Moreover, $\mu_{A}$ is a real positive function that we would like to
interpret as a “measure” on the space of functions. Alternatively, we may
apply the change of variable shown in Eq. (6.169) and define

$\langle\widetilde{\Psi}_{1},\widetilde{\Psi}_{2}\rangle=\operatorname{sign}(Q^{\prime})\int_{Q(0)}^{Q(+\infty)}{\rm
d}q_{0}\,\widetilde{\mu}(q_{0})\,\widetilde{\Psi}_{1}(q_{0})^{*}\,\widetilde{\Psi}_{2}(q_{0})\,,$
(7.193)

where $\widetilde{\mu}$ is related to $\mu_{A}$ so that the two Hermitian
forms are equivalent,

$\mu_{A}(a_{0})=|Q^{\prime}(a_{0})|\,\widetilde{\mu}(Q(a_{0}))\quad\Longrightarrow\quad\langle\Psi_{A1},\Psi_{A2}\rangle_{A}=\langle\widetilde{\Psi}_{1},\widetilde{\Psi}_{2}\rangle\,.$
(7.194)

####  7.1.1 Determining $\mu_{A}$ and $\widetilde{\mu}$

In order to find $\mu_{A}$, we are going to impose that the Hamiltonian
differential operator $H_{A}/N$ in Eq. (6.170) is Hermitian on a suitable
Hilbert space. To this end, we derive the following identity by integrating by
parts,

$\displaystyle\big{\langle}\Psi_{A1},{H_{A}\over N}\,\Psi_{A2}\big{\rangle}=$
$\displaystyle\;\big{\langle}{H_{A}^{\dagger}\over
N}\,\Psi_{A1},\Psi_{A2}\big{\rangle}$ $\displaystyle\,+{\hbar^{2}\over
12v_{3}}\left[\rho_{A}\\!\left({\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\,{{\rm d}\Psi_{A2}\over{\rm d}a_{0}}-{{\rm
d}\over{\rm d}a_{0}}\Big{(}{\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\Big{)}\Psi_{A2}\right)\right]_{0}^{+\infty}\,,$
(7.195)

where we have defined

${H^{\dagger}_{A}\over N}\,\Psi_{A{\cal C}}\equiv{\hbar^{2}\over
12v_{3}}\,{1\over a_{0}}\bigg{[}{a_{0}\over\mu_{A}}\,{{\rm d}\over{\rm
d}a_{0}}\Big{(}\rho_{A}\,{{\rm d}\over{\rm d}a_{0}}\Big{(}{\mu_{A}\over
a_{0}\rho_{A}}\,\Psi_{A{\cal C}}\Big{)}+\omega_{A}\Psi_{A{\cal
C}}\bigg{]}-3v_{3}V\Psi_{A{\cal C}}\,.$ (7.196)

For the Hamiltonian to be Hermitian, two conditions must be met:

$\bullet$ We must have

${H_{A}\over N}\,\Psi_{A{\cal C}}={H^{\dagger}_{A}\over N}\,\Psi_{A{\cal
C}}\,,$ (7.197)

which can be seen to be equivalent to

$\kappa^{\prime}\,{\Psi^{\prime}_{A{\cal C}}\over\Psi_{A{\cal C}}}=-{1\over
2}\,\Big{(}\kappa^{\prime\prime}+\frac{\rho_{A}^{\prime}}{\rho_{A}}\,\kappa^{\prime}\Big{)}\,,\quad\mbox{where}\quad\kappa={\mu_{A}\over
a_{0}\rho_{A}}\,.$ (7.198)

If $\kappa^{\prime}$ is non-vanishing, we obtain a separated differential
equation by dividing by $\kappa^{\prime}$, which leads upon integration to an
expression of $\Psi_{A{\cal C}}$ in terms of $\kappa^{\prime}\rho_{A}$.
However, this cannot be true since $\Psi_{A{\cal C}}$ is arbitrary. Hence,
$\kappa^{\prime}\equiv 0$ i.e. $\kappa$ is a constant. Because the
normalization of wavefunctions is irrelevant, we may take $\kappa=1$ so that

$\mu_{A}(a_{0})=a_{0}\,\rho_{A}(a_{0})\,,\quad\lambda a_{0}\geq
0\,,\quad\mbox{and}\quad\widetilde{\mu}(q_{0})=A(q_{0})\,A^{\prime}(q_{0})^{2}\,\widetilde{\rho}(q_{0})\,.$
(7.199)

As a result, since $\mu_{A}$ and $\widetilde{\mu}$ are positive, the functions
$\rho_{A}(a_{0})$ and $\widetilde{\rho}(q_{0})$ appearing in the
Wheeler–DeWitt equations must also be positive for all $\lambda a_{0}\geq 0$,
and not only in the range $0<\lambda a_{0}<1$ as shown in Sect. 6.2. This is
necessary for the whole picture to be consistent.

$\bullet$ Moreover, the boundary term in Eq. (7.195) must vanish for all
functions of the Hilbert space,

$0=\left[\rho_{A}\Big{(}\Psi^{*}_{A1}\,{{\rm d}\Psi_{A2}\over{\rm
d}a_{0}}-{{\rm d}\Psi^{*}_{A1}\over{\rm
d}a_{0}}\,\Psi_{A2}\Big{)}\right]_{0}^{+\infty}\quad i.e.\quad
0=\left[\widetilde{\rho}\,\Big{(}\widetilde{\Psi}^{*}_{1}\,{{\rm
d}\widetilde{\Psi}_{2}\over{\rm d}q_{0}}-{{\rm
d}\widetilde{\Psi}^{*}_{1}\over{\rm
d}q_{0}}\,\widetilde{\Psi}_{2}\Big{)}\right]_{Q(0)}^{Q(+\infty)}\,.$ (7.200)

In Appendix A.5, we recover the result $\mu_{A}=a_{0}\rho_{A}$ and derive for
any complex functions $\Psi_{A_{1}}$, $\Psi_{A2}$ the following identity valid
for all $\lambda a_{0}\geq 0$,

$\displaystyle\mu_{A}\,\bigg{\\{}\Big{(}{H_{A}\over
N}\,\Psi_{A1}\Big{)}^{*}\,\Psi_{A2}$ $\displaystyle-\Psi_{A1}^{*}\,{H_{A}\over
N}\,\Psi_{A2}\bigg{\\}}$ $\displaystyle\equiv-{\hbar^{2}\over 12v_{3}}\,{{\rm
d}\over{\rm d}a_{0}}\left\\{\rho_{A}\Big{(}\Psi^{*}_{A1}\,{{\rm
d}\Psi_{A2}\over{\rm d}a_{0}}-{{\rm d}\Psi^{*}_{A1}\over{\rm
d}a_{0}}\,\Psi_{A2}\Big{)}\right\\}.$ (7.201)

Applying this result to wavefunctions annihilated by the Hamiltonian, the left
hand side is identically zero and we obtain

$\rho_{A}\Big{(}\Psi^{*}_{A1}\,{{\rm d}\Psi_{A2}\over{\rm d}a_{0}}-{{\rm
d}\Psi^{*}_{A1}\over{\rm d}a_{0}}\,\Psi_{A2}\Big{)}={\rm cst.}\,,\quad\lambda
a_{0}\geq 0\,.$ (7.202)

As a result, the vanishing of the boundary term in Eq. (7.200) is trivially
true for all solutions of the Wheeler–DeWitt equations. To put it another way,
the two dimensional vectorial space of solutions of the Wheeler–DeWitt
equation is a Hilbert space equipped with the Hermitian product defined in Eq.
(7.192) (or Eq. (7.193)), under which the Hamiltonian operator is Hermitian.

####  7.1.2 Crosscheck with quantum mechanics

We can check our conclusions, by applying them to the case where the path
integral measure ${\cal D}q$ corresponds to a field $q$ with canonical kinetic
term. As seen in Sect. 6.3, this amounts to choosing $q=A^{-1}(a)$ given in
Eq. (6.184), for which we know that $\widetilde{\rho}(q_{0})=(6v_{3})^{3\over
4}$ when $0<A(q_{0})<1$. Indeed, using Eq. (7.199), we find that
$\widetilde{\mu}(q_{0})$ is a constant

$\widetilde{\mu}(q_{0})=(6v_{3})^{-{1\over 4}}\quad\mbox{when}\quad
0<A(q_{0})<1\,.$ (7.203)

This is the correct answer, as it reproduces the well known prescription in
quantum mechanics where the wavefunction norm involves a constant measure.

###  7.2 Universality at the semi-classical level

The considerations of the previous subsection are valid for any choice of
positive function $\rho_{A}(a_{0})$, $\lambda a_{0}\geq 0$, in the WDW
equation and corresponding measure $\mu_{A}$ in the Hilbert space of
wavefunctions. An immediate consequence is that at the semi-classical level,
the probability amplitudes $\sqrt{\mu_{A}}\,\Psi_{A{\cal C}}$ are universal,
since Eqs. (6.178) and (6.182) yield

$\sqrt{\mu_{A}(a_{0})}\Psi_{A{\cal
C}}(a_{0})=\left\\{\begin{array}[]{lr}\displaystyle\sum_{\epsilon=\pm
1}N_{{\cal C}\epsilon}\,{\exp\\!\Big{[}\displaystyle\epsilon
s\,{2v_{3}\over\hbar\lambda^{2}}\,\big{(}1-(\lambda a_{0})^{2}\big{)}^{3\over
2}\Big{]}\over\displaystyle\big{(}1-(\lambda a_{0})^{2}\big{)}^{1\over
4}}\,(1+{\cal O}(\hbar))\,,&0<\lambda a_{0}<1\,,\\\
\displaystyle\sum_{\epsilon=\pm 1}M_{{\cal
C}\epsilon}\,{\exp\\!\Big{[}\displaystyle i\epsilon
s\,{2v_{3}\over\hbar\lambda^{2}}\,\big{(}(\lambda a_{0})^{2}-1\big{)}^{3\over
2}\Big{]}\over\displaystyle\big{(}(\lambda a_{0})^{2}-1\big{)}^{1\over
4}}\,(1+{\cal O}(\hbar))\,,&\lambda a_{0}>1\,.\end{array}\right.$ (7.204)

In fact, the choices of functions $\rho_{A}$ and $\omega_{A}$ in the
Wheeler–DeWitt equation (6.170) can only affect corrections in $\hbar$ to the
Hermitian product defined in Eq. (7.192) beyond the semi-classical
level.282828 For completeness, one must also show that this fact remains true
at $\lambda a_{0}\simeq 1$ and $\lambda a_{0}\to 0$. This turns out to be the
case, as can be demonstrated by applying the WKB method to the Wheeler–DeWitt
equation with arbitrary $\rho_{A}$ and $\omega_{A}$, in neighbourhoods of
$\lambda a_{0}=1$ and $\lambda a_{0}=0$. The solutions in these cases can be
written in terms of Airy functions and parabolic cylinder functions,
respectively.

