Shock Tube Calculations for the Calibration of

Fast Pressure Probes

S. Messer F. D. Witherspoon M. Phillips

HyperV Technologies, Inc.

Chantilly, VA 20151

July 9, 2008

Abstract

We review the fluid dynamics describing shock propagation in the case

of a shock tube. The resultant equations may be used for calibrating fast

pressure probes. We detail the limits of applicability of these equations.

1

1 Introduction

Fast pressure probes are used to calibrate spacecraft thrusters[1] and to diagnost

plasmas in general. The standard procedure for calibrating a fast pressure probe

is to mount the probe inside a rigid “shock tube”[2]. A typical shock tube has a

high-pressure (“driver”) side and a low-pressure (“driven”) side. The boundary

between the two is established by a thin diaphragm. When the diaphragm is

suddenly burst, a shock wave is generated as gas flows from the driver side

to the driven. The shock front has a thickness of a few times the mean free

path of the gas molecules[3, p. 714]. (Because of this, very low pressures on the

driven side produce a thick shock front. Pressures comparable to 1 atm produce

shock fronts of less than a micron thickness.) We commonly use a thin foil or

plastic diaphragm which bursts when the pressure difference across it is about

one atmosphere.

2 Pressure Probes

2.1 Pressure probe construction and mounting

2.1.1 Electrical characteristics of piezoelectric sensors

Fast pressure probes may be used to measure transient pressures and mass

flux in plasma and fluid dynamics experiments. These probes are typically

based on a piezoelectric crystal which generates charge in response to applied

stress[4]. Since piezoelectric crystals are fundamentally current sources, their

output impedance is very high. A charge amplifier or source follower must be

used to convert the high-impedance signal to one that can be measured as a

well-defined voltage through typical cabling. In addition, the finite impedance

of the sensing circuit means that the probe’s signal will gradually decay. As

a result, the probe provides accurate readings only for pressure changes much

2

faster than the “discharge time constant” - typically seconds or longer[4].

2.1.2 Mounting a commercial pressure sensor in a glass probe stalk

We typically use a quartz or Pyrex tube to mount a commercial fast pressure

probe like those available from Kistler International and PCB Piezotronics. The

PCB fast pressure sensor model 113A21 has a physical size and response char-

acteristics suitable to current experiments. It outputs to a #10-32 coaxial mi-

crodot connector, which is attached to a PCB model 480D06 power unit. This

unit acts as a charge amplifier and provides a bias voltage to the probe head to

optimize its response. The model 113A21 probe head fits nicely inside a quartz

(or Pyrex) tube with 7 mm inner diameter (ID) and 9.5 mm outer diameter

(OD). (Such tubes are available in 4-foot lengths from Quartz Scientific.) To

mount it, first connect the probe head to its output cable and thread the cable

through the tube. Then wrap the head with a few layers of electrical tape at

the mounting collar and again a little further back. The electrical tape should

increase the probe head’s diameter so that it just fits into the tube. Apply

epoxy to the probe sides, saturating the electrical tape, but keeping the epoxy

clear of the electrical connection. Push the probe part of the way into the glass

tube. Next, put a thin layer of epoxy on one side of a polished quartz disc.

(The disc currently in use has a 10 mm diameter and is 0.5 mm thick. It was

purchased from Machined Glass Specialists of Springboro, OH. The diameter of

the disc should closely match the OD of the glass tube, but the chosen diameter

may be any convenient value.) Use the quartz disc to push the probe head into

place, and hold it centered on the end of the glass tube. (See figure 1.) Cajon

or Ultra-Torr style vacuum fittings may be used as a mechanical feedthrough

to support the probe. Additional supports may be necessary if the glass tube

(probe stalk) is longer than about a foot.

3

2.1.3 Mounting a pressure probe for calibration in a shock tube

The pressure sensor should be mounted flush with the downstream end of the

shock tube’s driven section, so that the shock wave travels directly to it from

the burst diaphragm. Mounting the sensor flush with the wall simplifies the

analysis substantially. If the pressure sensor were mounted in the middle of the

tube, we would have to consider the more-complex 2 dimensional dynamics to

describe flow around the probe. See figure 2.

