, has precluded much analytical
progress, and the demands of computing power are such that routine simulations of
large turbulent flows has not yet been possible. Thus, the progress in the field has
depended more on experimental input. This experimental input in turn points in part to a
search for optimal test fluids, and the development and utilization of novel
instrumentation. These aspects will be discussed in this lecture.
Turbulence exists in a wide range of contexts such as the motion of submarines,
ships and aircraft, pollutant dispersion in the earth's atmosphere and oceans, heat
and mass transport in engineering applications as well as geophysics and
astrophysics [See, e.g., D.J. Tritton, Physical Fluid Dynamics. Clarendon Press,
Oxford (1988)].
The problem is also a paradigm for strongly nonlinear systems, distinguished by
strong fluctuations and strong coupling among a large number of degrees of
freedom. [G. Falkovich, K.R. Sreenivasan, Phy. Today 59, 43 (2006)].
Let’s take a quick glance at Newton’s second law for
hydrodynamics, including turbulence (Navier-Stokes)
∂v
1
+ (v ⋅ ∇ )v = − ∇p + ν∇ 2 v
∂t
ρ
1
∂v′
+ (v′ ⋅ ∇′)v′ = −∇′p′ +
∇′ 2 v′
∂t
Re
Re = UL/ν
v' =
v
U
r' =
r
L
t' =
tU
L
p' =
p
ρU 2
While it is not possible to predict turbulent motion in all details as a function of both time and
position, turbulence can be well-characterized statistically, with reproducible average values
of certain quantities.
Flow past a circular cylinder. (a) Re = 26. (b) Re = 2000.
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1
The Reynolds number is a manifestation of dynamical similarity. That is, if we
consider two simple flows that are geometrically similar, then they are dynamically
identical if the corresponding Re is the same for both, regardless of the specific
velocities, lengths and fluid viscosities involved.
Q. Two flows are dynamically similar if---
Matching such parameters between laboratory testing of a model and the actual
full-scale object (the prototype) is the principle upon which aerodynamic modeltesting is based.
a) the fluids have the same kinematic
viscosity
b) the flows have the same Re
c) the flows have large Re>>1
Above left: wake behind a flat plate inclined 45 degrees to the direction of the flow (left to
right). Above right: A foundered ship in the sea inclined 45 degrees to the direction of the
current.
Where does the injected energy go?
Making Turbulence in the laboratory
Richardson Energy Cascade
Traditionally, large arrays of wake-producing “cylinders”-- such as we saw above-are placed in the flow in order to generate something approaching homogeneous
and isotropic turbulence.
grid
flow
Eddies produced by flow
(e.g., through a grid at
length scale M= mesh size)
Cascade of energy in inertial
range: local transfer of energy
(due to nonlinear inertial term in
NS eqn.).
Initial turbulent forcing
at scale of the mesh
size M.
Viscous dissipation at
small scales for which
local Re~1.
For “inertial range” between large energy injection scale L and the smallest dissipation scale
η the energy spectrum (in k-space) is E(k) = C ε2/3 k-5/3 where ε is the rate of energy
transfer per unit mass. (E(k) is the energy contained in the wavenumber shell between k and
k+dk.
Here, flow at speed U through a grid of crossed tines with mesh size M, generates a
turbulent wake, injecting kinetic energy initially on that scale.
Smallest scale given by η ∼ LRe-3/4.
If there are any universal statistical properties of turbulence, it is reasonable to look for
them as Re→∞, since the separation between the energy-injection scales and the
dissipative scales increases with Re.
100 years ago…
Q. As the viscosity gets smaller
.
a)The dissipation length scale grows
b)The dissipation length remains the same
Leiden, 1908: Kamerlingh Onnes succeeds in
liquifying helium, an element first identified
spectroscopically in India in 1868 during a total
eclipse of the sun. This leads to the discovery of
superconductivity a few years later, superfluidity a
few decades later, and many diverse applications
in science and engineering, including the
experimental study of fluid turbulence:
c) The dissipation length becomes smaller
Re =
Arbitrarily increasing U can introduce
Mach number (=U/c).
UL
ν
Increasing L has its obvious limits.
Helium has the lowest kinematic
viscosity of any fluid.
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Alternative approaches
It is reasonable to ask why nature’s “laboratories”, such as the atmosphere
and oceans, cannot be instrumented and studied for turbulence dynamics.
They can, but this is not a substitute for controlled laboratory studies when
questions become sharp and a deeper understanding is required.
Direct numerical simulations (DNS) can be performed in which the
appropriate equations are solved on a computer without making any
approximation. The range of scales needing to be well resolved,
however, grows as Re3/4 , and thus Re9/4 in 3 dimensions, severely
hampering DNS efforts. The state of the art in DNS is about
Re~104, or about 3-4 orders of magnitude lower than the Re
corresponding to a typically commercial jet aircraft, and the same
amount for most atmospheric and oceanic flows.
Under certain circumstances, large eddy simulations (LES)--which
compute only the large scales but model the small scales-- do better
in terms of providing useful information, but they are not satisfactory
as universal recipes. The state of the art in computer hardware is
years away from allowing us to address the most important problems
in natural and engineering fluid turbulence.
