MODELOWANIE INŻYNIERSKIE ISNN 1896-771X 32, s. 461-468, Gliwice 2006
NUMERICAL SIMULATION OF LEAKAGE FLOW BETWEEN MOVING ROTOR AND HOUSING
OF SCREW COMPRESSOR
JAN VIMMR
ONDŘEJ FRYČ
Department of Mechanics, Faculty of Applied Sciences, University of West Bohemia in Pilsen
Abstract. Mathematical modelling of transonic flow of viscous compressible fluids in very narrow channels and gaps is one of the very topical and demanding problems of fluid dynamics today. This article presents the numerical simulations of leakage flow in a narrow channel formed by a tooth of male rotor and housing in a screw compressor. The influence of the male rotor motion on the compressible viscous flow through the clearance gap for various male rotor angular velocities is observed. The dependence of simulation results on different numerical grids is rated and the sufficiently coarse grid for obtaining reasonable results is searched.
The influence of turbulence models on obtained numerical results is investigated.
1. INTRODUCTION
Screw compressors, Fig. 1, represent one of many types of machines used to compress air or other gases. The compression process is provided by complex shape of two screw rotors embedded together and by geometry of work chambers between screw rotors and compressor housing. Volume of these work chambers decreases during the rotors motion. The chambers can not be separated perfectly to allow the motion and to secure not jamming the machine. The chambers are connected by the system of clearance gaps. Three main types of gaps exist – frontal gaps at axial ends of rotors, inter-lobe gaps between rotors and rotor-housing gaps between rotors and housing. The processes, which take place in work chambers and especially in clearance gaps on their boundaries, have a significant influence on the performance of a screw compressor. The knowledge of details of the leakage flow through clearance gaps is essential to make reasonable estimates for mass flow rate and to define the loss of medium.
The flow can be reasonably modelled by computational fluid dynamics for so called dry compressors, Fig. 1, where no multiphase flow occurs. The dry compressors are used in applications where the cleanness of handled gas is essentially important as e.g. in medical applications.
The leakage flow in the gaps is studied by many compressor engineers, e.g. [7, 8, 9]. In [7]
and [8] many numerical simulations of compressible viscous fluid flow through a male rotor- housing gap in a screw compressor for various pressure ratios are presented. All of these computations were only performed by available commercial computational system Fluent for
various turbulence models and the rotary motion of the male rotor was not involved in the computations.
In [10], the numerical simulations of fluid flow in the two-dimensional model of 100μm wide male rotor-housing gap in a screw compressor for the given pressure ratio pinlet/poutlet = 2 has been performed using the own developed numerical code. This problem was solved as non- stationary turbulent compressible Newtonian fluid flow with ideal gas properties. Turbulent flow was considered to be statistically steady and the mathematical model was described by the non-linear conservative system of the compressible Navier-Stokes equations, [10, 11]. Its numerical simulation was based on the cell-centered finite volume formulation of the explicit two-step TVD MacCormack scheme proposed by Causon [2] which was defined on a structured quadrilateral grid. For the computation of turbulent viscosity, the algebraic turbulence model introduced by Baldwin and Lomax [1, 6] was implemented in the own developed numerical code. The original numerical results obtained and published in [10]
demonstrate unambiguously that the flow through this gap for the prescribed pressure ratio pinlet/poutlet = 2 is transonic but without shock waves, see Fig. 3, typical for transonic and supersonic flows at macroscales. In [4], these results are compared with results obtained for the same conditions in Fluent using various models of turbulence.
The aim of this contribution is to perform the numerical computations of a turbulent compressible viscous fluid flow through this clearance gap between moving male rotor and housing of screw compressor. The numerical computations are performed on a simplified two- dimensional geometry representing frontal cross-section of the compressor (the plane perpendicular to rotors axes) to get some important information about properties of numerical model. So, this contribution brings new information about effects of the male rotor motion to leakage flow in comparison to previous models not involving the rotor motion presented e.g. in [10, 4] or in [7]. For numerical computations the commercial computational fluid dynamics package Fluent was used. Computations were performed using two different models of turbulence on quadrilateral computational grids with various cell densities. The comparison of Spalart-Allmaras and RNG k-ε turbulence models is provided. The accuracy of results on different grids is also provided.