2.2 Pressure probe data interpretation

We interpret the calibration data (figure 3) based on the level of the approx-

imate flat-top voltage and the rise time of the initial slope. This rise time is

the time for the voltage to cross the interval of 10% to 90% of the full flattop

voltage. Note that the piezoelectric crystal, its housing, and the shock tube all

suffer mechanical vibrations when the shock front hits the end of the shock tube.

These accelerations and compressions drive current oscillations in the piezoelec-

tric crystal. In addition, the electronics necessarily have a finite response time.

Pressure probe signals involving transients comparable to the rise time should

be interpreted carefully. If the force sensor is mounted so that fluid flow presses

against it, the probe will measure the stagnation pressure Pstag. If the deceler-

ation onto the probe face is incompressible, which is the case for most liquids

with flow velocities much less than the sound speed, Pstag = P + 12ρv2. In the

case relevant to shock tubes, compressible gasses, and flow velocities faster than

or comparable to the sound speed, the deceleration is isentropic, with

Pstag = P

(1 +

γ − 12

v2

a2s

) γγ−1

. (1)

The speed of sound (squared) is a2 = γRTM , v is the fluid velocity, and γ is

the ratio of compressibilities. If the sensor is perpendicular to the fluid flow, it

4

will measure the regular fluid pressure P [3, p. 240,649]. Note, however, that

for probe-calibration purposes, the probe should be mounted flush with the rear

wall of the tube and thus measures the pressure in region 5, P5, which has no

fluid flow. (See section 3.2.1 for more details of assumptions and notation.)

3 Assumptions and Notation for Shock Tube

Dynamics

3.1 Regions of Different Flow Characteristics

The shock dynamics divides the gas into several distinct regions as follows[5, 6,

7]. See figures 2 and 4:

1 - The unshocked driven gas This gas retains the initial conditions of

the gas on the driven side of the diaphragm. It is divided from region 2 by the

shock front.

2 - The shocked driven gas This is matter that was originally on the

driven (downstream) side of the diaphragm, but which has been heated, com-

pressed and accelerated by the passage of the shock front. In extreme cases, the

shock may be sufficient to drive chemical reactions, but here we assume that

this is not the case. It is divided from region 3 by the “contact surface.”

3 - The expanding driver gas This gas was originally on the driver

(upstream) side of the diaphragm, but is flowing in response to the void left

by the shocked driven gas and the pressure of the still-compressed driver gas.

It separates region 2 from that still-compressed gas by a gradient of pressure,

velocity, and temperature.

5

4 - The still-compressed driver gas This gas is well upstream of the

diaphragm, and has not yet begun to flow. It retains the conditions of the initial

driver gas.

5 - The doubly-shocked driven gas Once the shock front reaches the

downstream end of the shock tube, it is reflected. Region 1 is no longer present,

and instead we deal with region 5. This gas was initially on the driven side of

the diaphragm, but was accelerated by the initial shock, then brought to rest by

the reflected shock. It is separated from region 2 by the reflected shock front.

The pressure in this region is what should be used to calibrate the pressure

probe.

Since the pressure probe is mounted flush with the downstream wall of the

shock tube, it is initially in region 1. It will briefly be in the shock front itself,

while the shock is in the process of reflecting off the wall and probe. Thereafter,

it will be inside region 5 until the reflected shock crosses the contact surface.

We will not consider dynamics after that point. We use numbered subscripts to

distinguish the thermodynamic variables in different regions. For example, P1

is the pressure in region 1; P4 is the pressure in region 4.

