Ahmed Kovacevic
email: a.kovacevic@city.ac.uk

Sham Rane

Nikola Stosic

COMPUTATIONAL FLUID DYNAMICS IN ROTARY POSITIVE DISPLACEMENT SCREW MACHINES

Keywords:

Positive displacement screw machines are classified as compressors, expanders, extruders, blowers, vacuum pumps, liquid and multiphase pumps. They are commonly used in industrial refrigeration, air conditioning, process gas compression and variety of liquid, multiphase and vacuum pumping applications. It is estimated that almost 20% of world energy consumption is used for various means of gas compression and pumping. Therefore, reliability and efficiency of such machines has large impact on economy and ecology important for every manufacturer of such machines.

Screw machines are normally designed with the aid of analytical tools, based often on one-dimensional flow models solved by numerical methods and validated by experiments. Continuing demands for improvements in accuracy when designing such machines for existing or new applications have generated a need to use advanced means of assessment of fluid flow and solid fluid interaction within these machines such as 3D Computational Fluid Dynamics and Computational Continuum Mechanics. The first use of CFD for analysis of screw machines was reported in late 1990's. Since then, significant developments have been made to allow use of CFD in the design process of such machines. The most significant development which enabled such calculation methods was in representing the flow domains with the numerical grids capable of accurate and fast CFD calculation. This paper reviews the state of the art in application of CFD in modelling of positive displacement screw machines and outlines benefits which the use of such techniques may bring. The paper also identifies challenges and future development trends in application of CFD for rotary positive displacement screw machines.

Figure 1. Typical example of oil injected screw compressor

Twin screw compressors are most common machines used for industrial compression of gases. These have been invented in 1878 by Heinrich Krigar in Germany but have not been in use until 1938 when Swedish company SRM licensed first screw compressor to Howden in Glasgow. Screw compressors can compress dry gasses in which case rotors are rotated through synchronising gears in order to maintain positive clearance between the rotors and to prevent their contact. More often twin screw compressors are operated with liquid injection to cool, seal and lubricate rotors which enables direct contact between the rotors and renders the need for expensive synchronising gears and internal seals. The most frequently used injection liquid in compressors is oil but it can be water or any other inert liquid. Consequently oil injected twin screw compressors comprise more that 95% of all screw compressors in use. Principle of operation and modelling of such compressors is extensively discussed in [START_REF] Stosic | Screw Compressors Mathematical Modelling and Performance Calculation[END_REF]. Typical example of twin screw compressor is shown in Figure 1. It is estimated that since the beginnings of industrial use of screw compressors, over 10 million of such compressors are produced worldwide. Japanese magazine for refrigeration compressors JARN reports that in 2013 some 137,000 refrigeration screw compressors are produced worldwide. Considering that a large proportion of screw compressors are smaller units used for air application and that the number of large gas screw compressors is significantly lower than refrigeration units, it is safe to say that over 500,000 of new screw compressors are put into operation every year. Although the authors could not identify a comprehensive source to estimate the size of the screw pump and vacuum pump market it is believed that the number of these machines will be several hundreds of thousands. With such large number of screw machines produced every year, the impact on environment and economy is significant and every effort should be made to make them operate more efficiently and reliably.

MODELLING SCREW MACHINES

Low order models

Early designs of screw compressors were based on the assumption of an ideal gas in a leak proof working chamber undergoing a compression process which could reasonably be approximated in terms of pressure-volume changes by the choice of a suitable value of the exponent "n" in the relationship pVn = Constant. The advent of digital computing made it possible to model the compression process more accurately and, with the passage of time, ever more detailed models of the internal flow processes have been developed, based on the assumption of one-dimensional non-steady bulk fluid flow and steady one dimensional leakage flow through the working chamber. Together with suitable flow coefficients through the passages, and an equation of state for the working fluid, it was thus possible to develop a set of non-linear differential equations which describe the instantaneous rates of heat and fluid flow and work across the boundaries of the compressor system. These equations can be solved numerically to estimate pressure-volume changes through the suction, compression and delivery stages and hence determine the net torque, power input and fluid flow, together with the isentropic and volumetric efficiencies in a compressor. In addition, the effects of oil injection on performance can be assessed by assuming that any oil passes through the machine as a uniformly distributed spray with an assumed mean droplet diameter. Such models have been refined by comparing performance predictions, derived from them, with experimentally derived data. A typical result of such modelling is the suite of computer programs described by [START_REF] Stosic | Screw Compressors Mathematical Modelling and Performance Calculation[END_REF]. Similar work was also carried out by many other authors such as Fleming and Tang 1998 and [START_REF] Sauls | An Analytical Study of the Effects of Manufacturing on Screw Rotor Profiles and Rotor Pair Clearances[END_REF]. In recent years reports on development such models are still being made, for example Read et al, 2014 and Papes et al, 2015. Due to their speed and relatively accurate results, such mathematical models are often used in industry. However, these neglect some important flow effects that influence compressor performance, mainly in the suction and discharge ports.

3D Computational Fluid Dynamics

Screw compressor performance can be estimated more precisely by use of Three-dimensional Computational Fluid Dynamics (CFD) or Computational Continuum Mechanics (CCM). Computational fluid dynamics (CFD) covers a broad area, which attracted the interest of many investigators at the beginning of the computer era. It is based on the numerical simulation of the conservation laws of mass, momentum and energy, derived for a given quantity of matter or control mass. Finite volume method is most commonly used in CFD as described by Ferziger and Perić, 1995. The particular interest for the analysis of screw compressors is in the unsteady calculation with moving boundaries. As the rotors of a screw machine turn during operation of the compressor, the fluid volume in between them is deformed and the CFD grid which represents the fluid volume also needs to deform. Without capturing this deformation it is not possible to capture the real three dimensional fluid behaviour inside the working chamber. The challenge is in capturing the deformation of the main flow domain as well as in the leakage gaps which play essential role in the efficiency of screw machines. General grid generators are not capable of fulfilling this task and therefore efforts started almost two decades ago on grid generation for screw machines. The first report on the application of CFD in analysis of screw machines can be found in [START_REF] Stošić | CFD Studies of Flow in Screw and Scroll Compressors[END_REF]. They presented a solution in both screw and scroll compressor geometry without full grid deformation but by the addition of a representative momentum source in the conservation equations. The results were encouraging for pressure distribution but did not provide full flow properties. A breakthrough was achieved in 1999 by Kovačević with the use of an analytical rack generation method, proposed by Stošić (1998), applied to generate an algebraic, adaptive, block structured, deforming grid calculation for twin screw rotors. Since then there have been several activities reported on the CFD analysis of twin screw compressors. The use of this method for screw compressor applications is justified by its ease of use and its speed. The analysis of the working chamber is transient in nature and requires a grid representing every time step rotor position and domain deformation. In this respect, algebraic methods can be used to recalculate the grid quickly. Kovačević, Stošić andSmith (1999 and2000) successfully used an algebraic grid generation method together with boundary adaptation and transfinite interpolation which has been implemented in the program SCORG -(Screw Compressor Rotor Grid Generator). This has been written in FORTRAN with a C# front end application. In his thesis, Kovačević (2002) presented the grid generation aspects in detail. Kovačević et al. (2002aKovačević et al. ( , 2002b[START_REF] Kovačević | Three Dimensional Numerical Analysis of Screw Compressor Performance[END_REF][START_REF] Kovačević | Boundary Adaptation in Grid Generation for CFD Analysis of Screw Compressors[END_REF][START_REF] Kovačević | Screw Compressors Three Dimensional Computational Fluid Dynamics and Solid Fluid Interaction[END_REF] has reported CFD simulations of twin screw machines to predict flow, heat transfer, fluid-structure interaction, etc. Kovačević et al. (2007) also published a textbook on the CFD analysis of screw compressors. implement a grid conversion algorithm for unstructured to block structured mesh from the solution of the Laplace equation for twin screw compressors and pumps using differential methods. The use of differential methods requires the PDE to be solved for every rotor position and then the grid generation calculation of the solution has to be repeated from the equipotential and gradient lines. In his thesis, Voorde (2005) presented the principles of solving the initial Laplace equation and then using it to construct a block structured deforming mesh. Based on this grid generation, flow in a double tooth compressor and a twin screw compressor was analysed and the results were compared with experimental data over a range of discharge pressures and rotor speeds.

Papes, Degroote and Vierendeels (2013 and 2014) presented a 3D CFD analysis of an oil injected twin screw expander using the same differential grid generation approach. A real gas model for the working fluid R245fa was used and results for the different leakage flows were presented together with chamber pressure variation curves and performance analysis at different pressure ratios.

Recently CFX Berlin (2014) has introduced a software tool called TwinMesh to link with ANSYS CFX solver and this tool generates a grid for twin rotor machines on similar lines to that of the algebraic and differential approaches. Hesse et al. ( 2014) have presented the application of such a deforming grid for the analysis of an oil free twin screw compressor. One of the major factors affecting the performance prediction of twin screw compressors by use of computational fluid dynamics (CFD) is the accuracy with which the leakage gaps are captured by the discretization methods. The accuracy of mapping leakage flows can be improved by increasing the number of grid points on the profile. However, this method faces limitations when it comes to the complex deforming domains of a twin screw compressor because the computational time increases tremendously. In order to address this problem, Rane et al (2012 to 2015) formulated an analytical grid distribution procedure that can independently refine the region of high importance for leakage flows in the interlobe space and produce single high quality numerical mesh for screw rotor domains. By using this method the flow domains in the gaps are refined which improves accuracy of leakage flow predictions and the calculation of multiphase screw machines with the variety of solvers is made possible, which is the latest breakthrough in analysis these machines.

