Dynamics of Unsteady Supercavitation Impacted by
Pressure Wave and Acoustic Wave Propagation in
Supercavitating Flow
B.C. Khoo and J.G. Zheng
K.M. Lim and S.S. Ramesh
Department of Mechanical Engineering
National University of Singapore
Outline
 Part 1: to simulate supercavitating flow based on compressible
Euler flow solver
 Part 2: to simulate acoustic wave propagation due to various
hydrodynamic sources present in the vicinity of subsonically
moving supercavitating vehicle using boundary element method
Part 1: numerical simulation of supercavitation
 Background
 Physical model and numerical method
 Numerical results and discussion
 Summary
Background on cavitation/supercavitation
Cavitation types
Bubble
cavitation [1]
Sheet
cavitation [1]
Supercavitation [1]
Vortex and
sheet
cavitation [2]
Cloud
cavitation [1]
[1] J.P. Franc, J.M. Michel. Attached cavitation and the boundary layer: Experimental and numerical
treatment. Journal of Fluid Mechanics. (1985) Vol. 154, pp. 63-90.
[2] G. Kuiper. Cavitation research and ship propeller design. Applied Scientific Research. (1998) Vol.
58, pp. 33-50.
What is supercavitation?
 Supercavitation is formation of gas bubble in a liquid flow arising
from vaporization of fluid.
 The flow pressure locally drops below the saturated vapour
pressure.
 The gas bubble is large enough to encompass whole object.
Supercavitation
image [3]
[3] J.D. Hrubes. High-speed imaging of supercavitating underwater projectiles. Exp. Fluids. (2001) Vol. 30, pp.
57–64.
Cavitation damages:
 erosion of devices
 noise
 vibration
 loss of efficiency
Benefits of supercavitation:
 drag reduction
eroded
propeller
drag
reduction
(The viscosity is much larger in
liquid water than in vapour.)
 Stability mechanism by tail
slapping
stability
effect [3]
Objective of part 1
 Our interest is focused on numerical resolution of supercavitation
bubble over an underwater object subjected to pressure wave.
 The supercavitating flow is quite complex due to its two-phase
and highly unsteady nature.
 Few works on this topic are found in the literature except for say
[4, 5].
Interaction of
pressure wave and
supercavitation
[4] J.G. Zheng, B.C. Khoo and Z.M. Hu. Simulation of Wave-Flow-Cavitation Interaction Using
a Compressible Homogenous Flow Method. Commun. Comput. Phys. (2013) Vol. 14, No. 2,
pp. 328-354.
[5] Z.M. Hu, B.C. Khoo and J.G. Zheng. The simulation of unsteady cavitating flows with
external perturbations. Computer and Fluids. (2013) Vol. 77, pp. 112-124.
Physical models and numerical methods
Available physical models
 Two-phase model:
 Both phases coexist at every point in flow field and one has
to solve separate governing equations for each phase.
 Model is complex and difficult to implement.
 One-fluid model with finite-rate phase transition [6]

 
   ( V )  0
t



 V  
   ( VV  pI )     0
t

 n  n 
   ( n  nVm )  n
t
Continuity and momentum
equations for mixture
Continuity equation for each phase
 The finite-rate phase change can be taken into account.
 It is difficult to determine the parameters associated with
phase transition a priori.
 One-fluid model with instantaneous phase change [4, 5]
 There are no empirical parameters in governing equations.
 It is easier to implement this kind of model.
[6] L.X. Zhang, B.C. Khoo. Computations of partial and super cavitating flows using implicit pressure-based
algorithm (IPA). Computer and Fluids. (2013) Vol. 73, pp. 1-9.
Physical model employed here
Axisymmetric compressible Euler equation
U f g


   i  1 H
t x y
where
U    ,  u ,  v,  E 
T
f    u ,  u 2  p,  uv,   E  p  u 
g    v,  uv,  v 2  p,   E  p  v 
T
T
H  1/ y   v,  uv,  v 2 ,   E  p  v 
T
with
   v v, sat  1   v  l , sat ,
E  e  u 2  v 2  2
 e   v v, sat ev, sat  1   v  l , sat el , sat ,
 v  0,1
 A homogeneous model is employed and liquid and vapour
phases are assumed to be in the kinematic and thermodynamic
equilibrium.
 The mixture density and momentum are conserved.
 Phase transition is assumed to occur instantaneously.
Two cavitation models (equations of state (EOS))
(a) Isentropic cavitation model[7]
  p  B  A 1/N
 0 
 ,
B

