T. Zhang- A study of thermal counterflow in HE II using particle image velocimetry (PIV) technique

8/3/2019 T. Zhang- A study of thermal counterflow in HE II using particle image velocimetry (PIV) technique

1/147

1

CHAPTER ONE

INTRODUCTION

1.1 He II Thermal Counterflow

Liquid helium can exist in one of two liquid phases separated by a second-order

phase transition, called the -transition. Above the -transition, liquid helium is referred

as He I, or normal helium, and behaves like a classical Navier-Stokes fluid obeying

conventional fluid dynamic and thermal hydraulic models. Below the -transition, liquid

helium enters the superfluid state, called He II. One unique fluid dynamic phenomenon

existing in He II is the thermal counterflow, which can be understood in terms of the

relative motion of two interpenetrating fluid components in the two-fluid model

developed by Tisza [1] and Landau [2].

As the dominant heat transfer mode in He II as well as an ideal system for

exploring the dynamics of superfluid turbulence, thermal counterflow has been studied

extensively since its discovery in 1950s, focusing on two major aspects: 1) hydrodynamic

characteristics of the two fluid components in He II, and the interaction between them

under dynamic conditions; and 2) fundamentals of superfluid turbulence, including the

generation and development of superfluid turbulence, and its effects on the

hydrodynamics of the two fluid components [3-9]. However, due to the restrictions of

measurement technology, past research mainly rely on the measurements of pressure and

temperature gradients to study the heat transport and hydrodynamics of thermal

counterflow, while direct measurements of the velocity fields of the two fluid

components remain unattainable. Furthermore, although the numerical studies are able to


2/147

2

provide some detailed information about the velocity fields [10, 11], these results have to

be verified by actual fluid dynamics measurements.

1.2 Second Sound and Heat Diffusion

Also unique to the fluid system of He II are the processes of second sound and

non-linear heat diffusion. Second sound, which is basically a wave motion of entropydriven by a temperature perturbation, was predicted by Tisza [1] and Landau [2] based on

the two-fluid model, and first discovered by Peshkov in 1944 [12]. Later in 1952, the

nonlinearity theory of second sound was developed by Khalatnikov, who concluded that

the second sound wave would become nonlinear and eventually form a thermal shock

when the temperature perturbation is finite [31]. Heat diffusion is generally induced by

the applying of a heat source, and develops when He II enters the turbulent state. Due to

the existence of mutual friction between the two fluid components of He II in turbulent

state, the heat diffusion equation of He II displays a unique non-linearity similar to the

ordinary diffusion equation derived from Fouriers law of conduction.

The generation and development of both second sound and heat diffusion are

intrinsically associated with the dynamics of superfluid turbulence. Therefore, the study

of these two processes can help to further understand the fundamentals of He II in

turbulent state. In previous research, second sound shock and heat diffusion have been

studied only through the measurements of the heat pulse induced temperature profiles.

While it is predicted that transient thermal counterflow may be induced by these two

processes, to date actual measurements of the corresponding velocity field have never

been performed. Undoubtedly, such measurements will provide a new means of studying

the propagation of second sound and heat diffusion, as well as the superfluid turbulence

associated with them.


3/147

3

1.3 PIV Technique

Particle Image Velocimetry (PIV) is one of the most important achievements of

flow diagnostic technologies in experimental fluid mechanics. It originated from a series

of particle-imaging techniques developed in 1980s, such as Particle Tracking

Velocimetry (PTV) and Laser Speckle Velocimetry (LSV) [13]. The basic principle of

PIV is quite simple: the fluid motion is visualized by seeding micron-sized tracer

particles and illuminating the flow field with a laser light sheet. Two images of theparticles in the sheet are recorded in a short time interval. From their positions at the two

instances of time, the particle displacements are calculated, and thus the velocity field of

the flow can be inferred.

Compared to other flow measurement techniques, PIV has many advanced

features, such as non-intrusive and instantaneous whole field measurement, high accuracy

and high spatial resolution. Due to these features, PIV has been applied in most areas of

fluid mechanics and aerodynamics research. But, in contrast to its vast and broad

applications to conventional fluids, such as water and gases, there is very little in the

literature concerning the application of PIV to cryogenic fluids, especially to liquid

helium [14, 15].

1.3 Research Objectives

This study attempts to apply the PIV technique to the unique fluid system of He

II. As an entirely new technique for the research of He II, PIV would, for the first time,

make it available to measure the spatial structure of the thermal counterflow velocity

field induced by steady heating, or by second sound and heat diffusion. These detailed


4/147

4

fluid dynamics measurements should enable us to further understand the hydrodynamics

of the two fluid components of He II and also the dynamics of quantized superfluid

turbulence. In this study, three specific research objectives need to be accomplished.

The first objective is to implant the PIV technique to liquid helium. Due to the

exceptional physical properties of liquid helium, i.e. low density and extremely small

viscosity, selection of tracer particles is tough work, which requires a careful study of the

particle dynamics in liquid helium [16]. The critical parameters for particle selection

must be determined. Based on those parameters, appropriate tracer particles for liquid

helium must be selected from a variety of candidate particles. Also, the unique

experimental environment, low-temperature and vacuum, of liquid helium requires the

development of a new seeding method to properly introduce the tracers particles into

liquid helium.

The second objective is to conduct PIV measurements of thermal counterflow at

steady state. In particular, the normal fluid velocity field at different applied heat fluxes

and bath temperatures must be measured to reveal the spatial structures of thermal

counterflow. Measurements of normal fluid velocity profile as a function of the heat flux

and bath temperature are also required in order to compare it with the results predicted by

thermal counterflow theory. Some research questions need to be answered, including how

the tracer particles interact with the two fluid components in He II, and how the

interaction will influence on the motion of particles.

The third objective is to study the propagation of second sound shock and heat

diffusion by measuring the induced transient thermal counterflow field. To accomplish

this goal, the instantaneous velocity fields need to be measured with a high temporalresolution. Also, the velocity profiles versus time must be measured and compared with

the temperature profile to evaluate the applicability of PIV technique in this case.


5/147

5

This dissertation is organized in five major parts: chapter 2 is the background of

liquid helium, which consists of the two phases of liquid helium, He I and He II, the two-

fluid model, hydrodynamics associated with the two-fluid model in laminar and turbulent

state, basics of thermal counterflow, and the two transient heat transfer processes, second

sound and heat diffusion, in He II; chapter 3 is a brief introduction to the PIV technique,

in which the principle and features of PIV are discussed, and several important technical

aspects of PIV, including the selection of tracer particles, image acquisition, and

correlation process, are reviewed; in chapter 4, the experimental techniques associated

with the application of PIV to liquid helium is presented. Challenges for applying PIV to

liquid helium are discussed. The particle dynamics in a fluid is thoroughly studied in the

context of three different kinds of flows. The tracking characteristics of a variety of tracer

particles are discussed to evaluate their potential application in liquid helium, and the

particle seeding technologies are discussed. At the end of this chapter is some detailed

information about the PIV experimental set-up, including the laser, camera, timing

control unit and programs, optical system, cryostat, and the counterflow channel; chapter

5 is the results and discussions, in which the PIV measurements of steady thermal

counterflow as well as the transient thermal counterflow induced by second sound shock

and heat diffusion are presented and discussed; chapter 6 is the conclusions and some

suggestions for potential improvements and future experiments.


6/147

6

CHAPTER TWO

BACKGROUND OF LIQUID HELIUM

2.1 Liquid Helium

Helium is the most difficult of all the permanent gases to liquefy due to the

weakness of attractive forces between helium molecules. The fluid has a critical

temperature of 5.2 K and a normal boiling point of 4.21 K. The first successful

liquefaction was accomplished by Kamerlingh Onnes in 1908. Helium is the only

element that exists in the liquid state at absolute zero temperature, and as a result liquid

helium has received extensive research as an ideal system for the study of quantum

fluids.