The boundary conditions used at the suction and discharge are specified pressure and temperature. In order to provide better location of the pressure specification, generally the ports are hypothetically extended and the boundary conditions are provided at the end that allow flow to oscillate into and out of the domain. This paper reviews some important aspects of grid generation procedures developed by the research group at City University in last two decades which enabled accurate prediction of screw compressor performance by use of CFD. Additionally, several examples of use of CFD for analysis of different types of screw machines are presented to show opportunities for use of CFD in analysis and design of screw machines in industry and academia.

EARLY 3D CFD MODELLING AND VALIDATION

Computational fluid dynamics for screw machines

The working process of positive displacement screw machines is based on the principle of changing the size and position of a working domain volume which consequently changes the pressure within the working domain. Quantities such as mass, momentum, energy etc. are required for modelling of screw machines. Conservation of these quantities can be represented by a general transport equation for a control volume ( 1), (Ferziger and Peric, 1996) Rotation of rotors deforms the working domain and the solution domain therefore changes with time due to movement of the boundaries. In the global coordinate system which is fixed, the only change in the conservation equation ( 1), caused by the movement of a boundary, is the appearance of the relative velocity velocity in the convective term, where is the velocity vector at the cell face. In such case, the general conservation equation can be written as (2),

The grid velocity and the grid motion are independent of the fluid motion. However, when the cell faces move and if the grid velocities are calculated explicitly and, in turn, used to calculate the convective fluxes, the conservation of mass and other conserved quantities may not necessarily be preserved. To ensure full conservations of these equations, the space conservation law needs to be satisfied as given by (3).

Space conservation can be regarded as mass conservation with zero fluid velocity. The unsteady terms in the governing equations involving integration over a control volume , which is now changing with time, need to be treated in a way consistent with the space conservation equation with a deforming and/or moving grid. The governing equations required for the solution form a closely coupled, time dependent set of partial differential equations (PDE's) and often employ a Finite Volume Method (FVM) to be solved. Resolving the mesh displacement is crucial for ensuring space conservation which requires the grid velocities and changes in CV volumes to be known at each new time step. Providing that the numerical mesh used for transient calculation of screw machine flow domains has structured topology with the constant number of computational cells for any rotor position, then the movement of vertices which define the mesh can be used for calculation of wall velocity ensuring that the space conservation and the entire solution are fully conservative, Kovacevic et all, 2006. However, if the numerical mesh does not keep the same topology throughout transient calculation, it is much more difficult to fulfil the conservation requirements. Demirdzic and Peric, 1998 showed that the error in mass conservation due to non-conformance of space conservation is proportional to the time step size for constant grid velocities and is not influenced by the gird refinement size. This has been investigated by Rane et al, 2012, to test if the user defined node displacement for structured mesh can be replaced by key-frame remeshing which is the most commonly used method for deforming unstructured meshes. It was found that many limitations to key-frame re-meshing exist which make it unsuitable for analysis of screw machines. Namely, it requires time consuming pre-processing, has limited applicability to complex meshes and leads to inaccuracies in conservation of calculated variables.

Grid Generation

Grid generation approach may be numerical, analytical or variational, as outlined in Kovacevic et al, 2006. Numerical grid generation is based on solving an elliptic partial differential equation on the unstructured mesh in order to firstly obtain the division line between the rotor flow domains and later to distribute grid points within the domain, Vorde et all, 2005. The method can produce good quality meshes but requires significant time for grid generation and a finite element or finite volume solver to obtain the mesh. Applying the principles of analytical grid generation through transfinite interpolation with adaptive meshing, the authors have proposed general, fast and reliable algorithm for automatic mapping of arbitrary twin screw machine geometry, Kovacevic, 2002. On that basis, the authors have developed a program called SCORG (Screw COmpressor Rotor Geometry Grid generator), which enables automatic rotor profiling, grid generation and direct connection with a number of CFD solvers. Kovacevic et al, 2006.

CFD calculation and validation

The numerical mesh of the screw rotor flow domains generated by SCORG does not require use of a numerical solver for grid generation and is directly transferable to variety of Computational Fluid Dynamics or Computational Continuum Mechanics code. This ensures a fully conservative mesh for moving, stretching and sliding rotor domains and robust CFD calculation with all commercially available CFD solvers (Kovacevic, 2006). Apart of the rotor domains, the solver requires suction port and the discharge port. These stationary parts can be generated using SCORG or any commercial grid generator. Figure 2 shows the complete numerical grid of a twin screw machine ready for CFD calculation. The structured numerical mesh of rotors changes for every time step. Its movement is handled by relocation of vertices by importing the new coordinates before each time step. Their coordinates are calculated for all required rotor positions in advance. The structured meshes of two sliding rotor subdomains and stationary ports are in the solver connected by non-conformal interfaces. Typical results obtained for oil free screw compressor using commercial CFD solver Ansys CFX is shown in Figure 3 These included variations in mesh size, turbulence model, differencing scheme and many other factors. The results of the overall compressor were very close to measurements which indicated that the procedure for CFD calculation of screw compressors can be used in industry. However it was also shown that the use of different differencing schemes and turbulence methods significantly influence local velocity and pressure values in some regions of the machine such as discharge port and leakage gaps. Although these local values have a low impact on the overall performance, their influence on flow development had to be further investigated. Very few authors have analysed local effects in screw compressors. For example Vimmr, 2006, following on Kauder et al, 2000, analysed flow through a static mesh of the single leakage flow path at the tip of the male rotor to conclude that rotor relative velocity does not affect flow velocities significantly and that neither of the turbulence models they used significantly change the outcome of modelling. That was in agreement with the findings of Kovacevic et al, 2006, but also confirmed that the need for further validation of full 3D CFD results could not be obtained by simplified numerical or experimental analysis. Instead, a full understanding of the local velocities in the suction, compression and discharge chambers of the machine was needed to further validate the existing methods. Laser Doppler Velocimetry (LDV) was used for this further validation.

LDV flow measurements in a screw compressor

In order to measure flow velocities inside a screw compressor, the experiment was set up at City University where this technique was used to obtain measured values. The extensive study was performed in measuring velocities in the compression domain and in the discharge chamber of the test air screw compressor as discussed by Guerrato et al, 2007. A transparent window for optical access into the rotor chamber of the test compressor was machined from acrylic to the exact internal profile of the rotor casing and was positioned at the pressure side of the compressor near the discharge port, as shown in Figure 5. After machining, the internal and external surfaces of the window were fully polished to allow optical access. Optical access to the discharge chamber was arranged through a transparent plate, 20 mm thick, installed on the upper part of the exhaust pipe. The optical compressor was then installed in a standard laboratory air compressor test rig, modified to accommodate the transmitting and collecting optics and their traverses. The laser Doppler Velocimeter operates in a dual-beam near backscatter mode. It comprised of a 700 mW argon-Ion laser, a diffraction-grating unit, to divide the light beam into two and provide frequency shift, and collimating and focusing lenses to form the control volume. A Fibre optic cable was used to direct the laser beam from the laser to the transmitting optics, and a mirror was used to direct the beams from the transmitting optics into the compressor through one of the transparent windows. The collecting optics were positioned around 25% of the rotor chamber and 15% of the discharge chamber to the full backscatter position and comprised collimating and focusing lenses, a 100 µm pin hole and a photomultiplier equipped with an amplifier. Coordinate systems shown in Figure 5 (a), (b) and (c) are applied to the rotors where αp and Rp are, respectively, the angular and radial position of the measured control volume and Hp is the distance from the discharge port centre. Taking the appropriate coordinate system, measurements were obtained at Rp=48, 56, 63.2mm, αp=27º and Hp=20 mm for male rotor, and at Rp=42, 46, 50 mm, αp=27º and Hp=20 for female rotor. The CFD calculation was performed using the numerical mesh generated from SCORG and the comparison with the measured data, obtained with the LDV technique. The rotor mesh consists of 935000 numerical cells. For the purpose of obtaining a grid independent solution, three different meshes were generated, the smallest consisting of 600000 numerical cells and the biggest with 2.7 million cells, which was the largest possible case that could be calculated by the single processor of the used desktop computer. Figure 6 shows a comparison of the axial mean velocities in the compression chamber close to the discharge port. This figure shows very good agreement throughout Zone (1) and Zone (2), as defined in Figure 6. In Zone (3) both the measured and calculated velocities increase but the increase in velocities obtained from CFD is larger as a consequence of the inability of the used k-e model of turbulence to cope with the flow near walls in rather large numerical cells. Such a configuration of the numerical mesh is the result of the methodology used for generating and moving of the numerical mesh, as explained in more detail in Guerrato et all, 2007. 3) The effect of opening of the discharge port on velocities is significant near the leading edge of the rotors and causes a complex and unstable flow with very steep velocity gradients. The highest impact of the port opening on the flow is experienced near the tip of the rotor with values decreasing towards the rotor root. The measurements confirm that turbulence plays a significant role in the flow between narrow leakage passages and the large main and discharge domains. It was assumed that the most probable reason for this slight disagreement in the CFD results with measurements is due to turbulence modelling. Therefore further research in the turbulence models for internal flow in the compressor ports is suggested. Favourably, the flow on both sides of that region appears not to be so turbulent. Due to that fact and because the internal energy in positive displacement machines is significantly larger than the kinetic energy, this does not greatly affect the overall estimation of performance. Despite this, further development and improvement of 3D CFD codes are needed.