3.2 Thermodynamic Assumptions

3.2.1 Calorically Perfect Ideal Gas

In the following, we assume that both the driver and driven gasses are “calori-

cally perfect” ideal gasses. An ideal gas has the equation of state

P = ρTRM . (2)

Here P is the pressure, ρ is the mass density, and T is the temperature. R =

8.314 JK - mol is the ideal gas constant[8]. M is the gas’s molecular weight.

(We use SI units throughout this document.) It is common to refer to the

6

quantity R = RM as the “specific gas constant.” In this document, however, we

prefer to explicitly present the ideal gas constant and molecular weight, as these

quantities are more easily found in reference tables.

Since the gas is calorically perfect, its enthalpy per mass h and internal

energy per mass e are functions only of temperature. For an ideal gas, we may

assume h and e are linear in T , with the following constants of proportionality:

h = CP T =γRT

(γ − 1)M =a2

γ − 1(3)

e = CV T =RT

(γ − 1)M = h − P

ρ(4)

The new symbols are the isobaric compressibility CP , the isochoric compressibil-

ity CV , and the speed of sound, a. The ratio of compressibilities, γ = CP /CV is

determined by the number of degrees of freedom which are thermally accessible

to an individual molecule of the gas. In reality, it is thus a weak function of

temperature. For reasonable temperatures and pressures[3, pp. 633-634], air

behaves ideally with γ = 75 = 1.40 to within 1% accuracy[6]. (Diatomic gasses

generally have γ = 7/5; monatomic gasses typically have γ = 5/3. Rotation

and translation are thermally-accessible degrees of freedom at room tempera-

ture, but vibration is not accessible[9, 10].)

3.2.2 Adiabatic Dynamics

We further assume there is no flow of mass, momentum, or energy to / from

the walls of the shock tube. Except for the action of the shock front itself, we

further assume that all changes to the gas are isentropic. These assumptions

are equivalent to claiming that there is no diffusion of mass, no conductivity,

no viscosity, and no friction, drag or adhesion between the gas and the walls of

the tube[11, 3]. If the shock tube is much longer than its diameter, the system

is approximately one-dimensional. We ignore all transverse dynamics.

7

4 Matching Conditions

The initial conditions in regions 1 and 4 are known. Our task is to extrapolate

from these measurements to the conditions in regions 2, 3, and 5. (Although

the pressure probe does not directly sense regions 2 and 3, determining the

thermodynamic parameters in those two regions is necessary for the calculation

of the region 5 conditions.) This involves three sets of matching conditions as

follows:

4.1 The shock front divides regions 1 and 2.

4.1.1 Quantities conserved across shock

Since the shock front is thin and violent, it is highly dissipative. We therefore

cannot assume adiabatic changes across the shock. Once the shock is formed,

however, it moves through the background gas with a constant speed. The shock

itself does not accumulate mass, momentum, or energy. Instead it transforms

the conditions on one side of the shock to those on the other. Because of this,

we note that from the shock’s frame of reference, we may apply the steady-state

versions of the mass, momentum, and energy conservation equations:

ρ1w = ρ2(w − u2) (5)

P1 + ρ1w2 = P2 + ρ2(w − u2)2 (6)

h1 +12w2 = h2 +

12(w − u2)2 (7)

Here we have introduced two new velocities. The speed of the shock (in the

laboratory frame) is w, and the velocity of the gas is u. In all cases, velocities will

be considered positive when they point in the shock tube’s overall “downstream”

direction - i.e. from the driver end toward the driven end. From the initial

conditions the original gas is motionless, u1 = u4 = 0. The speed at which

8

matter enters the shock from region 1 is therefore w, and the speed at which it

leaves the shock for region 2 is w − u2.

