Shock Waves (2000) 10: 377 387
Shock tube investigation of hydrodynamic issues
related to inertial confinement fusion
M.H. Anderson, B.P. Puranik, J.G. Oakley, P.W. Brooks, R. Bonazza
Fusion Technology Institute, Department of Engineering Physics, University of Wisconsin-Madison,
1500 Engineering Drive, Madison, WI 53706, USA
Received 5 January 1999 / Accepted 10 July 2000
Abstract. A shock tube investigation of two hydrodynamic issues related to inertial confinement fusion
(ICF) is undertaken. ICF is a promising source of energy for the future. There has been a considerable
increase in the interest in ICF with the development of the National Ignition Facility (NIF). However, much
remains to be investigated before a useful yield is obtained from a fusion reaction for power generation. The
physics involved in carrying out a fusion reaction combines hydrodynamics, plasma physics and radiation
effects superimposed on each other, at extremely small scales, making the problem very complex. One such
phenomenon occurring in the deuterium-tritium pellet implosion is the Richtmyer-Meshkov instability
occuring at each layer of the fuel which results in the mixing of the ablator with the fuel. This causes
dilution of the fuel and reduces the yield of the reaction. Another issue is the impulsive loading of ICF
reactor cooling tubes due to the shock wave produced as a result of the fusion reaction. These tubes must
withstand the impulse of the shock wave. A shock tube provides an ideal environment to study these issues
at large geometric scales with the isolation of hydrodynamics from other effects. A new vertical, square
shock tube has been designed specifically for the purpose of studying these fluid flow phenomena from
a fundamental point of view. The shock tube is vertical, with a large square inner cross-section and is
designed to allow for the release of a M = 5 shock into air at atmospheric pressure. In this paper, we
describe the new shock tube and related instrumentation in detail and present a few preliminary results
on the Richtmyer-Meshkov instability and shock-cylinder interactions.
Key words: Inertial confinement fusion, Richtmyer-Meshkov, Hydrodynamic instability, Shock diffraction
1 Introduction
The use of light ion, heavy ion, electron beam or laser
driven inertial confinement fusion (ICF) has received re-
cent interest with the development of the National Igni-
tion Facility (NIF). In addition to the many neutronic
hurdles inherent in ICF, there are several hydrodynamic
issues that need to be resolved for the ignition of the DT
fuel and for the cooling of possible ICF reactors, such as
SOMBRERO, HIBALL, LIBRA, LIBRA-LiTE, LIBRA-
SP, BLASCON, HYLIFE-I, OSIRIS (Kulcinski et al. 1994;
Moir 1996). These hydrodynamic issues are well suited for
exploration with a shock tube.
Fig. 1. Conceptual implosion of DT target. The outer shell is
ablated by laser or X-ray energy. This ablation propels the DT
fuel inward, compressing the inner gas until ignition occurs
1.1 Ignition of DT fuel
1995), see Fig. 1. If the solid surface of either the DT ice
The target capsules for NIF and other proposed ICF re-
or the ablator is too rough, the densities and tempera-
actors contain deuterium-tritium (DT) solid and gaseous
tures needed for ignition may not be attained because of
layers inside an ablative shell (typically polycarbonate or
the mixing of the fuel and ablator due to hydrodynamic
beryllium) that is several millimeters in diameter (Lindl
instabilities at their interface.
Correspondence to: M.H. Anderson The amplitude of surface perturbations between the
(e-mail: [e-mail]) two different density materials (ablator material and DT
378 M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion
ice) increases upon shock acceleration leading to a distor-
tion of the interface; this phenomenon, resulting in the
dilution of the fuel, is known as the Richtmyer-Meshkov
(RM) instability (Richtmyer 1961,Meshkov 1970). This is
similar to the more familiar Rayleigh-Taylor (RT) insta-
bility (amplitude growth due to constant acceleration), in
that the growth of the perturbations is due to a baroclinic
generation of vorticity consequent to the non-zero cross
product of the pressure and density gradients at the in-
terface (Samtaney and Zabusky 1994). However, the RT
perturbations grow unbounded only if the constant ac-
celeration is directed from the light to the heavy fluid,
whereas in the RM case, growth occurs regardless of the
direction of the shock propagation. The growth rate of
the interfacial perturbations in the linear stage of the
process (characterized by an amplitude · much smaller
Fig. 2. Design of the INPORT tubes. PbLi or other suitable
than the wavelength ) is proportional to the wavenum-
liquid metal flows through the tube and forms an ablative liq-
ber k of the perturbations (k =2Ä„/), the initial ampli-
uid layer by flowing through a porous weave of SiC, C or steel
tude ·0, and the Atwood number A = (Á2 - Á1)/(Á1 +
Á2) (Richtmyer 1961). The linear regime is well under-
stood; however, when the amplitude becomes comparable
protected concepts. Therefore, interactions between hy-
to the wavelength, the growth becomes non-linear exhibit-
drodynamic shocks and target chamber structures are is-
ing phenomena expected to be quite different than during
sues for most IFE power plant design concepts.