####  7.2.1  Normalizability

Another consequence is that none of the wavefunctions of this Hilbert space is
normalizable. Indeed, when $|M_{{\cal C}+}|\neq|M_{{\cal C}-}|$ we have

$\mu_{A}(a_{0})\,|\Psi_{A{\cal C}}|^{2}\underset{a_{0}\to+\infty}{\sim}{1\over
a_{0}}\,\bigg{|}\sum_{\epsilon=\pm 1}M_{{\cal
C}\epsilon}\exp\\!\Big{[}\displaystyle i\epsilon
s\,{2v_{3}\over\hbar\lambda^{2}}\,\big{(}(\lambda a_{0})^{2}-1\big{)}^{3\over
2}\Big{]}\bigg{|}^{2}>{1\over a_{0}}\,\big{|}|M_{{\cal C}+}|-|M_{{\cal
C}-}|\big{|}^{2}\,,$ (7.205)

while for $|M_{{\cal C}+}|=|M_{{\cal C}-}|\neq 0$,

$\mu_{A}(a_{0})\,|\Psi_{A{\cal
C}}|^{2}\underset{a_{0}\to+\infty}{\sim}{2|M_{{\cal C}+}|^{2}\over
a_{0}}\Big{(}1+\cos\\!\big{[}2v_{3}\lambda\,a_{0}^{3}\big{]}\Big{)}\,.$
(7.206)

In these cases the “norms” are infinite due to the logarithmic divergences of
the integrals at $a_{0}\to+\infty$, while for $M_{{\cal C}+}=M_{{\cal C}-}=0$
the norm is of course vanishing. Hence, in the simplest minisuperspace model
we are considering, there are no normalizable states in the Hilbert space,
including the no-boundary “ground state.” So at best we can use these
wavefunctions to define relative probabilities, in terms of ratios of the
probability densities evaluated at different points of minisuperspace.
Interesting work on defining probabilities and constructing observables in
quantum cosmology includes Refs. [30, 13, 31, 32, 33].

In more realistic cases, the minisuperspace models comprise extra matter
degrees of freedom, besides the scale factor $a$. DeWitt considers at least
one matter field associated with a dust filling universe [8], while Hartle and
Hawking include a conformally coupled scalar field [1]. One may also consider
including an inflaton field with appropriate potential, for applications to
inflationary cosmology. As a result, the Hilbert spaces of solutions of the
corresponding Wheeler–DeWitt equations become infinite dimensional. As
discussed in Ref. [8], one may choose a suitable basis labelled by quantum
numbers associated with the matter Hamiltonian, and define a proper Hermitian
product. Presumably, in the more involved general cases, one can construct
normalizable solutions of the Wheeler–DeWitt equation, in terms of
superpositions of the basis states. Moreover, there should be superpositions
exhibiting classical behavior, which can be interpreted as wave-packets
“moving” in minisuperspace [8]. It would be desirable to construct
normalizable solutions in terms of path integrals, implementing the no-
boundary proposal. Notice however that for the conformally coupled scalar
field of [1], the Wheeler–DeWitt equation can be cast into a separable form,
and the large $a_{0}$ asymptotic behavior of the solutions becomes independent
of the matter Hamiltonian quantum number. So in this case, the wavefunction
factorizes in the large $a_{0}$ limit, exhibiting similar asymptotic behavior
to the one described in this work. As a result, the solutions in this
conformally coupled scalar field case continue to be non-normalizable.

####  7.2.2 Comparison with the literature

We can compare our analysis with that of Ref. [8]. In this work, the
minisuperspace model does not implement a cosmological term but takes into
account, as outlined above, matter particles of dust characterised by their
own Hamiltonian. The author makes a choice of a Wheeler–DeWitt equation
associated with the coupled system “scale factor + dust” corresponding to

$\rho_{A}(a_{0})=a_{0}^{p}\quad\mbox{with}\quad
p=-1\,,\quad\mbox{and}\quad\omega_{A}(a_{0})={7\over 16}\,{1\over
a_{0}^{2}}\,.$ (7.207)

From our discussion below Eq. (7.191), this choice of $\rho_{A}$ is valid if
one chooses to define the wavefunctions with the measure ${\cal D}q$
associated with the field

$q=Q(a)=\sqrt{a}\,.$ (7.208)

However, we do not know what the correct expression of the function
$\omega_{A}(a_{0})$ should be in this case. Finally, the Hermitian product
used in Ref. [8] uses a trivial measure $\mu_{A}$, which is consistent with
our analysis since

$\mu_{A}(a_{0})=1=a_{0}\,\rho_{A}(a_{0})\,.$ (7.209)

It is also interesting to mention the discrepancy between our results and some
conclusions reported in Ref. [1], where the discussion of the measure
$\mu_{A}$ is presented in the case of the minisuperspace model we study. They
consider the Wheeler–DeWitt equation in a form where
$\rho_{A}(a_{0})=a_{0}^{p}$ and $\omega_{A}(a_{0})=0$. From our analysis, we
conclude that the Hilbert space measure must be $\mu_{A}(a_{0})=a_{0}^{p+1}$.
However, the authors of Ref. [1] take $\mu_{A}(a_{0})=a_{0}^{p}$, which leads
to the conclusion that all solutions of the Wheeler–DeWitt equation are
normalizable. The origin of the disagreement is that instead of imposing
hermiticity of the Hamiltonian $H_{A}/N$, they impose hermiticity of what they
call the “Wheeler–DeWitt operator,” which is $a_{0}H_{A}/N$. In practice, this
amounts to changing $\mu_{A}\to\mu_{A}a_{0}$ everywhere in our derivation of
Sect. 7.1, and thus to replacing Eq. (7.199) with $\mu_{A}a_{0}=a_{0}\rho_{A}$
i.e. $\mu_{A}=\rho_{A}$.

##  8 Conclusion

In this work, we have considered the minisuperspace model describing a
homogeneous and isotropic universe with positive cosmological constant. More
specifically, we focused on the wavefunction defined as a Euclidean path
integral satisfying the “no-boundary proposal.”

The invariance under redefinitions of the lapse function imposes a gauge
fixing of the reparametrization symmetry of Euclidean time. To do so, we have
applied a procedure from first-quantized string theory, where quantum
amplitudes are defined as path integrals of two-dimensional field theories. In
both cases, the gauged-fixed path integrals involve integrals over the moduli
spaces of metrics, and the results do not depend on the gauge choice. On the
contrary, the redefinitions of the scale factor lead to inequivalent choices
of diffeomorphism-invariant path-integral measures. All prescriptions yield
different forms of the Wheeler–DeWitt equation and thus a priori distinct
quantum theories. However, the quantum Hilbert spaces of wavefunctions lead to
universal predictions, at least at the semi-classical level. It would be
highly interesting to determine whether this remains true beyond this level of
approximation, or even exactly.

In our analysis, the ground-state wave functions are evaluated using the
steepest-descent method when the scale factor on the boundary 3-sphere
satisfies $0<\lambda a_{0}<1$. We hope to extend this result in the near
future to the case $\lambda a_{0}>1$, for which the instanton solutions are
complex. Computing the path integrals for $\lambda a_{0}=0$ is also
challenging since there exists a one-parameter family of instanton solutions
in this case, implying that a functional determinant possesses a vanishing
eigenvalue.

The techniques we have used to fix the gauge of the diffeomorphisms as well as
to compute the quantum fluctuations of the scale factor can be implemented in
richer setups. In particular, it would be very natural to consider
minisuperspace models involving scalar fields with potentials possessing non-
negative (local) minima, and possibly to account for the statistically natural
emergence of a long enough period of inflation. Taking into account extra
degrees of freedom also allows for the existence of wave-packets “moving” in
minisuperspace [8]. Hence, it would be very interesting to see whether
normalizable states constructed as path integrals obeying the no-boundary
proposal exist. In this work, we have considered the amplitude describing the
transition from “nothing”, the “space” reduced to the empty set, to a
3-sphere. This serves as the definition of the “ground state” wavefunction of
Hartle and Hawking. It would be worth investigating whether more general
transitions may be related to “excited states.” This may require insertions of
operators in the path integrals, in the spirit of insertions of vertex
operators in string amplitudes, which describe excited states of the string.

## Acknowledgements

The authors would like to thank Lihui Liu for useful inputs during the
realization of this work. H.P. would like to thank the University of Cyprus
for hospitality.

## Appendix A

For completeness, we display in this Appendix proofs of simple theorems and
technical details used in the core of the paper.

###  A.1 On the fiducial metric

######  Theorem 1.

Any metric $g_{00}$ in class $\ell$, defined in an arbitrary domain
$[x^{0}_{\rm Ei},x^{0}_{\rm Ef}]$, can be written as
$g_{00}=\hat{g}^{\xi}_{00}[\ell]$ for some $\xi\in{\rm
Diff}[\hat{g}_{00}[\ell]]$.