4.1.2 Dimensionless Parameters

It is convenient to define two more quantities: The Mach number of the shock

relative to the sound speed in region 1 is

MS ≡ w/a1. (8)

The pressure ratio across the shock is

χ ≡ P2/P1 (9)

4.1.3 Useful intermediate velocity equations

We rewrite equation 5 as

u2 = wρ2 − ρ1

ρ2(10)

and combine this with equation 6 to get

w2 =(

ρ2

ρ1

) (P2 − P1

ρ2 − ρ1

). (11)

We can write a similar equation for w − u2:

(w − u2)2 =(

ρ1

ρ2

) (P2 − P1

ρ2 − ρ1

)(12)

4.1.4 Ratios of temperature and density

We use the ideal gas law (Eq. 2) to convert the above densities into ratios of

pressure to temperature and rewrite our velocities. Because we assume there

are no chemical reactions resulting from the shock, M1 = M2. Equation 9

simplifies the result substantially:

w2 =χT 2

1R(χ − 1)M1(χT1 − T2)

(13)

(w − u2)2 =T 2

2R(χ − 1)χM1(χT1 − T2)

(14)

9

We replace the enthalpy and velocities in equation 7 using these forms of the

velocities and equation 3. After sorting through the algebra, we have:

T2

T1= χ

γ1(χ + 1) ± |χ − 1|γ1 − 1 + χ(γ1 + 1)

(15)

T2

T1= χ

γ1 ∓ 1 + χ(γ1 ± 1)γ1 − 1 + χ(γ1 + 1)

(16)

One of the solutions is just the no-shock condition in which state 2 and state

1 are identical. But by definition of state 2, a shock separates the two regions,

implying they have different conditions. Choosing this set of signs, we have

T2

T1= χ

γ1 + 1 + χ(γ1 − 1)γ1 − 1 + χ(γ1 + 1)

. (17)

A look back at the ideal gas law (eq. 2) lets us write the density ratio:

ρ2

ρ1=

γ1 − 1 + χ(γ1 + 1)γ1 + 1 + χ(γ1 − 1)

(18)

4.1.5 Velocity equations solved

Combining these formulae for density and temperature with our earlier expres-

sions for MS , u2, and w (equations 8, 10, and 13) lets us write explicit equations

for the velocities in terms of the pressure ratio. Since we are dealing with ve-

locities on the left-hand side of the following equations, we use equations 3 and

17 to replace all temperatures with a1 [5, p. 212]:

MS =√

1 +γ1 + 12γ1

(χ − 1) (19)

u2 =a1

γ1(χ − 1)

√2γ1

χ(γ1 + 1) + γ1 − 1(20)

Unfortunately, we have not yet managed to close the set of equations. In

particular, χ has not yet been fixed. This is not surprising, as the shock dy-

namics should depend on the initial conditions of the driver gas. To calculate

the pressure ratio χ, we’ll need to find equations describing region 2 in terms of

the initial conditions in region 4. This task is the subject of the next section.

10

4.2 Expansion wave propagates through region 3.

4.2.1 Nature of the contact surface

Since region 3 is defined by the gradient between the undisturbed gas of region 4

and the driven flows of region 2, the quantities u3, T3, P3, and ρ3 are all functions

of position. The contact surface divides region 2 from region 3 and provides a

straightforward matching condition. Since it is not a shock, the speeds must be

the same on either side - so the downstream end of region 3 flows with the same

velocity as that in region 2, u2. We note that this speed is steady - the contact

surface does not accelerate. By definition, no mass crosses it until diffusion sets

it. We assume adiabatic expansion, so there is no diffusion. Since the mass on

each side of the contact surface is constant and since the velocity u2 is constant,

the fluid’s momentum is also constant. Therefore there is no net force applied

at the contact surface and the pressures must be the same on either side: The

downstream end of region 3 has pressure P2. However, the temperature, density,

and even composition may change across the boundary.

4.2.2 Assumptions about the expanding gas

Since there is no real discontinuity between regions 3 and 4, all variables in region

3 should approach the values in region 4 at the upstream end. We assume the

expansion is sufficiently mild that we may approximate γ3 = γ4. We treat the

expansion as adiabatic since there is assumed to be no heat flux to or from the

wall and no diffusion, viscosity, or conductivity. We cannot use the conservation

equations from section 4.1.1 since those assume steady-state conditions. The

pressure gradient in region 3 is clearly not constant, but consists of a finite-

amplitude expansion wave that propagates upstream, setting mass in motion.