Fig.1. Example of dry screw compressor design
2. MODELLING OF FLUID FLOW THROUGH THE MALE ROTOR-HOUSING GAP
2.1 Problem formulation
We considered the turbulent flow of compressible, viscous and heat-conducting Newtonian fluid with ideal gas properties through a two-dimensional flow region representing male rotor-
housing gap. The flow region is shown in Fig.2. It is bounded by two adiabatic walls that represent the circular part of the housing and the head of male rotor tooth. The other boundaries represent the inlet from the higher pressure chamber and outlet to the lower pressure chamber. The width of gap in its narrowest section equals to 100µm. We studied the situation for the typical pressure difference between neighbouring chambers and assumed the pressure ratio to be 2, with atmospheric pressure in the lower pressure chamber.
Fig.2. Computational domain and computational grid with 178×68 quadrilateral cells
2.2 Numerical model and parameters settings
For numerical computations the commercial software package Fluent was used. This software is based on finite volume method. The simulations were performed on topologically similar structured quadrilateral grids obtained by merging cells of the mesh displayed in Fig.2.
For solving of fluid motion equations the segregated solver and implicit numerical scheme were chosen. The segregated solver was preferred due to its higher stability in situations, where both high- and low-velocity regions occur in the flow field. The discretisation of second order accuracy was chosen for all equations. To secure the higher stability of computational scheme the operating pressure, used internally by the solver, was set to 0 Pa.
2.3 Modelling of male rotor motion
To involve the effect of moving rotor into the computational model the numerical concept of moving mesh was used [3]. We considered that the whole computational domain representing the male rotor-housing gap rotates around the main rotor axes. It means that in the rotating frame of reference the boundary pressures remain constant during the simulation and the fluid velocities are zero on the rotor head and non-zero on the housing wall. The velocity of fluid on the housing has opposite direction than the direction of rotor movement, so it is zero in absolute frame of reference. Because the computations are performed in the rotating frame of reference using moving mesh concept, the governing equations describing the fluid flow need to be slightly modified, see [5, 3] for details.
When the governing equations of fluid motion are solved in the rotating frame of reference, the acceleration of the fluid is augmented by additional terms that appear in the momentum equations. Fluent allows solving the rotating frame problems using the relative velocity v as ir the dependent variable. The relative and absolute velocities are related by the following equation
k j ijk i r
i v ε ω r
v = − , (1)
where ε is permutation symbol, ijk ω is the angular velocity of the rotating frame and j r is the k position vector in the rotating frame.
3. RESULTS OF NUMERICAL SIMULATIONS
The contour plots of Mach number and static pressure distributions in the absolute frame of reference (see Fig.3 and Fig.4) show that no shocks are presented in the flow field. Then it is visible that the flow is transonic with maximum of Mach number (Mmax ≈ 1.25) situated between gap opening and the narrowest part of the channel, see Fig. 3. The obtained results are qualitatively very similar for all tested rotor angular velocities. The direction in which the male rotor moves can be clearly seen from velocity profiles plots in Fig.3.
Fig.3. Contours of Mach number and velocity profiles in the male rotor-housing gap obtained by segregated solver using Spalart-Allmaras turbulence model on the finest grid of 178×68
cells for the angular velocity of 9000 rpm
Fig.4. Contours of static pressure obtained by segregated solver using Spalart-Allmaras turbulence model on the finest grid of 178×68 cells for the angular velocity of 9000 rpm
All results presented in next sections relate to the flow centreline. The flow centreline is the line defined as a circular arc in the distance of 50µm from the housing wall, so it crosses the centre of flow channel in its narrowest part. It is important to note that all following plots display the quantities in the absolute frame of reference.
3.1 Comparison of results with simulation using Baldwin-Lomax turbulence model
In [10], the numerical simulation of compressible viscous fluid flow through a male rotor- housing gap using the own developed numerical code based on the algebraic Baldwin-Lomax turbulence model is presented. The results obtained in [10] and the new results presented in this article are compared for the rotor angular velocities of 0 rpm in Fig.5.