the linear stage. Among these differences are: a decrease
A couple of examples of liquid wall protection concepts
in the perturbation amplitude growth rate; the onset of
are the INPORT (INhibited Flow in PORous Tube) design
Kelvin-Helmoltz shear instabilities (Chandrasekhar 1961);
(Fig. 2) and the rigid PERIT (PErforated RIgid Tube)
the interaction between different modes (different hydro-
unit design (Kulcinski et al. 1994).
dynamic scales affect one another) (Haan 1991); the dis-
The INPORT design consists of an array of hollow
tortion of the interface and the departure of its shape from
tubes, on the inside of the first structural wall, which
a sinusoidal shape. Knowledge of this non-linear growth
carry a liquid PbLi eutectic alloy. These tubes are con-
rate along with that of the fairly well developed linear
structed out of a porous orthogonal weave of SiC, C or
growth of the perturbations is crucial in quantifying the
mixing of the ablator material and the cold fuel. This mix- steel which allows an ablative film of the PbLi to form on
the outer surface of the tube which absorbs X-rays and
ing rate is necessary to determine whether the density and
temperature of the hot center spot are sufficient for igni- target debris while the bulk of the liquid flowing through
the tube absorbs the thermal energy and mitigates the
tion. In an effort to study this growth rate, the problem
can be simplified to a two-dimensional perturbation be- isochoric heating by the neutrons. The first layer of tubes
in the PERIT system design have fan sprays, which create
tween two gases of densities Á1 and Á2, accelerated by a
shock wave generated in a shock tube. This simplifica- a liquid sheet of PbLi which performs essentially the same
task as the film on the INPORT tube design. These tubes
tion in geometry, and the isolation of pure hydrodynamic
must be able to withstand the high impact and diffraction
effects from radiation and plasma effects, allow a better
of the shock wave formed by the thermonuclear reaction of
understanding of the mixing due to the aforementioned
the DT fuel. To determine the pressure load and its distri-
instability.
bution around the cooling tubes it is possible to conduct
shock diffraction studies within a shock tube. Bryson and
1.2 Impulsive shock loading on cooling tubes Gross (1960) and Syschicova et al. (1967) conducted some
initial studies recording the shock formations at different
An additional hydrodynamic issue in ICF that is suitable times after an initial shock-cylinder interaction. This in-
for study within a shock tube is the protection of the first formation is quite valuable; however, only the formation
structural wall in reactors. Many types of inertial fusion and geometrical aspects of the reflected and diffracted
energy (IFE) target chambers experience significant hy- shocks were investigated while the pressure distribution
drodynamic motion. In gas-protected target chambers, the around the cylinder and its accelerations were not studied.
target X-rays and debris ions stop in the gas. Their energy Bishop and Rowe (1967) measured the pressure distribu-
generates a blast wave that consists of both a shock and tion around a cylinder in a blast channel with piezoelectric
a radiation wave, where the relative strength of each is a pressure transducers. However, due to the limitations of
function of the opacity of the gas. Liquid-protected tar- the blast channel, only experiments with low Mach num-
get chambers that are initially at low gas density (thick bers were conducted. The need for experimental pressure
liquid and wetted-wall concepts) produce significant gas data to compare with numerical solutions, and the lack of
densities by vaporization of some of the liquid. This va- a robust facility for directly measuring pressures at high
por exhibits many of the same features as the gas in gas- Mach numbers, has prompted several studies using holo-
M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion 379
graphic interferometry, e.g. (Heilig (1969)). This method
works well and an estimate of the pressure distribution can
be obtained from the experiments using the density gra-
dients and an assumed equation of state. Although these
types of measurements are helpful for comparing with nu-
merical simulations, they are not direct measurements of
the pressure distribution and are an indirect method of
calculating the impulsive force on the cylinder from the
shock wave. In an effort to increase the extent of the exper-
imental database, accelerations and pressure distributions
around cylinders for strong shocks are directly measured
in a large, square inner cross-section shock tube able to
achieve Mach numbers of the order of 5 into atmospheric
air. The time history of the horizontal and vertical ac-
celerations and pressure distributions are recorded with
piezoelectric transducers mounted flush on a cylindrical
Fig. 3. Schematic of the primary features of the shock tube.
surface.
The shock wave is created by high pressure gas in the driver
section rupturing a diaphragm. The shock wave travels down
the tube and interacts with either an interface between two
2Experimental facility
different gases or with an object in the test section
2.1 The Wisconsin shock tube
A new shock tube has been fabricated especially for the
shock-interface interaction and shock diffraction studies.