Pressure waves travel at the local sound speed, a3(x, t), relative to the fluid

flow, u3(x, t). The lab-frame velocity is dxdt = u3 ± a3.

11

4.2.3 Microscopic fluid equations

We guarantee microscopic conservation of mass,

Dρ

Dt+ ρ∂xu = 0, (21)

and one-dimensional force balance,

ρDu

Dt+ ∂xP = 0. (22)

(We assume there are no applied external body forces.) Note that both these

equations use the total derivative,

D

Dt= ∂t + u∂x (23)

Making this substitution into equation 22 lets us rewrite it as follows:

∂xP + ρ∂tu + uρ∂xu = 0 (24)

It is tempting to use equations 2 and 3 to convert the DρDt in equation 21 to

a derivative of pressure, since this would give us derivatives only of P and u.

This would be incorrect, since it would implicitly assume that temperature was

held constant. We have assumed the expansion to be adiabatic, and therefore

isentropic, but not isothermal. For isentropic changes,(dP

dρ

)s

= a2. (25)

Substituting this into equation 21 gives

DP

Dt+ ρa2∂xu = 0. (26)

4.2.4 The Method of Characteristics

We rewrite equations 24 and 26 in similar forms to make their relationship clear:

0 =1a(a∂xP ) + ρ(∂tu + u∂xu) (27)

0 =1a(∂tP + u∂xP ) + ρ(a∂xu) (28)

12

We consider the sum and difference of these two equations:

0 =1a(∂tP + (u ± a)∂xP ) ± ρ(∂tu + (u ± a)∂xu) (29)

In principle, these equations could be used to solve for the pressure and

velocity fields throughout region 3, but at the moment, we are only interested

in using the known conditions in region 4 to constrain those in region 2. Recall

from the discussion that began this section: 1) the two ends of region 3 are

connected by a pressure wave, and 2) all pressure waves travel at velocities

dxdt = u±a. We note dP = ∂tPdt+∂xPdx = (∂tP + dx

dt ∂xP )dt and similar for u.

We can then multiply equation 29 by dt and eliminate the explicit dependence

on (x, t). Since we are concerned with a pure expansion wave, the wave travels

upstream; u opposes a, and we choose the negative sign:

du3 =dP3

ρ3a3(30)

If a counter-propagating wave mode is present, dxdt won’t have a simple form,

and we must find another route to a solution. The differential equation 30 holds

only along the path specified by dxdt = u − a. This trick is called the “Method

of Characteristics,” and the specified path is called the “characteristic line” [5,

p. 228].

4.2.5 Integration along the characteristic line

To integrate this equation, we note that so long as the system is adiabatic, the

thermodynamic quantities in any two states S and S’ are related by

P

P ′ =(

ρ

ρ′

)γ

=(

T

T ′

) γγ−1

. (31)

We can use this and equation 3 to write the right-hand side of equation 30

completely in terms of pressure and a reference state:

ρ = ρ′(

P

P ′

) 1γ

(32)

13

a =

√γRT

M (33)

a = a′(

P

P ′

) γ−12γ

(34)

dP3

ρ3a3=

dP3

ρ4a4

(P4

P3

) γ4+12γ4

(35)

This lets us set up an integral:

u2 − u4 =P

γ4+12γ4

4

ρ4a4

∫ P2

P4

P3

−(γ4+1)2γ4 dP3 (36)

We note that u4 = 0 and carry out the integration:

u2 =P

γ4+12γ4

4

ρ4a4

2γ4

γ4 − 1

(P

γ4−12γ4

2 − Pγ4−12γ4

4

)(37)

4.2.6 Ratios of variables in regions 2 and 4

To clean up equation 37, we note that equations 2 and 3 together give us

ρ =Pγ

a2. (38)

As a result, [5, p. 234]

P2

P4=

P3(CS)P4

=(

1 +γ4 − 1

2

(u2

a4

)) 2γ4γ4−1

. (39)

Since region-3 variables are functions of position, we use the notation P3(CS)

to refer to the value of P3 at the contact surface. From this and equation 31,

we have

ρ3(CS)ρ4

=(

1 +γ4 − 1

2

(u2

a4

)) 2γ4−1

(40)

T3(CS)T4

=(

1 +γ4 − 1

2

(u2

a4

))2

. (41)

As noted in section 4.2.1, it density and temperature are discontinuous across

the contact surface, so we cannot equate ρ2 and T2 with their region-3 counter-

parts at the contact surface.