(a) (b)
Fig.5. Distributions of Mach number (a) and static pressure (b) obtained by different turbulence models on the finest grid of 178×68 cells along the flow centreline for the angular velocity of
0 rpm
3.2 Influence of male rotor motion
The influence of male rotor motion on the flow field in the model of male rotor-housing gap was tested for typical operating conditions of a screw compressor. The angular velocities of moving rotor were selected in the range of 6000 – 12000 rpm and results obtained in this range were compared with the older results published in [4] and obtained on the model of gap with non-moving rotor. The distributions of Mach number and static pressure for various angular velocities of male rotor can be seen in Fig.6. The distributions of static pressure are nearly
identical for all tested situations. The distributions of Mach number show that with growing male rotor angular velocity the Mach number maximum decreases and its position moves toward the end of the gap and that the absolute velocities at the end of gap for various rotor velocities are identical, see Fig. 6.
(a) (b)
Fig.6. Distributions of Mach number (a) and static pressure (b) obtained for various angular velocities by segregated solver using Spalart-Allmaras turbulence model on the finest grid of
178×68 cells along the flow centreline
3.3 Influence of grid density
In order to resolve the boundary layer with sufficient accuracy, the computational grids used for numerical computations were refined in the direction normal to the walls, see Fig. 2.
The density of computational grid is one of the most significant factors influencing the numerical results of simulations. It is very important to use very fine grid to obtain accurate results, but reasonable results can be also obtained on coarser grids as shown in Fig.7. The computational grids consisting of 28×12, 56×22 and 178×68 quadrilateral cells were used in simulations.
(a) (b)
Fig.7. Distributions of Mach number (a) and static pressure (b) obtained on various grids along the flow centreline by segregated solver using Spalart-Allmaras turbulence model for
the angular velocity of 9000 rpm
3.4 Influence of turbulence model selection
Tested turbulence models included in computational system Fluent, see [3], differ in the way how they compute the turbulence related variables. Fig.8 shows the difference in results.
Spalart-Allmaras turbulence model predicts higher velocities in the flow field than the RNG k-ε model. Spalart-Allmaras model brings into results less dissipation caused by viscous effects.
(a) (b)
Fig.8. Distributions of Mach number (a) and static pressure (b) obtained for two different turbulence models by segregated solver on the finest grid of 178×68 cells along the flow
centreline for the angular velocity of 9000 rpm 4. CONCLUSIONS
In our simulations of fluid flow in narrow channel (represented by the male rotor-housing gap of the screw compressor with moving rotor wall) we have investigated several factors having influence on the flow character. We have studied the influence of growing angular velocity of the male rotor on the flow field in the male rotor-housing gap. Then we have tested the dependence of numerical results on the density of used numerical grid and last we have compared the results obtained by commonly used turbulence models included in Fluent.
This contribution has two main benefits. The simulations show that the rotor motion has no significant influence on the resulting fluid velocity field and that for the male rotor angular velocity of 0 rpm the results obtained by Fluent are comparable to those obtained by the own developed software and published in [10].
For numerical computations of fluid flow in the narrow channel of our type is more suitable to use the segregated solver because of co-existence of low- and high-velocity regions in the flow field. Then we have shown that for accurate prediction of extreme values of observed quantities it is important to use very fine grid because the extremes are relative sharp. The usage of very coarse grid in gap flow simulations is thus not totally proper.
The choice of turbulence model influences in our numerical simulations the character of the wake. Tested Spalart-Allmaras turbulence model predicts higher velocities in wake than the RNG k-ε model. The extreme values do not differ significantly.
The rotor motion has for typical operating conditions of screw compressor slight influence on the flow through clearance gap. The flow is only influenced in the area between velocity maximum and channel opening. In absolute frame of reference the velocity of flow remains nearly constant at the end of gap.
This paper is further contribution to the fluid mechanics modelling in screw-type machine applications.
ACKNOWLEDGEMENTS
This work was supported by the research project MSM 4977751303 of the Ministry of Education, Youth and Sports of the Czech Republic to which we express our grateful thanks.
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