APPLICATION OF EARLY 3D CFD MODELLING TO ANALYSIS OF SCREW MACHINES

Fluid-solid interaction

Interaction between compressor components and fluid flow within screw machines plays significant role on its performance and reliability. Kovacevic 2002, reported on investigation of this influence by use of Computational Continuum mechanics to determine fluidsolid interaction in screw compressors. In Computational Continuum Mechanics (Ferziger and Peric, 1995) The figure shows pressure distribution and deformation vectors and magnitude. In both cases the interlobe clearance gap is increased during the operation. For the medium pressure application these deformations are small, order of magnitude smaller than the clearance gap. In the high pressure application deformations are of the same order of magnitude as clearances. In order to make the results visible, the deformations are presented 20000 times enlarged.

In the oil free air compressor, due to the lack of cooling, the air temperature rise is significant.

Use of the CFD and coupled models for noise prediction

Identification of sources of noise in screw compressors and its attenuation becomes an important issue for the majority of applications. Pressure fluctuations in the discharge port affect not only aero acoustics in that domain but also a mechanically generated noise due to rotor rattling. It is believed that adequate porting can decrease the level of noise and increase the performance of the machine. Mujic, 2009 presented comparison of results obtained by the chamber model, combined model and full 3D model to estimate pressure oscillations as a function of the shape of the port. The comparison of results obtained by modelling and measurements is shown in Figure 10. A method of improving the profile accuracy is by increasing the number of grid points on the profile. However, this method faces limitations when it comes to the complex deforming computational domain of the twin screw compressor because the grid quality deteriorates and computational time increases tremendously. In order to address this problem, an analytical grid distribution procedure has been developed (Rane et al, 2015) that can simultaneously refine the region of high importance i.e. the interlobe space and produce the conformal interface between the rotors.

Figure 13

Numerical mesh generated using rotor-to-rack boundary distribution and background blocking By this means, the surface mesh on the casing is of the highest quality with regular quadrilateral cells. The surface mesh on the interlobe interface mostly follows axial grid lines with only small transverse movements in the vicinity of the top and bottom cusp's which are cyclically repeating. The current implementation allows for a fully conformal interface with the equal index of the top and bottom CUSP points which ensures straight line in the axial direction. This also accurately captures blow-hole area. The features of the newly developed single domain mesh for two rotors are shown in Figure 13.The total grid size can be controlled by independent refining of the interlobe area between two cusp pints and therefore reducing the overall mesh size but improving the quality of mesh in interlobe gap and blow-hole area.

Twin screw machines with variable rotor geometry

Gardner, 1969 patented screw compressor with variable pitch rotors. However such machines are rarely used as compressors due to the lack of efficient and economical manufacturing techniques. The variable pitch rotors are more common in vacuum technology but the rotor profiles are simple to produce. The conformal numerical mesh generated by SCORG, 2014 could be used for the CFD analysis of performance of compressors with variable geometry rotors. Rane et al, 2012 reported the analysis of variable rotor geometry using SCORG and Ansys -CFX solver. Figure 14 shows the comparison of the sealing lines obtained for the uniform rotors and variable rotors. The sealing line on the uniform pitch rotors has the same length for each interlobe space along the rotor. The variable pitch rotors have the sealing line 20% longer at the suction and 20% shorter on the discharge end compared to the mid section which has the same sealing line length as the uniform pitch rotors. This indicates that variable pitch rotors have smaller leakage area towards discharge and higher volumetric efficiency compared to the constant pitch rotors. The performance of this oil free compressor of 128 mm 3/5 rotors was obtained using SST Turbulence model in Ansys CFX solver on the grid of nearly 2 million cells. Table 1 shows a comparison of predicted volumetric and adiabatic efficiencies. The research confirmed that the variable geometry rotors have advantage in high pressure applications and that CFD can be effectively used to estimate performance and benefits of variable geometry screw machines

Conical internally geared rotary compressor

The uniform pitch and variable section compressor with internally geared rotors is invented by start-up company Vertrotor, 2011. The 3/4 lobe compressor has cycloidal profiles as shown in Figure 15. In operation, the outer rotor is positioned on a central axis while the inner rotor rotates about an eccentric axis with varying centre distance from the suction to the discharge ends. Both axes are stationary in space. The numerical mesh is generated using SCORG, 2014. CFD analysis revealed interesting features of the machines such as flexibility for use with various pressure ratios and distribution of the pressure on one side of the machine as shown in Figure 15.

Figure 15

Conical internally geared screw compressor, numerical mesh and pressure on the rotor surface

Multiphase CFD simulation of oil injected compressor

The cross section of the oil injected screw compressor with 4/5 configuration of "N" profile rotors, axis distance 93 mm and 108 mm male rotor diameter is shown in Figure 16. The oil injection is simulated by use of Ansys-CFX with full multiphase Euler-Euler scheme in the coupled pressure based solver. The turbulence model used is SST. The single domain numerical mesh for rotors was produced by SCORG while the stationary mesh of the ports was produced using ICEM. Figure 17 shows the comparison of measured and calculated data. On the left the integral volume flow rate was compared at two different speeds. The diagram on the right shows the comparison of internal pressures measured by use of 4 piezoelectric probes. The whole process of generating numerical mesh lasts for just few minutes. The calculation required 3 full cycles to stabilise which took around 4 days of calculation on the mesh ranging around 1 million cells. The results match very well and demonstrate that the single domain rotor grid is today the most suitable for multiphase flows in twin screw machines.

CONCLUSIONS

The last two decades saw growing interest for modelling and design of screw machines using 3D Computational Fluid Dynamics. Following early developments of the grid generation algorithms, extensive study was performed to validate accuracy of the 3D CFD results. Today CFD is still research tool used by academia and industry. To become widely used by industry the grid generation procedure and CFD simulation must become faster and more reliable. Coupled chamber and CFD models will find their use more often in industry. Further developments are also expected in grid generation especially addressing grid quality and leakage flows. Also more work is required in implementing real fluids and management of rotor clearances to better reflect reality.

  flux source in general transport equation volume source in general transport equation cell surface fluid velocity surface velocity control volume cell face normal vector

Figure 2 .

 2 Figure 2. Typical representative of a numerical mesh for CFD calculation of screw compressors Mujić et al. (2008) presented an optimisation of the discharge port area based on flow behaviour in the discharge chamber. They used CFD for relative comparison of port geometry modifications and their influence on predicted pressure pulsations has been used to judge sound spectrum and noise level from the compressor. These noise levels predicted by CFD solutions have been used for designing discharge ports with reduced noise levels. Mujić (2009) in his thesis presented a 3D CFD coupled model in which the boundary conditions for the discharge port were obtained as time varying data from 1D thermodynamic chamber models. Such an approach simplified the numerical analysis and also provided faster results from the CFD models. The procedure was implemented for Star CCM+ solver. It was found that the results predicted by the coupled model for sound pressure levels were closer to the full 3D CFD models and also in close agreement with the experimental measurements. Kethidi et al. (2011) conducted further studies on the influence of turbulence modelling on the CFD prediction of local velocity fields in twin screw compressors. Nouri et al. (2006) carried out cycle-resolved velocity measurements within a screw compressor using LDV and these data have been used by Kethidi et al. (2012) to compare with the results from 3D CFD models. Sauls and Branch (2013) used the results from CFD calculations to develop an improved one-dimensional thermodynamic model for refrigerant screw compressors, by extracting calibration coefficients that influence the pressure variation during the discharge process. Recently, Arjeneh et al. (2014) presented the analysis of flow through the suction port of a compressor with water injection. It was reported that it was difficult to stabilise a full 3D analysis with both deforming rotor domains and multiphase models. A simplified suction port model, with time varying boundary conditions, obtained from a full 3D single phase analysis was successful in predicting pressure variation in the suction port that agreed closely with experimental results. Also wetting of the port walls by the injected water was well predicted. Voorde, Vierendeels and Dick. (2004 and 2005) were the first to

  . Full details of that study are presented in Rane et al, 2013 and 2015.

Figure 3

 3 Figure 3 Mass flow rate variations in the ports (left) and contours of pressure variation (right)The first experimental verification of the numerical results obtained using SCORG and commercial CFD solver Comet was reported in Kovacevic et all, 2002. This study was performed on the oil injected screw compressor with 'N' type rotors of a 5/6 lobe configuration and 128 mm male rotor outer diameter. The numerical mesh contained very moderate number of just over half of a million of cells of which around 200,000 were used to map the moving parts of the grid. The calculation was performed using segregated pressure based solver Comet of Adapco-CD with Euler-Lagrange treatment of multiphase flow calculating concentration of dispersed oil in air with average Souter diameter of oil droplets assumed at 20 micrometres. The turbulence model was K-omega. The converged solution obtained on an office PC was achieved with 120 time steps in approximately 30 hours of computing time. The results were compared with the measurements of the identical screw air compressor. Four piezo-resistive transducers were positioned in the housing to measure pressure fluctuations across the compressor.

Figure 4

 4 Figure 4 Oil injection and Comparison of measured and calculated pressures Obtained results were compared for various discharge pressures of 6, 7, 8 and 9 bar. Good agreement was obtained both for the integral parameters and the instantaneous pressure values, as shown in Figure 4. The mentioned report showed for the first time the distribution of oil in the working domain and discussed the effects of various factors and setup parameters on the calculation accuracy.

