B.C. Khoo and J.G. Zheng

K.M. Lim and S.S. Ramesh

Department of Mechanical Engineering

National University of Singapore

Dynamics of Unsteady Supercavitation Impacted by Pressure Wave and Acoustic Wave Propagation in

Supercavitating Flow

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

Background Physical model and numerical method Numerical results and discussion Summary

Part 1: numerical simulation of supercavitation

Background on cavitation/supercavitation

Cavitation types

Sheet cavitation

[1]

Bubble cavitation

[1]

Super-cavitation

[1]

Cloud cavitation

[1]

Vortex and sheet

cavitation [2]

[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.

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.

[3] J.D. Hrubes. High-speed imaging of supercavitating underwater projectiles. Exp. Fluids. (2001) Vol. 30, pp. 57–64.

Supercavitation image [3]

What is supercavitation?

Cavitation damages: erosion of devices noise vibration loss of efficiency

Benefits of supercavitation: drag reduction (The viscosity is much larger in liquid water than in vapour.) Stability mechanism by tail slapping

eroded propeller

drag reduction

stability effect [3]

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].

Objective of part 1

[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.

Interaction of pressure wave and supercavitation

Physical models and numerical methodsAvailable 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]

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.

0)(

Vt

0)(

pIVVt

V

nmnnnn V

t

)(

Continuity and momentum equations for mixture

Continuity equation for each phase

[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

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.

where

with

1U f g

i Ht x y

2

2

2

, , ,

, , ,

, , ,

1 / , , ,

T

T

T

T

U u v E

f u u p uv E p u

g v uv v p E p v

H y v uv v E p v

2 2, , ,

, , , , ,

1 2

1 0,1

v v sat v l sat

v v sat v sat v l sat l sat v

E e u v

e e e

1/N

0 sat

cav cavg l

sat1/N 1/

cav cav

, ,

, ,

p B Ap p

B

kp p

p B A pk

p B A p

1

2

2 2sv v sw w

1

. .a

a a

Sound speed model of Schmidt: This model is mathematically sound and physically reasonable. The pressure is the implicit function of density. The energy equation can be neglected.

Two cavitation models (equations of state (EOS))

[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.

Tait EOS

Isentropic model

0 0/(1 )k with

(a) Isentropic cavitation model[7]

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

Ideal gas EOS for vapour

For mixture of liquid and vapour in cavitation region, the pressure is set to be saturated pressure,

with

[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.

,

, 1 ,

N

satl sat

p p T B p TT

p RT

Tv = Tl = Tsat and pv = pl = psat.

1.5 3 3.5 4 7.51 2 3 4 5 6

1 2 5 16 43 110

3 3 3 3 3 3, 1 2 3 4 5 6

2 4 8 18 37 71

6 6 6 6 6 6, 1 2 3 4 5 6

ln / /

/ 1

ln /

sat cr cr sat

l sat cr

v sat cr

p p T T a a a a a a

b b b b b b

c c c c c c

1 / crT T where

Numerical method

The inviscid fluxes are numerically discretized using the cell- centered 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.

(p)

U 1(F +G )

t A pkx pkyk Ki

n n dl

(1) ( ) ( )

( 1) ( ) (1) (1)

U U U

1 1U U U U

2 2

n n

n n

tL

Lt

Semi-discrete form:

Time marching:

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

t=0.2ms

Case1: 1D single-phase (liquid) shock tube problem82.5 10LP Pa 2500RP Pa

293 , 0L R L RT T K u u Initial condition:

The results from Saurel’s and isentropic models are in good agreement. The shock and rarefaction are well captured.

t=0.2ms

Case2: 1D cavitation bubble

100 /Ru m s

100 /Lu m s810 , 293L R L RP P Pa T T K

Initial condition:

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)

3000 / ,U m s Here, 510 .P Pa The underwater body consists of three parts: a nose cone with half-

angle 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)

40 / ,U m s Here, 510 .P Pa The cylinder has radius of 10mm and length of 150mm.

Transient density field and its close-up view near cylinder with streamlines.

