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Evaluation of the turbulence model inuence on the
numerical simulations of unsteady cavitation
Olivier Coutier-Delghosa, Regiane . Fortes Patella, Jean-Luc Reboud
To cite this version:
Olivier Coutier-Delghosa, Regiane . Fortes Patella, Jean-Luc Reboud. Evaluation of the turbulence
model inuence on the numerical simulations of unsteady cavitation. Journal of Fluids Engineer-
ing, American Society of Mechanical Engineers, 2003, 125 (1), pp.38-45. �10.1115/1.1524584�. �hal-
00211202�
Evaluation of the Turbulence Model Influence on the
Numerical Simulations
of Unsteady Cavitation
Unsteady cavitation in a Venturi-type section was simulated by two-dimensional compu-tations of viscous, compressible, and turbulent
cavitating flows. The numerical model used an implicit finite volume scheme (based on the SIMPLE algorithm) to solve Reynolds-
averaged Navier-Stokes equations, associated with a barotropic vapor/liquid state law that strongly links the density variations to the
pressure evolution. To simulate turbulence effects on cavitating flows, four different models were implemented (standard k- RNG;
modified k- RNG; k- with and without compressibility effects), and numeri-cal results obtained were compared to experimental
ones. The standard models k- RNG and k- without compressibility effects lead to a poor description of the self-oscillation behavior
of the cavitating flow. To improve numerical simulations by taking into account the influence of the compressibility of the two-phase
medium on turbulence, two other models were implemented in the numerical code: a modified k- model and the k- model including
compressibility effects. Results obtained concerning void ratio, velocity fields, and cavitation unsteady behavior were found in good
agreement with experimental ones. The role of the compressibility effects on turbulent two-phase flow modeling was analyzed, and it
seemed to be of primary importance in numerical simulations.
Introduction
Cavitating flows in turbomachinery lead to performance losses
and modifications of the blades load. These consequences can be
related to the quasi-steady cavitation effects, mainly depending on
the time-averaged shapes of the vaporized structures. Moreover,
turbomachinery under cavitating conditions is also submitted to
transient phenomena, such as compressibility effects, flow rate
fluctuations, noise, vibration, erosion, in which the unsteady be-
havior and the two-phase structure of the cavitating flow are in-
fluent. The study and modeling of the unsteady behavior of cavi-
tation are thus essential to estimate the hydraulic unsteadiness in
turbomachinery and the associated effects.
In this context, some numerical models taking into account the
two-phase structure and the unsteady behavior of cavitation have
been developed. Most of them are based on a single fluid ap-
proach: the relative motion between liquid and vapor phases is
neglected and the liquid vapor mixture is treated as a homoge-
neous medium with variable density. The mixture density is re-
lated directly to the local void fraction and is managed either by a
state law 共Delannoy and Kueny 关1兴, Song and He 关2兴, and Merkle
et al. 关3兴兲, using a supplementary equation relating the void frac-
tion to the dynamic evolution of bubble cluster 共Kubota et al. 关4兴,
and Chen and Heister 关5兴兲, or by a multiple species approach with
a mass transfer law between liquid and vapor 共Kunz et al. 关6兴兲.
The last model can be used in a full two-fluid approach, including
relative motion between the phases 共Alajbegovic et al. 关7兴兲.
Main numerical difficulties are related to the strong coupling
between the pressure field and the void ratio and to the coexist-
ence of the strong compressibility of the two-phase medium with
the quasi-incompressible behavior of the pure liquid flow. More-
over, the influence of the turbulence on unsteady two-phase com-
pressible flows is not yet well known.
This paper presents unsteady cavitating flow simulations per-
formed with a two-dimensional code. It is based on the model
developed by Delannoy and Kueny 关1兴 for inviscid fluids. Several
physical modifications have been investigated by Reboud and
Delannoy 关8兴, Reboud et al. 关9兴, and Coutier-Delgosha et al. 关10兴
to increase the range of applications and improve the physical
modeling.