Figure 2.1 shows the P-T phase diagram of helium. From it, several unique

characteristics of helium can be noted [17, Chap. 3]:

a) The solid state is not obtainable at any temperature unless an external pressure

in excess of 2.5 MPa is applied. This is due to the large zero point energy of the helium

molecule, which causes liquid as the lowest entropy state.

b) There is no triple point of coexistence between liquid, vapor and solid because

the solid state can exist only under a certain pressure.

c) Liquid helium can exist in two very different phases, normal fluid phase (He I)

and superfluid phase (He II). The line that separates these two phases is termed the -

line, which was adopted because the specific heat near the transition is discontinuous and

has the shape of the Greek letter. The -transition between He I and He II is classified

as a second-order phase transition, which means it has a discontinuous slope in the


7/147

7

temperature dependence of the entropy and there is no latent heat resulted from the

transition. The physical significance of-transition is that the two liquid phases cannot

coexist in equilibrium. The -transition temperature, T , is 2.176 K at saturated vapor

pressure and decreases gradually along the line with increasing pressure until it

intersects with the solid phase.

Figure 2.1 P4 PHe phase diagram [17].

The two liquid phases of helium, He I and He II, show substantially different

behaviors. He I is essentially a classical Navier-Stokes fluid with associated static and

thermodynamic properties, and obeys the hydrodynamics described by conventional


8/147

8

Navier-Stokes equations. However, He II, referred as superfluid helium, displays many

exceptional physical and transport properties, and thus will be discussed in detail below.

2.2 He II as a Quantum Fluid

Below -transition, liquid helium enters the He II phase and displays several

unique transport properties, such as a very high thermal conductivity, many orders of

magnitude larger than that of other liquids and even pure metals, and an ideal inviscid

flow behavior discovered by Allen and Misener in 1938. In their experiment (Fig. 2.2 a),

Allen and Misener determined that the viscosity of He II is vanishingly small (on the

order of 10P-12

P Pas) through measuring the He II flow through a capillary channel [18].

However, another measurement done by Keesom, who used a rotating cylinder

viscometer (Figure 2.2 b) to measure the viscosity of He II, yielded a finite viscosity of

the order of5

10

Pas, which is very close to that of He I [19]. To interpret this so calledviscosity paradox, a two-fluid model of He II has been developed by Tisza [1] and also

by Landau in a slightly different form [2].

According to this model, He II consists of two interpenetrating fluid components,

the normal fluid component and the superfluid component. The normal fluid component

behaves like the ordinary Navier-Stokes fluid with a density n , viscosity n and

specific entropy ns . In contrast, the superfluid has only the density s but no entropy

( 0=ss ) and viscosity ( 0=s ). In the two-fluid model, the above viscosity paradox can

be explained as follows. For Allen and Miseners experiment, the normal fluid

component was held back by its viscous interaction with the capillary wall so that only

the superfluid component can flow through the narrow capillary. Thus the measured


9/147

9

viscosity is that of the superfluid component 0=s , so the flow is inviscid. For the

damping experiment of a rotating cylinder, however, the normal fluid component

interacts with the rotating disc and produces the damping force. Therefore, the viscosity

measured in this experiment is actually the normal fluid viscosity n , which is similar in

magnitude to that of He I.

(a) (b)

Figure 2.2 Two different methods of measuring the viscosity of He II(a) flow through a capillary channel [18]; (b) damping of a rotating cylinder [19].

UTwo-fluid model of He II

From the two-fluid model, the density and total mass flow rate of He II can be

written as,

sn += (2.1)

ssnnvvvJvvvv

+== (2.2)


10/147

10

where is the bulk density, vv

is the bulk flow velocity, nvv

and svv

are the velocities of

normal fluid and superfluid components respectively.

While is nearly constant (145 kg/m P3

P) at saturated pressure, the normal fluid

and superfluid densities,n

ands

are strongly temperature dependent

6.5

=

T

Tn and

=

6.5

1

T

Ts for TT (2.3)

Figure 2.3 shows the relative ratio of normal fluid and superfluid density to the bulk

density of He II at different temperatures.

Figure 2.3 Ratio of normal fluid and superfluid density in He II [17].

Since the velocity fields of the normal fluid and superfluid components are

independent of each other, the momentum equations of He II should be written for the


11/147

11

two components separately. For the superfluid component, the flow is driven by the

gradient of chemical potential rather than the pressure gradient, so one can get

=Dt

vD sr

(2.4)

where is given by

PTs +=

1 (2.5)

Here, P is the fluid pressure. Combining equations (2.4) and (2.5), the momentum

equation of superfluid can be written as,

TsPDt

vDs

ss

s+=

r

(2.6)

To compare with classical fluids, the term superfluid pressure is introduced as

ssP = . Then, equation (2.6) can be rewritten in the form of

s

s

s PDt

vD=

r

(2.7)

This equation has exactly the same form as the momentum equation for non-dissipative Euler fluid, indicating that superfluid itself can be treated as an ideal Euler

fluid.

For the normal fluid component, since it possesses viscosity like an ordinary

Navier-Stokes fluid, its momentum equation is identical to the classical Navier-Stokes


12/147

12

equation as below,

nnn

n

n vPDt

vD rr

2+= (2.8)

where nP is termed as normal fluid pressure and

PPP sn =+ (2.9)

Substituting equation (2.9) into equation (2.6), the momentum equation of normal

fluid component can be finally written as:

nns

nn

n vTsPDt

vD rr

2+=

(2.10)

From equations (2.7) and (2.10), He II can be viewed as a mixture of an ideal

Euler fluid and a classical Navier-Stokes fluid.

2.3 He II in Turbulent State

As we know from the two-fluid model, there is no dissipative term for the

superfluid component of He II. Therefore, it can be predicated that, when a vessel

initially at rest is filled with He II and driven to rotate, the superfluid would stay at rest

and only the normal fluid component could be brought into rotation by the viscous force.

This predication has been verified by Andronikashvilis experiment [20], in which the

normal fluid density n measured by a group of oscillating discs is in agreement with the


13/147

13

value obtained through second sound measurement (Figure 2.4a). However, another

experiment done by Osborne [21] (Figure 2.4b), shows that the superfluid does take part

in the rotation. This rotation paradox can be interpreted by the turbulent theory of He II,

which is first developed by Hall and Vinen based on a series of experiments of measuring

the rotation of He II under different conditions [22-27].

(a) (b)

Figure 2.4 Rotation paradox arising from two different experiments

(a) Measurement of n by Andronikashvili [20]; (b) Rotating bucket by Osborne [21].

According to Vinen, the superfluid component may begin to rotate when the

relative velocity between the normal fluid and superfluid reaches a certain value, termed

as the critical relative velocity rcV . Above this critical velocity, He II enters the turbulent


14/147

14

state, and a mutual friction force between the two components of He II will be generated,

driving the superfluid to rotate. This mutual friction force, which was first introduced by

Gorter and Mellink [28] to explain the heat transport data, is further described by Vinen

as the scattering of normal fluid thermal excitation by a tangled mass of quantized vortex

lines in superfluid. In an effort to determine the mutual friction force, the force per unit

length of the vortex line f is first given based on the classical turbulent interactive

mechanism,

rn

ns Vf

2

= (2.11)

wheresnr

vvVrr

= is the relative velocity between superfluid and normal fluid. Then the

next step is to determine the vortex line length per unit volume, termed vortex line

density.

Assuming a homogeneous distribution of vortex lines that at any time have a

density L and an average interline spacing 2/1= Ll , the vortex line density will grow

with the expansion of vortex rings due to the relative velocity between the normal fluid

and superfluid turbulent, and the growth rate is given by

r

n

g

VLB

dt

dL 2/31

2

=

(2.12)

where 1 and B are both empirical parameters. At the same, the line density will also

decrease due to the breakdown of interacting vortex lines into small rings. Those small

rings contract in size and eventually dissipate their energy as thermal excitation. The rate

of decay can be approximately described by


15/147

15

2

2

2Lk

dtdL

d

=

(2.13)

where 2 is another empirical parameter, andm

hk= ( h the Plancks constant and m the

mass of a helium atom) is the quantum of circulation.