 

k  gcav   lcav
 
,
1/ N
1/ 

 p 
 p  B  A 

k


  pcav  B  A 
p
 cav 
p  psat ,
Tait EOS
p  psat ,
Isentropic model
with k  0 /(1  0 )
Sound speed model of Schmidt:  This model is mathematically sound
and physically reasonable.
 The pressure is the implicit
1
2

1  

function of density.
  

a   

2
2 
The energy equation can be

.
a

.
a


sv
v
sw
w




neglected.
[7] T.G. Liu, B.C. Khoo, W.F. Xie. Isentropic one-fluid modeling of unsteady cavitating flow. J. Comput.
Phy. (2004) Vol. 201, pp. 80–108.
Equations of state and sound speed model: (a) Tait EOS and isentropic
cavitation model; (b) speed of sound versus void fraction.
 In liquid phase modeled by Tait EOS, the pressure is highly
sensitive to small change in the density. This poses challenges to
numerical simulation.
 The sound speed varies dramatically between liquid phase and
cavitation region.
(b) Model of Saurel based on modified Tait EOS[8]
 Temperature-dependent Tait EOS for liquid water
N




p  p   , T   B 
  1  p sat T  ,

  l , sat T  



 Ideal gas EOS for vapour
p   RT
 For mixture of liquid and vapour in cavitation region, the pressure
is set to be saturated pressure,
Tv = Tl = Tsat and pv = pl = psat.
with
ln  psat / pcr   Tcr / Tsat  a1  a2 1.5  a3 3  a4 3.5  a5 4  a6 7.5 
1
3
2
3
5
3
l , sat /  cr  1  b1  b2  b3  b4
2
6
4
6
8
6
16
3
ln   v , sat /  cr   c1  c2  c3  c4
 b5
18
6
43
3
 c5
 b6
37
6
110
3
 c6
71
6
where   1  T / Tcr
[8] R. Saurel, J.P. Cocchi and P.B. Butler, Numerical study of cavitation in the wake of a hypervelocity
underwater projectile. J. Propul. Power. (1999) Vol. 15, No. 4, pp. 513.
Numerical method
Semi-discrete form:
U
1

(Fn pkx +Gn pky )dl


t
A kKi (p)
Time marching:
U(1)  U( n )  tL  U( n ) 
U( n1) 
1 (n)
1
U  U(1)  
L  U(1) 