In the numerical code, the two-phase aspects of cavitation are
treated by introducing a barotropic law that strongly links the fluid
density to the pressure variations. From the numerical point of
view, this physical approach is associated with a pressure-
correction scheme derived from the SIMPLE algorithm, slightly
modified to take into account the cavitation process. The model
has been validated on numerous cases, such as Venturi, 关9,10兴,
foils, 关8,11兴, or blade cascades, 关12,13兴. The numerical results
showed a good agreement with experiments. Mainly, the code
leads to a reliable simulation of the cyclic self-oscillation behavior
of unsteady cavitating flows.
The complex unsteady mechanism that governs this cyclic cavi-
tation behavior is strongly affected by the applied turbulence
model, 关9,10兴. The aim of this paper is to study the influence of
different models 共standard k- RNG; modified k- RNG; k-
including or not compressibility effects兲 on the numerical simula-
tion of cavitating flows. Indeed, because of the barotropic state
law adopted in the model, the vaporization and condensation pro-
cesses correspond mainly to highly compressible flow areas.
Therefore, it is of primary importance to take into account the
compressibility of the vapor/liquid mixture in the turbulence
model.
The paper describes the turbulence models applied in the nu-
merical simulations and presents comparisons between different
numerical results obtained in a two-dimensional Venturi-type sec-
tion, whose experimental behavior has been already studied by
1
Currently at ENSTA UME/DFA, chemin de la Huniere, 91761 Palaiseau Cedex,
France.
O. Coutier-Delgosha, R. Fortes-Patella, LEGI-INPG, BP 53, 38041 Grenoble Cedex 9, France
J. L. Reboud, ENISE-LTDS, 58, rue Jean Parot, 42023 St. Etienne, France
1
Stutz and Rebouch 关14,15兴. In this geometry, the flow is charac-
terized by an unstable cavitation behavior, with almost periodical
vapor cloud shedding.
In addition to the standard k- model, three other turbulence
models were applied, to investigate their influence on the two-
phase turbulent flow. Mainly, the turbulence model proposed by
Wilcox 关16兴, which includes compressibility considerations, was
applied, and promising results are obtained. Comparisons with
experimental measurements of void ratio and velocities inside the
cavity, 关9兴, are presented.
The ability of these turbulence models to predict complex two-
phase flow is discussed, and the role of the compressibility effects
is studied.
Physical Model
The present work applies a single fluid model based on previ-
ous numerical and physical studies, 关1,8–10兴. The fluid density
varies in the computational domain according to a barotropic state
law
(P) that links the density to the local static pressure 共Fig. 1兲.
When the pressure in a cell is higher than the neighborhood of the
vapor pressure (P⬎ P
v
⫹ (⌬P
v
/2)), the fluid is supposed to be
purely liquid. The entire cell is occupied by liquid, and the density
l
is calculated by the Tait equation, 关17兴. If the pressure is lower
than the neighborhood of the vapor pressure (P⬍ P
v
⫺ (⌬P
v
/2)), the cell is full of vapor and the density
v
is given by
the perfect gas law 共isotherm approach兲. Between purely vapor
and purely liquid states, the cell is occupied by a liquid/vapor
mixture, which is considered as one single fluid, with the variable
density
. The density
is directly related to the void fraction
␣
⫽ (
⫺
l
)/(
v
⫺
l
) corresponding to the local ratio of vapor con-
tained in this mixture.
To model the mixture state, the barotropic law presents a
smooth link in the vapor pressure neighborhood, in the interval
⫾ (⌬P
v
/2). In direct relation with the range ⌬P
v
, the law is
characterized mainly by its maximum slope 1/A
min
2
, where A
min
2
⫽
P/
. A
min
can thus be interpreted as the minimum speed of
sound in the mixture. Its calibration was done in previous studies,
关10兴. The optimal value was found to be independent of the hy-
drodynamic conditions, and is about 2 m/s for cold water, with
P
v
⫽ 0.023 bar, and corresponding to ⌬ P
v
⬇0.06 bar 共green chart
on Fig. 1兲. That value is applied for the computations presented
hereafter.