In fact, any diffeomorphism $\xi$ defined on the domain $[\hat{x}^{0}_{\rm
Ei},\hat{x}^{0}_{\rm Ef}]$ and satisfying the separable differential equation

${{\rm d}\xi\over{\rm d}\hat{x}^{0}_{\rm
E}}=\sqrt{\hat{g}_{00}[\ell](\hat{x}_{\rm E}^{0})\over g_{00}(\xi(\hat{x}_{\rm
E}^{0}))}$ (A.210)

yields $\hat{g}_{00}^{\xi}[\ell](\xi(\hat{x}^{0}_{\rm
E}))=g_{00}(\xi(\hat{x}^{0}_{\rm E}))$ thanks to Eq. (2.10). Choosing

$\xi(\hat{x}^{0}_{\rm E})=\int_{\hat{x}^{0}_{\rm Ei}}^{\hat{x}^{0}_{\rm
E}}{\rm d}\hat{x}_{\rm E}^{0\prime}\,\sqrt{\hat{g}_{00}[\ell](\hat{x}_{\rm
E}^{0\prime})\over g_{00}(\xi(\hat{x}_{\rm E}^{0\prime}))}+x^{0}_{\rm Ei}$
(A.211)

also leads to $\xi(\hat{x}^{0}_{\rm Ei})=x^{0}_{\rm Ei}$. Applying Eq. (2.12)
for $\hat{g}_{00}[\ell]$ and using the above results, we have finally

$\ell=\int_{\xi(\hat{x}^{0}_{\rm Ei})}^{\xi(\hat{x}^{0}_{\rm Ef})}{\rm
d}\hat{x}^{\xi 0}_{\rm E}\,\sqrt{\hat{g}_{00}^{\xi}[\ell](\hat{x}^{\xi 0}_{\rm
E})}=\int_{x_{\rm Ei}^{0}}^{\xi(\hat{x}^{0}_{\rm Ef})}{\rm d}\hat{x}^{\xi
0}_{\rm E}\,\sqrt{g_{00}(\hat{x}^{\xi 0}_{\rm E})}\,,$ (A.212)

which implies that $\xi(x^{0}_{\rm Ef})=x^{0}_{\rm Ef}$ since $g_{00}$ is in
class $\ell$.

###  A.2 Isometries of the line segment

######  Theorem 2.

The isometry group of a line segment with metric $g_{00}$ is $\mathbb{Z}_{2}$
generated by orientation reversal.

Any isometry ${\cal I}$ satisfies $g^{\cal I}_{00}=g_{00}$. Since $g_{00}$ in
class $\ell$ can be obtained by acting on $\hat{g}_{00}[\ell]$ with a
diffeomorphism $\xi$, this equality can be written as $\hat{g}^{{\cal
I}\circ\xi}_{00}[\ell]=\hat{g}^{\xi}_{00}[\ell]$. Applying on both sides the
diffeomorphism $\xi^{-1}$, we obtain $\hat{g}^{\xi^{-1}\circ{\cal
I}\circ\xi}_{00}[\ell]=\hat{g}^{\xi^{-1}\circ\xi}_{00}[\ell]=\hat{g}_{00}[\ell]$.
Using Eq. (2.10), this means that

${1\over[(\xi^{-1}\circ{\cal
I}\circ\xi)^{\prime}]^{2}}\,\hat{g}_{00}[\ell]=\hat{g}_{00}[\ell]\,.$ (A.213)

Since we are free to choose the form of the fiducial metric, let us make for
this proof the particular choice

$\hat{g}_{00}[\ell](\tau)=\ell^{2}\quad\mbox{defined for}\quad\tau\in[0,1]\,.$
(A.214)

In that case, Eq. (A.213) becomes $(\xi^{-1}\circ{\cal
I}\circ\xi)^{\prime}=\pm 1$, which leads to $(\xi^{-1}\circ{\cal
I}\circ\xi)(\tau)=\pm\tau+{\rm cst.}$. Imposing the sets
$\\{(\xi^{-1}\circ{\cal I}\circ\xi)(0),(\xi^{-1}\circ{\cal I}\circ\xi)(1)\\}$
and $\\{0,1\\}$ to be equal for the domains of definition of both sides of Eq.
(A.213) to be the same, we are left with only two possibilities:
$(\xi^{-1}\circ{\cal I}\circ\xi)(\tau)=\tau$ or $(\xi^{-1}\circ{\cal
I}\circ\xi)(\tau)=1-\tau$. As a result we have ${\cal I}=\xi\circ\xi^{-1}={\rm
Id}$ or ${\cal I}=\xi\circ(\tau\to 1-\tau)\circ\xi^{-1}$. The isometry group
is thus

$\mathbb{Z}_{2}=\big{\\{}{\rm Id},{\cal
R}\big{\\}}\,,\quad\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{where}\quad\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal
R}=\xi\circ(\tau\to 1-\tau)\circ\xi^{-1}\,,\quad{\cal R}\circ{\cal R}={\rm
Id}\,.$ (A.215)

###  A.3 Modulus of the line segment

######  Theorem 3.

The moduli space of a line segment is of real dimension 1.

The moduli correspond to the deformations of the metric that are “orthogonal”
to those obtained by diffeomorphisms. Let us consider any metric $g_{00}$ in
class $\ell$ and defined on $[x^{0}_{\rm Ei},x^{0}_{\rm Ef}]$. We look for an
infinitesimal deformation, denoted $\delta^{\prime}g_{00}$, that is orthogonal
to all deformations $\delta_{\xi}g_{00}$, where $\xi\in{\rm Diff}[g_{00}]$.

Using Eq. (2.11) and the definition (3.28), we have to solve

$\displaystyle 0$
$\displaystyle=\big{(}\delta^{\prime}g,\delta_{\xi}g\big{)}_{\ell}=\big{(}\delta^{\prime}g,-2\nabla\delta
x_{\rm E}\big{)}_{\ell}$ $\displaystyle=\int_{x^{0}_{\rm Ei}}^{x^{0}_{\rm
Ef}}{\rm d}x^{0}_{\rm
E}\,\sqrt{g_{00}}\;\delta^{\prime}g_{00}\,(-2)\nabla_{0}\delta x^{0}_{\rm
E}\;g^{00}$
$\displaystyle=\Big{[}\sqrt{g_{00}}\;\delta^{\prime}g_{00}\,(-2)\,\delta
x^{0}_{\rm E}\;g^{00}\Big{]}_{x^{0}_{\rm Ei}}^{x^{0}_{\rm
Ef}}+\int_{x^{0}_{\rm Ei}}^{x^{0}_{\rm Ef}}{\rm d}x^{0}_{\rm
E}\,\sqrt{g_{00}}\;2\nabla^{0}\delta^{\prime}g_{00}\,\delta x^{0}_{\rm E}$
$\displaystyle=\Big{[}\sqrt{g_{00}}\;\delta^{\prime}g_{00}\,(-2)\,\delta
x^{0}_{\rm E}\;g^{00}\Big{]}_{x^{0}_{\rm Ei}}^{x^{0}_{\rm
Ef}}+\big{(}2\nabla\delta^{\prime}g,\delta x_{\rm E}\big{)}_{\ell}\,.$ (A.216)

Let us restrict for a moment to the diffeomorphisms obeying

$\delta x^{0}_{\rm E}(x^{0}_{\rm Ei})=0\,,\qquad\delta x^{0}_{\rm
E}(x^{0}_{\rm Ef})=0\,,$ (A.217)

so that the boundary term in Eq. (A.216) is absent. Since $\delta x^{0}_{\rm
E}$ is still arbitrary on $(x^{0}_{\rm Ei},x^{0}_{\rm E{f}})$, we obtain

$0=2\nabla^{0}\delta^{\prime}g_{00}=2\,\partial_{0}\big{(}g^{00}\,\delta^{\prime}g_{00}\big{)}\,,$
(A.218)

whose solution can be written as

$\delta^{\prime}g_{00}=2\,{\delta\ell\over\ell}\,g_{00}\,,$ (A.219)

where $\delta\ell$ is any integration constant.

Turning back to a generic diffeomorphism, Eq. (A.216) becomes

$0=\Big{[}\sqrt{g_{00}}\;2\,{\delta\ell\over\ell}\,(-2)\,\delta x^{0}_{\rm
E}\Big{]}_{x^{0}_{\rm Ei}}^{x^{0}_{\rm
Ef}}=-4\,\delta\ell\,\big{[}\delta\tau\big{]}_{0}^{1}\,,$ (A.220)

where we have used the fact that $\sqrt{g_{00}}\,\delta x^{0}_{\rm E}$ is
diffeomorphism invariant and, therefore, can be expressed in the coordinate
system associated with the metric in class $\ell$ given in Eq. (A.214).
However, we know that

$\delta\tau(1)-\delta\tau(0)=0\,,$ (A.221)

since otherwise a diffeomorphism would change the proper length of a segment
of constant metric. As a result, Eq. (A.220) is trivial in the sense that it
leaves $\delta\ell$ arbitrary.

Integrating Eq. (A.219), one obtains

$g_{00}[\ell]=\ell^{2}\,g_{00}[1]\,,$ (A.222)

where the classes are indicated in brackets. Thus the only moduli deformation
of a line segment is that associated with the variation of its length $\ell$.

Notice that the above equation also shows that the diffeomorphisms acting on
$g_{00}[\ell]$ are actually acting only on $g_{00}[1]$. Therefore, the
reparametrization group of the line segment is independent of its length. As a
result,

${\rm Diff}[g^{\xi}_{00}[\ell]]={\rm Diff}[g_{00}[\ell]]={\rm
Diff}[g_{00}[1]]\equiv{\rm Diff}\,,$ (A.223)

where the last equality means that the group depends only on the topology of
the line segment.