2
2t
 The inviscid fluxes are numerically discretized using the cellcentered finite volume MUSCL scheme.
 The time-marching is handled with the two-stage
Runge-Kutta scheme.
 The geometric source terms are dealt with separately.
Schematic of mesh.
Ghost cell on a wall boundary.
Boundary conditions
Implementation of boundary conditions is important to the
simulation of cavitation/supercavitation.
 Supersonic inlet: all flow variables on boundary are
determined by freestream values.
 Supersonic outlet: all variables are extrapolated from solution
inside the computational domain.
 Subsonic inlet: velocity is specified whereas other quantities
are extrapolated from interior of the domain.
 Subsonic outlet: background pressure is given and remaining
variables have to be extrapolated from interior of physical
domain.
Numerical results and discussion
Case1: 1D single-phase (liquid) shock tube problem
Initial
condition:
PL  2.5108 Pa
PR  2500Pa
TL  TR  293K , uL  uR  0
t=0.2ms
 The results from Saurel’s and isentropic models are in good agreement.
 The shock and rarefaction are well captured.
Case2: 1D cavitation bubble
uL  100m / s
Initial
condition:
PL  PR  108 Pa, TL  TR  293K
uR  100m / s
t=0.2ms
Case 3: cavitating flow over a high-speed underwater
projectile (isentropic model)
Axisymmetric subsonic flow at U∞=970m/s and P∞=105Pa.
(a)
(b)
(c)
Results for the subsonic projectile: (a) experimental image of Hrubes [3]; (b)
density map with the isentropic cavitation model; (c) comparison between
axisymmetric (upper half) and planar (lower half) supercavitation.
The comparison of supercavity profiles between the theoretical prediction,
experimental measurements and numerical simulation.
 The numerical results concur well with experimental data.
 The supercavity size is larger in the planar flow than in the
axisymmetric flow.
Transonic projectile travelling at speed of Mach 1.03.
(a)
(b)
The comparison of the experimental shadowgraph (a) and computed
density contour map (b) for the transonic projectile.
 The detached bow shock in front of cavitator, supercavity and wake
are all well resolved numerically.
 The calculated shock and cavity wake agree well with their
counterparts in the experimental shadowgraph.
Case 4: 2D supersonic supercavitation (Isentropic model)
 Here, U  3000m / s, P  105 Pa.
 The underwater body consists of three parts: a nose cone with halfangle of 45o and base radius of 1cm, a cylinder of length 1cm and a
rear cone with semi-vertex angle of 45o.
Comparison of calculated cavity half widths, L=1cm.
 For the supersonic flow simulation, Saurel’s model failed. Isentropic
model is more stable and robust.
 The resolved flow features including detached shock shape and standoff
distance and cavity half width are quantitatively consistent with those
reported in [9].
[9] D.M. Causon and C.G. Mingham. Finite volume simulation of unsteady shock-cavitation in
compressible water. Int. J. Numer. Meth. Fluids (2013) Vol. 72, pp. 632–649.
Case 5: 2D axisymmetric supercavitation (Isentropic model)
 Here, U  40m / s, P  105 Pa.
 The cylinder has radius of 10mm and length of 150mm.
Re-entrant jet
Transient density field and its close-up view near cylinder with
streamlines.
Interaction between pressure wave and supercavitation
 The pressure wave is introduced by increasing freestream velocity
suddenly, i.e. U  U  U  45m / s.
Pressure
wave
Density and pressure fields at 0.2ms after the abrupt freestream
velocity increase.
Density and pressure fields at 0.4ms.
Density field showing supercavity collapse.
 Cavitation bubble is large enough to envelop the whole cylinder, forming
a supercavity.
 Re-entrant jet is formed behind trailing edge of cylinder.
 When impacted by pressure wave, the supercavity locally shrinks from
its leading edge and eventually collapses.
Impingement of pressure wave on supercavitation
 Here, U  45m / s, U  10m / s.
Density field images.
 The higher the freestream velocity, the longer the supercavity.
 The supercavitation is unstable with respect to perturbations.
Case 6: supercavitation subjected to sudden freestream
velocity increase (Isentropic model)
 The initial freestream flow state is U∞=100m/s, P∞=105Pa.
 After a steady supercavity is formed, the freestream velocity
is suddenly increased to U∞=120m/s.
 The radius of cylinder is 10mm.
 The flow is assumed to be axisymmetric.
 To save computational cost, only part of supercavitation is
resolved.
Pressure wave
supercavitation
The schematic of simulation setup.
The density field evolution with ∆U=20m/s. Here, =0.1ms.
The pressure distribution along the cylinder surface at three different times.
 The supercavity is completely destroyed by the pressure wave
due to sudden freestream velocity increase.
 The pressure wave is relatively weak and not visible in the
density field.
 The collapse of cavity is followed by huge pressure pulse.
 It takes a relatively long time for the cavity to appear again and
eventually envelop the cylinder.
Case 7: smooth freestream velocity increase (Isentropic model)
 The initial freestream flow speed is U∞=100m/s.
 The freestream pressure is set to P∞=105Pa.
 After a steady supercavity is formed, the freestream velocity
is changed.
Three scenarios are considered:
 Scenario 1: the upstream velocity is suddenly increased by 10% (∆U=10m/s).
 Scenario 2: the upstream speed is linearly increased to 110m/s via
100+at, 0  t  10 / a
U = 
t>10 / a
110,
The acceleration is a=10/(nT) with T=Rc/aw where Rc and aw denote
radius of cylinder and sound speed in water, respectively. Here, n is
set to 50.
 Scenario 3: the acceleration is reduced by setting n=100.
The supercavity evolution process. Column 1: sudden freestream velocity
increase of ∆U=10m/s; column 2: linear velocity increase with n=50;
column 3: linear velocity increase with n=100. Here, =0.1ms.
Animation for density field evolution
 sudden
freestream
velocity
increase,
∆U=10m/s.
 constant
acceleration,
n=50.
 constant
acceleration,
n=100.
Case 8: supercavitation subjected to freestream
velocity perturbation (isentropic model)
 The freestream flow speed is U∞=100m/s.
 The freestream pressure is set to P∞=105Pa.
2π