Figure 5

 5 Figure 5 Position of coordinate systems and optical access windows the female (a) and male rotor (b); axial position (c)

Figure 6

 6 Figure 6 Flow zones identified in the results (left) and comparison of the LDV and CFD axial velocities Conclusions derived from the measurements, are as follows: (1) Chamber-to-chamber velocity variations were up to 10% more pronounced near the leading edge of the rotor. (2) The mean axial flow within the working chamber decreases from the trailing to the leading edge with velocity values up to 1.75 times larger than the rotor surface velocity near the trailing edge region. (3) The effect of opening of the discharge port on velocities is significant near the leading edge of the rotors and causes a complex and unstable flow with very steep velocity gradients. The highest impact of the port opening on the flow is experienced near the tip of the rotor with values decreasing towards the rotor root. The measurements confirm that turbulence plays a significant role in the flow between narrow leakage passages and the large main and discharge domains. It was assumed that the most probable reason for this slight disagreement in the CFD results with measurements is due to turbulence modelling. Therefore further research in the turbulence models for internal flow in the compressor ports is suggested. Favourably, the flow on both sides of that region appears not to be so turbulent. Due to that fact and because the internal energy in positive displacement machines is significantly larger than the kinetic energy, this does not greatly affect the overall estimation of performance. Despite this, further development and improvement of 3D CFD codes are needed.

  the same generic transport equation (2) can be used to simultaneously define behaviour of various materials in the working domain including fluids and solids. If the variable Φ in that equation is fluid velocity vi and the material properties of such domain could be defined by Stoke's law, this will form Newtonian fluid momentum equation. If on the other hand the variable Φ assumes rate of change of the deformation of solid and the material properties cold be describe by Hooke's law for thermoelastic body then this equation represents solid momentum equation. Providing that both solid and fluid domains of a screw machine can be defined by a combined numerical mesh then Finite volume method could be applied simultaneously within the same solver to solve fluid flow and solid deformations. Kovacevic 2002, reported on the investigation of change in clearances for three common applications of screw compressors, namely an oil-injected air compressor of moderate pressure ratio, a dry air compressor, of low pressure ratio, and a high pressure oil flooded compressor. In all cases the rotors are of 'N' type with a 5/6 lobe configuration. The solver used for this investigation was Comet and the numerical mesh for the rotors and flow domains was generated by use of SCORG.

Figure 7

 7 Figure 7 Oil injected screw compressor for moderate pressures (left) and high discharge pressures (right) The rotor deformations in the ordinary oil injected screw compressor are shown in Figure 7 (left). The high pressure oil injected refrigeration CO2 compressor with suction at 30 bar and 0 o C and discharge at 90 bar and 40 o C is shown in Figure 7 (right).The figure shows pressure distribution and deformation vectors and magnitude. In both cases the interlobe clearance gap is increased during the operation. For the medium pressure application these deformations are small, order of magnitude smaller than the

  For 3 bar discharge pressure the exit temperature has an average value of 180 o C. The deformations of the rotor are presented in left Figure 8. The fluid temperature in the immediate vicinity of the solid boundary changes rapidly, as shown in the right diagram of the same figure. However, the temperature of the rotor pair is lower due to the continuous averaging oscillations of pressure and temperature in the surrounding fluid. This is shown in the right diagram of Figure8, where the temperature distribution is given in cross section for both the fluid flow and the rotor body. The deformation, presented in the figure, is increased 5000 times in order to make it visible. The rotor deformation has the same order of magnitude as the rotor clearance.

Figure 8

 8 Figure 8 Rotor displacement vectors and temperature distribution for an oil free compressor 4.2 Coupled CFD and Chamber Model It is possible to increase the speed of calculation of screw machines retaining accuracy of 3D models. Mujic et al, 2008 used the decomposition of the compressor geometry shown in Figure 9 to introduce the idea of coupled models. According to that, the components of the system of greater interest are modelled with full 3-D CFD model while the components of secondary concern are simulated with a thermodynamic model. This can combine the advantages of the fast computation of as chamber model as in Stosic et all, 2005 with the enhanced capabilities of the 3-D model, described in detail in Kovacevic et all, 2006.

Figure 9

 9 Figure 9 3D domain of the screw compressor discharge chamber and full 3D model showing the interface The inlet and outlet flows are the means by which a compressor chamber exchanges energy and mass with its surroundings. These occur through openings which generally change both in size and shape with time. More than one of them can be connected to the compression chamber at the same time. In the mathematical models mentioned earlier, these flows are introduced through the enthalpy and mass contributions to the fluid in the compression chamber. If all three chambers are simulated by a quasi one dimensional thermodynamic model, as discussed by Stosic et al, 2005, both the mass and energy flow estimates are based on the assumption of adiabatic flow through the suction and discharge cavities.However, if a one dimensional model of the compression chamber needs to be integrated with a three dimensional model of the suction and discharge chambers the exchange of mass and energy must be calculated by the summation of the boundary flows that occur in the three dimensional domains. Once the integrated flows are added to those of the one-dimensional model in the compression chamber, they can be used to calculate the thermodynamic properties in that chamber in the form of pressure,

Figure 10

 10 Figure 10 Comparison of results provided by numerical modelsThe 3-D and coupled models offer better accuracy than the thermodynamic model, especially for higher harmonics of the gas pulsations. The discharge chamber geometry certainly does influence the gas pulsations and therefore the accuracy of the prediction is thereby improved in the case of the 3-D and coupled models. Additionally, since the thermodynamic model assumes uniform distribution of fluid properties across the control volume the value used for comparison of the results is that obtained at the centre of the discharge chamber. In the case of the 3-D model, the compared pressure values are those taken at the identical position

Figure 11

 11 Figure 11 Accuracy and speed of numerical modelsThe accuracy and computational time for obtaining solution with each numerical model are shown in Figure11. The chamber model requires modest computer resources and its computational time is much shorter than that required for 3-D computations. The accuracy of the 3-D and coupled models is better than that of the simple chamber model. The coupled model requires one order of magnitude lower computational time than the full 3-D model, for negligible loss in accuracy. Mujic et al, 2010, used the coupled model to investigate the influence of the discharge port shape on the pressure pulsations in the discharge plenum (i.e. noise generation) and its effects on

Figure 12

 12 Figure[START_REF] Kovačević | CFD Integrated Design of Screw Compressors[END_REF] Influence of the port shape on noise generation5. RECENT DEVELOPMENTS IN CFD MODELLING OF SCREW MACHINES 5.1 Single domain mesh with interlobe refinement for improved leakage flow capturingOne of the major factors affecting the performance prediction of twin screw compressors by use of computational fluid dynamics is the accuracy of representing the leakage by the discretization methods in grid generation Rane, et al, 2012. A method of improving the profile accuracy is by increasing the number of grid points on the profile. However, this method faces

Figure 14

 14 Figure 14 Comparison of interlobe sealing line lengths for constant and variable rotor geometry compressor

Figure 16 Figure 17

 1617 Figure[START_REF] Papes | Numerical simulation of a twin screw expander for performance prediction, 9th Int conf on compressors and their systems[END_REF] Oil injected screw compressor (left) and CFD simulation of oil distribution (right)

Table 1 .

 1 Comparison of adiabatic and volumetric efficiencies of uniform and variable geometry rotor compressors


Philipp Mattern
email: mattern@kit.edu

Martin Gabi

Jochen Kriegseis

S Y Mpo

O Tatin

A Plane-to-plane comparison of Common Averaged vs. POD Patterns of HS-SPIV Data within a pump

Keywords:

The purpose of the present study addresses to identify and compare different coherent flow patterns within the side channel of a regenerative flow pump (RFP). Common Average (CA) approaches, as well as the hidden beauty (Aubry [1991]) of a Proper Orthogonal Decomposition (POD) are applied on multiple 2D planes spanning over the third dimension for a single Operating Point. By comparing the strongest patterns for different measurement planes in terms of structure and energy, it is possible to gain new insights into the complex flow dynamics within the side channel, enabling to uncover the interplay of different planes and therefore the derivation of the 3D character of the flow dynamics. Multiple planes along the height of the side channel of the pump are discussed.

INTRODUCTION

Regenerative flow pumps (RFP) are characterized by very high specific pressure heads and low specific flow rates. As such, they cover a special field of application, which to date has been rarely documented in public literature. They are self-priming and their working principle is "quasi multistage". Since their efficiency is relatively low compared to the more common axial or radial types, advanced optimization strategies are of utmost importance for the development of such pumps. As the basic working principle of the machine is based on multiple interactions between side channel and impeller per single revolution, regenerative pumps generate periodical, highly complex flow fields (see Fig. 1a). These complex flow fields led to various theories regarding the working principle of the machine. A review of the theories was made by [START_REF] Senoo | A comparison of regenerative-pump theories supported by new performance data[END_REF] who concluded that the two main theories (momentum vs. circulation) are compatible. [START_REF] Grabow | Übersicht über die theorien zur klärung der strömung in seitenkanalmaschinen[END_REF] gave a detailed overview and discussion of both theories as well, however stating that although the circulation theory neglects drag and shear stress effects, it is widely accepted and gives better results for performance predictions as well as the general geometric configuration of a RFP. It has to be mentioned, that neither Senoo nor Grabow had yet access to CFD-results, nor modern flow field visualization techniques. Nowadays, with modern techniques being available, the momentum theory is regarded as the more flexible and precise one. Therefore recent approaches focused on improving the momentum theory [START_REF] Song | A modified theory for the flow mechanism in a regenerative flow pump[END_REF], [START_REF] Meakhail | An improved theory for regenerative pump performance[END_REF], [START_REF] Yoo | Improved momentum exchange theory for incompressible regenerative turbomachines[END_REF]). However, there was an ongoing demand for comprehensive experimental data. [START_REF] Quail | Design optimisation of a regenerative pump using numerical and experimental techniques[END_REF] concluded, that substantial efficiency and performance improvement would be attained with a better understanding of the flow field in the regenerative pump. Recently, [START_REF] Böhle | Evaluation of the flow inside a side channel pump by the application of an analytical model and cfd[END_REF] stated, that "the composition of a side channel pump is simple, but the flow inside this machine is very complex and all details have not been understood until now". [START_REF] Hollenberg | An investigation of regenerative blowers and pumps[END_REF] said, that to understand RFP, "the most fruitful approach is therefore from overall performance data on specific units and corresponding flow visualization studies. [START_REF] Quail | Study of a regenerative pump using numerical and experimental techniques[END_REF] sum it up : "There is a need for significant developments in instrumentation technology and novel approaches which enable detailed data to be acquired over large regions at higher accuracy, (particularly for flow in the current study), but at a reasonable cost." To understand the complex flow phenomena within a RFP and to provide the demanded data, the side channel of a generic model (see Figure 2) is examined by means of 2D-3C High-Speed Stereo Particle Image Velocimetry (2D-3C HS SPIV).