Suppose the growth and decay processes are independent, the steady state vortex

line density, 0L , can be obtained by setting these two rates equal,

dgdt

dL

dt

dL

=

(2.14)

Thus 0L can be written as,

2

0 rVL = (2.15a)

where,

2

1

k

B n= (2.15b)

Combining equation (2.11) and (2.15), the mutual friction force per unit volume is

given by

)(20 snrnsns vvVALfF

rrr== (2.16)


16/147

16

where A is called the Gorter-Mellink coefficient, which can be determined empirically

by experiments.

Considering the existence of a mutual friction force nsFr

between the normal fluid

and superfluid components, the momentum equations of He II introduced before

(equations (2.6) and (2.10)) can be rewritten as,

nss

ss

s FTsP

Dt

vD rr

+=

(2.17)

nsnns

nn

n FvTsPDt

vD rrr

++= 2

(2.18)

Also, the equation of mass conservation is given by

0)( =+

vt

r

(2.19)

and the equation of entropy conservation is

0)()(

=+

n

vst

s r

(2.20)

The above group of equations (2.17-2.20) is termed the Navier-Stokes equations for

He II in turbulent state.

Since there are two fluid components in He II, each component should display its

own form of turbulence. As the normal fluid component behaves like a classical viscous


17/147

17

fluid, its transition to the turbulent state occurs when the Reynolds number exceeds an

ordinary critical Reynolds number of

=n

nc

nc

vdv

)Re( 1200 to 2000 (2.21)

where d is the hydraulic diameter of the channel. This equation yields a critical velocity

ncv , called normal fluid critical velocity, above which the normal fluid component enters

the turbulent state.

For the superfluid component, the turbulence that takes the form of tangled vortex

lines occurs when a superfluid critical velocitysc

v is reached. According to the quantum

mechanics theory,sc

v can be estimated as [17, Chapt. 4]

dmd

hv

sc

21010

(cm/s) (2.22)

Actual experimental measurements of scv follow a4/1

d law as

4/1 dvsc (cm/s) (2.23)

where d is in units of centimeters. When this empirical equation is applied to a flow

channel with diameter on the order of 1 cm, the estimated superfluid critical velocity is

about 10 mm/s, which is very small and can be easily reached. For the thermal

counterflow that will be discussed next, the applied heat flux corresponding to this

critical velocity is only about 2.6 kW/mP2

P at 1.80 K. In case that the superfluid velocity is

abovesc

v , the superfluid component can no longer be regarded as an inviscid fluid.


18/147

18

2.4. Thermal Counterflow in He II

Consider a flow channel that connects two He II reservoirs (as shown in Figure

2.5), when a steady heat is applied to one end of the channel, there exists a temperature

difference T between the two ends. Since only the normal fluid component has the

entropy and can carry the heat flow, it will move away from the heat source (left

reservoir) to the right reservoir and then give up the heat. At the same time, the superfluid

component must counter-flow from right to left to conserve the mass. When it arrives atthe left reservoir, part of the superfluid component will be converted to normal fluid by

absorbing heat, making the local densities of the two components obey the relationships

given by equations (2.1) and (2.3). Thus, a relative counterflow between the normal fluid

and superfluid components is established, and this internal convection process is termed

thermal counterflow.

Figure 2.5 He II thermal counterflow [29].

Since only the normal fluid can carry the entropy, the heat transport equation for

thermal counterflow takes the form,


19/147

19

av

nsTvq = (2.24)

where q is the applied heat flux, and avn

v represents the spatially averaged normal fluid

velocity over the cross section perpendicular to the heat flow. Also, the zero net mass

flow condition requires that

0=+av

nn

av

ss vv (2.25)

Combining equations (2.24) and (2.25), the averaged relative velocityr

V can be written

as

sT

qvvV

s

av

s

av

n

av

r

== (2.26)

Whenrc

av

rVV < , the thermal counterflow is in laminar state, and there is no mutual

friction force between the normal fluid and superfluid components. Considering the

simple case of steady state, equations (2.6) and (2.10) can be combined, yielding

nn vPr2= (2.27)

s

PT

= (2.28)

Equation (2.28) is called Londons equation. Equation (2.27) is equivalent to the

Poiseuille equation in classical fluid dynamics, indicating that the normal fluid velocity


20/147

20

will display a parabolic profile in laminar state. Given a channel of constant cross section,

equation (2.27) can be further solved to relate the normal fluid velocity nv to the pressure

gradient [17],

2d

vP nn

= (2.29)

where is a numerical constant: 12= for parallel plates or large aspect ratio

rectangular cross section, and 32= for circular tubes. Combining equations (2.29),

(2.28) and (2.24), the heat conduction equation for thermal counterflow in laminar state

can be obtained as below,

TTsd

qn

=

222(2.30)

which indicates that the temperature gradient is a linear function of the applied heat flux

(the region of2QQ

&& < as shown in Figure 2.6).

Whenrc

av

rVV , thermal counterflow is in turbulent state, and the mutual friction

force must be taken into account. In this case, equations (2.17) and (2.18) are applied. For

the case of steady state, these equations reduce to

nn vPr2= (2.31)

s

F

s

PT

s

ns

+

= (2.32)


21/147

21

Figure 2.6 Temperature and pressure difference data from Tough [3].

It can be seen that the first term in equation (2.32) has the same form as in

equation (2.28), while the second term represents an additional temperature gradient,

s

FT

s

ns

= , due to the mutual friction force. Replacing nsF by equations (2.16) and

(2.26), the temperature gradient T in turbulent state can be finally written as,

3

343222 qTs

A

qTsdTs

nn

= (2.33)

which is a nonlinear function of the applied heat flux (the region of 3QQ&& as shown in

Figure 2.6). The second term in this equation, due to its dependence on cubic power of


22/147

22

the heat flux, will dominate the temperature gradient for moderate heat flux or in a fully

developed turbulent state. In this case, equation (2.33) can be reduced to

3)( qTfT = (2.34)

where

)/()(343

TsATf sn = (2.35)

is a temperature- and pressure-dependent parameter. Equation (2.34) is called the Gorter-

Mellink relation, and )(1 Tf can be looked at as the effective thermal conductivity of He

II.

2.5 Transient Processes in He II

Generally, transient heat transfer processes in He II are initiated by an impulse or

a stepwise heat input. Unlike the steady state heat transfer process that only depends on

one characteristic parameter, applied heat flux q , transient heat transfer processes also

depend on the characteristic time s . Based on these two parameters, transient heat

transfer processes in He II can be classified into several different regimes [30].

As shown in Figure 2.7, the applied heat flux in regime I is very small, where

crqq < and 01.0

crq W/cmP

2P. In this case, the heat transfer process for any

s can be

simply described by equations (2.27), (2.28) and (2.30). For an applied heat flux in the

range of


23/147

23

Figure 2.7 Diagram of transient heat transfer regimes in He IIin the coordinates tBsB: time of applying heat and q: heat flux [30].

2/3)(

= qTav (2.36)

where v is the characteristic development time of superfluid turbulence, and )(Ta is a

temperature-dependent coefficient. In regime III, where vs the superfluid turbulence

is fully developed, and the transient heat transfer process can be characterized by second

sound attenuation and/or heat diffusion. Typically, in the upper part of regime III, where

vs >> , the heat transfer process reaches the steady state and thus can be simply


24/147

24

described by equation (2.32). In regime II, where vs < , the superfluid turbulence is

not fully developed, and the heat is transported in second sound shock and/or the heat

diffusion mode. When q 1.0 W/cmP2P, three different regimes are possible: in regime IV,

where s is relative small, second sound attenuation is the main characteristic of the heat

transfer; while in regime VI, where s is large, film boiling heat transfer may occur.

Below, the two main transient heat transfer processes, second sound and heat

diffusion are discussed.

2.5.1 Second Sound

As a result of having two fluid components, He II is able to transmit more than

one type of sound. In addition to the ordinary or first sound, there is a process known as

second sound, which can be described as the propagation of entropy in wave motion.