100+10sin(
t), 0  t  nT

U = 
.
nT
100,
t>nT
The value of n is taken to be 5, 10 and 30, respectively. The larger
value of n results in a perturbation with longer period.
The supercavity evolution subjected to the freestream velocity perturbation. The
three columns (from left to right) correspond to n=5, 10 and 30, respectively.
Here, =0.1ms.
Animation for density field evolution
 sinusoidal
perturbation in
freestream
velocity, n=5.
 sinusoidal
perturbation in
freestream
velocity, n=10.
 sinusoidal
perturbation in
freestream
velocity, n=30.
Case 9: supersonic supercavitation impacted by Mach 3.1
shock wave (Isentropic model)
 Here, U  3000m / s, P  105 Pa.
The time evolution of supercavitation impacted by a Mach 3.1 shock wave.
 The supercavity experiences deformation but quickly recoveries to
its original profile. It is relatively stable at high freestream speed.
Animation for density field evolution
Case 10: 2D partial cavitation (Saurel’s model)
 Cavitation number:   2( p  psat ) / ( V2 )
 Here, U  25m / s, P  105 Pa, Psat  2300Pa, T  293K.
Streamline
Density
Void fraction
Cavitation shedding
Flow recirculation
 The trailing edge of cavity is characterized by an unsteady re-entrant jet.
 The re-entrant jet pinches off bubble and leads to cavitation shedding.
Case 11: 2D unsteady supercavitation (Saurel’s model)
 Steady cavity: U  500m / s, P  105 Pa
 The pressure wave is generated by
suddenly increasing freestream velocity,
Numerical setup
Density
U  U  U  550m / s.
Pressure along cylinder surface
Flow recirculation due to adverse pressure gradient
 Local collapse of supercavity is accompanied by large pressure surge.
 The pressure increase associated with left cavity collapse is high
enough to create an adverse pressure gradient at trailing edge. This
leads to flow recirculation and re-entrant jet, which cause cavitation
shedding and full collapse.
Animation for density field evolution
Case 12: 2D unsteady supercavity impacted by a weaker
pressure wave (Saurel’s model)
 Smaller velocity increase: U  U  U  530m / s.
Density
Pressure along cylinder surface
 The left partial cavity breakup is accompanied by a weaker pressure
surge.
 There is no re-entrant jet formed and the left cavity expands
downstream, developing into a new supercavity.
Animation for density field evolution
Summary
 The isentropic model is proved to be more stable and robust than
Saurel’s model.
 It is found that the re-entrant jet is responsible for complete collapse of
upstream cavity. However, if the introduced pressure wave is not relatively
strong, the partial cavity can grow into a new supercavity.
 When impacted by a weak shock, the supercavitation at high freestream
speed undergoes deformation.
 The higher the freestream flow speed is, the more stable the
supercavitation is.
Part 2: Acoustic wave propagation in supercavitating
flows
 Supercavity inception/development by means of ‘natural cavitation’
and its sustainment through ventilated cavitation (caused by
injection of gases into the cavity) result in turbulence and
fluctuations at the water-vapour interface
 Consequently, three main sources of hydrodynamic noise are
(1) Flow generated noise  turbulent pressure fluctuations around
the supercavity
(2) Flow generated small scale pressure fluctuations at the vaporwater interface
(3) Pressure fluctuations due to direct impingement of ventilated
gas-jets on the supercavity wall
 These sound sources interfere with high frequency acoustic
sensors (mounted within the nose region) that are crucial for the
underwater object’s guidance system
Objective
 To simulate acoustic wave propagation due to various
hydrodynamic sources present in the vicinity of subsonically
moving supercavitating vehicle
By using flow data from an unsteady CFD solver developed in
Part 1 of the present research, BEM based acoustic solver has
been developed for computing flow generated sound.
Numerical model and method
Axisymmetric Boundary Integral Equation (BIE) for Subsonically
Moving Surface
 To study flow generated sound caused by turbulent pressure
fluctuations (quadrupole/volumetric sources) present in the cavity’s
vicinity, the convective Helmholtz equation is modified to include
double divergence of Lighthill’s stress tensor Tij
2
2