The tube is vertical, 9.2 m long, with a large square in-
ner cross-section (25.4 cm × 25.4 cm). The square cross-
section provides parallel walls suitable for flow visualiza-
tion without resorting to a structurally weak tube exten-
sion (Meshkov 1970). It has a structural capability to with-
stand a 20 MPa pressure load. Thus, strong shocks can be
fired into a driven section initially at atmospheric pres-
sure. High initial pressure in the driven section is useful
for reducing the effects of the wall vortices created as a
Fig. 4. Details of tube construction: the outer circular cross-
result of the interaction of the reflected shock from the
section is steel with a stainless steel square cross-section in the
tube s end wall with the boundary layer that is formed be-
center. The volume between the outer steel tube and the inner
hind the incident shock (Brouillette and Sturtevant 1994).
wall is filled with concrete
These vortices exert a strain on the interface and thus in-
troduce artifacts in the amplitude growth measurements.
of a cross, are placed just below the diaphragm to fa-
These effects are shown to be significant in shock tubes
cilitate its rupture in the form of four petals. After the
with small cross-sections and also if the driven section
diaphragm is ruptured, the four petals remain attached
is initially at sub-atmospheric conditions, which is often
to the diaphragm along its circumference. Table 1 lists di-
necessary, to produce strong shocks without creating pres-
aphragm materials, diaphragm thickness, driver gases and
sure loads that exceed the structural capability of exist-
the Mach numbers of the shock wave generated upon rup-
ing shock tubes (Houas and Chemouni 1996). The vertical
turing them. In all cases listed in the table, the driven gas
orientation of the tube allows preparation of a continuous
is air. The repeatability of the shock strength is typically
interface between gases of different densities utilizing grav-
to within Ä…0.4%.
itational stratification. Figure 3 shows the tube assembly.
The driver section is made from a circular, chrome- Figure 4 shows the cross-section of the driven section.
plated carbon steel pipe, 46 cm OD, 1.9 cm wall thick- An internal liner, consisting of four stainless steel plates
9.5 mm thick, welded together, is supported by a concrete
ness and 2 m long. It is equipped with four ignition tubes
matrix contained in a circular carbon steel pipe, 46 cm OD
mounted inside the driver section, capable of igniting a
to form a segment of the driven section. These segments
stoichiometric mixture of oxygen and hydrogen diluted in
are then capped with a class 300# A-105 flange welded
helium to produce high pressure and temperature driver
to the top and bottom of the tube. This allows any of the
conditions. To release the shock into the driven section,
nine sections to be bolted together in a variety of config-
a metal diaphragm is ruptured either by pressurizing the
urations.
driver section using compressed gas bottles, or if the
hydrogen-oxygen combustion approach is used, by deto- The test section consists of four steel plates, chrome
nation of an explosive charge placed around the circum- plated on their inside surface, 7 cm thick, welded together
ference of the diaphragm. Sharp knife edges, in the form to form a box with a 25.4 cm square inside cross-section.
380 M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion
Table 1. Diaphragm rupture characterization data
Material Description Rupture Driver gas Mach number
pressure, kPa in air
Al 5052 One sheet, 0.381 mm thick 232 Air 1.25
Al 5052 Two sheets, 0.381 mm thick 275 Air 1.28
Al 5052 One sheet, 1.524 mm thick 958 Air 1.66
Steel A366 One sheet, 0.864 mm thick 1528 Air 1.85
Al 5052 Two sheets, 1.524 mm thick 1718 Air 1.86
Al 5052 Four sheets, 1.524 mm thick 2571 Air 2.00
Steel A366 One sheet, 1.397 mm thick 2571 Air 2.00
Steel A366 Two sheets, 1.397 mm thick 3979 Air 2.15
Steel A366 Three sheets, 1.397 mm thick 4655 Air 2.23
Steel A366 One sheet, 3.038 mm thick 5282 Air 2.28
Steel A366 One sheet, 1.397 mm thick 2620 He 2.77
Steel A366 Two sheets, 1.397 mm thick 3873 He 3.08
The test section contains two circular ports 28 cm in di-
ameter, on opposite sides. Fused quartz windows, 24 cm in
diameter and 9 cm thick, are mounted in these ports for
optical access.
The tube also contains an  interface section specifi-
cally designed for the study of shock-interface interactions.
It consists of four steel plates, chrome plated on their in-
side surface, approximately 7 cm thick, welded together
to form a box with a 25.4 cm square inside cross-section.
An interface between two gases of different densities is
formed in this section either by retracting a thin metal
plate initially separating the gases or by placing a thin
nitrocellulose or mylar membrane (thickness of the order
of one micron), held in a support frame, between the two
gases. The retraction technique gives rise to a diffuse, con-
tinuous interface while the membrane produces a sharp,
discontinuous interface. The membrane can also be given
a sinusoidal shape by placing it in a suitable supporting
frame. Such an interface provides the initial perturbation
for the shock-interface interaction.