Differences between first and second sound are illustrated in Figure 2.8. It can be seenthat first sound is driven by a pressure perturbation, and during its propagation, the local

fluid density oscillates but the local temperature remains constant; while second sound is

driven by a temperature perturbation, and during its propagation, the local density

remains constant but the local temperature oscillates out of phase due to the variation of

relative concentration of normal and superfluid components.

When the amplitude of temperature perturbation T is very small, the linear

theory of wave motion is applied, and an amplitude-independent second sound velocity is

given by [17],

pn

s

c

Tsc

2

20

= (2.37)


25/147

25

Figure 2.8 Illustration of (a) first sound and (b) second sound. The portion of n and srepresents the relative concentration of normal and superfluid component [29].

wherep

c is the constant pressure specific heat. Though this equation shows that the

second sound velocity is temperature dependent, it is almost constant at about 20 m/s for

bath temperatures between 1.0 and 2.0 K.

When the temperature perturbation is finite, the second sound displays

nonlinearity. In this case, the second sound velocity is amplitude-dependent and given by

[31]

( )[ ]TTBcc /1202 += (2.38)

where B is called the nonlinear steepening coefficient


26/147

26

=

T

cc

TTB

p

p

320

ln (2.39)

The nonlinear steepening coefficient B is strongly temperature-dependent, and, as

shown in Figure 2.9, can be both positive and negative [32]. When it is positive, the

nonlinearity causes the points with higher amplitude on a pulse to travel faster than those

with lower amplitude, and thus makes an initially rectangular pulse evolve into a front-

steepened shock wave with an expansion fan at the tail (Figure 2.10 (a)). In contrast,

when it is negative, the nonlinearity causes a back-steepened shock wave with an

expansion fan at the front (Figure 2.10 (b)).

The second sound shock velocity su can be obtained by averaging the velocities

at the front and tail of the shock,

( )

+= TTBcus

/2

11

20(2.40)

From this expression, the shock Mach numberM can be determined as,

( )TTBc

uM /

2

11

20

+== (2.41)

One unique characteristic of second sound is that it primarily induces thermal

counterflow between the normal fluid and superfluid. Turner suggested that this process

only takes a very short time, one microsecond or less [33]. For the induced counterflow,

equation (2.26) gives the relative velocity between the two components. As discussed

before, when the applied heat flux q is small and the relative velocity is less than the


27/147

27

critical relative velocity rcV , there is no mutual friction and the counterflow is in laminar

state. In this case, the heat flux that can be transported by second sound shock is

approximately equal to the applied heat flux, and the shock wave generated by a

rectangular heat pulse has a trapezoidal temperature versus time profile (waveforms 1 and

2 in Figure 2.11).

Figure 2.9 Experimental measurement results of the nonlinear steepening coefficientB .The solid line represents the value given by equation (2.39) [32].

Figure 2.10 Development of second sound shock from a rectangular pulse (the arrowshows the movement of pulse); (a) front-steepened shock; (b) back-steepened shock.

(a)

(b)

time t


28/147

28

Figure 2.11 Deformation of second sound shock waves ( pq : applied heat flux; Ht : pulse

heating time; bath temperature is 1.70 K and the distance from heater is 30 mm) [34].

As q is increased and the relative velocity is larger than rcV , the quantized

vortices in the superfluid start to develop, and the mutual friction force between the

normal fluid and superfluid is generated. The second sound shock wave then interacts

with the quantized vortices, causing the shock waves to deform. It can be seen from the

waveforms 3 and 4 in Figure 2.11, that the deformation is characterized by the tilting of

the flat top and an elongated shock tail. In this case, the shock amplitude, i.e. T ,

continues to increase with increasing heat flux. However, the heat flux that can be

transported by second sound shock is no longer equal to the applied heat flux, and the

ratio of transported to input heat flux decreases markedly with increasing q [33].

A further increasing of heat flux q makes the shock wave eventually reach the

limiting profile (waveforms 5 and 6 in Figure 2.11). It can be seen that the whole

waveform is much shorter than the initial heat pulse, and the two forms almost coincide

with each other irrespective of different heating times. As long as the limiting profile is


29/147

29

attained, further increasing the heat flux cannot generate a stronger shock, or higher T ,and, in some cases, may even cause T to decrease.

In summary, for the cases of waveforms 3, 4, 5 and 6, only a fraction of the

applied heat flux can be transported by second sound shock. The rest of the energy

accumulates in a thin thermal boundary layer, and is then transported by the heat

diffusion process that will be discussed next [33, 34].

2.5.2 Heat Diffusion in He II

Heat diffusion is the other important transient heat transport phenomenon in He

II. It may occur in many cases once the turbulent state is established. The two common

cases that have been broadly analyzed and studied are stepwise heating and pulse heating.

For the stepwise heating problem, the characteristic time s is defined as the time

from the heater is on to the moment at which the measurement is conducted. When

vs > , where v is defined by equation (2.36), the superfluid turbulence is fully

developed, and there exists a mutual friction force between the normal fluid and

superfluid. In this case, the heat transfer process can be described by the Gorter-Mellink

relation, yielding a non-linear heat diffusion equation,

3/1

)(

1

=

TTft

TCp (2.42)

Compared to the ordinary diffusion equation, Tt

T 21 =

, based on classical Fouriers

law of conduction, this heat diffusion equation is quite difficult to be solved analytically


30/147

30

because of the existence of 1/3 power. While numerical methods are the most popular

way to treat this equation, an elegant approach has been developed by Dresner to give a

similarity solution under different boundary conditions [35, 36].

From equation (2.42), the resulting temperature profile during the process of heat

diffusion is a function of elapsed time. Figure 2.12 shows a simple one-dimensional case

in which a stepwise heating is applied at the bottom of a channel filled with He II. The

top of the channel is open to helium bath. At the initial stage of heat diffusion (point A),

the temperature gradient at the bottom is much steeper than that at the top. As time

increases, the temperature profile develops along the channel and finally reaches the

steady state (point B), where the temperature gradient is approximately constant along the

channel, and can be determined by equation (2.34).

Heat diffusion may also be generated by a heat pulse when the applied heat

energy is greater than that can be transported by the second sound. Figure 2.13 shows an

example of the temperature profile measurement after a heat pulse is applied (applied

heat flux is 40=q W/cmP2 Pand pulse duration is 1 ms) [34]. It can be seen that the first

temperature rise with a sharp front is due to the arrival of second sound shock, while the

second temperature rise results from the heat diffusion. Clearly, compared to second

sound, the development of heat diffusion is much slower. However, the energy

transported by heat diffusion, which is proportional to the area under the temperature

profile, is much larger than that can be transported by second sound. From this viewpoint,

in the case that either the applied pulse heat flux is very large or the pulse duration is very

long, the transient heat transfer process is dominated by the heat diffusion.

In contrast to the heat diffusion under stepwise heating that can be simply

described by equation (2.42), heat diffusion induced by pulse heating may become more

complicated. When the duration of a heat pulse w are much larger than the characteristic


31/147

31

Figure 2.12 Development of one-dimensional temperature profile during heat diffusion.

Figure 2.13 Profiles of temperature rise due to pulse heating (TBbB=1.7 K, q=40 W/cmP2

P,pulse duration=1 ms; z represents the distance away from the heater) [34].


32/147

32

development time for superfluid turbulence, v , the density of the quantized vortex line,

L , would have reached its equilibrium value (equation (2.15)) with respect to the relative

velocity, rV . Thus, the Gorter-Mellink relation (equation (2.34)) is valid and the heat

diffusion can still be described by equation (2.42) [37]. However, when w is comparable

to or shorter than v , the vortex line density may not always reach the fully developed

equilibrium value. In this case, the vortex developing process is fully coupled with the

hydrodynamic processes [38]. Regarding the heat diffusion, the Gorter-Mellink relation

based on a steady state assumption may not be valid, and therefore the application of

equation (2.42) should be somewhat limited.