Tij

p

p
2
2
2
 p  k p  2 ikM
M

z
xi x j
z 2
 Assuming linear acoustic source region and neglecting viscosity
effects of water, the Lighthill’s stress tensor Tij is expressed in
terms of Reynold’s stress tensor
Tij  0 uiu j
 M = VS / c denotes Mach number of moving surface
 By adopting Prandtl-Glauret transformation, the convective
Helmholtz equation is transformed to the standard form
(corresponding to the stationary problem)
Axisymmetric BIE (contd.)
 The axisymmetric BIE for transformed Helmholtz equation is given
by


~ ~
 2 Gˆ P
 ~
,Q
~
~
~

 C P pˆ P   
d Q~  pˆ Q ~
r Q dˆ Q~ 


nˆQ~
ˆ  0

 
 
 
~
 2 ˆ ~ ~
 pˆ Q ~ ~
ˆ  0 G P , Q dQ~  nˆ ~ r Q dˆ Q~

Q


 2 ˆ ~ ~
    G P , Q  d Q~
Sˆ _ rz  0


~ ~ ~ ~
 r Q dr dz



~
~
Q
P
denotes
source
point,
where
denotes field point
~
~
P
C (P ) - a constant whose value depends on location of source point
~ ~
Gˆ ( P, Q) denotes free space Green’s function
 involves gradients of Lighthill’s stress tensor
 Discontinuous Constant boundary elements are employed for
approximating acoustic variables p (sound pressure) and dp/dn
(normal derivative of sound pressure). Quadratic boundary
elements are used to model the geometry
Acoustic wave propagation due to subsonically moving
cylindrical projectile
Cylindrical projectile of radius a = 10 mm and length L = 15a
moving at uniform subsonic speeds, 150 m/s, 300 m/s, 450 m/s,
600 m/s, 750 m/s and 1050 m/s (corresponding to Mach
numbers 0.1, 0.2, 0.3, 0.4, 0.5 and 0.7 respectively)
The input data to the BE acoustic solver (namely pressure
fluctuations, density and velocity components) are obtained
from compressible Euler flow solver developed in the first part.
Objective: To determine self noise at the vehicle nose due to
various acoustic sources such as quadrupole sources (flow
generated sound sources) and dipole sources (pressure
fluctuations at the gas-water interface, ventilated gas jet
impingement)
The problem geometry considered in the BEM model is based on the
supercavity profile predicted by the CFD solver at steady state
The present problem has been studied for 3 cases
 Case A - acoustic wave propagation due to flow generated sound
sources – i.e. volumetric/quadrupole sound sources whose strength
per unit volume is the Lighthill’s stress tensor - Tij. For low Mach
number flows, Tij represents the Reynold’s stress (0vivj)
 Case B - acoustic wave propagation due to pressure fluctuations at
the gas-water interface
 Case C - acoustic wave propagation due to ventilated gas jet
impingement on the gas-water interface at a distance of 2a from the
face of cylindrical projectile. The jet impact diameter dJ = 6.4 mm
Case A
Case B
Case C
M = 0.1
M = 0.2
Case A – Flow generated noise
Case A – Flow generated noise
Case B – Pressure fluctuations at
vapor-water interface
SPL = 20 log (pac/10-6 ) dB
Case C – Impact of ventilated gas-jets
Case C – Impact of ventilated gas-jets
SPL = 20 log (pac/10-6 ) dB
Case B – Pressure fluctuations at
vapor-water interface
SPL = 20 log (pac/10-6 ) dB
M = 0.3
Case A – Flow generated noise
Case B – Pressure fluctuations at
vapor-water interface
Case C – Impact of ventilated gas-jets
2
0.5
4
3
3
2
0.5
4
M = 0.4
3
2
0.5
k0a
k0a
4
k0a
M = 0.5
M = 0.7
Case A – Flow generated noise
Case B – Pressure fluctuations at
vapor-water interface
SPL = 20 log (pac/10-6 ) dB
SPL = 20 log (pac/10-6 ) dB
SPL = 20 log (pac/10-6 ) dB
Case C – Impact of ventilated gas-jets
Case A – Flow generated noise
Case A – Flow generated noise
Case B – Pressure fluctuations at
vapor-water interface
Case B – Pressure fluctuations at
vapor-water interface
Case C – Impact of ventilated gas-jets
0.5
k0a
Case C – Impact of ventilated gas-jets
2
3
4
2
0.5
k0a
3
4
2
0.5
3
k0a
Variation of sound pressure level (dB) at vehicle’s nose with respect to nondimensionalized wavenumbers – effect of various sound sources – Hard nose
case
4
 In general, acoustic pressure due to volumetric sources (case A) are