Mass fluxes resulting from vaporization and condensation pro-
cesses are treated implicitly by the barotropic state law, and no
supplementary assumptions are required. Concerning the momen-
tum fluxes, the model assumes that locally velocities are the same
for liquid and for vapor: In the mixture regions vapor structures
are supposed to be perfectly carried along by the main flow. This
hypothesis is often assessed to simulate sheet-cavity flows, in
which the interface is considered to be in dynamic equilibrium,
关3兴. The momentum transfers between the phases are thus strongly
linked to the mass transfers.
Numerical Model
To solve the time-dependent Reynolds-averaged Navier-Stokes
equations associated with the barotropic state law presented here
above, the numerical code applies, on two-dimensional structured
curvilinear-orthogonal meshes, the SIMPLE algorithm, modified
to take into account the cavitation process. It uses an implicit
method for the time-discretization, and the HLPA nonoscillatory
second-order convection scheme proposed by Zhu 关18兴. The nu-
merical model is detailed in Coutier-Delgosha et al. 关10兴: A com-
plete validation of the method was performed, and the influence of
the numerical parameters was widely investigated. The present
paper describes mainly the different turbulence models applied.
Boundary Conditions. In the code, the velocity field is im-
posed at the computational domain inlet, and the static pressure is
imposed at the outlet. Along the solid boundaries, the turbulence
models are associated with laws of the wall.
Initial Transients Conditions. To start unsteady calculations,
the following numerical procedure is applied: First of all, a sta-
tionary step is carried out, with an outlet pressure high enough to
avoid any vapor in the whole computational domain. Then, this
pressure is lowered slowly at each new time-step, down to the
value corresponding to the desired cavitation number
. Vapor
appears during the pressure decrease. The cavitation number is
then kept constant throughout the computation.
The Geometry
Numerical simulations have been performed on a Venturi-type
section whose convergent and divergent angles are, respectively,
about 18 deg and 8 deg 共Fig. 2共a兲兲. The shape of the Venturi
bottom downstream from the throat simulates an inducer blade
suction side with a beveled leading edge geometry 共Kueny et al.
关19兴兲 and a chord length L
ref
⫽ 224 mm.
According to experimental observations, in this geometry the
flow is characterized by unsteady cavitation behavior, 关14兴, with
quasi-periodic fluctuations. Each cycle is composed of the follow-
ing successive steps: The attached sheet cavity grows from the
Venturi throat. A re-entrant jet is generated at the cavity closure
and flows along the Venturi bottom toward the cavity upstream
Fig. 1 Barotropic state law
„
P
…. Water 20°C.
Fig. 2 „
a
… Curvilinear-orthogonal mesh of the Venturi-type sec-
tion „160Ã50 cells….
L
ref
Ä224 mm. „
b
… Zoom in the throat
region.
2
end. Its interaction with the cavity surface results in the cavity
break off. The generated vapor cloud is then convected by the
main stream, until it collapses.
For a cavitation number
of about 2.4 共based on the time-
averaged upstream pressure兲 and an inlet velocity V
ref
⫽ 7.2 m/s,
vapor shedding frequency observed experimentally is about 50 Hz
for a cavity length of 45⫾5 mm, 关14兴.
The standard computational grid is composed of 160⫻50 or-
thogonal cells 共Fig. 2共a兲兲. A special stretching of the mesh is ap-
plied in the main flow direction just after the throat, so that the
two-phase flow area is efficiently simulated: about 50 grid points
are used in this direction to model the 45-mm long mean cavity
obtained hereafter 共Fig. 2共b兲兲. In the other direction, a stretching is
also applied close to the walls, to obtain at the first grid point the
non dimensional parameter y
⫹
of the boundary layer varying be-
tween 30 and 100 and to use standard laws of the walls. The grid
is finer in the bottom part of the Venturi section than in its upper
part, to enhance the accuracy in the cavitation domain: Cavities
obtained in the following sections contain about 30 cells across
their thickness.