33/147

33

CHAPTER THREE

BASICS OF PIV TECHNIQUE

3.1 Introduction to PIV

Figure 3.1 shows a typical experimental set-up of a PIV system [39], which

consists of a laser, light sheet optics, image capture device (either a regular camera to

record the images on photographic film or a CCD camera to record the images on CCD

sensor), and a synchronizer to control the camera and laser. Firstly, small tracer particles

are seeded into the flow field. Then the seeded flow field is illuminated twice by two

pulses of laser sheet separated by a certain time delay t . The light scattered by the

tracer particles is recorded and two successive images are captured. Each of the images is

then subdivided into an array of small size region, called interrogation window, and for

each of such windows, a numerical correlation algorithm (either auto-correlation or cross-

correlation) is applied to statistically determine the local displacement vector of particles

between the first the second illumination. It is assumed that all particles within one

interrogation window have moved homogeneously between the two illuminations.

Further assuming that the tracer particles move with the local flow velocity, the velocity

vectors in the whole flow field can be obtained by dividing the particle displacement by

t .

Compared with traditional flow diagnostic techniques, such as pressure tubes or

hot wires, PIV technique has the following features:

1) Non-intrusive and indirect velocity measurement. PIV works nonintrusively by

employing an optical signal as the probe. This allows the application of PIV in high speed

flows with shocks or in measuring boundary layers, where the flow may be disturbed by


34/147

34

Figure 3.1 A typical PIV experimental set-up [39].

the presence of other probes. Also, PIV measures the velocity of a fluid element

indirectly by measuring the velocity of tracer particles within the flow [39].

2) Statistical method. The density of particle images in PIV is mediated to be

medium so that the images of individual particles can be detected while it is no longer

possible to identify particle pairs by visual inspection of the image. Therefore, rather than

tracking individual particles, the PIV technique follows a group of particles through

statistical correlation of sampled numbers of the image field.

3) Whole field technique. Depending on the size of imaged flow field and

interrogation window, the PIV technique allows the velocity information at hundreds or

thousands of points be extracted out of the images. This is a very unique feature of PIV


35/147

35

compared with other velocity measurement techniques, such as Laser Doppler

Velocimetry (LDV) that only allows the measurement of velocity at a single point. Also,

this feature results in a high spatial resolution of PIV, and makes it possible to detect the

very fine spatial structure even in unsteady flow fields. However, as a trade-off of the

high spatial resolution, the temporal resolution of PIV is limited due to technical

restrictions, such as the frame rate of CCD camera and the firing rate of laser.

As a widely used quantitative flow diagnostic tool, many aspects of the PIV

technique, including the selection and seeding of tracer particles, image recording, image

processing, correlation analysis, improvement of accuracy, error correction and so on,

have been extensively studied [40-45]. Below, only several important technical aspects of

PIV are selected for discussion.

3.2 Tracer Particles

It is clear from the principle of PIV that it relies on the scattering of light from

tracer particles to determine the displacement of particles within a certain time delay.

Since PIV indirectly measures the flow velocity by means of measuring the particle

velocity, the traceability of particles has to be checked in order to avoid significant

discrepancies between the fluid and particle velocity. Basically, when the density of

tracer particles is not the same as that of the fluid, the influence of gravitational force will

cause a primary error in the PIV measurements. In order to reduce the error, the tracer

particles must be as small as possible to faithfully follow the fluid flow. On the other

hand, if the tracer particles are neutrally buoyant in the fluid, namely the density of

particles is equal to that of the fluid, it seems that the particle size is not limited by its

ability to track the flow. However, in this case, one more aspect of the fluid dynamic


36/147

36

properties of tracer particles, the relaxation time s , has to be considered. According to

Raffel [39], the relaxation time is a critical measure for the tendency of particles to attain

velocity equilibrium with the fluid, which is given by

f

pp

s

d

18

2

= (3.1)

where p is particle density, pd the diameter of particle, and f the viscosity of fluid.

Since a large relaxation time may cause a delay for the particles to reach the flow

velocity, and a continued phase lag when the fluid velocity is fluctuating, it has to be

limited. Thus, the particle size has to be limited even for the neutrally buoyant particles

because the relaxation time depends on the particle density rather than the density

difference between particles and fluid. This issue becomes more important when PIV is

applied to measure the turbulent or unsteady flows with large velocity fluctuations.

As described above, the tracking characteristics of particles require that theparticle size should be very small to ensure good tracking of the fluid motion. On the

other hand, the optical characteristics of particles require that the particle diameter should

not be too small so that there is sufficient light scattered from the particles for image

acquisition. The light scattering capability of particles can be measured as the SNR

(signal-to-noise ratio) of the scatted light. According to Mies scattering theory, the SNR

is a complex function of the power, wavelength and frequency of the light, and the

diameter and refractive index of the particles [39]. For simplicity, Melling gave a

convenient measure of the light scattering capability, the scattering cross section sA ,

which is defined as the ratio of the total scattered power to the laser intensity [40]. The

variation of sA as a function of the particle diameter and laser wavelength is shown in

Figure 3.2. It can be seen that larger particles generally scatter more light and give higher


37/147

37

SNR; when the particle diameter is less than 1 m (~ 2 for green light with = 532nm) the scattering capability decreases drastically with the particle diameter; for particles

of diameter from 1 to 10 m, they have a good SNR and are appropriate for PIV

experiments. In addition to the scattering capability or SNR, it is also required that the

particle size should be as uniform as possible because the excessive intensity of light

scattering from larger particles will induces a high background noise to the light

scattering from the smaller particle.

In summary, selecting the appropriate tracer particles for PIV applications is

really a case of compromise between the tracking and optical characteristics of particles.

Optimum tracer particles should be small enough to faithfully follow the flow and large

enough to scatter sufficient light intensity.

Figure 3.2 Scattering cross section as a function of the particle size [40].


38/147

38

3.3 Particle Image and Image Acquisition

When the seeded flow field is illuminated by a laser sheet, the light scattering

from particles is captured by the camera, and an image is produced on the recording

medium. The image diameter of a particle is given by Adrian as [41],

2/1222 )( spi ddMd += (3.2)

where id is the image diameter, pd the particle size, M the magnification number of the

camera, and sd is the point response function resulting from diffraction of the lens,

#)1(44.2 fMds += (3.3)

Here #f is the f -number of the lens, and the wavelength of light.

The image size of particles has a great deal to do with the accuracy of PIV

measurements. Westerwell, Dabiri & Gharib show that the uncertainly of using 2-pixel

large particle image is half the uncertainty of using 4-pixel large particle image when the

32 by 32 pixels interrogation window is used [46]. The image size also has an effect on

estimating the centroid of correlaton peak. It is suggested that the error of peak estimation

is minimal when the image size is on the order of the size of pixel [47]. In addition,

Raffel et al. asserted that, for a 32 by 32 pixels interrogation window, the optimum

particle image size should be around 2.2 pixels to achieve a minimal uncertainty [39].

The two successive particle images acquired from two laser shots separated by

time t can be recorded either on two individual image frame (single-exposed mode) or

on a single image frame (double-exposed mode). The double-exposed mode has an


39/147

39

inherent disadvantage called directional ambiguity, which means one cannot determine

which particle image is from the first exposure and which one is from the second

exposure. However, this mode is superior for high-speed flow because a small exposure

interval t can be easily obtained since there is no need to switch the recording medium.

For the single-exposed mode, the directional information is preserved since the sequence

of the frames is known. But it requires the exposure interval t not to exceed the frame

rate of cameras (for CCD camera, the typical maximum rate is 30 frames/second).

Because of the large t , the application of single-exposed mode is limited and only

suitable for the measurements of low-speed flow. To overcome this limitation, a so-called

frame straddling technique was developed. This technique used a high-performance

progressive-scan-interline CCD chip as the sensor. The chip consists of an array of

photosensitive cells and an equal number of storage cells. After the first laser pulse is

triggered, the first image is acquired and immediately transferred from the photosensitive

cells to the storage cells. Later, when the second laser pulse is triggered, the

photosensitive cells are available to store the second image. In this case, the storage cells

contain the first image and the photosensitive cells contain the second image. Then both

of them are transferred sequentially from the camera to the computer for permanent

storage. Using this technique, the exposure interval t can be reduced to less than 1

microsecond. As a result, the application of single-exposed mode is able to be extended

for measuring very high-speed flow.