14

Equations 9, 20 and 39 form a closed set when we include the initial condi-

tions describing regions 1 and 4. Numerical root-finding methods are necessary

to precisely solve this system of equations. The equations are approximately lin-

ear over reasonable intervals, so advanced search techniques are not necessary.

Linear interpolation is usually sufficient.

4.3 The reflected shock front divides regions 2 and 5.

4.3.1 Assumptions and definitions

Once the initial shock front reaches the downstream wall of the driven section,

it is reflected. We assume that we may treat the wall as stationary. Then the

reflected shock must be of sufficient strength to bring the flowing gas to exactly

zero velocity (u5 = 0). We note that the shock is now moving into region 2,

which has a finite lab-frame velocity u2. Our assumptions and equations are

otherwise similar to those for the initial, incident shock front (see section 4.1.1):

ρ2(wR + u2) = ρ5wR (42)

P2 + ρ2(wR + u2)2 = P5 + ρ5w2R (43)

h2 +12(wR + u2)2 = h5 +

12w2

R (44)

MR ≡ wR + u2

u2(45)

4.3.2 Characteristic velocities

Our strategy is to gradually replace the χ and other variables with Mach num-

bers. As intermediate steps, we will use equations 3 and 38 to replace thermo-

dynamic variables h, T , P , and ρ with sound and fluid speeds. We first use

equation 3 to rewrite equation 17 as a ratio of sound speeds:

(a2

a1

)2

= χγ1 + 1 + χ(γ1 − 1)γ1 − 1 + χ(γ1 + 1)

. (46)

15

We then rewrite equation 19 in several forms that will be more directly useful:

γ1 − 1 + χ(γ1 + 1) = 2γ1M2S (47)

χ =2γ1M

2S − γ1 + 1γ1 + 1

(48)

γ1 + 1 + χ(γ1 − 1) =2γ1

γ1 + 1(2 + M2

S(γ1 − 1)) (49)

Equations 47 through 49 transform equation 46 into

a2

a1=

1MS(γ1 + 1)

√(2γ1M2

S − γ1 + 1)(2 + M2S(γ1 − 1)). (50)

We now put equation 20 in terms of Mach numbers:

u2 =2a1

γ1 + 1

(M2

S − 1MS

)(51)

This combines with equation 45 to yield

wR = a2MR − 2a1

MS

(M2

S − 1γ1 + 1

)(52)

4.3.3 Pressure increase across the reflected shock

We use equation 38 to rewrite continuity (eq. 42) in terms of pressures and

sound speeds:

P5 = P2a25(wR + u2)

a22wR

(53)

We also use continuity to eliminate ρ5 from equation 43:

P5 = P2 + ρ2u2(wR + u2) (54)

Equation 38 can be used once again to remove density from this equation:

P5 = P2

[1 +

γ1

a22

u2(wR + u2)]

(55)

The two equations for the ratio P5/P2 can now be combined to yield an equation

in which only velocities appear:

a22wR + γ1u2wR(wR + u2) = a2

5(wR + u2) (56)

16

4.3.4 Mach number of the reflected shock

Next, we use equation 3 to write energy conservation (eq. 44) purely in terms

of velocities:

a25 = a2

2 +γ1 − 1

2u2(wR + u2) (57)

This form of the energy conservation equation can be used to eliminate a5 from

equation 56:

γ1wR(wR + u2) = a22 +

γ1 − 12

(wR + u2)(2wR + u2) (58)

Combining this with our definition of MR (eq. 45) and equation 52 gives

MR

M2R − 1

=a2

a1

MS

M2S − 1

. (59)