###  A.4 Independence on the fiducial metric

In this Appendix, we check explicitly that the Faddeev–Popov Jacobian and the
contributions $Z_{\epsilon}(a_{0})$ to the semi-classical expression of the
wave function $\Psi(a_{0})$ are independent of the choice of fiducial metric,
i.e. that that they are invariant under diffeomorphisms.

To this end, let us consider the change of coordinate
$\tau(\hat{y}^{\hat{0}}_{\rm E})$ that results in the change of lapse function

$\ell\,{\rm d}\tau=\sqrt{\hat{g}_{\hat{0}\hat{0}}[\ell](\hat{y}^{\hat{0}}_{\rm
E})}\;{\rm d}\hat{y}^{\hat{0}}_{\rm E}\,.$ (A.224)

The commuting tensors and the ghosts in Sect. 3.2 transform as

$\displaystyle\beta^{\hat{0}\hat{0}}(\hat{y}^{\hat{0}}_{\rm E})$
$\displaystyle=\sum_{k\geq
0}\boldsymbol{\beta}_{k}\,\chi^{\hat{0}\hat{0}}_{k}(\hat{y}^{\hat{0}}_{\rm
E})\,,\qquad\delta x_{\rm E\hat{0}}(\hat{y}^{\hat{0}}_{\rm E})=\sum_{k\geq
1}\boldsymbol{\gamma}_{k}\,\sigma_{\hat{0},k}(\hat{y}^{\hat{0}}_{\rm E})\,,$
$\displaystyle b^{\hat{0}\hat{0}}(\hat{y}^{\hat{0}}_{\rm E})$
$\displaystyle=\sum_{k\geq 0}\boldsymbol{\rm
b}_{k}\,\chi^{\hat{0}\hat{0}}_{k}(\hat{y}^{\hat{0}}_{\rm
E})\,,\qquad\,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \,c_{0}(\tau)=\sum_{k\geq 1}\boldsymbol{\rm
c}_{k}\,\sigma_{\hat{0},k}(\hat{y}^{\hat{0}}_{\rm E})\,,$ (A.225)

where the expansion modes are given by

$\chi^{\hat{0}\hat{0}}(\hat{y}^{\hat{0}}_{\rm
E})={\ell^{2}\over\hat{g}_{\hat{0}\hat{0}}[\ell](\hat{y}^{\hat{0}}_{\rm
E})}\,\chi^{00}(\tau(\hat{y}^{\hat{0}}_{\rm
E}))\,,\qquad\sigma_{\hat{0},k}(\hat{y}^{\hat{0}}_{\rm
E})={\sqrt{\hat{g}_{\hat{0}\hat{0}}[\ell](\hat{y}^{\hat{0}}_{\rm
E})}\over\ell}\,\sigma_{0,k}(\tau(\hat{y}^{\hat{0}}_{\rm E}))\,.$ (A.226)

However, Eqs. (3.36), (3.37), (3.38) remain unchanged, since they are derived
from the reparametrization-invariant inner product given in Eq. (3.28). Hence,
the computation of the Faddeev–Popov Jacobian displayed in Eq. (3.39) is valid
in any gauge.

Moreover, denoting $\xi(\tau)\equiv\tau(\hat{y}^{\hat{0}}_{\rm E})$, the
scalar field $\delta a$ in Sect. 4.3.1 is transformed into $\delta a^{\xi}$,
which can be expanded as

$\displaystyle\delta a^{\xi}(\hat{y}^{\hat{0}}_{\rm E})$
$\displaystyle\equiv\delta a(\tau(\hat{y}^{\hat{0}}_{\rm E}))=\sum_{k\geq
1}\boldsymbol{\delta\rm a}_{k}\,\phi^{\epsilon\xi}_{k}(\hat{y}^{\hat{0}}_{\rm
E})\,,$
$\displaystyle\mbox{where}\quad\phi^{\epsilon\xi}_{k}(\hat{y}^{\hat{0}}_{\rm
E})$ $\displaystyle\equiv\phi^{\epsilon}_{k}(\tau(\hat{y}^{\hat{0}}_{\rm
E}))\,,\quad\leavevmode\nobreak\ \leavevmode\nobreak\
(\phi^{\epsilon\xi}_{k},\phi^{\epsilon\xi}_{k^{\prime}})_{\bar{\ell}_{\epsilon}}=\delta_{kk^{\prime}}\,.$
(A.227)

The modes $\phi_{k}^{\epsilon\xi}$ are eigenvectors of the operator ${\cal
S}_{\epsilon}^{\xi}$ in the new coordinate system, with eigenvalues
$\nu^{\epsilon}_{k}$, since

${\cal S}^{\xi}\phi_{k}^{\epsilon\xi}(\hat{y}^{\hat{0}}_{\rm E})\equiv({\cal
S}_{\epsilon}\phi^{\epsilon}_{k})^{\xi}(\hat{y}^{\hat{0}}_{\rm E})=({\cal
S}_{\epsilon}\phi^{\epsilon}_{k})(\tau(\hat{y}^{\hat{0}}_{\rm
E}))=\nu_{k}^{\epsilon}\phi^{\epsilon}_{k}(\tau(\hat{y}^{\hat{0}}_{\rm
E}))=\nu_{k}^{\epsilon}\phi_{k}^{\epsilon\xi}(\hat{y}^{\hat{0}}_{\rm E})\,.$
(A.228)

As a result, the argument of the exponential in the definition of
$Z_{\epsilon}(a_{0})$ in Eq. (4.68), along with the norm of the scalar field
and its path-integral measure are invariant,

$\displaystyle(\delta a^{\xi},{\cal S}^{\xi}_{\epsilon}\delta
a^{\xi})_{\bar{\ell}_{\epsilon}}$ $\displaystyle=(\delta a,{\cal
S}_{\epsilon}\delta
a)_{\bar{\ell}_{\epsilon}}=\nu^{\epsilon}_{k}\,(\boldsymbol{\delta\rm
a}_{k})^{2}\,,$ $\displaystyle||\delta a^{\xi}||^{2}_{\bar{\ell}_{\epsilon}}$
$\displaystyle=(\delta a^{\xi},\delta a^{\xi})_{\bar{\ell}_{\epsilon}}=(\delta
a,\delta a)_{\bar{\ell}_{\epsilon}}=||\delta
a||^{2}_{\bar{\ell}_{\epsilon}}\leavevmode\nobreak\ \leavevmode\nobreak\
\Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal D}\delta
a^{\xi}={\cal D}\delta a=\bigwedge_{k\geq 1}{\rm d}\boldsymbol{\delta\rm
a}_{k}\,.$ (A.229)

This shows the gauge invariance of $Z_{\epsilon}(a_{0})$.

###  A.5 Hermiticity of the Hamiltonian

In this subsection, we retrieve the results of Sect. 7.1 using refined
arguments that provide insight. The following proofs include a generalization
of some treatment given in Ref. [8] in order to take into account a positive
measure $\mu_{A}(a_{0})$ to be determined and an unknown function
$\rho_{A}(a_{0})$ in the definition of the operator $H_{A}/N$.

Let us consider the integral

${\cal J}=\int_{0}^{+\infty}{\rm
d}a_{0}\,\mu_{A}\,\Xi\,\Big{\\{}\Big{(}{H_{A}\over
N}\,\Psi_{A1}\Big{)}^{*}\,\Psi_{A2}-\Psi_{A1}^{*}\,{H_{A}\over
N}\,\Psi_{A2}\Big{\\}}\,,$ (A.230)

where $\Psi_{A1}$, $\Psi_{A2}$ are arbitrary complex functions. Moreover,
$\Xi$, which is also complex, is an arbitrary test function. This means that
it is identically vanishing outside some closed interval $[a_{\rm min},a_{\rm
max}]\subset\mathbb{R}_{+}^{*}$. Using the notation of Eq. (7.192), we can
write the first term as follows

$\int_{0}^{+\infty}{\rm d}a_{0}\,\mu_{A}\,\Xi\,\Big{(}{H_{A}\over
N}\,\Psi_{A1}\Big{)}^{*}\,\Psi_{A2}=\big{\langle}{H_{A}\over
N}\,\Psi_{A1},\Xi\,\Psi_{A2}\big{\rangle}=\big{\langle}\Psi_{A1},{H_{A}\over
N}\,(\Xi\,\Psi_{A2})\big{\rangle}\,,$ (A.231)

where we have assumed that Eq. (7.197) holds. Indeed, the last equality is
obtained by integrating by parts and by noticing that no boundary term arises
thanks to presence of the test function which vanishes identically in
neighborhoods of $a_{0}=0$ and $+\infty$. Hence, we obtain

$\displaystyle{\cal J}$ $\displaystyle=\int_{0}^{+\infty}{\rm
d}a_{0}\,\mu_{A}\,\Psi_{A1}^{*}\,\Big{\\{}{H_{A}\over
N}\,(\Xi\Psi_{A2})-\Xi\,{H_{A}\over N}\,\Psi_{A2}\Big{\\}}$ (A.232)
$\displaystyle={\hbar^{2}\over 12v_{3}}\int_{0}^{+\infty}{\rm
d}a_{0}\,\mu_{A}\,\Psi_{A1}^{*}\,{1\over a_{0}\rho_{A}}\,\Big{\\{}{{\rm
d}\over{\rm d}a_{0}}\Big{(}{{\rm d}\Xi\over{\rm
d}a_{0}}\,\rho_{A}\,\Psi_{A2}\Big{)}+{{\rm d}\Xi\over{\rm
d}a_{0}}\,\rho_{A}\,{{\rm d}\Psi_{A2}\over{\rm
d}a_{0}}\Big{\\}}\phantom{\\!\\!\overset{\displaystyle|}{|}}$
$\displaystyle=-{\hbar^{2}\over 12v_{3}}\int_{0}^{+\infty}{\rm
d}a_{0}\,\Xi\,{{\rm d}\over{\rm d}a_{0}}\Big{\\{}\rho_{A}\Big{(}{\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\,{{\rm d}\Psi_{A2}\over{\rm d}a_{0}}-{{\rm
d}\over{\rm d}a_{0}}\Big{(}{\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\Big{)}\Psi_{A2}\Big{)}\Big{\\}}\,,\phantom{\\!\\!\overset{\displaystyle|}{|}}$
(A.233)

where the second line follows from the definition of $H_{A}/N$ given in Eq.
(6.170), while the last one is obtained by integrating by parts. Again, no
boundary term arises thanks to the test function. Comparing the last
expression with Eq. (A.230) and remembering that the test function is
arbitrary,292929One can choose Dirac distributions centered at an arbitrary
point in $\mathbb{R}_{+}^{*}$. we conclude that the integrands are identical,
namely