The modular structure of the shock tube allows the
placement of the interface section, the test section and
the end wall at various mutual distances, allowing for the
visualization of the interface after the initial shock accel-
eration at various delay times. The end wall of the shock
tube contains a circular, fused quartz window, 2.5 cm in
Fig. 5. Photograph of the University of Wisconsin shock tube.
diameter and 3.8 cm thick. A laser sheet can be projected
Shown in the photo is the driven section of the shock tube being
upward, for planar imaging of the interface. The whole lowered down onto the base (three solid steel legs fastened to
shock tube assembly is supported on three cylindrical, a 8 m3 thick concrete isolation pad). The square section in the
center of the photograph is the test section with the diagnostic
solid steel legs, 18 cm diameter, 30 cm long. The legs, in
turn, are anchored in an isolated 8 m3 concrete founda- ports used for optical access for the shock cylinder and RM
images
tion. A photograph of the shock tube, test section and
support structure is shown in Fig. 5.
release of the shock. The driven section has pressure trans-
ducer ports at various locations along its length. Piezo-
2.2 Instrumentation
electric pressure transducers are mounted in these ports,
flush with the inner wall of the tube, to measure the shock
The driver section is equipped with a strain gauge pres- speed and trigger the flow imaging diagnostic hardware.
sure sensor and a type E thermocouple to measure the The driver and driven sections can be evacuated to 13.3
pressure and temperature of the driver gas just before the kPa.
M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion 381
Since the velocities of the shock and the accelerated in-
terface are of the order of 1000 m/s, imaging times on the
order of tens of nanoseconds are required to obtain sharp
images. In the first series of flow visualization experiments
using the shadowgraph technique, either a modified con-
tinuum (Surelite II-PIV) pulsed Nd:YAG laser or an arc
discharge lamp is used as a light source for the imaging.
The laser consists of two laser cavities capable of deliv-
ering 250 mJ/pulse (and hence 500 mJ/pulse when the
beams are superimposed) at  = 532 nm. The pulse width
Fig. 6. Optical setup for shadowgraph diagnostics. The view-
is 10 ns making it suitable for imaging high speed flows.
ing area in the test section is 212 mm, however in the current
The Xenon Inc. Model-437B short duration 20 ns broad
setup the illuminating light source is only expanded to 140 mm.
spectrum, arc discharge lamp is used in some experiments
As the light rays pass through the test section they are re-
to attain a more spatially uniform collimated light source fracted by changes in the density gradient creating an image
than can be achieved with the laser. A 16-bit CCD cam- on the screen
era (Spectra Video Series, by Pixel Vision) is used to cap-
ture the flow field images. It has a back-lit, 1024 × 1024
pixel array, a thermo-electrically cooled sensor and a low
speed (40 kHz) transfer rate to minimize the dark current
and readout noise (1 electron/pixel/s and 5 8 electrons
per readout rms respectively, at -45ć%C). One image is
obtained per event. An HP-Infinium 4 channel digital os-
cilloscope with a sampling rate of 1 GHz per channel is
used to record the pressure traces from the piezoelectric
pressure transducers.
2.3 Diagnostics
In the case of shock-interface interaction experiments, typ-
ically four wall mounted pressure transducers are used in
each experiment. The transducer closest to the diaphragm
triggers the laser to pulse, through a variable delay tim-
Fig. 7a,b. Schematic of the pressure transducer locations. The
ing box. The delay is set such that the laser pulses when
cylinder can be rotated 90ć% so that the pressure distribution
the interface is in the test section. The delay is calculated
around the cylinder can be measured
from one-dimensional gas dynamics and previous experi-
ments correlating the thickness of the diaphragm to the
shock speeds it generates upon rupturing. The shock speed
180 degrees as measured from the topmost point on the
is measured from the traces obtained by two other wall
cylinder); see Figs. 7a,b.
mounted pressure transducers. The last pressure trans-
In addition, two accelerometers are mounted on the
ducer is mounted in the end wall to measure the highest
cylinder, one in the direction of shock propagation and the
pressure generated behind the reflected shock.
other perpendicular to it, to measure the acceleration of
A shadowgraph system is used to image the interface.
the cylinder due to the shock loading. The flow is imaged
Figure 6 shows the optical setup for the experiment.
with the shadowgraph technique. The delay for the laser
The laser beam is spatially filtered before it is ex-
pulse is set such that the shock is captured at different
panded by a plano-concave lens (focal length 15 mm). It
locations along the cylinder so the temporal evolution of
is then collimated by a plano-convex lens (focal length
the flow field behind the shock and around the cylinder
500 mm) into a parallel beam of diameter 140 mm. The
can be recorded.
collimated beam is passed through the test section win-
dows and projected on a screen. The CCD camera is fo- At the time of an experiment, the whole optical setup
cused on the screen and captures the image.
and a portion of the shock tube containing the test section
In the case of shock-cylinder interaction experiments, are surrounded by an enclosure made from thick theater
in addition to the transducers that trigger the laser and curtains so that the area surrounding the test section is
measure the shock speed, four more are flush mounted on completely dark. The camera shutter is opened and the
the cylinder at 0, 30, 60 and 90 degrees, as measured from CCD exposure is started prior to the rupture of the di-
the topmost point on the cylinder. The cylinder can be aphragm (the diaphragm rupture pressure is known from
rotated by 90 degrees and pressures at three more loca- the previous experiments). After the diaphragm is rup-
tions can be measured, thus giving a total of seven lo- tured, the shock triggers the light source and an image of
cations around the cylinder (0, 30, 60, 90, 120, 150 and the flow field of interest is obtained with the CCD camera.