higher by 50 dB to 60dB compared to effects due to pressure
fluctuations at supercavity wall caused either by flow over the
supercavity (case B) or by ventilated gas-jet impingement on
supercavity wall (case C)
 Sound Pressure Level (SPL) due to case B increases with speed
upto M = 0.3. At speeds corresponding to M = 0.4 and 0.5, SPL
decreases relative to M = 0.3 due to the initiation of laminar
separation (near M = 0.3)
 This transition to turbulent regime which occurs close to projectile’s
edge, increases pressure fluctuations at the vehicle’s nose. With
further increase in M, these instabilities are swept downstream of the
supercavity thereby minimising their effects at the vehicle nose.
 The acoustic effects are more pronounced at higher wavenumbers
owing to the presence of rigid projectile face which causes diffraction
and interference of acoustic waves in the vicinity of supercavity
k0a
M = 0.1
M = 0.3
M = 0.4
M = 0.5
0.25
1
3
SPL distributions (dB) due to pressure fluctuations at water-vapour interface
for various subsonic Mach numbers - hard nose section
 Pressure fluctuations increase at M = 0.3 near projectile face (where
laminar separation initiates) and are observed to drift downstream with
increase in Mach number
M = 0.4
M = 0.5
Hard nose
Compliant
Hard nose
M = 0.3
Compliant
k0a = 3
k0a = 2
M = 0.1
Results of Case A showing comparisons of SPL distributions (dB) for hard and
compliant nose sections. Regions highlighted in pink show the extent of volume
source region.
 Acoustic pressure levels predicted by the hard nose section are higher by 10 dB as
compared to the compliant nose sections owing to very low surface impedance for the
latter case
r
0o
10a
6a
Layout showing distances
from projectile nose at which
sound pressure directivities
(dB) are plotted; k0a = 3.5
4a
a
20a
z
90o
Legend:
Red line – Case A
Blue line – Case B
Black line – Case C
M
180o
M = 0.1
4a
20a
M = 0.3
M = 0.4
M = 0.5
Summary
 Axisymmetric boundary element solver was developed to study
acoustic wave propagation in a subsonically moving supercavitating
vehicle due to three main hydrodynamic sources viz. quadrupole
sources due to turbulent pressure fluctuations outside supercavity,
dipole like sources – caused by flow generated pressure instabilities
at vapor-water interface and impact of ventilated gas-jets
 In general acoustic pressure increases with increase in Mach
number of supercavity. The sound pressure directivity is
characterized by the presence of marked side lobes at higher Mach
numbers as compared to low vehicle speeds.
 Flow generated self noise dominates at higher wavenumbers,
whereas noise due to ventilated gas jet impingement dominates at
lower wavenumbers.
 Acoustic pressure levels predicted by the hard nose sections are
higher by 10 dB as compared to the compliant nose sections owing
to very low surface impedance for the latter case
 In general pressure fluctuations at the gas-water interface (case B)
contribute very less to self noise at the vehicle nose, particularly in
low wavenumber range upto M = 0.2. Around M = 0.3, laminar
separation occurs close to rigid projectile face (transition to turbulent
flow regime) and increases pressure fluctuations near the projectile
edge. With further increase in Mach number, these fluctuations are
swept downstream of the supercavity and their influence is
minimised at vehicle’s nose.
 At higher subsonic Mach numbers (say M = 0.7), the effect of
turbulent instabilities build up again and thereby amplify the acoustic
effects due to increase in pressure and turbulent fluctuations in the
vicinity of supercavity.
 Sound pressure radiates in specific directions at higher
wavenumbers and is characterized by distinct side lobes at higher
Mach numbers.
 In general, SPLs decrease approximately by 6 to 7 dB for cases B
and C at the far field (i.e., 20a) and by 12 to 15 dB for case A
(quadrupole sources).
Thank for your attention!