$\displaystyle\mu_{A}\,\bigg{\\{}\Big{(}{H_{A}\over
N}\,\Psi_{A1}\Big{)}^{*}\,$ $\displaystyle\Psi_{A2}-\Psi_{A1}^{*}\,{H_{A}\over
N}\,\Psi_{A2}\bigg{\\}}$ $\displaystyle=-{\hbar^{2}\over 12v_{3}}\,{{\rm
d}\over{\rm d}a_{0}}\left\\{\rho_{A}\\!\left({\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\,{{\rm d}\Psi_{A2}\over{\rm d}a_{0}}-{{\rm
d}\over{\rm d}a_{0}}\Big{(}{\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\Big{)}\Psi_{A2}\right)\right\\},\phantom{\\!\\!\overset{\displaystyle|}{|}}$
(A.234)

which is the local version of Eq. (7.195).

Since $\Psi_{A1}$, $\Psi_{A2}$ are arbitrary complex functions, we can
specialize to $\Psi_{A1}=\Psi_{A2}$ real, which yields

$0={{\rm d}\over{\rm d}a_{0}}\Big{\\{}\rho_{A}\Psi_{A1}^{2}\,{{\rm d}\over{\rm
d}a_{0}}\Big{(}{\mu_{A}\over a_{0}\rho_{A}}\Big{)}\Big{\\}}\,.$ (A.235)

The choice $\Psi_{A1}\equiv 1/\sqrt{\rho_{A}}$ leads to

${\mu_{A}\over a_{0}\rho_{A}}=\kappa+\kappa_{1}a_{0}\,,$ (A.236)

where $\kappa$, $\kappa_{1}$ are constants. Taking instead $\Psi_{A1}$ not
identically equal to $1/\sqrt{\rho_{A}}$ yields finally $\kappa_{1}=0$, i.e.
$\mu_{A}=\kappa a_{0}\rho_{A}$. Hence we obtain Eq. (7.201).

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# Cavitating Flow Behind a Backward Facing Step

Anubhav Bhatt, Harish Ganesh, S.L. Ceccio  University Of Michigan, Ann Arbor

###### Abstract

Flow topology and unsteady cavitation dynamics in the wake of a backward
facing step was investigated using high speed videography and time-resolved
X-Ray densitometry, along with static and dynamic pressure measurements. The
measurements are used to inform the understanding of underlying mechanisms of
observed flow dynamics as they are related to shock wave induced
instabilities. The differences in cavity topology and behaviour at different
cavitation numbers are examined to highlight flow features such as the pair of
cavitating spanwise vortices in the shear layer and propagating bubbly shock
front. The shock speeds at different cavitation conditions are estimated based
on void-fraction measurements using the X-T diagram and compared to those
computed using the one-dimensional Rankine Hugoniot jump condition. The two
speeds(computed and measured) are found to be comparable. The effect of
compressibility of the bubbly mixture is determined by estimating the local
speed of sound using homogeneous ‘frozen model’ and homogeneous ‘equilibrium
model’. The current estimation of sound speed suggests that expression used to
determine the local speed of sound is closer to the classic ‘frozen model’
than the ‘equilibrium model’ of speed of sound.

###### keywords:

Shockwaves, speed of sound, X-Ray densitometry

††journal:  International Journal of Multiphase Flows

##  1 Introduction

Developed cavitation in reattaching shear flows can result in unwanted noise,
erosion and vibration various hydrodynamic applications. Despite being
extensively studied and reported, the complexity of these turbulent, high void
fraction flow makes direct interrogation quite challenging. Given the
geometric simplicity and the availability of previous results detailing the
single phase phenomena, the flow behind a backward facing step is an ideal
candidate for experimental investigation of the effect of cavitation on
turbulent shear layers.

A schematic of the single-phase flow topology downstream of the backward
facing step is shown in Figure 1. The flow has a sharp separation point, fixed
to the edge of the step. As the single phase flow separates from the edge, the
free shear flow rolls up to create large scale spanwise vortices, similar to
those of mixing free shear layer structures (Troutt et al. [1984]). This shear
flow will ultimately reattach to form a recirculation bubble underneath the
shear layer. Further downstream the wall effects start to dominate (Bradshaw &
Wong [1972]) as shear layer curves sharply downward in the reattachment region
and impinges on the wall. In the reattachment region, the shear layer is
subjected to distorting forces due to effects of stabilizing curvature,
adverse pressure gradient and strong interaction with the wall (Eaton &
Johnston [1981]). Downstream of the reattachment a new sub-boundary layer
begins to form. It can be clearly seen that this geometrically simple flow has
almost all the features of separating and reattaching shear flows including a
mixing layer, recirculation, re-attachment and a redeveloping boundary layer.
In addition, for backward facing step flow configurations, the boundary of the
flow above the step may be in close proximity to the step to influence the
step flow, making the expansion ratio also a parameter of interest (Biswas et
al. [2004]).

Bernal & Roshko [1986] developed a plane mixing-layer model highlighting the
streamwise vortex structures and noted structures of dominant spanwise
vortices.They reported the ‘braid’ region between adjacent spanwise vortices
to contain an array of streamwise vortices which snake between neighboring
eddies. The spanwise vortex structures shed near the separation point can be
subjected to vortex pairing interaction upstream of reattachment region,
indicating that the growth of the separated layer may be strongly dependent on
the pairing process (Troutt et al. [1984]). Using PIV measurements and pattern
recognition Scarano et al. [1999] presented details of the flow structures
observed within the shear layer.

From a dynamics perspective, a low frequency “flapping motion” in the whole
separated region has been observed (Eaton & Johnston [1981]). One possible
explanation for this motion is attributed to the mass imbalance between shear
layer entrainment from recirculation region and re-entrant flow near the
reattachment point(Eaton & Johnston [1981]). This imbalance is said to be
caused by short term breakdown of spanwise vortices in shear layer that leads
to decrease in the entrainment rate, causing a change in volume of
recirculating fluid. Another explanation is attributed to a reduction in the
amount of mass engulfed by the separation bubble (Driver et al. [1987])
causing the bubble to momentarily collapse. Yet another explanation of
flapping motion states that the re-attachment point is unsteady and it moves
upstream intermittently in bursts, that splits shear layer into two halves and
compresses the fluid beneath the shear layer (Hasan [1992]). This compressed
fluid is subsequently ejected, pushing shear layer outward and causing self-
sustained flapping. The flapping motion occurs at a non-dimensional frequency
of O(1) (Spazzini et al. [2001], Wee et al. [2004]).

Liquid flows in the wake of the step can experience cavitation if the local
pressure in the shear layer falls below the vapor pressure. For turbulent
shear flows, cavitation inception(defined as the first occurrence of
observable cavitation) is known to first occur in streamwise braids as they
are stretched between the spanwise vortices of the shear layer (Katz & O’hern
[1986] ; O’hern [1990]). At this stage, cavitation bubbles are hypothesized to
decouple the stretching and rotation rates of streamwise vortices (Belahadji
et al. [1995]) and known to not significantly alter bulk flow topology (Iyer &
Ceccio [2002]).

Figure 1: Topology of the non-cavitating flow behind a backward facing step.

As the extent of the cavitation increases, it can influence the vorticity in
the shear layer, and gas can begin to fill the separated region of the flow.
For cavitating wakes, the presence of developed cavitation has a significant
impact on large scale structure dynamics like vortex spacing, shedding
frequency and convective speeds of spanwise structures (Franc & Michel [1983];
Young & Holl [1966]; Aeschlimann et al. [2011]; Wu [2019]).

Classical re-entrant jet driven shedding and propagating bubbly shock-wave
driven shedding are two known mechanisms that govern shedding dynamics in
developed partial cavities (Ganesh et al. [2016]; Wu et al. [2019]). The re-
entrant liquid flows are primarily driven by adverse pressure gradient at the
cavity closure (Callenaere et al. [2001]). The bubbly shockwaves on the other
hand are known to occur due to the effect of compressibility of the vapor-
liquid mixture that reduces the local speed of sound, thereby driving flow to
go locally supersonic. These features have also been observed in separated
cavity flows formed behind obstacles (Barbaca et al. [2019]). A proper
expression for the speed of sound is necessary to accurately estimate the
underlying dynamics of a rich bubbly shock, thereby explaining shock-wave
induced instabilities that may effect cavity shedding.

In the current study, we will examine the flow features of developed cavities
formed in the wake of a backward facing step, and quantitatively understand
the underlying cavity dynamics through high speed videography and time
resolved X-ray densitometry. Our goal is to both gain a better understanding
of the flow physics as well as to develop a data set that can be used for the
validation of computational models. The manuscript is divided into five parts.
First, we present the experimental setup; we then define three flow regimes of
interest (Type A, B and C) and describe their flow dynamics (cavity shedding
frequency and pressure inside cavity). Finally we relate these dynamics to the
underlying effect of flow compressibility.