382 M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion
3 Experimental results
Typical shock tube data are presented for the RM in-
stability study and the ICF reactor cooling tube study.
Data include pressure and accelerometer summaries for
the cooling tube study and shadowgraph images for both
the cooling tube and interface growth studies.
3.1 Richtmyer-Meshkov instability experiments
As stated in the introduction, to achieve ignition of the DT
fuel in the ICF reactor, the mixing of the ablator material,
the DT ice and the hot inner gas after shock acceleration
must be minimized. In an effort to study this mixing rate
and the growth of the Richtmyer-Meshkov instability, an
interface between two different density gases is created.
Fig. 8. Shadowgraph of a flat interface between air and argon
The goal is to try to quantify the hydrodynamic mixing
with an incident Mach number of 1.931 in air, 1.12 ms after
between these gases in a quasi two-dimensional system
shock acceleration. This shadowgraph is taken with a Nd:YAG
and then apply this knowledge to the more complex ICF
laser as the light source
capsule, where hydrodynamic, radiation and plasma ef-
fects affect the instability. The initial interface between
two gases has previously been created in several different
dark diffuse line where the interface is calculated to be.
manners, which fall into two general categories: continu-
The shock accelerated interface appears to be flat as ex-
ous (diffuse) and discontinuous (sharp) interfaces. There
pected, since there is no initial perturbation imposed on
are certain advantages and disadvantages to studying both
the interface. (Any perturbations of the film are too small
types of interfaces and there have been many variations
to be resolved with our optical technique; however, small
in the methodology used to create them (Rightley et al.
perturbations in the film, along with diffusion, will result
1998, Houas et al. 1988). The discontinuous interface al-
in thickening of the initial interface as observed.) Addi-
lows the study of the growth of perturbations on a sharp
tional experiments performed with series of wires to sup-
interface with a known initial condition. The disadvantage
port the membrane across the frame indicate that the
of this method is that some type of membrane is required
thick line shown in Fig. 8 is most likely a combination
to separate the two gases. This membrane adds another
of an intact membrane that is ripped from the sides of
component to the system which may affect the flow and
the support frame and accelerated down the tube and the
growth rate. The continuous interface allows the study
interface between the two gases. This explains the small
of the growth rate without the addition of a membrane,
bump in the middle of the interface; since, as the mem-
however, the initial conditions are less well known and
brane travels down the tube it experiences aerodynamic
more difficult to impose. In the Wisconsin shock tube it
drag resulting in a parachute effect. From these results it
is possible to conduct experiments with both continuous
is concluded that it is necessary to use several wires to
and discontinuous interfaces; however, this paper presents
aid in the rupture of the membrane in order to reduce the
results using only discontinuous interfaces. As stated ear-
effects of the membrane.
lier, the shock tube is constructed in a modular form so
A second interface with an initially imposed sinusoidal
that the distance between the interface and the test sec-
perturbation (with an amplitude of 0.318 cm and a wave-
tion where the growth of the interface is measured can
length of 6.35 cm) is shown in Fig. 9. The imposed pertur-
be changed. This allows the investigation of the growth
bation of the initial interface allows the investigation of
at different times after the interface is accelerated by the
the growth of a known sine wave.
shock wave for a given Mach number. The data presented
The amplitude and wavelength of this initial pertur-
below are taken with the interface section at a distance
bation also initially sets the interface in the non-linear
of 45.4 cm above the centerline of the test section viewing
regime so that non-linear growth rates can be studied a
window. In all RM unstable interface images to follow, the
short time after initial shock-acceleration. This has been
shock is traveling from the top to the bottom and is below
done for a few different post-shock Atwood numbers (0.24-
the field of view.
0.87). Figure 10 shows the growth of the sinuous interface
Figure 8 shows a representative shadowgraph of an ini-
between air and CO2 1.149 ms after the acceleration by a
tially flat interface between air and argon when acceler-
shock with a Mach number 1.999.
ated by a shock wave with a Mach number of 1.931, 1.12
ms after the initial shock acceleration. The interface is created by separating air and CO2 with
Two different membranes are used to create the in- a 0.94 µm mylar film supported on the sine wave frame.
terface: mylar (0.94 µm thick) and nitrocellulose (approx- A series of 0.23 mm diameter nylon monofilament wires
imately 0.5 µm thick). The shadowgraph images for both approximately 1.27 cm apart (located at the peak, trough
of the different membranes are similar and indicate a thick and approximately halfway between the peak and trough
M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion 383
Fig. 9. Schematic of sine wave
frame and initial sine wave pertur-
bation imposed on the interface. A
mylar film is inserted in the frame
and separates the two gases to pre-
vent mixing before the shock ar-
rival. The inside of the frame is
25.4 x 25.4 cm2. The viewing area
is approximately 14 cm in diame-
ter
Fig. 11. Shadowgraph of a sinuous interface between helium
Fig. 10. Shadowgraph of a sinuous interface between air and
and CO2 with an incident Mach number of 1.909 in helium,
CO2 with a Mach number of 1.999 in air, 1.149 ms after shock
0.892 ms after shock acceleration. Note the large amplitude
acceleration. This image and the following interface images are
spikes of CO2 into the helium. The dark, thin region in the
taken with a 20 ns pulse duration arc lamp as the light source.