Figure 2. CAD Model of the regenerative flow pump

The model has been specifically designed with focus on full optical accessibility in order to obtain the desired flow information. The RFP is integrated in a common pump test rig equipped with multiple common measurement systems to gather data, such as torque, volume flow, pressure rise, temperature and speed. This common setup enables the RFP to be run at every operating point possible along the characteristic curve of the machine. First approaches [START_REF] Mattern | Investigations in a side channel pump using high speed stereo particle image velocimetry[END_REF][START_REF] Mattern | Flow characteristics beyond time and phase a modal analysis of the patterns in a regenerative pump[END_REF]) on analyzing the measured flow fields showed that common methods such as temporal and/or spatial averaging oversimplify the fluid structures thus rendering a proper interpretation of the results difficult (Later on Figure 5 shows these limitations, as in the averaged flow field any dynamics suggested by the instantaneous field is smoothed out). Combined approaches like phase averaging (Fig. 1b) are more beneficial, however depending on a specific frequency. To overcome the above mentioned shortcomings, POD has been used to decompose the raw data into modal patterns superimposed to the mean flow (e.g. see Fig. 7 right hand side). This approach provides a new insights into the complex flow development within the machine.

EXPERIMENTAL SETUP

A double flow impeller with 24 blades on each side and a diameter of 200 mm is covered by an acrylic casing with the side channel integrated. The machine is running at a speed of 500 rpm. The common layout of the test rig allows full control over the flow regime and therefore the operation point (OP) of the machine. Flow direction through the machine, field of view and the chosen coordinate system are indicated in Figure 3 ax defines the axial coordinate, while r and θ are the radial and circumferential coordinates, respectively. Note that in-and outlet are separated by a stripper (hatched area), with its center being used as the origin for the θ coordinate. The rotation axis defines the origin for r, while the axial tip of the blades defines the origin for ax. The light sheet of the laser enters the acrylic casing from the left and illuminates the area marked in red. At its widest point, the measured area covers approxi-mately 70°, starting from θ = 190°up to θ = 260°, all with respect to the stripper. Within this regime, the flow is considered developed, as influences of both, inand outlet should be minimal. Even though not shown in the present work, it is worth to note, that all PIV experiments were simultaneously monitored with more common methods as well, all synced to the same time signal. Pressure probes were installed to take the static wall pressure near the in-and outlet of the machine. Additionally, torque, as well as rotational speed and angular position of the impeller were measured. Although the whole measurement campaign covered six different operating points for three different impeller designs and for five different measurement planes along the height of the side channel, this paper is focusing on the 24 blade impeller and a single operating point in mild overload to illustrate the axial development of the flow, taking all five measurement planes into account as can be seen in Figure 4. In addition, the simplified transport mecha- nism of the machine, described by an axial interaction between the flow in the side channel and the blades of the impeller (secondary flow) is shown in red as well.

The SPIV System at hand consists of two Photron SA4 and a Quantronix Darwin Duo, set up in a standard forward/backward scattered configuration, with an angle of 65°between both cameras and a distance of 340 mm between the cameras and the light sheet. The flow is captured at 2000 fps with a resolution of 1 megapixel. More details of the experimental setup can be found in [START_REF] Mattern | Investigations in a side channel pump using high speed stereo particle image velocimetry[END_REF]).

Post Processing

For PIV Analysis PIVTEC's PIVview (V3.5.9.1) is used. Common methods of dewarping and disparity correction are applied. The cross correlation has an interrogation area of 16x16 pixels and an overlap of 50%. A strict dynamic mean filter of 2*mean velocity ± 1*standard deviation leads to a small amount of around 0.5% invalid vectors, which could nearly all be replaced by their second highest correlation peaks. Separation time (pulse distance) between the two pulses is 100 µs. During 1.4 seconds of measurement, N = 2771 velocity maps for an area of 110x50 mm each containing around 3000 three component vectors are captured. Further post-processing and analysis of the velocity data was done using MAT-LAB.

AVERAGED FLOW FIELDS

To understand the axial flow development, Figure 5 is introduced. In-plane velocities are given by planar Although the mathematical definition of the positive direction is counter-clockwise, out of comprehensibility, θ is defined in rotating direction of the impeller (ω), therefore clockwise. The axial out-of-plane velocity is given as contour plot, with the colorbar indicating the levels. Zero crossing of the contour variable is indicated by a dotted magenta line. All components are normalized by the average side channel velocity c k = V /A sidechannel , according to the current volume flow and therefore operating point shown together with the characteristic curve in the lower left corner (as all planes were measured for the same operating point, they were normalize with the same factor as well). The present plane is illustrated in the upper left corner. The outer limits of the side channel are indicated by thick black arcs. To clarify the positions within the flow field, geometrically characteristic points along the circumference, as well as the radius are given by dotted gray lines for distinct θ and r i . It is clearly visible that the measured area does not cover the whole side channel and changed from plane to plane. This is a common problem of SPIV and the corresponding challenging optical access. Due to the complexity of the flow, missing areas are spared instead of extrapolated, as this would be too speculative. It is observable, that the in-plane components in the 2mm plane are more or less following the curvature of the side channel. For every plane more distant to the impeller, the streamlines are more and more bend to the inner side wall. Additionally, the profile along the radius for planes near the impeller is slower at the inner wall and increases up to the outer wall. For upper planes, in-plane magnitude increases, while the profile gets more homogeneous. For the axial component it is visible that the zero crossing is moving towards the inner side wall for more distant planes. Regarding the flow development along the side channel, the contour lines still follow the curvature considerably better than the in-plane components. To compare quantitatively, radial splines are extracted at θ = 225 for c ax and c u and shown in Figure 6. For the back-flow into the impeller, magnitudes of the normalized axial component (c ax ) are sorted according to the corresponding height of the plane. and are similar in trend. As already observed, the zero crossing moves from outer to inner radius with increasing plane height (channel width 0.75 → 0.6). Besides being significantly steeper in general, the splines do not share a distinct trend or order after the zero crossing, therefore out-flow of the impeller, anymore. Generally, planes [2/6/10mm] are very similar in trend and magnitude, with only planes [14/18mm] showing clear differences. The circumferential component (c u ) shows also varying behavior: Regarding only the radial development, planes [2/6/10mm] share more or less the same tendency, with their magnitude differing more distinct than in case of c ax . For big radii they start to overlap. Planes [14/18mm] are switched, regarding a monotonic order and show tendencies of a profile of a developed flow along the circumference. Along the height of the side channel one can observe an increase of the circumferential velocity up to 1.3 times the mean channel velocity in case of plane 14 mm. At the 18mm plane velocities start to decrease again. This is reasonable, as it is the closest plane to the upper wall. With the the in-plane components being nearly one order of magnitude higher than the axial velocities, the volume flow seems to dominate the flow direction for this OP. All planes have a small velocity gradient for a wide span along r, except the steep gradients at the inner and outer rim. However, with this being said, there's not much more information to extract from the time averaged data. One can get a basic idea about the flow structure, but per definition, time averaging levels out the dynamics. To identify those dynamics, a more decent way of analyzing the same pool of data is necessary.

COHERENT FLUCTUATIONS

As already shown in [START_REF] Mattern | Flow characteristics beyond time and phase a modal analysis of the patterns in a regenerative pump[END_REF], a common way to identify dynamics is to do phase averaging (exemplary shown in Figure 1b). Here it is done in steps of 1.5°, dividing the angular distance of 15°between two blade into 10 phase averaged results consisting of only ∼ 280 flow fields (therefore only 10% of the available data pool of ∼ 2800 flow fields). The axial component reveals pattern-like structures, which are obviously somewhat related to the underlying blades of the impeller. On the other hand, as the in-plane components already have been identified as the predominant ones, the phase averaged in-plane velocities only differ slightly from the ones of the

c i := c i (θ, r) + c i (θ, r, t) + c i (θ, r, t) (1)
As stated in Pechloff and [START_REF] Alexander | Small disturbance navier-stokes method: Efficient tool for predicting unsteady air loads[END_REF] and [START_REF] Oberleithner | Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction[END_REF], due to the fact that the phase average depends on a chosen periodic quantity (here: phase angle) it can also be expressed as:

c i (θ, r) = c i (θ, r) + c i (θ, r, t). (2)
Combining ( 1) and (2) this leaves the coherent fluctuations c i (here: for each blade position):

c i = c i (θ, r) -c i (θ, r) (3)
This combined approach is shown in Figure 7a (7b will be introduced later on). Now that the predominant average flow is suppressed, the working principle of the machine is uncovered. While the impeller is rotating, the fluid has to evade the blade by streaming into the side channel in axial direction (warm colors), as well in-plane in radial direction. In addition, there's also an in-plane flow visible from the pressure side (warm colors) to the suction side (cool colors) above each blade channel. Furthermore, a flow over the blades from pressure-to suction side in circumferential direction becomes obvious. Due to the nature of the flow, complex vortices at the outer diameter, as well as near the hub can be assumed. Taken the upper planes into account as well, it shows, that those vortices become more obvious (10mm), however disappearing again when close to the upper side channel wall (18mm). Note that while the in-plane velocities stay within the same order of magnitude, the axial components differ drastically with the values of the 18mm plane being nearly one order of magnitude lower than the 2mm plane.