Re-entrant jet

45 / .U U U m s The pressure wave is introduced by increasing freestream velocity

suddenly, i.e.

Interaction between pressure wave and supercavitation

Density and pressure fields at 0.2ms after the abrupt freestream velocity increase.

Pressure wave

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

Density field images.

Here, 45 / ,U m s 10 / .U m s

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)

Pressure wavesupercavitation

The schematic of simulation setup.

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.

The density field evolution with ∆U=20m/s. Here, =0.1ms.

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.

The pressure distribution along the cylinder surface at three different times.

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 / aU =

110, t>10 / a

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.

sudden freestream velocity increase, ∆U=10m/s.

constant acceleration, n=50.

constant acceleration, n=100.

Animation for density field evolution

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.

2π100+10sin( t), 0 t nT

U = .nT100, 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.

sinusoidal perturbation in freestream velocity, n=5.

sinusoidal perturbation in freestream velocity, n=10.

sinusoidal perturbation in freestream velocity, n=30.

Animation for density field evolution

The time evolution of supercavitation impacted by a Mach 3.1 shock wave.

Case 9: supersonic supercavitation impacted by Mach 3.1 shock wave (Isentropic model)

The supercavity experiences deformation but quickly recoveries toits original profile. It is relatively stable at high freestream speed.

3000 / ,U m s Here, 510 .P Pa

Animation for density field evolution

Case 10: 2D partial cavitation (Saurel’s model)

Density Void fraction

Streamline

Cavitation number:2

sat2( ) / ( )p p V

25 / ,U m s Here, 510 ,P Pa 2300 ,satP Pa 293 .T K

Flow recirculationCavitation shedding

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)

Numerical setup

Density Pressure along cylinder surface

The pressure wave is generated by suddenly increasing freestream velocity,

550 / .U U U m s

500 / ,U m s 510P Pa Steady cavity:

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)

Density Pressure along cylinder surface

530 / .U U U m s Smaller velocity increase:

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 vapor-

water 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.

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

ji

ij

xx

T

z

pM

z

pkMpkp

2

2

2222 i2

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

jiij uuT 0

Numerical model and methodAxisymmetric Boundary Integral Equation (BIE) for Subsonically Moving Surface

M = VS / c denotes Mach number of moving surfaceBy adopting Prandtl-Glauret transformation, the convective Helmholtz equation is transformed to the standard form (corresponding to the stationary problem)

The axisymmetric BIE for transformed Helmholtz equation is given by

Axisymmetric BIE (contd.)

zrQrQPG

Qrn

QpQPGQrQp

n

QPGPpPC

rzSQ

QQ

QQQQ

~d~d~~d

~,

~ˆ

ˆd~~

ˆ

~ˆ

d~

,~ˆˆd

~~~ˆd

ˆ

~,

~ˆ~ˆ

~

_ˆ

2

0

~

~~ˆ

2

0

~~

ˆ

2

0

~~

where P~

denotes source point,Q~

denotes field point)

~(PC - a constant whose value depends on location of source pointP

~

)~

,~

(ˆ QPG 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 M = 0.3

M = 0.4 M = 0.5 M = 0.7

k0a

SP

L =

20

log

(pac

/10-6

) d

B

0.5 2 3 4

Case A – Flow generated noise

Case B – Pressure fluctuations at vapor-water interface

Case C – Impact of ventilated gas-jets

k0a0.5 2 3 4

Case A – Flow generated noise

Case B – Pressure fluctuations at vapor-water interface

Case C – Impact of ventilated gas-jets

SPL

= 2

0 lo

g (pac

/10-6

) d

B

Variation of sound pressure level (dB) at vehicle’s nose with respect to non-dimensionalized wavenumbers – effect of various sound sources – Hard nose case

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

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

M = 0.1 M = 0.3 M = 0.4 M = 0.5

4a

20a

Legend: Red line – Case A Blue line – Case B Black line – Case C

r

M

0o

z

a

4a6a10a

20a

90o

180o

Layout showing distances from projectile nose at which sound pressure directivities (dB) are plotted; k0a = 3.5

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!