upper right of the field of view is a defect that developed in
The calculated one-dimensional gas dynamics location is used
one of the windows. The size of the spikes and bubbles are
as the center of the interface and the spike (·s) and bubble
estimated in the figure in a similar way as was done in Fig. 10
(·b) growth are measured from this position
of the sine wave), are used to help initially support and of the interface. The higher Atwood number is indicative
maintain the mylar film in the sine wave shape and to help of two gases of substantial density differences.
break apart the membrane upon shock interaction. This
Figure 12 shows a sinusoidal interface between nitro-
produces an interface with a post-shock Atwood number
gen and CO2, 0.648 ms after acceleration by a shock of
of 0.248. The diameter of the field of view in the figure is
Mach number 2.834. This experiment is similar to the
13.54 cm and contains approximately two wavelengths of
air/CO2 experiment discussed earlier, however, it is at a
the initial perturbation ( = 6.35 cm). As can be seen, the
much higher Mach number and much earlier time. The
amplitude has grown significantly from the initial condi- growth of the amplitude of the sine wave is therefore sig-
tion (approximately by a factor of ten) while the wave- nificantly smaller at this point. The higher Mach number
length has remained relatively constant.  Spikes of the
results in a higher compression of the initial interface and
heavy fluid (CO2) can be seen to protrude into the lighter
seems to lead to a slower growth rate. The higher Mach
air above, whereas the lighter fluid forms bubbles into the
number also resulted in visible secondary shocks off of the
heavier fluid.
joints in the tube walls. In the previous figures the lower
Mach number resulted in weaker secondary reflections and
Figure 11 shows the growth of the instability with a
post-shock Atwood number of 0.867, incident Mach num- were not as prominent as in this Fig. 12.
ber 1.909, 0.892 ms after shock interaction. This inter- In an effort to estimate an experimental growth rate
face is created by separating helium and CO2 using the of the initial perturbation from the shadowgraph images
mylar membrane. The amplitude of the CO2 spikes into discussed above, it is decided to use the distance traveled
the lighter helium has grown significantly as compared to by the interface for a given incident Mach number, based
Fig. 10 and the bubbles of helium into CO2 are much flat- on one-dimensional gas dynamics calculations as the refer-
ter. This is primarily due to the different Atwood numbers ence location. This is somewhat of an arbitrary reference,
384 M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion
where Vshock is the incident shock velocity. As can be seen
in Table 2, the calculated growth rates (obtained from
the linear theory) for Figs. 10 and 12 seem fairly consis-
tent with the experimentally measured growth rates. The
prediction for test 119, where there is significant observed
non-linear growth (see Fig. 11), however, is substantially
higher than the measured growth rate. This observation
of a decrease in growth rate (as compared to the predic-
tion of the linear theory) for highly non-linear perturba-
tions is consistent with numerical and theoretical studies
conducted by Holmes et al. (1995) and Zhang and Sohn
(1996).
The above shadowgraphic images yield data of the
growth rate of perturbations and are currently being used
to qualitatively validate numerical predictions from hy-
drodynamic codes. Future experiments are planned to use
planar Rayleigh scattering, Mie scattering, PLIF (planar
laser induced fluorescence) and other techniques that will
result in more quantitative measurements of the growth
Fig. 12. Shadowgraph of a sinuous interface between nitro-
rate. In these types of diagnostics a laser sheet (approxi-
gen and CO2 with an incident Mach number of 2.834 in N2,
mately 7 cm wide and 0.3 mm thick) will be projected from
0.648 ms after shock acceleration. The amplitude of the spikes
the bottom of the tube. The Rayleigh scattered light or
and bubbles is reduced (as compared to Fig. 10) due to shock
fluorescence can be used to quantitatively determine the
compression and the shorter time of travel for the interface.
density field of one of the two gases forming the inter-
Below the interface are secondary reflections off of the shock
face. Using a thin light sheet also reduces errors due to
tube walls
boundary layers and other three-dimensional effects as-
sociated with averaging the refracted light through the
but serves as a convenient theoretical location of the center
entire 25.4 cm wide shock tube.
