DOI 10.1007/s10891-015-1309-0

Journal of Engineering Physics and Thermophysics, Vol. 88, No. 5, September, 2015

NUMERICAL INVESTIGATION OF FLOW IN A CENTRIFUGAL COMPRESSOR Yu. A. Grishina and V. N. Bakulinb

UDC 621.43.031.3:519.6

With the use of the domestic software suite of computational hydrodynamics FlowVision based on application of the method of control volumes, numerical simulation of air composition and delivery by a centrifugal compressor employed for supercharging a piston engine has been carried out. The head-flow characteristics of the compressor, as well as the 3D fields of flow velocity and pressure distributions in the elements of the compressor flow passage, including the interblade channels of the impeller, have been obtained for various regimes. In the regimes of diminished air flow rate, surging phenomena are identified, characterized by a return flow. The application of the technique of numerical experiment will make it possible from here on to carry out design optimization of the compressor flow passage profile and thus to improve its basic characteristics — the degree of pressure increase, compressed air flow rate, and the efficiency — as well as to reduce the costs of the development and production of compressors. Keywords: computational hydrodynamics, method of control volumes, FlowVision suite, centrifugal compressor, head-flow branches, velocity and pressure fields. Introduction. At the present time centrifugal compressors (CFC) are widely applied as both driving units and parts of turbo-compressors for supercharging internal combustion piston engines. This makes it possible to substantially increase the power of engines with simultaneous reduction in the specific fuel consumption and improvement of ecological characteristics — decrease in the toxicity of exhaust gases and reduction of the exhaust noise. The effective parameters of a power plant with such a supercharging unit depend substantially on the efficiency of the compressor, i.e., on the possibility of obtaining a high pressure head at a required compressed air flow rate and a high CFC performance. To raise the CFC efficiency, it is necessary, with the aid of rational profiling of the flow passage parts, i.e., a correct choice of the internal geometry of all the components such as the inlet pipe, impeller, diffuser, collector (scroll casing), and the outlet pipe, to provide for the maximal uniformity of the fields of gas-dynamical parameters (primarily of velocities and pressures) and to maximally decrease the gas-dynamical losses. The corresponding adjustment of the geometry of the flow passage parts is usually made experimentally by means of multiple tests with various versions of the construction of compressor elements. This is a very complex and expensive procedure. It is especially important to adjust the CFC elements operating as the supercharging units of piston engines, since, first, the working process, including the gas exchange of these engines, is of nonstationary and pulsating character and, second, these engines mounted on cars, light aeroplanes, ships, diesel locomotives, and in other engineering structures operate under the conditions of variable regimes. Therefore recommendations regarding the structure should stand up against the integral characteristics obtained in a wide range of flow parameters rather than for one working point. This introduces additional difficulties for carrying out experimental investigations. The present time is characterized by intense development of computation engineering and methods of numerical simulation of gas-dynamical processes in various objects. The possibility of carrying out computational experiments has evolved. They allow one to obtain rational constructive recommendations aimed at increasing the effectiveness of the elements of various energy plants in relation to the gas dynamics without carrying out expensive natural experiments [1–11]. In the present work, to carry out numerical simulation we applied the advanced domestic software suite of the threedimensional computational hydrodynamics FlowVision HPC. It employs a high-performance numerical method of control volumes, an automatic generator of a rectangular computational grid, and a unique technology of the subgrid resolution of the complex geometry of computational domain. This technology allows one to import geometry from the documentation a

N. É. Bauman Moscow State Technical University, 5 2nd Baumanskaya Str., Moscow, 105007, Russia; email: [email protected]; bInstitute of Applied Mechanics of the Russian Academy of Sciences, 7 Leningradskii Ave., Moscow, 125040, Russia; email: [email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 88, No. 5, pp. 1232–1236, September–October, 2015. Original article submitted November 20, 2014. 1274

0062-0125/15/8805-1274 ©2015 Springer Science+Business Media New York

of the System of Computer-Aided Design and to exchange information with the systems of finite-element analysis. The setting up of this problem envisages the mating of the rotating part of the computational domain (impeller) with the immobile parts (at the inlet, with a branch pipe, at the outlet from the impeller, with a diffuser). The calculations were carried out on a PC with a 4-core processor Intel Core-i7 2.0 GHz, RAM 4 GB. About 250,000 cells were located in the computational domain, i.e., in the CFC flow passage. Description of the Mathematical Apparatus. In the FlowVision package use is made of the program code of computational hydrodynamics based on partial differential equations (continuity, Navier-Stokes momentum, energy, and diffusion equations). These equations can be presented in the form of a generalized dependence as

∂ ∂ ∂ ρW j Φ ) = (ρΦ) + ( ∂τ ∂x j ∂x j

⎛ ∂Φ ⎞ ⎜⎜ Γ Φ ⎟ + SΦ , ∂x j ⎟⎠ ⎝

j = 1, 2, 3 ,

(1)

where the source term SΦ is equal to the difference between the generation SΦg and the annihilation SΦa of flows, i.e., SΦ = SΦg – SΦa. The specific forms of ΓΦ and SΦ, as well as of SΦg and SΦa, depend on the physical meaning of the variable Φ. According to (1), the sum of nonstationary and convective flows (the left side of the equation) is equal to the sum of the diffusion and source terms (the right side of the equation). Assigning certain values to the variable Φ, we obtain the fundamental mass, energy, diffusion, and momentum conservation equations. According to the Reynolds approach, the instantaneous value of any parameter is represented as a sum of its timeaveraged value of Φ and the pulsating value of Φ′, i.e., Φ = Φ + Φ′, where Φ =