3.4 Correlation Process

PIV images are typically processed by subdivision into an array of overlapping

interrogation windows. For each window, a correlation process is performed to produce a

table of correlation values over a range of displacements. The overall displacement of


40/147

40

particles in the window is represented by a peak in this correlation table [45].

Corresponding to the two image acquisition modes, there are two different correlation

methods, auto-correlation and cross-correlation.

For images acquired in double-exposed mode, auto-correlation is applied. An

auto-correlation function is defined by

++=..

),(),(),(WI

dxdydyydxxIyxIdydxACOR (3.4)

where ),( dydx is the displacement vector, ),( yxI is the luminous intensity distribution

of the interrogation window. The overall displacement of particles is determined by

),( 00 dydx that makes the function ),( dydxACOR reach its peak value.

For auto-correlation, a primary correlation error is from the loss of image pairs

because particles move out of the interrogation window. When the particle displacement

increases, this so called velocity bias becomes worse, and the auto-correlation peak

becomes smaller and less likely to be determined. In order to reduce the correlation error,

one can either increase the seeding density of particles or increase the size of

interrogation window. As the number of particles increases, the probability of obtaining

an accurate measure of the displacement of a set of particles increases [45]. It has been

demonstrated by Adrian that the number of spurious vectors that appears in PIV data drop

dramatically as particle seeding density increases to an average of about 10 particle

images per interrogation window [41]. However, the seeding density cannot be too high

otherwise the characteristics of the flow being measured will be altered. An alternative isto increase the interrogation window size, which will increase the number of tracer

particles in an interrogation window without increasing the seeding density. In this case,

the velocity bias associated with particles entering and exiting the interrogation window

during the exposure time interval t is also reduced. But one drawback of increasing the


41/147

41

interrogation window size is that the spatial resolution of PIV measurements decreases.

For auto-correlation, it is suggested that the optimal interrogation window size should be

about six times of the local particle image displacement [48].

When images are acquired in single-exposed mode, cross-correlation is applied. A

cross-correlation function is defined by

++=..

21 ),(),(),(WI

dxdydyydxxIyxIdydxCCOR (3.5)

where ),(1 yxI is the luminous intensity distribution of the first image and ),(2 yxI is the

intensity distribution of the second image. The statistical overall displacement of the

particles is determined by a vector originating from the geometrical center of the first

interrogation window to the point where the ),( dydxCCOR function reaches its peak

value.

There exist a rule for standard cross-correlation processing, which means the

particle displacements between two frames must be less than length of the

interrogation window. Limited by the rule, the spatial resolution of cross-correlation is

at least four times of the particle displacements. To improve the resolution, a sub-region

shifting technique has been developed, where the second interrogation window is

spatially shifted with respect to the first interrogation window by an amount equal to the

mean flow displacement between the two exposures. Using this technique, most of the

particles within the first interrogation window will also be in the second interrogation

window, making it possible to use an interrogation window much smaller than that prescribed by the rule. In addition to the image shifting technology, another way to

improve the spatial resolution, called adaptive cross-correlation, has been developed by

Adrian [41], who suggested that cross-correlation can be performed between a small

interrogation window on the first image and a larger interrogation window on the second


42/147

42

image. Just like the image-shifting, this technique also ensures that most of the particles

within the first interrogation windows will be in the second interrogation windows.

For both the auto-correlation and cross-correlation process, the vital step is to find

the position of correlation peak in sub-pixel level. This step may greatly affect the

accuracy of correlation results. Typically, correlation results without a special peak-

finding scheme are accurate within +/- half pixel. In contrast, when a sophisticated peak-

finding scheme is applied, the accuracy may be improved up to 0.01 pixel. Many

different peak-finding schemes have been developed. Among them, the Gaussian three-

point curve fit scheme developed by Westweel is claimed to produce the least uncertainty

because the cross-correlation peak actually displays a Gaussian intensity profile [43].

Above is a brief summary of the basics of PIV technique. Those technical aspects

that have been discussed, including tracer particle selection, image acquisition, and

correlation process, are generally applicable to most of the PIV systems. But, when

applying the PIV technique to the unique fluid system of liquid helium, some more and

new aspects should also be considered, which are going to be discussed in the next

chapter.


43/147

43

CHAPTER FOUR

EXPERIMENTAL TECHNIQUES

4.1 Challenges for Applying PIV to Liquid Helium

Flow visualization studies provide micro-scale details of the flow field under

study, and when applied to liquid helium may help to further understand the fluid

dynamics of this unique fluid system. The first attempt to visualize liquid helium fluid

dynamics was reported by Chopra and Brown, who used suspended tracer particles to

observe and measure acoustic streaming in He II [50]. Since then, there have been a

number of visualization studies in liquid helium, including the qualitative visualization of

Taylor-Couette flow in He II [51] and a series of quantitative studies of He II thermal

counterflow jet using Laser Doppler Velocimetry (LDV) technique [52-54]. Compared to

these techniques that have been employed (either direct observation or LDV), PIV is a

more advanced flow diagnostic technique, having the ability to make quantitative whole-

field velocity measurements with high accuracy and spatial resolution. Using PIV for

visualization studies of liquid helium allows one to directly measure the velocity

distribution, and thus more thoroughly explore the fluid dynamic and heat transport

processes on small scales.

Despite the fact that PIV has been an established technique and extensively

applied for thermal-fluid studies of conventional fluids, such as water, air and various

gases, the technique has not been previously applied to liquid helium experiments due to

the following challenges:

1) The unique experimental environment of liquid helium, namely extremely low

temperature and in some cases low pressure, make it difficult to conduct PIV


44/147

44

measurements. As discussed before, PIV is a nonintrusive and indirect measurement

technique, which relies on tracking the motion of tracer particles to acquire the velocity

of flow. So, tracer particles play a critical role in PIV. While there is a wide variety of

tracer particles available for PIV measurements of conventional flows, many of them,

such as gas bubbles and oil drops, are not applicable for liquid helium because of the

experimental environment.

2) Liquid helium is one of the lowest density condensed fluids with a density

about 1/8th of that of water at its normal boiling point ( 2.4=T K), increasing by about

15% as the temperature is lowered to 2.2 K. In the superfluid state (He II phase), the

density of saturated liquid helium is nearly constant at 145 kg/mP3

P. This low density

makes it difficult to find neutrally buoyant tracer particle for liquid helium. Also, it is

more difficult to find tracer particles to accurately follow the flow because of the

extremely low dynamic viscosity of liquid helium, which is around 310P-6P Ns/mP2P for He

I at 4.2 K and 1.410P-6P Ns/mP2P for He II at 1.8 K.

3) The seeding of particles has been assumed as a trivial issue for PIV

experiments on conventional fluids. However, this task is not so easily accomplished for

liquid helium. Due to its unique experimental environment, most of the traditional

particle generation and seeding technologies that involve in using aerosol generator,

atomizer, or Laskin nozzle, are no longer applicable. In addition, the particles have an

increased tendency for coagulation in liquid helium. This places a stringent requirement

on particle seeding.

Because of these challenges, two major issues, particle selection and particle

seeding, have to be addressed in order to successfully apply PIV to liquid helium.


45/147

45

4.2 Particle Dynamics in a Fluid

As indicated before, the selection of appropriate tracer particles is two-fold: on

one hand the particles must be small enough to faithfully follow the flow; while on the

other hand, they must be large enough to scatter sufficient light for image acquisition.

Since it has already been pointed out that particles with diameter larger than 1 m are

generally able to generate enough SNR for PIV measurements, selecting particles in this

size range would primarily depends on their tracking characteristics, which can bedescribed by two parameters, slip velocity and relaxation time. For an in-depth view of

these two parameters, the particle dynamics in a fluid must be studied.