Intermediate Conclusions

Common approaches are indeed capable to uncover some salient patterns and corresponding dynamics from flow fields. Time averaging reveals the underlying secondary flow: The high positive axial velocities at the outer rim on the pressure side of the blade can be explained, as the fluid is blocked by the casing. Consequently, it cannot be transported furthermore along the radius and therefore has to evade upwards into the side channel. From there on, it starts to decelerate in axial direction (see Figure 6a) up until the upper wall, where it is deflected in direction of the hub again. Finally it enters the blade channel again near the hub. This defines the secondary flow (as sketched in Figure 4). Phase averaging refines the flow field even further: Blade related structures can be identified. It becomes obvious that the flow itself is quite complex and that in addition to the secondary flow there are further periodic flow mechanism within the machine, which are hard to identify. This is especially the case for the more distant planes. A huge disadvantage is the necessity of additional information: For each PIV result, the exact angular position of the impeller has to be known. Therefore the PIV System as well as the rotary encoder have to be in sync, both with a sufficient high enough measurement frequency (at least the 2000 Hz from the PIV System) and no temporal deviation. Another limitation is the reduced size of the data pool: with only being able to use 10% of the available data to do a phase average there remain considerable amounts of noise. Furthermore, as phase averaging is only suitable for periodic phenomena, it is necessary to assume a characteristic frequency. In case of fluid machinery the blade passing frequency is sufficient in most of the cases. However, by doing so only periodical phenomena with this distinct frequency can be observed, where all other frequencies are smeared out by definition (see [START_REF] Aubry | On the hidden beauty of the proper orthogonal decomposition[END_REF] and [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF] for details of this method). Another more flexible approach to understand the complex flow situation furthermore is the POD, as illustrated below. This method advances beaond the above-discussed limitations of a characteristic frequency, and takes the full data pool into account.

POD RESULTS

The analysis of the POD results for each plane can be split into two steps. First, the modal structures and their energy contributions to the flow are investigated: 2771 discrete velocity fields per measurement result in the same amount of POD modes. Subsequently, the weighting coefficients are calculated and their time dependent behavior is investigated.

Modal Analyses

To streamlines for the in-plane and as contour plot for the out-of-plane component. The first two modes, as well as Mode 3 and 4, reveal very similar flow patterns. Both pairs show blade related flow structures in plane and out-of-plane. Modes 3&4 represent the first harmonics of Modes 1&2. They reveal twice as many, but accordingly thinner patterns. As a Mode is generated from all measured flow fields at once, the position of the impeller (and therefore the blades), cannot be drawn. In addition, modal patterns are to be compared qualitatively, so no quantitative values are given (However, quantification can be determined from the reconstructed coefficients).

For the first five Modes these coefficients are given with Figure 9 for a small period of 50 time-steps (the first 50 PIV snapshots). It is important to understand that directions in modal patterns are depending on the sign of the coefficient. Therefore a negative coefficient switches all directions in a given modal flow pattern plot. A distinct difference between Mode 1 and 2 is the fact, that the structures are shifted by a certain angle in flow direction. This 90°-phase shift becomes more obvious from the reconstructed coefficient of the respective modes (Fig. 9). The coefficient oscillation of Modes 1&2 reveals the blade passing frequency of 200 Hz. The 90°-phase shift at identical frequency is verified by the circular shape of the corresponding Lissajous figure of Modes 1&2 (Figure 10a). The rather small spread indicates quasi-constant amplitudes of both coefficients (Fig. 9). This 90°-phase shift can also be observed for Modes 3&4 (Fig. 10b).

The amplitude fluctuation now is clearly wider, leading to a more disturbed characteristic. The reason for that is the variation in amplitude as it can be seen in Figure 9. The mentioned harmonic characteristic of Modes 3&4 with respect to Modes 1&2 is verified by the doubled frequency in coefficient oscillation in Figure 9, as well as through the Lemniscate shape of the Lissajous figure of Modes 2&3 shown in Figure 10c. This means, that also the second strongest Mode pair is related to the same dynamic phenomenon. The energy distribution for Nevertheless they show the blade related patterns at a doubled frequency (first harmonics) and therefore are not to be neglected completely. Mode 5 (Fig. 8e) already contains only about 1% of the total energy. This Mode is added as the strongest of all low-energy bearing Modes, showing no structures obviously related to any parameter of the machine, as well as no distinct oscillation of the coefficient (unlike Modes 1-4) as shown in Figure 9. However, as the circular shape of Figure 10d suggests, oscillating machine related information still exist (here: Modes 6&7), but as the modal energies asymptotically approach a constant value, higher modes only make a low contribution to the total dynamic energy. This indicates, that the influence of noise and turbulence increases.

Plane Comparison

According to the above explained scheme, the [10/18mm] planes are analyzed as well. If one compares the strongest Modes 1&2 of the 2mm and 10mm planes with their respective combined approach results shown in Figure 7 the similarity is obvious. One has to keep in mind that all POD Modes (7b) were purely generated out of the raw PIV vector fields. No additional information, such as angular position of the impeller, or rotational speed was necessary. This is unlike the combined approach (7a), were both information were mandatory. To further understand the physical meaning of the shown modal structures, the retransformation to the temporal domain has to be analyzed. It explains the similarity between Modes 1 and 2. By alternating between the two shifted structures with blade passing frequency, the visible blade structures and vortex systems in either modes are moved through the temporal domain, following the blades. It indicates the driving force behind mass transportation between side channel and impeller. This explains the similarity between the strong Modes and their respective combined approach. If an average flow field is subtracted from phase results (thus the combined approach), the coherent dynamic structure of the flow reveals. POD on the other hand extracts the dynamic of the flow and sorts flow patterns according to their energy contribution to the total dynamic energy. As for example Modes 1&2 for the 2mm plane contribute already approximately 20% to the dynamic flow energy, it has to have a visible influence on the phase average results. This is also the case for the higher planes. Even though the energy contribution of the strongest Modes 1&2 for the 10mm plane to the total energy is already significantly lower (combined only about ∼ 8%; see Figure 12) compared to the 2mm plane, they stand out clearly above the following Modes regarding their energy. Additionally, they also show difference in energy for the first two Modes is not visible anymore (Fig. 15) and neither is their phase dependency (Fig. 16a). Besides Mode 1, which stands out a little, related structure. Additionally, there is also no obvious phase relationship between the first two Modes, shown as the rather random distribution in Figure 16a. However, by taking a closer look at the coefficients of Modes 3&4 in Figure 18, a blade related oscillation can be assumed (Note that the shown time period is extended to 200 time-steps to take the fluctuation of Mode 1 into account as well). The 90°-phase shift can be assumed as well in Figure 16b (The appearance of a filled circle, instead of being ring-shaped like the Figures from the planes before can be explained by the strong fluctuation of the amplitude again). Therefore, despite being dominant in energy, it can be suggested that the same dynamic phenomenon as already seen in the other planes is the driving force behind Modes 3&4.

CONCLUSION

The understanding of the highly unsteady and complex flow structure within a regenerative pump is crucial for further formulating the energy exchange between the impeller and the fluid. Even high-speed stereo PIV measurements analyzed in a classic approach provide only limited information on the complex three dimensional swirl flows within the machine. The interacting flow patterns, which are the driving force behind the momentum exchange and therefore the working principle of the machine, are concealed by the highly energetic main flow. A simple subtraction of the average flow field from the instantaneous flow field gives a first impression of the vortex structures in game. However, without filtering the results are noisy and can even be misleading in case of sudden random disturbances in the flow field. By using additional information such as angular positions to do phase averaging, it is possible to switch the point of view from the fixed laboratory system to the rotating blade related system furthermore revealing blade related patterns. However, as mentioned before, phase averaging limits the possibility to discover phenomena to preset frequencies. With the POD approach this limitation is lifted. Once the predominant average flow field is subtracted, the POD method massively adds to a simpler description of the flow within the machine. Particularly, the entire flow scenario can be approximated in good accuracy with only a few modes of salient flow patterns, making it an excellent tool for filtering random noise. Therefore, it is now possible to focus on those structures and, in an upcoming approach, to identify the parameters of the machine generating those specific flow structures.

A deeper investigation of those phenomena goes beyond the scope of the present paper and is the content of upcoming projects. However, a short example shall be given: The full power of POD is revealed when trying to interpret the first Mode for the 18mm plane. The extended time period of 500 time-steps in Figure 18 shows that the coefficient of Mode 1 actually is oscillating, but with a frequency of ∼ 16Hz, which as much lower than the blade passing frequency of 200 Hz. The pattern itself (Fig. 17) shows smooth, partly homogeneous structures. In combination with its coefficient, the underlying phenomenon can be interpreted: The fluid is oscillating in main flow direction with a frequency of 8.3Hz. This frequency is the rotary frequency of the machine at 500 rpm. Therefore, the underlying phenomena is rather related to the impeller itself than the blades. Common approaches would not have been able to uncover this.

  Figure 1. Velocity flow fields [2mm]

Figure 3 .

 3 Figure 3. Close up of the acrylic casing. PIV measurement area marked in red

Figure 4 .

 4 Figure 4. Secondary flow (red) and acquired measurement planes (green)

Figure 5 .

 5 Figure 5. Time averaged velocity flow fields (c i ) along the height of the side channel [2/10/18mm]

Figure 6 .

 6 Figure 6. Time averaged flow components (c i ) along the side channel width for all planes

( a )

 a Coherent fluctuation flow fields ( ci) (b) POD Mode

Figure 7 .

 7 Figure 7. Comparison between the Combined Approach and the corresponding POD Modes [2/10/18mm]

Figure 8 .

 8 Figure 8. POD results as flow pattern Modes 1 to 5 [2mm]

Figure 9 .