##  2 Experimental Setup

Experiments were carried out in the Michigan 9” recirculating cavitation
channel. The upper leg of the tunnel has 6:1 contraction leading to 228.6 mm
diameter section that leads to 209.6 mm by 209.6 mm square section with
filleted corners. To minimize the baseline attenuation in X-ray densitometry
for non-cavitating flow, the channel is further reduced to 78.4 mm by 78.4 mm
test section. A fifth order polynomial profile (f(x)) leads to the flat ramp
of the backward facing step. Step height (H) was chosen to match the blockage
used by researchers at Johns Hopkins(agarwal2018cavitating). The test section
profile is shown in Figure 2.

The inlet velocity ($u_{0}$) was measured by measuring the pressure drop
across the 6:1 contraction using Setra 230, 0-68 kPa, differential pressure
transducer with an accuracy of 0.25% of full scale (FS). From this measured
$u_{0}$ velocity at step($u_{H}$) is determined using mass
conservation($h*h*u_{0}$ = $(h-H)*h*u_{H}$) For current dataset $u_{H}$ of 8,
10 and 12 m/s are maintained for $Re_{H}$ = $8.3\cdot{10^{4}}$,
$10.6\cdot{10^{4}}$ and $12.7\cdot{10^{4}}$ respectively to operate the water
tunnel within practical inlet pressures ($p_{0}$). Inlet pressure ($p_{0}$)
was varied from inception to before super-cavitation for a $\sigma_{0}$ range
of 0.5 to 1.0. $p_{0}$ was measured using Omega PX409-030A5V, 0-206 kPa static
pressure transducer (0.08 % FS accuracy). Pressure $p_{1}$ was measured using
differential pressure transducer Omega PX2300, 0-34 kPa (0.25 % FS accuracy)
between pressure taps at $p_{0}$ and $p_{1}$. Omega PX409-005AI-EH, 0-34 kPa
(0.08 % FS accuracy) static pressure transducer measures pressure at X = 1.5H
($p_{2}$); while Omega PX203, 0-206 kPa (0.25 % FS accuracy) static pressure
transducer measures pressure at X = 8H ($p_{3}$). Dynamic pressure transducer
PCB 113M231, 7.04 $\mu A$/kPa sensitivity is used to measure dynamic pressure
dp. The signal is acquired at a sampling frequency of 50 kHz for 3 seconds.
Finally the overall pressure drop across the test section is measured using
Omega RX2300, 0-34 kPa (0.25 % FS accuracy). All pressure taps on optically
accessible acrylic windows have 1.6 mm diameter. All the static and dynamic
pressure measurement locations are summarized in Table 1. The dissolved oxygen
was maintained below 20 % of saturation for all experiments.

2-D planar PIV was done to characterize the single phase flow of the setup by
determining the boundary layer thickness at the step and reattachment length.
High speed cinematography (HS) was performed using time synchronized Phantom
710 cameras at 5000 fps for 1.5 s with both top and east side view.
Cinematographic X-ray densitometry system (Ganesh et al. [2016],
makiharju2013time) was used to make spanwise-averaged quantitative void
fraction ($\alpha$) measurements at 1000 fps for 0.787 s exposure time and
0.125 mm spatial resolution.

Figure 2: Schematic of the backward facing step mounted in the water tunnel.
Top and side field of view configurations for the high speed videography are
shown. For X-ray densitometry a 115 mm by 28 mm view, roughly similar to the
highlighted side view is taken (see Figure 5)

.

Table 1: Inlet parameters, transducer locations and geometry of the step.

##  3 Results

The experiments were conducted by keeping the velocity at step $u_{H}$, fixed
at 8, 10 or 12 m/s and reducing inlet pressure ($p_{0}$) from inception (near
$\sigma_{0}=1$) to near super-cavitation (near $~{}\sigma_{0}=0.6$). As seen
in previous literature (O’hern [1990]), inception first occurs in streamwise
braids, away from the separation point. The inception condition is shown in
Figure 3. On reducing $p_{0}$ further, greater extent of cavitation occurs
primarily between X = -3H to -5H, classified as Type A or ‘developing
cavitation’ phase of the flow. Developing cavitation is characterized by the
presence of vapor bubbles in spanwise vortices of the shear layer. A high-
speed video snapshot of developing cavitation is depicted in Figure 4(a).

Figure 3: Cavitation inception in the streamwise direction is highlighted
(streamwise ’braids’ Katz & O’hern [1986]). Intermittent cavitation occurs at
$\sigma_{0}=0.98$ for $u_{H}$ = 10 m/s. The red lines highlight the edge of
the step.

With a reduction in inlet pressure $p_{0}$, unsteady cavitation designated as
Type B is observed. Type B self-sustained shedding cavitation is the regime
where periodic oscillation of the cavity length occurs at comparatively higher
frequency (Typical cavity fluctuations for Type A occur below 10Hz). As
$p_{0}$ is dropped even further, Type C cavitation is observed. This is the
stage right before super-cavitation where shedding still occurs albeit at a
much lower rate and cavity is filled with vapor. The instantaneous cavity
lengths for Type C start to extend beyond the field of view of the X-ray
densitometry system (roughly the side view shown in Figure 2). The three
phases of cavitation are shown in Figure 4 and Figure 5.

Figure 4: High speed snapshots of three different cavitation regimes (a, b, c
for $\sigma_{0}$=0.78, 0.72, 0.60 respectively) at $u_{H}$ = 10 m/s. Type A
comprises of cavitation in shear layer mostly near reattachment region(away
from separation point). Types B and C shedding regimes are of primary interest
in this study. Figure 5: The mean void fraction fields (snapshots) of the
three cavitation regimes (a, b, c for $\sigma_{0}$=0.78, 0.72, 0.60
respectively) at $u_{H}$ = 10 m/s. The mean length of cavity $<L_{C}>/H$ and
distance to maximum void fraction value (in the mean field, $<L_{M}>/H$ are
shown. The actual densitometry data captures entire 0-100 % void fraction
scale, but color scale artificially limited to 0-2% for Type A and 0-25% for
the rest.

For inception and developing cavitation regimes the measured mean void
fraction values are less than 10%. The X-ray densitometry system yields 2%
uncertainty in the reported void fraction values. This is the main reason why
higher void fraction Type-B and Type-C regimes are of primary interest in the
current study (acceptable signal to noise ratio (SNR)).

###  3.1 Average Cavity Topology

We use the mean void fraction flow fields to determine the downstream location
where void fraction values ($\alpha_{MEAN}$) are 0.05 in the region of cavity
closure, defining the mean cavity length $<L_{C}>$. The mean void fraction
field is estimated as :

$\alpha_{MEAN}=\ \sum\limits_{N=0}^{786}\dfrac{\alpha(x,y,N)}{787}$ (1)

The distance to the maximum void fraction ($max(\alpha_{MEAN})$) in the field
is recorded as $<L_{M}>$. Both these distances are measured from the
separation point on the step, as shown in Figure 5(b) and plotted in Figure 6.

Figure 6: Mean cavity length($<L_{c}>$) and location of maximum void
fraction($<L_{M}>$) for $u_{H}=8m/s(\blacksquare,\square)$,
$10m/s(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\blacktriangle,\triangle)$,
$12m/s(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ,\bullet)$
for different $\sigma_{0}$. Closed symbols are for $<L_{c}>$ and open symbols
for $<L_{M}>$.

It is observed that higher the Re, for a given $\sigma_{0}$, the lower the
mean cavity length. Conversely, $L_{M}$ lies between 3 to 4H, regardless of
the Re. This can be juxtaposed with the shear stress distribution reported in
the previous literature (Jovic & Driver [1995]). The shear stress maximizes
roughly at 0.67 reattachment length, which is 3.7H for their specific single
phase flow configuration.

###  3.2 Typical Shedding Cycle

To better identify specific flow features during the shedding cycle observed
in Type B and Type C cavities, high speed cinematographic frames of the flow
are shown in Figure 7. The primary cavitating spanwise vortex (green) and
secondary vortex (blue) are seen interacting in Figure 7(c). This interaction
can be seen through streamwise vortex bridges formed between the two spanwise
vortices. Both these vortices have the same sense of rotation(anticlockwise).
During the filling period (Figure 7(f)-(g)) the side-view of the cavity
appears quasi-steady. The ejected spanwise vortex at the end of a shedding
cycle can be visualized in Figure 7(l).

Figure 7: The timeseries of high speed snapshots of a periodic Type B shedding
cycle. Case at $\sigma_{0}$=0.72 for $u_{H}$ = 10 m/s. Cycle starts with
visible cavitation in spanwise vortex(b). The recirculation bubble and shear
layer continue to fill with vapor(e)-(g), until an adverse flow front(i)
causes the cavity to disappear(shed cavity in (k)).

The X-Ray densitometry image frames of a typical Type-B shedding cycle are
shown in Figure 8. The cycle begins with cavitation in the core of a spanwise
vortex (vortex 1), highlighted in green between X = -0.5H to X = -2H (Figure
8(a)). As this structure convects downstream, it grows in size and interacts
with another vortex (vortex 2, highlighted in blue). Subsequently the vortex
pair becomes indistinguishable and a filled cavity starts to emerge between X
= -2H to X = -6H seen in Figure 8(d)-(f). The seemingly quasi-steady snapshots
shown in Figure 7(f)-(g) can be compared to Figure 8(e)-(f) to indicate
additional flow features captured through densitometry. The cavity
consistently grows from the aft, i.e. it fills from roughly around the
reattachment zone until a filled cavity state is achieved.