Continuum Approximation
An interesting question that has been raised at various times is whether
the increasingly small scales that develop as the Reynolds number is increased
render tenuous the continuum approximation of classical hydrodynamics; i.e.,
does one have to worry about aspects of molecular motion? U. Frisch has
addressed this question in a general context and demonstrated that the ratio of
the dissipation scale η to the molecular mean free path grows with increasing
Reynolds number. The hydrodynamic approximation thus should become better
applicable at higher Re. However, there are no definitive experiments yet to
confirm this.
.
Thermally-generated turbulence
Like other flows, convection is often turbulent, and in fact, turbulent
thermal convection plays a prominent role in the energy transport within
stars, atmospheric and oceanic circulations, the generation of the earth's
magnetic field, and also innumerable engineering processes in which
heat transport is an important factor. It is possibly the most ubiquitous
fluid flow in the universe.
106
109
1012
1015
1018
1021
Ra
Thermal convection transports and mixes
heat from the bottom of a cooking pot to
the top.
Rayleigh-Benard Convection
RBC near onset: linear temperature gradients, up/down symmetry, steady patterns
T
α fluid thermal expansion coefficient
H
Fluid
ν fluid kinematic viscosity
α, ν, κ
κ fluid thermal diffusivity
T+∆T
D
The control parameters for convection:
Ra =
gα∆TH 3
νκ
Rayleigh number
Pr =
ν
κ
Prandtl number
Γ=
D
H
Aspect ratio
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Convective patterns in Nature
Some of the motivating examples of thermal convection at
limiting values of the control parameters in Nature
Sun
Ra ~1022
10-3<Pr<10-10
atmosphere
Mantle
Often there is no up/down symmetry
in natural convective systems due to
non-Boussinesq conditions, differing
boundary conditions on top/bottom
Ra ~ 106
Pr~1021
Solar granulation
RBC at very high Ra: thermal boundary layers at the upper and lower
walls are highly stressed regions giving rise to “plumes.”
The temperature gradient is all at
the wall! At high Ra in experiments
the boundary layer can be of order
100 micrometers!
A cartoon of Turbulent RBC: self-organization of the
fluctuations into a coherent mean wind
Plumes in water
Sparrow, Husar & Goldstein J.
Fluid Mech. 41, 793 (1970)
Global heat transfer: Nusselt number
Nu ≡
(Pr~103,magma)
(from L. Kadanoff, Physics Today, August 2001)
Can we push experiments using new fluids?
measured heat flux
heat flux due to pure conduction
= ratio of effective turbulent thermal conductivity to its normal conductive value
Nu = f (Ra; Pr; Γ; ….. )
Nu = C Raβ +…. as Ra tends to infinity
“classical” results for high Ra
β=1/3: (assumes diffusive boundary layers on rigid surfaces)
β=1/2: (assumes entirely turbulent values for transport coefficients)
⎛α ⎞
Ra = g ⋅ ⎜ ⎟ ⋅ ∆T ⋅ H 3
⎝νκ ⎠
4.4 K , 2 mbar:
5.25 K, 2.4 bar:
α / νκ = 5.8 × 10−3
α / νκ = 6.5 × 109
Compare:
Water: α /νκ ~ 10
Air: α /νκ ~ 0.1
The shaded region gives an approximate area of operation with the P-T plane.
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4
Turbulence laboratory located at Elettra
Turbulent Heat Transfer
Conductivity enhancement by 20,000!
10000
Γ=1/2
Pressure Relief
and Sensing
Liquid Nitrogen
Reservoir
Nucorr
1000
Cryocooler
100
0.32
Nucorr=0.088Ra
Liquid Helium
Reservoir
Top Plate
(Fixed Temperature)
Soft Vacuum
Heat transfer
Cell Fill
Tube
10
6
Multilayer
Insulation
7
8
9
10
11
12
13
14
15
16
17
10 10 10 10 10 10 10 10 10 10 10 10
Convection Cell
(Cryogenic Helium Gas)
Ra
Thermal
Shields
Log-log plot of the Nusselt number versus Rayleigh number
4.5 K
Bottom Plate
(Fixed Heat Flux)
20 K
(2) J.J. Niemela & K.R. Sreenivasan, J. Fluid Mech., 557 411-422 (2006).
77 K
V(t), cm/s
Irregular reversals of a
convective wind
How high can Ra be pushed? 10m high convection cell-- at
one time proposed for the SSC-- capable of Ra~1021 .
Segment of continuous
5.5-day time series.
10
VM
0
VM
-10
0
2000
4000
6000
(1) J.J. Niemela, L. Skrbek, K.R. Sreenivasan & R.J. Donnelly, Nature, 404, 837 (2000)
8000
Features:
ƒAbrupt changes
ƒNon-periodic
ƒConstant magnitude
Inside cell dimensions
D = 5m, L = 10m,
Max volume ~ 25,000
gallons of liquid helium
equivalent, Minimum ~1 gal.
10000
t, sec
Geomagnetic polarity reversals: range of time scales~ 103-105 years.
Outside dimensions
~7 m dia and ~20 m high
Refrigeration needed
< 200 W
RHIC, BNL
Glatzmaier, Coe, Hongre and Roberts Nature 401, p. 885-890, 1999
Huge accelerator
facilities have large
quantities of liquid
helium on hand, used
to cool superconducting
magnets.
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