of the interface. The one-dimensional calculations place
the mean location of the interface approximately halfway
between the tips of the spike and bubble in all images
shown. The age of the interface given above is a direct
3.2 Shock-cylinder interaction experiments
measure of the time between shock interface interaction
and the pulse of the light source. The fact that the one-
The cooling tubes in an ICF reactor are subject to re-
dimensional calculation of the interface location ends up
peated shocks from the ignition of DT fuel reactions that
in the center of the spike and bubble is an indication that
occur in rapid succession. To properly design them, it is
the gases were relatively pure. The spike height (·s) and
necessary to understand the force load history on the cool-
bubble height (·b) are then measured from this reference
ing tubes. The cooling tubes are long cylinders that have
location and a growth rate is estimated by the following
a slight curvature along the azimuthal axis. The ICF re-
expression:
actor cooling tube is modeled with a 25.4 cm long cylin-
der, 6.35 cm in diameter, placed in the center of the test
d· 0.5(·s + ·b) - ·o
section. The 25.4 cm length completely spans the 25.4 cm
= , (1)
dt tage
cross-section of the shock tube and permits a two-
expt
dimensional fluid dynamics study of shock-induced tran-
where tage is the measured age of the interface.
sient flow around a cylinder. Two 2.54 cm diameter, fixed
Table 2 shows the experimental growth rates along
rods are in place beneath each end of the tube to ver-
with the linear theory, calculations of the growth rate ob-
tically support it during the experiment. Accelerometers
tained using Richtmyer s (1961) expression:
are mounted vertically and horizontally on the interior di-
ameter (1.27 cm) of the cooling tube model. The hollow

d·
portion of the tube also serves as a cable-way for the trans-

= kVintA ·o, (2)
ducers cables. Several experiments are conducted with the
dt
theory
cylinder in each position so that statistically consistent
where k is the wave number (k = 2Ä„/), Vint is the in- measurements are made, and also, a shock history image
series is obtained.
terface velocity, A is the post-shock Atwood number and

·o is the initial amplitude corrected for shock compres- The presented data are for a Mach number 1.85 shock
sion using the expression initially suggested by Markstein in air at atmospheric pressure and temperature. The peak
(1957): initial pressure is shown in Fig. 13a,b along with the pres-
sure history.

Vint The circular figure (Fig. 13b) presents the maximum

·o = 1 - ·o, (3)
initial pressure distribution around the cylinder while the
Vshock
M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion 385
Table 2. Experimental results and linear theory calculations of the growth rate, where ·o = 0.318 and  = 6.35 cm

Test Mach Vshock, Vint, Age, A ·o, (d·/dt)expt , (d·/dt)theory ,
number number m/s m/s ms cm m/s m/s
110 (Fig. 10) 1.999 697.4 396.5 1.149 0.248 0.137 16.20 13.30
119 (Fig. 11) 1.909 1946.7 531.7 0.892 0.867 0.231 27.17 105.17
116 (Fig. 12) 2.834 1000.9 662.2 0.648 0.267 0.107 15.39 18.75
Fig. 14. Vertical and horizontal acceleration of the cylinder
during shock diffraction
supported. However, numerical studies of the structural
response of the tube model to impulsive loading are in
progress and the results will be compared against the ex-
perimental data shown in Fig. 14. Upon such validation,
the numerical model will represent a useful design tool for
the final cooling tube prototype.
Visual diagnostics for these experiments use the shad-
Fig. 13a,b. Pressure traces from the transducers located on
owgraph technique (optically sensitive to the second deri-
the cylinder during shock diffraction. The separation in time
vative of the density gradient). Three shadowgraph results
indicates the relative position of the pressure transducers. The
are shown in Figs. 15a c.
first trace is from the pressure transducer on the top of the
The thick black vertical structures between the cylin-
cylinder. The lower figure shows the maximum pressure gener-
der and the bottom of each of the images are the two
ated at the different measurement locations around the cylin-
(inline) support rods. The optical diagnostic provides a
der
history of the incident shock moving around the cylinder.
The initial shock is planar (it appears as a horizontal dark
graph (Fig. 13a) shows the time history of the pressure dis- line above the cylinder in Fig. 15a.) A most significant fea-
tribution. Qualitatively, the figure reveals that the maxi- ture of the shadowgraph images shown in Figs. 15b,c is the
bulbous shock reflection from the surface of the cylinder.
mum pressure occurs at the top of the cooling tube model
The incident shock remains planar as it travels vertically
(normal to the shock) and then decreases as the angle
downward (except where it is attached to the cylinder and
(relative to the upwards vertical) increases. For impulsive
a Mach stem is formed); however, the reflected shock off
force measurements needed for cooling tube design, it is
of the cylinder surface is curved (as expected). The radius
necessary to integrate the pressure history data for each
of the reflected shock grows in time as the incident shock
angular segment of the cylinder as a function of time.
travels down the cylinder. The time series of photographs
The maximum pressure occurs  instantaneously at the
reveal the behavior of both the planar shock and gas flow
top of the tube, however, the force on the tube must
be determined by integrating the seven pressure time se- around the cylinder.
ries over the surface of the tube. The maximum vertical The combination of dynamic pressure history, accel-
acceleration on the tube is 13,800 g s. This result can- eration data and optical diagnostics provide a unique ex-
not be used directly because of differences between the perimental characterization of the ICF cooling tube model
cooling tube prototype and the cooling tube model, both when subjected to shock loading. This new experimental
in the size of the tube and the manner in which it is apparatus is crucial to the understanding of the structural
386 M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion
have been conducted successfully and discussed briefly
above. The results suggest that, in the future, more quan-
titative data can be obtained using advanced diagnostic
techniques that can serve as a set of benchmarking data
for the numerical codes that simulate such hydrodynamic
phenomena. It is hoped that these studies will help ad-
dress issues regarding the target designs and the cooling
tube designs for possible IFE reactors along with unravel-
ing some of the hydrodynamic physics of flow instabilities.