Before the discussion of particle dynamics, for PIV measurements, one should

first consider the effect of particle concentration. As we know, from the viewpoint of

correlation process, the particle concentration should be as high as possible because more

particles in an interrogation window considerably reduces the velocity bias resulting from

particles entering or leaving the interrogation window. However, from the viewpoint of

the flow field itself, the particle concentration cannot be too high otherwise the particles

would interact with each other and affect the characteristics of the flow. Thus, the particle

concentration must be in a certain range to achieve accurate correlation results without

altering the flow field. Adrian has suggested an ideal particle concentration at about 10

particles per interrogation window [41]. Also, a more practical formula to approximate

the seeding concentration, sC , required for PIV measurements has been determined by

Gray [55],

2

224

I

sdz

MC

(4.1)


46/147

46

where M is the magnification number of the camera, z the thickness of laser sheet, and

Id the length of a square interrogation window. These parameters can vary depending on

the applied PIV system. In our liquid helium PIV experimental set-up that will be

discussed later, a 48 48 pixels interrogation window is used with the pixel size of 6.7

m, giving the length of interrogation window 32.0=Id mm. The camera has a

magnification number of 455.0=M and the thickness of laser sheet is 2 mm. From

equation (4.1), the desirable particle concentration is around 7.710P9P particles/mP3P. If we

further assume that the particles are dispersed homogeneously within the flow field, the

average distance sd at which the particles are separated with each other can be simply

estimated from the relation that,

s

s

Cd

=

+

3

11

(4.2)

From this equation, the average particle separation at the desirable particle concentration

is about 0.5 mm. For the tracer particles with mean diameter pd less than 10 m, the

ratio of ps dd / is greater than 50, which is large enough to ensure that both the

interaction between particles and their influence on flow field are negligible. For tracer

particles of significantly larger diameter than 10 m, however, the interaction between

particles may influence the flow field unless the particle concentration is reduced from

the optimum value given by equation (4.1).

Assuming that the influence of particles on the flow and the interaction between

them can be ignored, the movement of a spherical particle with a diameter of pd and

density of p in a fluid with density f and dynamic viscosity f along the direction of


47/147

47

the gravitational field can be described by the following one-dimensional partial

differential equation [56],

( )( )

dt

vvdd

dt

dvdgdvvCd

dt

dvd pffp

f

fpPfPfpfDpPPP

+++=

33322

3

126)(

686

(4.3)

where pv is the particle velocity, fv the fluid velocity, and DC is the viscous drag

coefficient. This equation is known as the Basset-Boussinesq-Oseen (BBO) equation. The

left side gives the acceleration force acting on the particle. On the right side of the

equation, the first term represents the viscous drag force; the second term is the body

force due to the gravitational field and is assumed to be in the direction of the flow; the

acceleration of the fluid leads to a pressure gradient in the vicinity of the particle and

hence generate an additional force given by the third term; and the acceleration force of

the virtual mass, which is equal to half the fluid mass displaced by the particle, is

accounted for in the fourth term. In this equation, we have neglected the so-called Basset

history term, which takes into account the unsteadiness of the flow field. This is because,

for the dynamics of solid particles in liquid flow, the effect of Basset term need not

always be significant [40].

When the particle Reynolds number pRe is very small ( 1Re


48/147

48

f

pfpf

pvvd

)(Re = (4.5)

Equation (4.4) is a good approximation for most conventional flows (water and gas

flows) due to the combination of extremely small particle size and moderate viscosity of

the fluids. However, for liquid helium flows, the particle Reynolds number given by

equation (4.5) may not meet the requirement of 1Re


49/147

49

4.2.1 Motion of Particles in a Steady Flow

Considering a simple condition that the flow is in steady state, i.e. 0=dt

dvf,

equation (4.8) can be simplified as,

)()(18)(

2 pfpf

p

ffpgvv

ddt

vvd

+=

(4.9)

Since is also a function of )( pf vv , this equation is a nonlinear partial differential

equation and may be solved numerically.

If we consider the case that the particle velocity also reaches steady state, which

means 0=dt

dvp, the ultimate velocity difference between the particle and the fluid,

termed slip velocity, can be solved for as,

f

fpp

slip

gdv

18

)(2 = (4.10)

When the particle density is not equal to that of the fluid, the particle velocity deviates

from the actual fluid velocity. Since the PIV technique actually measures the velocity of

particles rather than that of fluid, this slip velocity principally determines the uncertainty

of PIV measurements. Also, the slip velocity represents the terminal velocity that a

particle can reach in a stationary fluid, termed the settling velocity sv . From equation

(4.10), measuring the settling velocity of particles in a fluid provides an easy way to

indirectly estimate the particle size [58].


50/147

50

When the particle Reynolds number is very small, 1 , equation (4.9) can be

solved analytically, giving the velocity difference as a function of time

+

=

sf

fpp

pf

tC

gdvv

exp

18

)(1

2

(4.11)

where s is the relaxation time and in this case is

f

p

s

d

18

2

= (4.12)

and constant 1C in equation (4.11) can be determined using the initial condition of the

particle. Assuming the particle velocity is piv when injected into the flow, the solution of

particle motion can be written as

+

=

s

pif

s

slippf

tvv

tvvv

exp)(exp1)( (4.13)

From this equation, the velocity difference between particle and fluid is changing with

time. The relaxation time describes the rate at which the particle catches up with the flow.

When st >> , the effect of initial particle velocity is negligible, and the velocity

difference is equal to the slip velocity given by equation (4.10). For the specific case that

the particle is neutrally buoyant in the fluid, i.e. fp = , the slip velocity between

particle and fluid is zero. And the relaxation time becomesf

pp

s

d

12

2

= .


51/147

51

When the deviation from Stokes drag law, equation (4.7), has to be considered,

there is no analytical solution for equation (4.9). But in this case the relaxation time can

be approximated as,

f

p

s

d

18

2

= (4.14)

Since is always greater than 1, equation (4.12) actually gives a conservative estimationof the relaxation time when the particle Reynolds number is finite. As the same, a

conservative estimation of the slip velocity can be derived from equation (4.10), which is

f

pfp

slip

dgv

18

)(2

= (4.15)

4.2.2 Motion of Particles in an Accelerating Flow

Now consider the situation with the particle initially at rest while the fluid

undergoes an accelerating flow described by,

tatvf =)( (4.16)

where a is the acceleration rate. The initial condition of the flow and the particle is

0)0()0( ==== tvtv pf (4.17)


52/147

52

Substituting equation (4.16) into equation (4.8), one can get

agta

vdt

dv fpf

s

p

s

p +

+

=+

2

31(4.18)

where s is either from equation (4.12) or (4.14) depending on the particle Reynolds

number. The solution of this equation consists of two parts: a homogeneous solution hpv ,

from the equation

01

,

, =+ hps

hpv

dt

dv

(4.19)

and a particular solution ppv , . Equation (4.19) can be easily solved as,

=

s

hp

tCv

exp1, (4.20)

where 1C is a constant. And the particular solution ppv , is assumed to have the following

form,

BtAv pp +=, (4.21)

Substituting this equation into equation (4.18), constants A and B can be determined as,

aA = (4.22)


53/147

53

s

pffgaB

+

= 1

2

3(4.23)

Then the analytical solution of equation (4.18) can be written as,

s

pff

s

p gatat

Ctv

+

++

= 1

2

3exp)( 1 (4.24)

Now, 1C can be determined using the initial condition, equation (4.17). The final solution

becomes,

tat

gavs

s

pff

p +

+

=

exp11

2

3(4.25)

When st >> , the slip velocity between particle and fluid in an accelerating flow is,

s

f

slippf avvv

+=

2

31 (4.26)

where slipv is given by either equation (4.10) or (4.15) depending on the particle

Reynolds number.

Compared with the results for particle in a steady flow, the particle in an

accelerating flow has the same relaxation time, but there exists an additional slip velocity

between the fluid and particle resulting from the acceleration of flow. Usually, )( sa is

small because s is small, so the additional slip velocity can be neglected. But for some


54/147

54

flows that are associated with abrupt velocity changes, such as oblique shock in transonic

flow and second sound shock induced counterflow that will be discussed later, a is a

large number so that the additional slip velocity must be considered. It should be noted

that these results can also be applied to decelerating flows.