The ratio a2/a1 is eliminated by equation 50:

MR

M2R − 1

=2γ1M

4S(γ1 − 1) − M2

S(γ21 − 6γ1 + 1) − 2(γ1 − 1)

(γ1 + 1)(M2S − 1)

, (60)

An alternate form is [5, p. 216]

MR

M2R − 1

=MS

M2S − 1

√1 +

2(γ1 − 1)(M2S − 1)

(γ1 + 1)2

(γ1 +

1M2

S

). (61)

5 Requirements for Validity of Shock Tube Model

We have obtained a description of the shock tube which is valid after the di-

aphragm opens. What are the conditions for validity? The principal assump-

tions were:

Adiabatic expansion We assume that mass diffusion, viscosity, and ther-

mal diffusion are negligible.

Closed system We assume that there is negligible flux of mass, momen-

tum, and energy to/from the environment

17

No qualitative changes of state We can allow separate gasses in regions

1 and 4, but the gasses in regions 1, 2, and 5 are assumed to be the same

- or at the very least to have the same γ and M. Similarly, the gasses in

regions 3 and 4 must be the same. This severely limits any chemical reactions,

and the derivation should be re-examined if the shock produces region 2 or 5

temperatures near those necessary for reactions or ionization. Similarly, γ and

the equations of state will change substantially if the adiabatic expansion cools

region 3 to near the dew point.

Qualitatively simple wave dynamics Equation 30 assumes a single

expansion wave in region 3. Once this wave reflects off the upstream end of

the tube, there will be mixing of wave modes. Both expansion and compres-

sion waves will influence the driver gas. Before this disturbance can affect the

matching conditions at the contact surface, it must propagate downstream. It

moves at the local speed u3 + a3, and will eventually overtake the contact sur-

face - even if u2 > a4. It will also overtake the shock front if a2 + u2 > w.

The second temporal constraint is that our equations describing the reflected

shock assume a uniform region u2, and so the equations for the reflected shock

fail when it reaches the contact surface or the expansion wave’s reflection. For

shock tubes with total length of about two yards near atmospheric pressure and

room temperature, this condition is valid for about a millisecond after the di-

aphragm bursts. Fast pressure probes may have time responses as short as one

microsecond. For L4 and L1 being the lengths of the driver and driven sections

respectively and t = 0 denoting the diaphragm bursting, the expansion wave

reaches the upstream end of the tube at time L4/a4. The shock wave reaches

the downstream end at L1/w. The reflected shock reaches the contact surface

at L1w

(w+wR

u2+wR

).

18

Thin shock front The equations we use do not model the dynamics inside

the shock front itself, but rather match conditions on either side of the shock

front. As a result, it ignores interesting dynamics that take place on the scale

length of the shock front’s thickness. Detailed models of the shock front are

necessary if the mean-free path of the gas is comparable to or longer than

physical dimensions of the system. Such models are also necessary to understand

probe signals with time scales comparable to the mean-free path divided by the

shock velocity. For scale, the mean-free path of air at STP is on the order of

0.8µm. The time a stationary point spends in a shock traveling at the speed of

sound is therefore on the order of 2 ns. At 1 mTorr pressure, the mean-free path

is on the order of 60 cm, and the comparable transfer time is 2 ms. Laboratory

calibrations using these equations and meter-scale shock tubes should therefore

be done near or above atmospheric pressure.

One-dimensional dynamics More subtly, the above model assumes one-

dimensional dynamics and a complete, sudden opening of the diaphragm. The

shock tube should be designed so as designed so as to make this an accurate

assumption. There are two primary results in the (likely) event of incomplete

opening of the diaphragm[2]. First, pressure probes closer to the diaphragm

than about 12 times the tube diameter D will measure a weaker shock than

that predicted by the theory. Second, there will be oscillations with frequency

f∞ = a2D . Both effects result from the shock wave diffracting around the edges

of the partially-open diaphragm. Unless P4 >> P1 or careful design ensure full,

rapid opening of the diaphragm, the shock-tube designer should make sure it

is both much longer than 12 times its diameter and that f∞ is either above or

far below both the sampling frequency and all the probe’s resonant frequencies.