Acknowledgements. The authors would like to acknowledge the
financial support of the University of Wisconsin-Madison and
the Department of Energy (DE-FG03-98DP00207 and DE-
FG02-97ER54413).
References
Bishop VJ, Rowe RD (1967) The interaction of a long dura-
tion friedlander shaped blast wave with an infinitely long
right circular cylinder. Atomic Weapons Research Estab-
lishment, Aldermaston, Berkshire, England AWRE Report
No. 0 38/67
Brouillette M, Sturtevant B (1994) Experiments on the
Richtmyer-Meshkov instability: single-scale perturbations
on a continuous interface. J Fluid Mech 263: 271 292
Bryson AE, Gross RF (1960) Diffraction of strong shocks by
cones, cylinders and spheres. J Fluid Mech 10(1): 1 23
Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic
Stability. Oxford University Press, Oxford, pp 433
Haan SW (1991) Weakly nonlinear hydrodynamic instabilities
in inertial fusion. Phys Fluids B3(8): 2349 2355
Heilig WH (1969) Diffraction of a shock wave by a cylinder.
Phys Fluids 12, Supplement 1: 154 157
Houas L, Farhat A, Brun R (1988) Shock induced Rayleigh-
Taylor instability in the presence of a boundary layer. Phys
Fluids 31(4): 807 812
Holmes RL, Grove JW, Sharp DH (1995) Numerical investiga-
tion of Richtmyer-Meshkov instability using front tracking.
Fig. 15a c. Shadowgraph images of the shock diffraction
J Fluid Mech 301: 51 64
around a 6.35 cm diameter cylinder at a Mach number of 1.85.
Houas L, Chemouni I (1996) Experimental investigation of
The shock was formed from high pressure air into atmospheric
Richtmyer-Meshkov instability in a shock tube. Phys Flu-
air. The successive images show the diffraction and reflection
ids 8(2): 614 627
of the shock as it contacts the cylinder
Kulcinski GL, Peterson RR, Moses GA, Bruggink D, Cousseau
P, Engelstad RL, Lee YM, Khater HY, Lovell EG, Mac-
Farlane JJ, Mogahed EA, Rutledge S, Sawan ME, Svi-
response of the tube to shock-loading needed for the ICF
atoslavsky IN, Wang P, Wittenberg LJ (1994) Evolution of
cooling tube design.
light ion driven fusion power plants leading to the LIBRA-
SP design. Fusion Technology 26: 849 856
Lindl J (1995) Development of the indirect-drive approach to
4 Conclusion
inertial confinement fusion and the target physics basis for
ignition and gain. Phys Plasmas 2(11): 3933 3992
The physics involved with attaining useful energy from in-
Markstein GH (1957) Flow disturbances induced near a slightly
ertial confinement fusion reactions is quite complex and re-
wavy contact surface, or flame front, traversed by a shock
quires continued experimental investigation. A shock tube
wave. J Aerosol Science 24: 238
provides an ideal environment to study the hydrodynamic
Meshkov YY (1970) Instability of the interface of two gases
issues related to ICF reactions. The Wisconsin shock tube
accelerated by a shock wave. NASA Technical Translation
is designed with the purpose of studying these issues in de-
F 13 074
tail and extending the existing experimental database into
Moir RW (1996) Liquid wall inertial fusion energy power
strongly shocked flow regimes. Preliminary experiments
plants. Fusion Engineering and Design 32-33: 93 104
M.H. Anderson et al.: Shock tube investigation of hydrodynamic issues related to inertial confinement fusion 387
Richtmyer RD (1961) Taylor instablity in shock acceleration Syshchicova MP, Beryozkina MK, Semenov AN (1967) The
of compressible fluids. Comm on Pure and Appl Math 8: flow formation around a body in a shock tube (in Rus-
297 319 sian). In: Aerophysical Investigation of Supersonic Flows,
Rightley PM, Vorobieff P, Martin R, Benjamin RF (1998) Ex- Collection of Papers, Science Publishers, Moscow, pp 7 13
perimental observations of the mixing transition in a shock- Zhang Q, Sohn S (1996) Nonlinear theory of unstable fluid
accelerated gas curtain. Phys Fluids 11(1): 186 200 mixing driven by shock wave. Phys Fluids 9(4): 1106 1124
Samtaney R, Zabusky N (1994) Circulation deposition on
shock-accelerated planar and curved density-stratified in-
terfaces; models and scaling laws. J Fluid Mech 269: 45 78