4.2.3 Motion of Particles in Turbulent Flows

An extensive discussion of equation (4.8) and its solution for turbulent flow at

finite particle Reynolds number has been given by Mei [57]. Here, we will just consider a

simplified case, in which the turbulent flow is described as a time-averaged velocity u

plus sinusoidal velocity fluctuations,

)sin(0 tUuvf += (4.27)

where 0U and are the amplitude and frequency of the velocity fluctuation

respectively. Based on the previous discussion, equation (4.8) can be rewritten as,

s

slipff

s

fpp v

dt

dvv

t

v

dt

dv

+=+

2

3(4.28)

Same as equation (4.18), this equation can be solved analytically by finding the

homogeneous and particular solution. In this case, the homogeneous solution is the same

as given by equation (4.20), but the particular solution takes the following form,

32, )sin( CtCv pp ++= (4.29)


55/147

55

Substituting this solution into equation (4.28), we can get the constants 2C , 3C and .

Also, the constant 1C in equation (4.20) can be determined by applying the initial

condition of particle, 0)0( ==tvp . The results are summarized as below,

( ) 321 sinexp)( CtCt

Ctvs

p +++

=

(4.30a)

011

UtgSt

tguvC slip

= (4.30b)

0

2

21

)(1U

tgSt

tgC

+= (4.30c)

slipvuC =3 (4.30d)

21

1

Sta

aSttg

+

= (4.30e)

where the Stokes number St represents the ratio of particle relaxation time to the

characteristic time of the fluctuating flow,

sSt = (4.31)

and a is given by


56/147

56

fa

23= (4.32)

When the particle density is much larger than the fluid density, i.e. fp >> , a

is very close to zero. In this case, the solution (4.30a-e) can be simplified as,

+

+

+

+

++=

s

slipslipp

tuv

St

StU

St

tUvutv

exp

11

)sin()()(

2

0

2

0 (4.33)

and the phase lag can be determined as,

Sttg = (4.34)

From equation (4.33), when st >> , the particle velocity in a turbulent flow also

consists of two parts: a time-average velocity and a sinusoidal velocity fluctuation.

Comparing to the fluid velocity given by equation (4.27), the velocity difference between

the particle and fluid comes from two sources: one is a time-independent regular slip

velocity slipv as given by equation (4.10) or (4.15), and the other is the difference

between their fluctuating amplitudes, which is usually measured as an amplitude error

given by,

20

2

00

1

11

1

StU

StUU

+=+

= (4.35)


57/147

57

For high frequency fluctuating flows, i.e. a large , the amplitude error as well as the

phase lag may becomes very large. To ensure that the particle is still able to accurately

track the fluctuating behavior of turbulent flow, it must have an extremely small size,

which makes the relaxation time s very small.

Another special case is when the particle density is the same as that of the fluid,

i.e. fp = . From equation (4.32), we can get 1=a . Substituting into equations (4.30a-

e), the particle velocity is

)sin(exp1)( 0 tUt

utvs

p +

=

(4.36)

It can be seen that both the amplitude error and the phase lag are zero in this case.

The only velocity difference exists in the time-averaged velocity, but it is negligible when

st >> . Therefore, when PIV is used for measuring the turbulent flow, it is highly

recommended to use the neutrally buoyant particles.

4.3 Selection of Tracer Particles

In order to make successful PIV measurements, one critical task is to select the

appropriate tracer particles. Generally, the particles are required to have a small slip

velocity and a short relaxation time. A quantitative criterion to determine how small the

slip velocity should be can be set as,

fslipvv (4.37)


58/147

58

where is an arbitrary proportionality factor typically of order a few percent. For high

velocity flows, it is relatively easy to meet the above criterion. However, when the flow

velocity is low, this criterion may not be satisfied unless there is a very small slip

velocity. In this case, the slip velocity must be taken into account when doing quantitative

analysis of PIV measurements. For steady flows, the relaxation time does not affect the

accuracy of measurements, but it has to be considered for determining how long it takes

the particle to acquire the flow velocity. For transient or turbulent flows, the relaxation

time becomes more important since it also affects the measurement accuracy in addition

to just the response time (see equations (4.26) and (4.35)). Usually, it is required that the

particle relaxation time be much smaller than the characteristic time constant of the

transient or turbulent flows.

To select tracer particles for liquid helium, the previous discussion of particle

dynamics in a fluid is applied. For normal helium or He I, the above theory can be

applied because He I behaves like a classical Navier-Stokes fluid with associated static

and dynamic properties. However, for superfluid helium or He II, the inviscid nature of

the superfluid component of He II may bring into question the applicability of previous

discussion in this unique fluid system. To a certain extent this issue is addressed in the

dissertation research. However, there is evidence that the drag coefficient on a solid

sphere in He II obeys classical correlations, indicating that a sphere interacts with He II

flowas if it were a classical fluid [59]. Therefore, just for the purpose of selecting

potential tracer particles, we assume that the motion of particles in He II follows the same

dynamics equation as in classical flows, so that all the previous analysis results of particle

dynamics are also valid for He II. Since only the normal fluid component of He II hasviscosity, the fluid viscosity

f in previous discussion should be replaced with the

normal fluid viscosity of He II,n

. At the same time, one would expect for He II flows

that only the normal fluid velocity could be measured with PIV technique.


59/147

59

From equations (4.10), the slip velocity becomes zero if the tracer particles have

the same density as the fluid. However, the very low density of liquid helium makes it

hard to find neutrally buoyant particles. One practical approach is to select the particles

with density very close to that of liquid helium. This can reduce the slip velocity, but it

does not help to reduce the relaxation time, which is given by equation (4.14). From this

viewpoint, a better approach is to select particles with extremely small particles size,

which would make both the slip velocity and relaxation time small.

Table 4.1 Tracking Characteristics of Tracer Particles in Liquid Helium

He I (4.2 K) He II (1.8 K)

Type of

particlesSupplier

p

(kg/mP3P)

pd

(m) slipv (mm/s) s (ms) slipv (mm/s) s (ms)

Hollow glass

spheres3M 160 30 5.7 (3.9P*P) 3.7 (3.4P*P) 5.5 (3.1P*P) 9.0 (8.1P*P)

160 120 90.4 (20.0P

*P

) 59.4 (53.8P

*P

) 88.7 (13.8P

*P

) 143.2 (129.8P

*)P

PQ 200 20 5.4 (4.0P*P) 2.0 (1.8P*P) 9.2 (5.1P*P) 4.7 (4.2P*P)

200 100 135 (27.6P*P) 48.6 (44.1P*P) 230 (26.8P*P) 117 (106P*P)

TSI 1100 8 11.3 (8.6P*P) 1.4 (1.25P*P) 25.6 (13.8P*P) 3.2 (2.9P*P)

1100 12 25.5 (15.7P*P) 3.1 (2.8P*P) 57.6 (22.5P*P) 7.2 (6.5P*P)

Polymer

microspheres

Bangs

Laboratory1100 1.7 0.51 (0.42P*P) 0.06 (0.06P*P) 1.2 (1.0P*P) 0.15 (0.13P*P)

Solid deuterium 206 10 1.5 (1.26P

*P

) 0.50 (0.45P

*P

) 2.5 (2.0P

*P

) 1.2 (1.1)

Solid neon 1150 10 18.6 (12.6P*P) 2.3 (2.0P*P) 42.1 (19.0P*P) 5.2 (4.7P*P)

Solid HB2B/DB2B 140 10 0.27 (0.18P*

P) 0.38 (0.34P*

P) 0.2 (0.11P*

P) 0.9 (0.82P*

P)

Note: The values with a superscript * are calculated from equations (4.10) and (4.14), where the effect of

is considered. The values without * are from equations (4.12) and (4.15), where is regarded as 1.


60/147

60

We have tried both the above approaches, and many different kinds of particles

have been selected, including a variety of commercially available solid particles as well

as solid particles generated by freezing liquids or gases. The particle t

T. Zhang- A study of thermal counterflow in HE II using particle image velocimetry (PIV) technique

Documents

Transcript of T. Zhang- A study of thermal counterflow in HE II using particle image velocimetry (PIV) technique