A mathematical model of steady-state cavitation in Diesel injectors S. Martynov, D. Mason, M....

A mathematical model of steady-state cavitation in Diesel injectors

S. Martynov, D. Mason, M. Heikal, S. SazhinInternal Engine Combustion GroupSchool of EngineeringUniversity of Brighton


Structure

Introduction Phenomenon of cavitation Objectives Mathematical model of cavitation flow Model implementation into PHOENICS Test cases Results Conclusions Acknowledgements


Introduction

Cavitation in the hydraulic, lubrication and fuel injection systems of automotive vehicle.

Cavitation effects: noise and vibration, rise in the hydraulic resistance, erosion wearing, improved spray breakup


Introduction

Effects of cavitation are described via the boundary conditions at the nozzle outlet: injection velocity, effective flow area, and velocity fluctuations.


Phenomenon of cavitation

Hydrodynamic cavitation - process of growth and collapse of bubbles in liquid as a result of reduction in static pressure below a critical (saturation) pressure.

Similarity criteria:

l

lUD

Revpp

ppCN

2

21


Phenomenon of cavitation

Cavitation starts from the bubble nuclei Similarity at macro-level (Arcoumanis et al, 2000) Scale effects prevent similarity at micro-level

Real-size nozzle (Ø =0.176mm) Scaled-up model (20:1) Re = 12 600; CN = 5.5


Objectives of study

Development of a scalable model for the hydrodynamic cavitation

Validation of the model against measurements of cavitation flows in Diesel injectors


Mathematical model of cavitation flow Simplified bubble-dynamics theory

bubbles of initial radius Ro and fixed concentration n

pppp

dt

dRv

l

v

sgn3

2


Mathematical model of cavitation flow

The homogeneous-mixture approach. Conservation equations for the mixture:

0~

~~

~~

j

j

x

u

t

k

kij

i

j

j

iT

ljij

jii

x

u

x

u

x

u

xx

p

x

uu

t

u~

~

3

2~

~

~

~1~Re

1~

~

2

1~

~~~

~~~

initial and boundary conditions; turbulent viscosity model; closure equations for properties.



R – radius of bubbles (m);n – number density (1/m3 liquid)

Rliquid

bubble

vapour

Volume fraction of vapour:

334

334

1 Rn

Rn



Void fraction transport equation:

ppsignpCNfCx

u

t vk

k

~)(~

~~

1

3/1~ nLC – cavitation rate constant

3/23/1)1()( f

Properties of the mixture:

lv )1(

lv )1( ),,,( constvlvl

L – hydrodynamic length scale


Model implementation into PHOENICS PHOENICS versions 2.2.1 and 3.6 Steady-state flows Collocated body-fitted grids CCM solver with compressibility factor Up-winding applied to densities in

approximations for the mass fluxes Mass fraction transport equation was

solved using the standard procedure Super-bee scheme applied to the mass

fraction equation for better resolution of steep density gradients

Turbulence model – RNG k-


Test cases – steady-state cavitation in rectangular nozzles

Roosen et al (1996):

Tap water, 20oC L=1mm, H=0.28mm, W=0.2mm, rin=0.03mm

Winklhofer, et al (2001):

Diesel fuel, 30oC L=1mm, H=0.30mm, W=0.3mm, rin=0.02mm

Measurements: Images of

cavitation Inlet/ outlet

pressures Pressure fields Velocity fields Mass flow rates


Results – Cavitation flow of water )(m104 314 n

Photograph and visualised velocity field of cavitating flow (Roosen et al, 1996) in comparison with the results of computations by the model.CN = 2.87


Results – Cavitation flow of water )(m104 314 n

Photograph of cavitating flow (Roosen et al, 1996) in comparison with the results of computations of the vapour field.

Effect of cavitation number CN = 6.27


Scalable model of cavitation flow

n L3=idem: model for n Ro/L=idem: Ro / L → 0

j

ij

ij

jii

xx

p

x

uu

t

u~

~

Re

1~

~

2

1~

~~~

~~~

ppsignpCNfCx

u

t vk

k

~)(~

~~

1

Momentum conservation:

VF transport equation:

Similarity conditions: Re=idem CN=idem


Scalable model of cavitation flow

pv – pmin = maximum tension in liquid;

pv = vapour pressure;

n* = liquid-specific number density parameter.

Number density of cavitation bubbles versus liquid tension.

2/3min

*

v

v

p

ppnn

)m(102 310*

n


Effect of shear stresses on cavitation flow

Flowing liquid (Joseph, 1995):

Static liquid:

= maximal rate of strain, 1/s;= dynamic viscosity of liquid, Pa s;= turbulent viscosity, Pa s;= adjustable coefficient.

maxiiS

tCt

= maximal rate of strain, 1/s;= dynamic viscosity of liquid, Pa s;= turbulent viscosity, Pa s;= adjustable coefficient.

maxiiS

tCtmax12 ii

tt

vcr

SC

ppp

vcr ppp

vii pSp max2

Effect of turbulent shear stresses:


Results – cavitation flow of Diesel fuel

Measured (top, Winklhofer et al, 2001) and predicted (bottom) liquid-vapour fields.

10 );(m102 318 tCn

max12 iit

tvcr SCpp

Distributions of static pressure and critical pressure along the nozzle.

CN = 1.86


Conclusions A homogeneous-mixture model of

cavitation with a transport equation for the volume fraction of vapour has been developed

An equation for the concentration of bubble nuclei has been derived based on the assumption about the hydrodynamic similarity of cavitation flows.

Effect of shear stresses on the cavitation pressure threshold has been studied

The model has been implemented in PHOENICS code and applied for analysis of cavitation flows in nozzles


Acknowledgements

PHOENICS support team

European Regional Development Fund (INTERREG Project “Les Sprays” – Ref 162/025/247)

Ricardo Consulting Engineers UK


Thank You

A mathematical model of steady-state cavitation in Diesel injectors S. Martynov, D. Mason, M....

Documents

Transcript of A mathematical model of steady-state cavitation in Diesel injectors S. Martynov, D. Mason, M....