 9 Figure 9. Reconstructed coefficient Mode 1-5 [2mm]

Figure 10 .

 10 Figure 10. Lissajous figures [2mm]

Figure 14 .Figure 15 .

 1415 Figure 13. Lissajous figures [10mm]

Figure 16 .Figure 17 .

 1617 Figure 16. Lissajous figures [18mm]

Figure 18 .

 18 Figure 18. Reconstructed coefficient Mode 1, 3&4 [18mm]

ACKNOWLEDGMENTS

This paper is based upon work supported by the German Reasearch Foundation (DFG) under Grant No. 585385; Reference: GA 718/5-1.


Junho Kim

Han Ho Song

Seung Jin Song

Measurements of the Non-dimensional Thermal Parameter Effects on Cavitation in a Turbopump Inducer

Keywords:

Cavitation is sensitive to liquid temperature. Hence, temperature effects on cavitation in turbopump is an important issue. To quantify the temperature effects, the non-dimensional thermal parameter (Σ*) has been proposed. In this study, effects of non-dimensional thermal parameter on cavitation performance and rotating cavitation onset in a turbopump inducer have been experimentally investigated in water. Distinct from previous research, experiments have been performed at various temperatures and rotational speeds while maintaining a constant Reynolds number. Static pressure upstream and downstream of the inducer have been measured to determine cavitation performance, and unsteady pressures transducers and high speed camera have been used to identify cavitation instabilities. Two types of cavitation instabilities have been identified -rotating cavitation and asymmetric attached cavitation. Increasing non-dimensional thermal parameter lowers the critical cavitation number and the breakdown cavitation number at all values of non-dimensional thermal parameters. The onset cavitation number of rotating cavitation is decreased as the non-dimensional thermal parameter increases.

.

INTRODUCTION

In liquid rocket engine, turbopumps rotate at high speeds in low pressure liquid fuel and oxidizer to minimize size and weight. These conditions leads inevitably to cavitation, therefore, inducer is installed upstream of turbopumps to prevent cavitation occurrence at an impeller. However, the inducer also suffers from cavitation, and cavitation causes performance degradation and cavitation instabilities such as rotating cavitation, asymmetric attached cavitation, and cavitation surge. Rotating cavitation is a local cavitation instability in which an uneven cavitation region propagates at an either supersynchronous or (sometimes) subsynchronous speed in the direction of the inducer rotation. In asymmetric attached cavitation, an uneven cavitation attached attached to the blade rotates at the same speed as the inducer. Cavitation surge is a system instability where the cavitation oscillates with subsynchronous frequency in the axial direction.

Cavitation is sensitive to liquid temperature. According to Brennen [START_REF] Brennen | Hydrodynamics of Pumps[END_REF], cavitation growth is suppressed as the liquid temperature increases, and this phenomenon is called temperature effects. At higher liquid temperatures, the evaporation rate for the same bubble volume is larger than at lower temperatures. Therefore, at higher liquid temperatures, the total heat needed for evaporation at the bubble interface and the temperature drop within the bubble (due to evaporation) are both larger. Inside the bubble, as the temperature decreases, the vapor pressure also decreases. Thus, the 'effective' cavitation number is increased, and cavitation growth is delayed as the liquid temperature increases.

Stepanoff [START_REF] Stepanoff | Cavitation Properties of Liquids[END_REF] suggested the B-factor, or the ratio of the vapor volume to the liquid volume in the vaporization process, as a parameter for the temperature effects of various liquids and explain the difference in cavitation performance between water and other liquids. Brennen [START_REF] Brennen | Hydrodynamics of Pumps[END_REF] suggested the thermal parameter, Σ (m/s 3/2 ), from the simple heat transfer analysis of cavity bubble growth. This parameter represents the rate of energy transfer from the liquid to the vapor.

Σ = (ρ v L) 2 ρ l 2 C pl T ∞√ α l (1)
Subsequently, Franc et al. [START_REF] Franc | An Experimental investigation of Thermal Effect in a Caivtating Inducer[END_REF] derived a non-dimensional thermal parameter Σ * from the non-dimensional equation for heat transfer analysis of cavity bubble growth.

Σ * = Σ√C/U 3 (2)
The non-dimensional thermal parameter (Σ * ) considers the characteristic length of the pump (C) and reference velocity (U). As the temperature of the working fluid increases or the rotational speed decreases, the non-dimensional thermal parameter also increases.

Ruggeri and Moore [START_REF] Rugger | Method for Prediction of Pump Cavitation Performance for Various Liquids, Liquid Temperatures, and Rotative Speeds[END_REF] measured cavitation performance in an inducer with various liquids at different rotational speeds and temperatures. Cavitation performance is enhanced as a liquid temperature increases and a rotational speed decreases. Franc et al. [START_REF] Franc | Visualization of Leading Edge Cavitation in an Inducer at Different Temperatures[END_REF] measured leading edge cavity length by visualization in an inducer at different temperatures with refrigerant R-114 as the working fluid. Cavity length decreased and the onset of cavitation instabilities delayed as temperature increased. Yoshida et al. [START_REF] Yoshida | Thermodynamic Effect on Rotating Cavitation in an Inducer[END_REF][START_REF] Yoshida | Thermodynamic Effect on Subsynchronous Rotating Cavitation and Surge Mode Oscillation in a Space Inducer[END_REF] found that 1) increasing liquid nitrogen temperature suppresses the onset of both rotating cavitation and asymmetric attached cavitation and weakens the intensity of asymmetric attached cavitation. Kikuta et al. [START_REF] Kikuta | Influence of Rotational Speed on Theromdynamic Effect in a Cavitating Inducer[END_REF] performed experiments in liquid nitrogen to measure the influence of rotational speed on temperature effects. Cavity length was suppressed as the non-dimensional thermal parameter was increased by lowering the rotational speed at a given temperature. Torre et al. [START_REF] Torre | Experimental Characterization of Thermal Cavitation Effects on Space Rocket Axial Inducers[END_REF] found that the critical cavitation number1 was higher when the water temperature is lower at a constant rotational speed. On the other hand, the breakdown cavitation number 2 was not affected by the increasing water temperature.

The temperature effects with non-dimensional thermal parameter on cavitating inducer in water have been investigated recently. Ehrlich and Murdock [START_REF] Ehrlich | A Dimensionless Scaling Parameter for Thermal Effects on Cavitation in Turbopump Inducers[END_REF] derived a non-dimensional thermal scaling parameter in a different manner. They compared in cold water and hot water cases, and found that rotating cavitation was suppressed at a high non-dimensional thermal parameter. Kim and Song [START_REF] Kim | Measurment of Temperature effects on Cavitation in a Turbopump Inducer[END_REF] measured temperature effects on critical cavitation number and rotating cavitation onset using the non-dimensional thermal parameter. The critical cavitation number and the onset cavitation number of rotating cavitation decreased and became constant as the non-dimensional thermal parameter increases. However, their experiments were performed with a constant rotational speed at different water temperatures, so that, Reynolds number is increased as water temperature increases.

Despite such efforts, the effects of non-dimensional thermal parameter without Reynolds number effects on the cavitation performance and cavitation instability onset have not yet been investigated. Therefore, the present study provides detailed measurements of cavitation performance and cavitation instabilities in a turbopump inducer at varying non-dimensional thermal parameter. Distinct from the previous research, water temperature and an inducer rotational speed is controlled to maintain a constant Reynolds number for measuring non-dimensional thermal parameter effects. The specific research questions of the present study are as follows.

1. How does non-dimensional thermal parameter affect inducer cavitation performance without Reynolds number effects?

2. Howe does the characteristics of inducer cavitation instabilities change in non-dimensional thermal parameter variation with a constant Reynolds number?

Experimental Facility and Procedures

Figure 1 shows an inducer water test facility at Seoul National University [START_REF] Kim | Measurment of Temperature effects on Cavitation in a Turbopump Inducer[END_REF]. It is composed of water tank, rotating parts, a flow meter, a control valve, and a booster pump. Rotating parts include the test inducer and a motor. The test facility has been used for cavitating and non-cavitating inducer experiments. Pressure in the facility is regulated by either a vacuum pump or compressed air connected to the water tank. Water temperature is controlled by electrical heater installed in the water tank. An inducer rotates at a maximum 10,000 rpm with a 60-kW motor, and the rotational speed is controlled within ± 1 rpm by a variable frequency inverter.

Figure 2 is the test inducer designed by the Korea Aerospace Research Institute (KARI), and its specifications are listed in Table 1 [START_REF] Choi | Effects of a Bearing Strut on the Performance of a Turbopump Inducer[END_REF].

The mean flow rate has been measured downstream of the inducer with 0.25 % full scale (20 kg/s) electro-magnetic flow meter. Static pressure transducers with an accuracy of 0.04 % full scale (200 kPa (P1) and 400kPa (P2)) have been installed upstream (P1) and downstream (P2) of the inducer leading edge to measure static head coefficient. Cavitation instabilities have been identified with unsteady pressure transducers with 0.1 % full scale (350 kPa) installed upstream of the inducer leading edge. The flow coefficient has been fixed at the design flow coefficient within 0.067 % throughout the all experiments. The uncertainty of each parameter with 95 % confidence interval is as follows -± 0.00398 in ψ; ± 0.000422 in ϕ ; and ± 0.000759 in σ. 2 The breakdown cavitation number is defined to be the cavitation number at which the head coefficient value decreases to 75 % of the normal head coefficient value.

.