Figure 8: X-Ray timeseries of a typical Type B shedding cycle at
$\sigma_{0}$=0.72 for $u_{H}$ = 10 m/s. The void fraction ($\alpha(x,y,t)$)
range is set to 0-70%. The adverse flow front highlighted in (h) resembles the
bubbly shock front

Once a fully filled cavity is formed and reaches a high instantaneous void
fraction value($\alpha=0.5-0.8$), a void fraction front begins to move
upstream, resulting in a cavity shedding seen in Figure 8(g)-(h). As this flow
front reaches the step, the cavity disappears and a cavitating spanwise vortex
(green) is ejected into the oncoming flow. The shedding cycle then starts all
over again.

The Type C shedding case is depicted in Figure 9. Most features of the
cavitating cycle are similar to those of Type B albeit at a much slower rate
that makes it quasi-steady in direct high speed videos. The lower cavitation
number leads to fuller/longer cavity with a consistently filled shear layer.

Figure 9: X-Ray timeseries of a typical Type C shedding cycle at
$\sigma_{0}$=0.60 for $u_{H}$ = 10 m/s . The void fraction ($\alpha(x,y,t)$)
range is set to 0-85%. Compared to Type B, Type C cavities are longer, have
higher vapor content and collapse at a slower rate. High speed snapshots of
Type C appear quasi steady, unlike the X-Ray timeseries which clearly show
shedding.

###  3.3 Void Fraction

Based upon local pressure distribution and flow dynamics, void fraction values
can vary significantly. The mean void fraction profiles for the onset of Type
B cavitation at different Re are shown in Figure 10. It can be seen that the
mean void fraction profiles for the onset of shedding are almost identical
across different Re.

Figure 10: Mean void fraction contours at $u_{H}$ = (a) 8 m/s, (b) 10 m/s and
(c) 12 m/s at $\sigma_{0}=0.72$. 5 slices at $X/H=$ -1, -3, -4, -5, -7 are
taken to compare void fraction values across the three velocities shown in
Figures 11,12, 13. Figure 11: Mean void fraction values(2% uncertainty) along
slices Y/H = -1 to 1.5; $u_{H}=8m/s(\square)$, 10 m/s
($\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle$)
and
$12m/s(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ$)
at $\sigma_{0}=0.72$ (Type B). The mean void fraction values are comparable
across three Re.  Figure 12: Mean void fraction values for Type C; $u_{H}$ = 8
m/s ($\square$), 10 m/s (
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle$ )
and 12 m/s
($\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ$) at
$\sigma_{0}=0.60$. No significant change in void fraction values due to Re is
seen.

X-slices of the mean void fraction field for Type B shedding regime at X = -H,
-3H, -4H, -5H and -7H are shown in Figure 11. The mean void fraction values
are comparable across different Re. X-slices slices for the Type C shedding
are shown in Figure 12. The mean void fraction values, now higher in magnitude
are still similar across Re. Taking similar slices on the Root Mean Standard
Deviation (RMSD) void fraction field yields comparable RMSD values across the
three Re for Type B regime in Figure 13(a). The RMSD field is generated as
follows :

$\alpha_{RMSD}\ =\
\sqrt{\dfrac{{\sum\limits_{N=0}^{786}(\alpha(x,y,N)-\alpha_{MEAN})}^{2}}{787}}$
(2)

Comparatively larger RMSD values are seen in Type C shedding regime (Figure
13(b)). With decreasing cavitation number, void fraction fluctuations
increase. As mentioned earlier, the reported void-fraction values have 2%
uncertainty in measurements.

Figure 13: RMSD values for the three Re; $u_{H}=8m/s(\square)$, 10 m/s (
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle$ )
and
$12m/s(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ$);(a)
Type B;(b) Type C. RMSD values for Type B are comparable, meanwhile for Type
C, RMSD of cavity void fraction is sensitive to Re. This can be partially
attributed to different mean pressures at these Re that affect local pressure
fluctuations.

###  3.4 Cavity Shedding Frequency

The cavity shedding cycle highlighted in Figure 7 and Figure 8 occurs
periodically, and as the flow transitions from Type A all the way to Type C,
the shedding frequency, f, also changes. The Strouhal number (St) used to non-
dimensionalize the shedding frequency is given by $St=\frac{f\cdot{u_{0}}}{H}$

Figure 14: Dynamic pressure signal(a,$\cdot\cdot\cdot$ ) synchronised with
time varying void fraction signal(b,$-$) and corresponding power spectral
density(c) for a Type B cavitating case at $\sigma_{0}=$ 0.73 at $u_{H}$ = 10
m/s. The highest energy peaks are close. Figure 15: Dynamic pressure
signal(a,$\cdot\cdot\cdot$ ) synchronised with time varying void fraction
signal(b,$-$) and corresponding power spectral density for a Type C cavitating
case at $\sigma_{0}=$ 0.59 at $u_{H}$ = 10 m/s. Compared to Type B we can see
a lower shedding frequency and lower pressures dp.

The shedding frequency is determined from two time varying signals. First, the
spatially averaged local void fraction averaged over a 1.25mm by 0.875mm probe
area located at (-5.5H, -0.5H). The shedding frequency observed through the
void fraction signal does not change, for the probe located anywhere between X
= -6.5H to -0.5H (Y= -H to 0). Second, the shedding frequency is recorded with
the dynamic pressure transducer (dp). The dynamic pressure signal(dp), the
time varying void fraction signal and corresponding power spectral densities
for a Type B and Type C case are shown in Figure 14 and Figure 15
respectively. The primary shedding frequencies lie between 10-70 Hz. The
resulting Strouhal number is defined with the strongest frequency peak, and it
monotonically decreases with decreasing $\sigma_{0}$, shown in Figure 16. Note
that St is not strongly influenced by changes in Re.

Figure 16: The change in strouhal number(shedding frequency) with decreasing
$\sigma_{0}$. $u_{H}=$ 8 m/s ($\square\blacksquare$), 10 m/s (
$\triangle\blacktriangle$) and $12m/s$ ( $\circ\bullet$). The strouhal numbers
obtained through the void fraction signals($\alpha$) are shown in filled
symbols, while open symbols depict strouhal number from dp signal.

###  3.5 Cavity Static Pressure

The static wall pressure is measured at different streamwise locations beneath
the cavity. The inlet pressure $p_{0}$ is used as the basis to find non-
dimensional coefficient of pressure
$Cp_{i}=\frac{p_{i}–p_{0}}{{\textstyle\frac{1}{2}}\cdot{\rho}\cdot{{u_{0}}^{2}}}$
where i=1, 2, 3 correspond to the pressure measured at the three static
pressure taps beneath the cavity .

Figure 17: The values of $-Cp_{1}$ (filled,
$\blacksquare\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\blacktriangle\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet$)
and $-Cp_{2}$ (open,
$\square\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ$
) as $\sigma_{0}$ is reduced across three Re; $u_{H}=$ 8 m/s
($\square\blacksquare$), 10 m/s (
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle\blacktriangle$)
and $12m/s$ (
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ\bullet$).
Magnitude of $Cp_{2}$ is higher than $Cp_{1}$ for all three Re indicating that
behind the step, the pressure drops between pressure taps at location 1 and 2.

The Cp values at different $\sigma_{0}$ for $p_{3}$ and $p_{2}$ are shown in
Figure 17. $-Cp_{2}$ values are higher than $-Cp_{1}$ values across all three
Re. Both $Cp_{1}$ and $Cp_{2}$ values peak around $\sigma_{0}$ = 0.71-0.78,
which corresponds to onset of Type B shedding. As cavity fills with vapour
($\sigma_{0}$ is reduced), the $-Cp$ value is seen to decrease. We can define
a cavitation number based on cavity pressure as ($\sigma_{c}$ ; where
$\sigma_{c}$ = $-Cp_{1}$. In Figure 18 we see that the cavity pressure
approaches vapor pressure (–) as $\sigma_{0}$ is reduced. This means that even
during Type B and Type C shedding cycles, the cavity is not homogeneously
filled with vapor, as also evidenced by the void fraction measurements.

Figure 18:  $\sigma_{c}$ with decreasing $\sigma_{0}$ (Closer look at
-$Cp_{1}$). $u_{H}=8m/s(\blacksquare$ ),
$10m/s(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\blacktriangle$),
$12m/s(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet$).
At lower cavitation numbers, pressure inside the cavity approaches vapor
pressure.

-Cp for $p_{3}$ and $p_{4}$ are shown in Figure 19. $Cp_{4}$ remains comparatively unchanged across different cavitation regimes. At high cavitation numbers $Cp_{3}$ is negative, indicative of higher pressure in the re-attachment region compared to the inlet pressure as suggested in previous literature(Jovic & Driver [1995], ramamurthy1991some). $-Cp_{3}$ increases drastically as flow transitions from Type B to Type C. This is due to drop in $p_{3}$ as the cavity grows larger ($p_{3}$ approaches vapour pressure).

Figure 19: The values of $-Cp_{3}$
($\blacksquare\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\blacktriangle\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet$)
and $-Cp_{4}$
($\square\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ$)
as $\sigma_{0}$ is dropped across three Re; $u_{H}=8m/s(\square\blacksquare)$,
$10m/s(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle\blacktriangle$)
and
$12m/s(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ\bullet$
)

###  3.6 Shockwave Properties

The flow front shown in Figure 8(g) primarily drives the shedding(collapse) of
the cavity, and this flow front is analogous to the propagating shockwaves in
rich bubbly mixtures reported in previous literature (Ganesh et al. [2016]).
As $\sigma_{0}$ is reduced (flow transitions from Type A to B) the vapor
content in shear layer and recirculation bubble increases. The bulk modulus of
the vapor liquid mixture decreases with decreasing $\sigma_{0}$ leading to
higher compressibility in the shear layer and the recirculation bubble. Also
with increasing vapor content the local speed of sound in the mixture
decreases (Brennen & Brennen [2005]), and at optimal void fraction values the
flow becomes locally supersonic. This makes the flow susceptible to formation
of bubbly shock fronts that can govern overall the cavity shedding mechanism.