(Although Persico et al.[2] don’t explicitly give the source of this factor of 12,

their data suggests that 12D is the distance at which the shock front pressure

19

difference has risen to 1 − 1/e times the value predicted by one-dimensional

theory. Also note that their data and simulations have been restricted to P1 ≈1 atm and P4 ≈ 2 atm.)

Acknowledgments The authors would like to acknowledge help by Tom

York, Richard Bomgardner, Andrew Case, David Neil VanDoren, and David

Neal VanDoren.

20

References

[1] T. M. York, C. Zakrzwski, and G. Soulas. “Diagnostics and Performance of

a Low-Power MPD Thruster with Applied Magnetic Nozzle.” J. Propulsion

and Power 9, 4, pp. 553-560 (July - Aug. 1993).

[2] G. Persico, P. Gaetani, and A. Guardone. “Dynamic Calibration of fast-

response pressure probes in low-pressure shock tubes.” Meas. Sci. Technol.

16 9, pp. 1751-1759 (2005).

[3] R. W. Fox and A. T. McDonald Introduction to Fluid Mechanics, Fourth

Edition. John Wiley & Sons, Inc.: New York, 1992.

[4] “PCB Piezotronics, Inc. - Tech Support” HTTP://www.pcb.com/ techsup-

port/tech pres.php Downloaded Sept. 12, 2007.

[5] J. D. Anderson, Jr. Modern Compressible Flow with Historical Perspective,

2nd ed. McGraw-Hill: New York, 1990.

[6] R. J. McMillan. “Shock Tube Investigation of Pressure and Ion Sensors

Used in Pulse Detonation Engine Research.” AFIT/GAE/ENY/04-J07 Air

Force Institute of Technology: Wright-Patterson Air Force Base, Ohio

(June 2004).

[7] J. D. Anderson, Jr. Computational Fluid Dynamics: The Basics with Ap-

plications. McGraw-Hill: New York, 1995.

[8] NRL Plasma Formulary. ed. J. D. Huba. NRL/PU/6790-02-450 Naval Re-

search Laboratory: Washington, DC (2002).

[9] “Wikipedia: Diatomic” HTTP://en.wikipedia.org/wiki/Diatomic Down-

loaded Sept. 13, 2007.

21

Figure 1: Pressure probe mounted in its housing. We attach a special cable

(black) to the output of a commercial probe head (black), and slide the cable

through a glass tube (white). Epoxy and electrical tape (gray) bind the probe to

the inner wall of the tube. One side of a thin quartz disc (white) is smeared with

epoxy and this is used to push the probe head into its final position, assuring

intimate contact between the probe head and the quartz disc. The disc protects

the head from the plasma’s charged particles.

Figure 2: The gas inside the shock tube is divided into four regions after the

diaphragm bursts. Regions 1 and 4 are undisturbed gas. Behind the shock

front is region 2, which contains flowing driven gas. Behind the contact surface

is region 3, which contains expanding driver gas. The pressure probe should be

mounted flush with the wall terminating region 1.

4 3 2 1

[10] “Wikipedia: Monatomic” HTTP://en.wikipedia.org/wiki/Monatomic

Downloaded Sept. 13, 2007.

[11] A. H. Shapiro. The Dynamics and Thermodynamics of Compressible Fluid

Flow. John Wiley & Sons: New York, 1953.

22

Figure 3: Raw pressure probe data. A step-wise increase in applied pressure

(green) results in a delayed, oscillating voltage from the probe (red). The os-

cillations are due in part to pressure waves traveling through the piezoelectric

crystal and its housing.

23

Figure 4: Simulation of gas dynamics shortly after the diaphragm burst. All

variables show a steep increase at the shock front, and a more gradual, changing

slope within region 3. Note the cooling in region 3 due to the rapid expansion.

24