Turbulence Models
„a… Standard k-
RNG Model. The first model applied in
the numerical code is a standard k- RNG model, associated with
laws of the wall, 关9兴.
In this model, the effective viscosity applied in the Reynolds
equations is defined as
⫽
t
⫹
l
where
t
⫽
C
k
2
/ is the
turbulent viscosity and C
⫽ 0.085 共Yakhot et al. 关20兴兲.
This model is originally devoted to fully incompressible fluids,
and no particular correction is applied here in the case of the
highly compressible two-phase mixture. Thus, the fluid compress-
ibility is only taken into account in the turbulence equations
through the mean density
changes.
With this model, the unstable cavitating behavior observed ex-
perimentally is not correctly simulated: After an initial transient
fluctuation of the cavity length, the numerical calculation leads to
a quasi-steady behavior of the cavitation sheet, which globally
stabilizes 共Fig. 3兲.
The resulting cavity length is much too small, compared with
the experimental observations 共in this case, the error is larger than
50 percent兲. Moreover, comparisons with experimental data ob-
tained by double optical probes by Stutz and Reboud 关14,15兴
show that the numerical mean void ratio is overestimated in the
main part of the cavity. Calculations give a high time-averaged
void ratio in the upstream part of the cavitation sheet 共⬎90 per-
cent兲, abruptly falling to 0 percent in the wake, while the mea-
sured void ratio never exceeds 25 percent and decreases slowly
from the cavity upstream end to its wake.
This poor agreement with the real configuration seems to be
related to an overprediction of the turbulent viscosity in the rear
part of the cavity. The cyclic behavior of the cloud cavitation
process is strongly related to the re-entrant jet development from
the cavity closure, 关17兴. As a matter of fact, the main problem in
the turbulent flow simulations consisted in the premature removal
of the reverse flow along the solid wall: the re-entrant jet was
stopped too early and it did not result in any cavity break off.
It is worth noting that numerical tests reported in 关10兴 confirm
that the mesh size, spatial scheme, and the time discretization
applied in the model do not modify the results. Computations
performed with a finer mesh (264⫻ 90) and first or second-order
accurate time discretization schemes still lead to the same com-
plete stabilization of the cavitation sheet.
„b… Modified k-
RNG Model. To improve the turbulence
modeling and to try to better simulate the re-entrant jet behavior
Fig. 4 Modification of the mixture viscosity „
n
Ä10…
Fig. 3 Time evolution of the cavity length. The time is reported in abscissa, and the
X
position in the tunnel of cavitation is graduated in ordinate. The colors represent
the density values: white for the pure liquid one and from red to dark blue for the
vapor one. At a given point in time and position, the color indicates the minimum
density in the corresponding cross section of the cavitation tunnel. Calculation
conditions:
É2.4;
V
ref
Ä7.2 mÕs; meshÄ160Ã50-time-step ⌬
t
Ä0.005
T
ref
„
T
ref
Ä
L
ref
Õ
V
ref
….
3
and the vapor cloud shedding, we modified the standard k- RNG
model simply by reducing the mixture turbulent viscosity, mainly
in the low void ratio areas:
t
⫽ f
共
兲
C
k
2
/
where
f
共
兲
⫽
v
⫹
冉
v
⫺
v
⫺
l
冊
n
共
l
⫺
v
兲
nⰇ 1.
Indeed, according to the experimental results 关15兴, the re-
entrant jet seems to be mainly composed of liquid (
␣
⬃0), and
thus the reduction of the mixture turbulence viscosity leads to
Fig. 5 Transient evolution of unsteady cavitating flow in the Venturi-type duct. „
a
… Tem-
poral evolution „in abscissa… of the cavity length „graduated in ordinate…. Instantaneous
density distribution of attached and cloud cavities are drawn on the left at
T
Ä11
T
ref
„velocity vectors are drawn only 1 cell over 2 in the two directions…. „
b
… Time evolution of
the upstream pressure.
4
