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Vapor production in cavitation extracts the latent heat of evaporation from the surrounding liquid, which decreases the local temperature, and hence the local vapor pressure in the vicinity of cavity. This is called thermodynamic/thermal effect of cavitation and leads to the good suction performance of cryogenic turbopumps. We have already established the simple analysis of partially cavitating flow with the thermodynamic effect, where the latent heat extraction and the heat transfer between the cavity and the ambient fluid are taken into account. In the present study, we carry out the analysis for cavitating inducer and compare it with the experimental data available from literatures using Freon R-114 and liquid nitrogen. It is found that the present analysis can simulate fairly well the thermodynamic effect of cavitation and some modification of the analysis considering the real fluid properties, that is, saturation characteristic, is favorable for more qualitative agreement.

Vapor production in cavitation extracts the latent
heat of evaporation from the surrounding liquid, which decreases the local
temperature, and hence the local vapor pressure in the vicinity of cavity. This
is called a thermodynamic/thermal effect of cavitation. The thermodynamic
effect of cavitation could be ignored for usual applications especially with
water at ambient temperature, but is much more important for cryogenic fluids
such as liquid oxygen and liquid hydrogen. For example, the suction performance
of turbopump inducer for liquid propellant rocket engine is much better if
operated with cryogenic fluids than cold water (Yoshida et al. [

Many theoretical/numerical studies have been done to clarify
the thermodynamic effect of cavitation. Focusing on the recent studies, Kato [

We have developed a simple analysis of unsteady
cavitating flow combining a free streamline theory and a singularity method,
and succeeded in simulating the cavitation instabilities of hydrofoil (Watanabe
et al. [

In the present study, we apply our analysis for a cavitating flow
of two different working fluids, Freon R-114 and liquid nitrogen, in a cascade and compare the results
with the experimental ones in a turbopump inducer reported by Franc
et al. [

We consider a flat plate cascade with the chordlength

Model for present analysis.

We divide the velocity components into the uniform
velocity

In the present study, we linearize the equations under
the assumptions of the small angle of attack

We assume that the cavity is sufficiently thin so
that all boundary conditions are applied on the blade suction surface. In the
following sections, boundary and complementary conditions applied are described
for

We assume that the pressure on the
cavity surface is equal to the vapor pressure, which is locally different due
to the temperature depression around the cavity surface under the presence of
the thermodynamic effect of cavitation. Integrating the linearized momentum
equation in the

We employ the
following flow tangency condition on the wetted blade surfaces:

We assume that
the pressure difference across the blade vanishes at the trailing edge. This
condition is simply expressed as follows:

We employ the closed cavity model for its
simplicity. The cavity thickness

In the previous section, we have described about the kinematic and dynamic boundary conditions, assuming that the local vapor pressure distribution along the cavity surface is known. To close the problem, we have to obtain the local vapor pressure distribution. Here, we model the thermodynamic effect of cavitation, using the following heat conduction model for the liquid flow around the cavity and the evaporation model expressing the heat flux across the cavity surface due to evaporation.

We assume that
the heat conduction in the main flow direction (

Temperature
difference

We assume that
the velocity inside the cavity is uniform in the

Control volume for continuity equation.

From this equation, we can calculate the local
evaporation velocity

We have assumed
that the pressure on the cavity surface is equal to the vapor pressure, which is
locally different due to the temperature depression around the cavity surface
under the presence of the thermodynamic effect of cavitation. In order to
relate the vapor pressure

Discretization
of singularities distributed along the blades and cavities are made in the same
manner as Horiguchi et al. [

We define the
normalized strength of singularities as follows:

Discretizing
the boundary and complementary conditions (

The present
analysis treats the cavity on each blade individually, so that it can be
applied to the analysis of cavitating flow with different cavity shapes for
each blade such as alternate blade cavitation, which is known to occur for
inducers with even blade count (Horiguchi et al. [

Figure

Experimental results in the case of R-114 and cold water (Franc et al. [

Cavity length

Temperature depression

Figure ^{°}, the solidity of 2.0, and the dimensional chord
length is 0.203 mm as a numerical configuration. The incidence angle and the
main flow velocity are chosen as ^{°} and

Numerical results in the case of R-114 and cold water.

Cavity length

Temperature depression

Figure

Experimental results in the case of LN2 and cold water (Yoshida et al. [

Cavity length

Temperature depression

Figure ^{°}, the solidity of 2.0. The incidence angle is set to
be 5^{°} and the turbulence diffusion factor

Numerical results in the case of LN2 and cold water.

Cavity length

Temperature depression

Figure

Saturation curve for nitrogen.

Modified results in the case of LN2 and cold water.

Cavity length

Temperature depression

In the present
study, we have carried out the singularity analysis considering the
thermodynamic effect of cavitation for the cavitating cascade in Freon R-114
and liquid nitrogen. Through the detailed comparisons with the existing experiments
done by Franc et al. [

The development of the cavity is suppressed due to the thermodynamic effect of cavitation. This effect is more significant for higher temperature.

The thermodynamic effect becomes more apparent, and then the temperature depression becomes larger as the cavity becomes longer, because the larger amount of latent heat of evaporation is needed.

When the cavity trailing edge reaches the throat section of the inducer, the temperature depression slightly decreases. This is probably due to the interaction between the cavity trailing edge and the flow around the leading edge of the adjacent blade.

In the cases with the fluids with narrower liquid temperature range and/or the cases where the large temperature depression is expected, the temperature depression might be limited by the triple point. To simulate this effect, the real saturation curve should be at least taken into account.

The present
study analytically models the thermodynamic effect of cavitation, whereas some
parts such as the determination of the turbulence diffusion factor

Coefficient matrix in (

Thermal diffusivity

Constant vector in (

Chordlength

Normalized strength of singularities

Specific heat

Normalized temperature increase

Kernel function, defined by (

Blade spacing

Imaginary unit

Latent heat of evaporation

Cavity length

Number of blades

Blade index

Vapor pressure

Unknown vector in (

Source representing cavity

Heat flux on cavity surface

Temperature

Main flow velocity

Local evaporation velocity

Flow velocity components in

Velocity deviation on cavity surface

Velocity deviations in

Complex conjugate velocity

Coordinates

Complex coordinate,

Angle of attack

Stagger angle

Bound vortices representing blade

Temperature depression

Turbulent diffusion factor

Cavity thickness

Thermal conductivity

Densities of liquid and vapor phases

Cavitation number.

Blade index

Upstream infinity.

This study is partly supported by the Grant-in-Aid for Scientific Research for the Ministry of Education, Science, Sports and Culture (no. 19760119).