Experimental Results and Discussions 2.1. Cavitation Performance

Figure 3 shows the cavitation performance at 298 K and the design flow coefficient [START_REF] Kim | Measurment of Temperature effects on Cavitation in a Turbopump Inducer[END_REF]. The head coefficient remains constant at 0.218 for cavitation number from 0.3 to 0.071. For cavitation number lower than 0.0702, the head coefficient starts to decrease, thus, the critical cavitation number for this condition is 0.0702. Repeatability of the result is confirmed by 3 separate experiments showing similar values and trends.

Cavitation Performance

To know the cavitation instabilities occurred at the test inducer, unsteady pressures have been measured to identify cavitation instabilities at design flow coefficient and ambient temperature as a reference. Fast Fourier Transform (FFT) analysis and the cross-correlation of signals from two unsteady pressure transducers have been used to determine the characteristics of the instability. Visualization using high speed camera has been employed to confirm the cavitation instability. For axial instabilities, the phase of the crosscorrelation between the two unsteady pressure signals is equal to 0°. For rotating instabilities, the ratio of the phase of the cross-correlation (φ) and the angular distance between the pressure transducers (Δθ) is equal to the number of the rotating cells. (ncell) ncell = φ /Δθ

(3)

The frequency of instability (f r ) is then determined by dividing the detected frequency ( f d ) by the rotating cell number (ncell) 𝑓 𝑟 = 𝑓 𝑑 / ncell (4)

Figure 4 shows the power spectral density peaks from unsteady pressure transducers along the cavitation number. 3 dominant peaks at different cavitation number range occur corresponding to f/Ω = 3.0 (A), 1.14 (B), and 1.0 (C) where f/Ω = 3.0 is the blade passing frequency. According to the previous research [START_REF] Kim | Measurment of Temperature effects on Cavitation in a Turbopump Inducer[END_REF], the types and characteristics of each dominant peak have been identified using the power spectral density, phase, and coherence analysis. The peak at f/Ω = 3.0 (A) blade passing and symmetric cavitation, the peak at f/Ω = 1.14 (B) is rotating cavitation, and the peak at f/Ω = 1.0 (C) is asymmetric attached cavitation. Visualization with high speed camera has been confirmed and identified the propagation speed of each cavitation instability. Figure 5 shows the cavitating inducer for cavitation number is 0.069 where the rotating cavitation occurs. The red circles indicate the smallest cavity at each rotation. The smallest cavity (uneven cavity) propagates at a speed of 1.14 Ω which is same frequency with the peak B. Thus, the peak B is rotating cavitation propagating faster than rotational speed. Figure 6 is the visualization of the asymmetric attached cavitation where cavitation number is 0.051. The red circles indicates the smallest cavity at each rotation. Uneven cavities attached to the inducer blades and rotating with the blades, thus peak C is asymmetric attached cavitation.

Temperature Effects on Cavitation Performance and Cavitation Instabilities

In the previous study [START_REF] Kim | Measurment of Temperature effects on Cavitation in a Turbopump Inducer[END_REF], Reynolds number is increased from 2.54 × 106 to 6.41 × 106 as non-dimensional thermal parameter increases (Σ * ) from 0.0116 to 1.78. To investigate the non-dimensional thermal parameter effects with a constant Reynolds number, Reynolds numbers at all test conditions are adjusted to maintain constant at 4.08 × 106. Tests have been conducted at water temperatures of 306 K, 313 K, 323 K, 328 K, 337 K, and 347 K with rotational speeds of 6,600 rpm, 6,000 rpm, 5,000 rpm, 4,500 rpm, 4,000 rpm, 3,500 rpm, and the corresponding Σ * are listed in Table 2. In this paper, the inducer diameter (0.047 m) and tip velocity (24.67 m/s) are considered as the characteristic length and reference velocity. Σ * is increased as water temperature increases and the inducer rotational speed decreases. The dependence of the critical cavitation number, breakdown cavitation number, and the rotating cavitation onset on Σ * with a constant Reynolds number experiment are clearly shown in Fig. 9 comparing with a varying Reynolds number experiment [START_REF] Kim | Measurment of Temperature effects on Cavitation in a Turbopump Inducer[END_REF]. The critical cavitation number trends in a constant Reynolds number case is similar with a varying Reynolds number case for Σ * < 0.605. Distinct from the previous measurements, the critical cavitation number keep decreasing as Σ * increases for Σ * > 0.605 with a constant Reynolds number case. Also, as shown in Fig. 10, the breakdown cavitation number decreases as Σ * increases in a constant Reynolds number case. On the other hand, in a varying Reynolds number case, the breakdown cavitation number is independent of Σ * . Ruggeri and Moore [START_REF] Rugger | Method for Prediction of Pump Cavitation Performance for Various Liquids, Liquid Temperatures, and Rotative Speeds[END_REF] performed at various rotational speed in cryogenic flow. They found that cavitation performance was enhanced as a rotational speed increases, which means that increasing Reynolds number increases the

Conclusions

The influence of non-dimensional thermal parameter in water on the cavitation performance and cavitation instability onset for a turbopump inducer has been experimentally

Figure 1 .

 1 Figure 1. SNU turbopump inducer experimental facility.

Figure 2 .

 2 Test inducer.

Figure 3 .

 3 Figure 3. Repeatability test for cavitation performance (𝛟 𝐝 =0.096, 𝐓 ∞ = 298 K (Σ * =0.0116)).

Figure 4 .

 4 Figure 4. The power spectral density of inducer inlet pressure fluctuations (𝛟 𝐝 =0.096, ∑*=0.0116 (𝐓 ∞ =298 K)).

Figure 5 .

 5 Figure 5. High speed images of rotating cavitation propagation. Red circles indicate small cavity, and small cavity propagates faster than an inducer rotational speed (σ=0.069, 𝛟 𝐝 =0.096, Σ * =0.0116 (T=298 K)).

Figure 6 .

 6 Figure 6. High speed images of asymmetric attached cavitation propagation. Red circles indicate small cavity, and small cavity is attached to blade with an inducer rotational speed (σ=0.051, 𝛟 𝐝 =0.096, Σ * =0.0116 (T=298 K)).

Figure 7

 7 Figure 7 shows plots of the head coefficient versus cavitation number for various values of Σ * with a constant Reynolds number values. Arrows show the critical cavitation number for each Σ * values. The critical cavitation number and breakdown cavitation number decrease as Σ * increases at all values of Σ * . Figure 8 presents the power spectral density for Σ * from 0.0176 to 1.58. The onset cavitation number and amplitude of unsteady pressure fluctuation of rotating cavitation are decreased as Σ * increases.

Figure 7 .

 7 Figure 7. Cavitation performance of various nondimensional thermal parameters with a constant Reynolds number (𝛟 𝐝 =0.096).

Figure 8 .

 8 Figure 8. The power spectral density of unsteady pressure fluctuation at various non-dimensional thermal parameter ranges (Σ * = 0.0176 (306 K, 6,600 rpm) ∼ 1.58 (347K, 3,500 rpm)).

Figure 9 Figure 9 .

 99 Figure 9 also shows that the rotating cavitation onset cavitation number decreases by 19.7 % as Σ * increases from 0.0176 to 1.58. On the other hand, the rotating

Figure 10 .

 10 Figure 10. The breakdown cavitation number versus the non-dimensional thermal parameter in case of a constant Reynolds number and a varying Reynolds number at 𝛟/𝛟 𝐝 =1.0.

= 1 2=

 1 blade inlet tip angle 𝛽 2𝑏𝑡 = blade outlet tip angle σ = cavitation number, = (p 1 -p v )/ Inducer rotational speed, rotational frequency

Table 1 .

 1 Design characteristics of the test inducer.

	Item		Value
	Diameter (D)		0.094 m
	Design flow coefficient (ϕ d )	0.096
	Blade number		3
	Solidity at tip		2.7
	Blade tip angle	Inlet (β 1bt )	9.6 deg
		Outlet (β 2bt )	15.0 deg
	Non-dimensional tip clearance (t/D)	0.0106
	Angle variation on line normal to hub tan (β b ) = (r t r ⁄ )tan (β bt )

Table 2 .

 2 Test conditions and thermal parameter values.

	Temperatur e [K]	Rotational speed [rpm]	Reynolds number	Σ * = Σ√C/U 3
	306	6,600	4.03 × 10 6 1.76 × 10 -2
	313	6,000	4.09 × 10 6 3.79 × 10 -2
	323	5,000	4.08 × 10 6 1.25 × 10 -1
	337	4,000	4.08 × 10 6 6.05 × 10 -1
	347	3,500	4.08 × 10 6	1.58

  investigated for Σ * from 0.0176 to 1.58 while maintaining a constant Reynolds number. The followings are the new conclusions from the investigation. 1. Cavitation performance and the onset of rotating cavitation trends with varying Σ * are different between constant Reynolds number and varying Reynolds number experiments for Σ * is larger than 0.604. 2. As Σ * increases, the critical cavitation number and breakdown cavitation number decrease at all ranges of Σ * while maintaining a constant Reynolds number. Increase of Reynolds number degrades cavitation performance. 3. The amplitude of the rotating cavitation peak oscillations is diminished and the onset cavitation number of the rotating cavitation is decreased at all values of .Σ * . The amount of degradation of rotating cavitation onset cavitation number is larger in constant Reynolds number experiments for Σ * > 0.604. Thus, increasing Reynolds number degrades hydro dynamic stability for Σ * is larger than 0.604. 4. The critical cavitation number trends are similar to the trends of cavitation number for the onset of rotating cavitation at all values of Σ * with a constant Reynolds number.

The critical cavitation number is defined to be the cavitation number at which the head coefficient value decreases to 95 % of the normal head coefficient value.

ACKNOWLEDGMENTS

Financial support from the National Research Foundation of Korea (NSF) (Project Number: 0420-20140064), BK21+ Program, and SNU-IAMD are gratefully acknowledged by the authors.