As the cavitating spanwise vortices (highlighted in Figure 8(c),(d)) and
subsequent bubbly shock fronts propagate over the differential pressure
transducer (dp) corresponding pressure rise can be recorded. This is shown in
Figure 20, which highlights the different pressure rises over the dynamic
pressure transducer (dp) and the local void fraction probe.

Between stages I and II, a pressure rise is recorded, with the differential
pressure dp rising with the passage of the front. When the flow front impinges
on the step, another larger pressure pulse is created (III). The collapse of
the shed vortices cause the remaining pressure spikes (IV and V).

The speed at which this flow front propagates ($U_{RH}$) can be predicted by
the one dimensional Rankine-Hugoniot jump condition found by simple mass and
momentum conservation across a 1-D shockwave in the shock-wave frame of
reference:

$U_{RH}=\frac{d\rho}{\rho_{L}}\left[\frac{(1-\alpha_{2})}{(1-\alpha_{1})*(\alpha_{1}-\alpha_{2})}\right]$
(3)

where, $\rho_{L}$=density of liquid(water) and $\alpha_{1}$, $\alpha_{2}$ are
void fraction values pre and post shock. Since we have measured these values,
we can compare the predicted and measured front propagation speed. The
$\alpha_{1}$ and $\alpha_{2}$ values are found by spatially averaging void
fraction values in void fraction probe as the front convects over the dynamic
pressure transducer, creating the corresponding change in pressure, dp.

Figure 20: The differential pressure (dp) and corresponding X-ray densitometry
snapshots as the shock-front moves past the transducer and hits the step
during a shedding cycle. $\sigma_{0}$=0.72 for $u_{H}$ = 10 m/s.

The propagation speed $U_{s}$ is independently measured using the X-T diagrams
of near wall void fraction, as shown in Figure 21. Comparing the estimated Us
from X-T diagrams and $U_{RH}$ computed using Figure 3 shows a very good
agreement.

Figure 21: X-T diagram at Y/H = -0.25; $\sigma_{0}$=0.72 for $u_{H}$ = 10 m/s.
The shock propagation is assumed to be linear and inverse of the slope of X-T
diagram gives shockspeed.  Figure 22: Comparison between computed shock-speed,
$U_{RH}(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\square$)
and shock-speed estimated from X-T diagrams, $U_{S}(\circ)$ for varying
$\sigma_{0}$ shows a good agreement between $U_{S}$ and $U_{RH}$.

###  3.7 Speed of Sound Estimation

The compressibility of the vapour-liquid mixture affects the periodic shedding
of the cavity, evidenced by the formation of the propagating bubbly shock
fronts. Quantifying the local speed of sound of the bubbly mixture helps
elucidate the tendency of the flow to become supersonic at a given
$\sigma_{0}$.

Due to complex nature of the rich bubbly mixtures seen in type B and type C
flows, and absence of local velocity/pressure fields it is difficult to make
precise experimental measurement of the local speed of sound. Instead we can
estimate the local speed of sound by using simplified expressions suggested in
the literature (Brennen & Brennen [2005]). By considering homogenously
dispersed phase (V: vapour) in carrier fluid (L: water) and absence of any
relative motion between the phases the governing conservation equations
simplify to their single phase form. Imposing additional thermodynamic
constraints leads to ‘the homogeneous equilibrium model’ and ‘the homogeneous
frozen model’ for the speed of sound. The equilibrium model assumes
instantaneous heat transfer between phases and mixture existing in
thermodynamic equilibrium. The frozen model on the other hand considers zero
heat transfer between phases (i.e. no mass transfer). The real two-phase
mixture response should lie somewhere between these two extreme assumptions
(Brennen & Brennen [2005]). If $\epsilon_{L}$ fraction of the liquid phase
exists in thermodynamic equilibrium, then $(1-\epsilon_{L})$ fraction will be
insulated (similarly for $\epsilon_{V}$ fraction of the vapour phase). The
general qualitative expression that can be used for both the models to
estimate the speed of sound ($c_{i}$) is given by:

$\frac{1}{\rho_{m}{c_{i}}^{2}}=\frac{\alpha}{p_{i}}\left[(1-\epsilon_{V})f_{V}+\epsilon_{V}g_{V}\right]+\frac{1-\alpha}{{p_{i}}^{1+\eta}}\epsilon_{L}g^{*}{p_{c}}^{\eta}$
(4)

where, i=1,2 & 3, corresponds to the pressure measured at the three static
pressure taps beneath the cavity. Mixture density
$\rho_{m}=\alpha\rho_{V}+(1-\alpha)\rho_{L}$. Quantities f and g define
thermodynamic properties of respective phase. For water $g^{*}$ = 1.67, $\eta$
= 0.73, $g_{V}$ = 0.91 and $f_{V}$ = 0.73 (Brennen & Brennen [2005]). The
pressure $p_{c}$ ( = 22.06 MPa) is the critical pressure of water.
$\epsilon_{V}=\epsilon_{L}=1$ corresponds to ‘homogeneous equilibrium model’
while $\epsilon_{V}=\epsilon_{L}=0$ gives the ‘homogeneous frozen model’ from
Figure 4. The Mach number is then calculated using $u_{H}$ as:

$M_{i}=\frac{u_{H}}{c_{i}}$ (5)

Figure 23: $M_{i}$ estimated using the ‘homogeneous equilibrium model’
($\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bigstar\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbin{\blacklozenge}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\blacktriangleright$)
and ‘homogeneous frozen model’
($\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\scalerel*{\stackinset{c}{-.125pt}{c}{}{\scalebox{0.7}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{$\bigstar$}}}{$\bigstar$}}{\bigstar}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbin{\Diamond}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rhd$)
for speed of sound; $M_{1}$
($\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bigstar\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\scalerel*{\stackinset{c}{-.125pt}{c}{}{\scalebox{0.7}{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{$\bigstar$}}}{$\bigstar$}}{\bigstar}$
) $M_{2}$
($\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbin{\blacklozenge}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbin{\Diamond}$)
& $M_{3}$
($\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\blacktriangleright\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rhd$).
$u_{H}=10m/s$.
($\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}-\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$)
depicts M = 1 and indicates when the flow becomes supersonic

Figure 23 highlights that the experimentally observed onset of shockwaves
(Type B shedding regime shown in Figure 8) approximately coincides with the
qualitative estimation of sound-speed using the ‘homogeneous frozen model’.
The flow is seen to transition from subsonic regime at Type A to supersonic
($M>1$) at the onset of Type B regime for $M_{1}$ and $M_{2}$. $M_{3}$ being
subsonic($<1$) can be attributed to the selection of $u_{H}$ instead of the
local mean velocity near $p_{3}$ (mean flow velocity at $p_{3}$ is not
directly measured). The ‘homogeneous equilibrium model’ is seen to
consistently overestimate $M_{i}$ and does not capture the subsonic to
supersonic transition suggested by the formation of shock fronts (Type A to
Type B transition region). Thus, this suggests that the actual speed of sound
is closer to that given by the ‘homogeneous frozen model’.

Figure 24: The variation of $M_{i}$ with decreasing $\sigma_{0}$ across three
different Re estimated using ‘homogeneous frozen model’;
$u_{H}=8m/s(\square\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\square\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\square\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}),\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}10m/s(\triangle\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\triangle\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\triangle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
&
$12m/s(\circ\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\circ\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0});\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}M_{1}(\square\triangle\circ),\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}M_{2}(\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\square\triangle\circ\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})\
\&\
\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}M_{3}(\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\square\triangle\circ\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})$.
(-) depicts M = 1.

The Mach number values estimated using the ‘homogenous frozen model’ for the
three Re are shown in Figure 24. It is seen that $M_{i}$ decreases
monotonically with increasing $\sigma_{0}$. $M_{1}$, $M_{2}$ and $M_{3}$ do
not have a clear Re dependency.

##  4 Conclusions

In the current study on a backward facing step we have identified three
cavitating regimes observable under the x-ray densitometry and high speed
cinematographic system. Throughout the study we determine the effect of change
in Re on various cavity properties.

Mean cavity lengths have a Re dependency as seen in Figure 6. During a
shedding cycle the stronger spanwise vortices incept and grow in size as they
convect away from the separation point. The vortex-pair interaction dominates
the cavity filling part of the shedding cycle as visualized in Figure 7. An
adverse flow front (shockwave) moving towards the step eventually diminishes
the cavity (Figure 8(g)-(i)). The shedding frequency decreases monotonically
with decreasing $\sigma_{0}$ and does not have Re dependence. The mean void
fraction values in the cavity are comparable across the three Re.

Static pressure $p_{1}$, $p_{2}$ and $p_{3}$ inside/around the cavity are
measured as highlighted in Table 1. The local pressure first drops and then
recovers between $p_{1}$ and $p_{3}$. Looking at $Cp_{1}$ in Figure 18 shows
that the cavity is not homogenously filled with vapour. As $\sigma_{0}$ is
dropped (Type C and below) $p_{1}$ is seen to approach vapour pressure.

The dynamic pressure (dp) gives insight into the pressure fluctuations induced
due to convecting shock-fronts. The shock speed computed using a simple 1-D
Rankine Hugoniot jump condition lines well with the shock-speed estimated
through the X-T diagram. The simple 1D assumption holds true even for
shockwaves in the current complex flow system.

Finally a simplified expression is used to qualitatively estimate the local
speed of sound and corresponding Mach number are discussed. The ‘homogeneous
frozen model’ appears to be consistent with the observed formation of shock
fronts with decreasing cavitation number.

##  5 Acknowledgements

Current study has been funded by Office of Naval Research as a part of
Multidisciplinary University Research Initiative(MURI) titled ‘Predicting
turbulent multi-phase flows with high fidelity’ (Grant: N00014-17-1-2676)
under program manager Dr. Ki-Han Kim.

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