1 t

τ0 + t

∫

Φ (τ) d τ . Here, the period of the

τ0

averaging of the value of t is greater than the period of turbulent pulsation in a flow, but it is small as compared with the time constant for any slow variation of the flow field due to the flow nonstationarity. Just as in other methods of through counting with a fixed grid, in the applied method of control volumes the computational domain with a three-dimensional step Δx, Δy, Δz is divided into a certain, quite definite number of nonintersecting volumes, that is, cells with central nodes. The system of the initial equations of gas dynamics can be represented in a generalized differential form as

∂f + ∇ ( vf ) = ∇ ( D ∇f ) + Q , ∂t

(2)

where f is the computed variable. The equations are integrated through each control volume (i cell, Fig. 1) of the computational grid and over the time interval from t = t n to t n+1 (t n+1 – t n = Δt is the time step of calculation). In the case of the absence of an additional source Q of the variable f we may write

∫

Vi

fdV

t n +1

− ∫ fdV Vi

tn

+

∫ v∫

Δt Ai

f vdAdt =

∫ v∫ D∇fdAdt ,

(3)

Δt Ai

where Vi is the cell volume (Fig. 1), and Ai = Aei + Awi + Ani + Asi + Ahi + Ali is the cell surface (the sum of the areas of all "free faces" that border on their cells). Thereafter the finite-difference form of representation is applied. To account for the small-scale "subgrid" turbulence, we used the standard k–ε model of turbulence in our calculations [5, 12]. Modeling of Flow in a Centrifugal Compressor. The aim of the first stage of the work was to master the possibilities of the FlowVision software suite and to numerically simulate a three-dimensional air flow in a CFC used in a highly overaccelerated composite engine. Figure 2 presents the initial division of the computational domain (until the adaptation of the computational grid). For fine reflection of the complex configuration fragments (in particular, of the impeller interblade channels), an adaptation was made with additional fractionation of the cells along the complex boundaries. Under the "solid wall" boundary conditions we applied the well-known approach with logarithmic near-wall functions. Under the "inlet" boundary condition, we specified the total pressure, and under the "outlet" condition — the normal mass velocity that corresponds to the compressor stage output in this regime. The time step was assigned by recognizing that during one interaction the rotor surface must not move through a distance larger than half the size of the computational cell.

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Fig. 1. Computational cell of the method of control volumes.

Fig. 2. Initial division of the computational domain. Fig. 3. Computational velocity fields in the transverse cross section of a CFC.

Results of Numerical Simulation of Flow in a Centrifugal Compressor. Figures 3–5 present the results of numerical calculations in the form of velocity and pressure fields in the transverse and meridian cross sections of the flow passage, where the circumferential nonuniformity of distribution is seen. It is obvious that here the influence of the insufficiently adjusted

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Fig. 4. Computational pressure fields in the transverse cross section of a CFC.

Fig. 5. Computational pressure fields in the meridian cross section of a CFC.

geometry of the composite scroll casing and outlet pipe is felt. These results show the necessity of carrying out further improvements in the design of the CFC flow passage elements. The pressure head branches of the compressor characteristics (Fig. 6) were calculated showing the dependence of the degree of pressure increase πk on the compressed air flow rate G at the impeller rotational speed n = 67,000, 55,000, and 45,000 rpm. Figure 6 also shows several points (4) obtained with the aid of the simplified one-dimensional technique for regimes close to the nominal one [13]. At the air flow rate G = 0.05 kg/s and on its further decrease, the flow nonuniformity and instability originate and become stronger. The zones of return flow, especially noticeable along the forepart of the casing, appear periodically in the diffuser, interblade channel, and in the inlet pipe. Fluctuations of the integral values of pressure and of the rate of flow through the compressor are noted. This attests to the appearance of the surge phenomena. The corresponding velocity fields obtained at the air flow rate G = 0.03 kg/s are shown in Fig. 7. It is seen from the figure that the zones of reverse drop occupy a considerable part of the transverse cross-sectional area of the flow passage. The corresponding calculation by the simplified technique with the aid of gasdynamical functions [14] also indicate the onset of the surge regime. Conclusions. The computationally obtained nonuniformity in the pressure distribution over the interblade channels of the impeller, in the diffuser, and composite scroll casing identifies the region in the flow passage of the scroll casing that needs refinement of its profile. 1277

Fig. 6. Characteristics of a CFC at different rotational speeds of the rotor: 1) 67,000 rpm; 2) 55,000; 3) 45,000; 4) corresponding values of πk obtained with the aid of the simplified one-dimensional technique for regimes close to the nominal one. Fig. 7. Velocity field on the occurrence of surge. It is considered that in further stages of the work one is to perform a detailed computational analysis of the influence exerted by the geometry of individual elements of the compressor, including the pressurization line, on the compressor characteristics in various operational conditions. It is also planned to obtain the integral characteristics of a CFC operating as part of a composite piston engine. The authors are grateful to Rodin Alexei Olegovich for the calculations and registration of the calculation results. This work was carried out with financial support from the Russian Foundation for Basic Research (project 14-08-01030).

NOTATION Ai, cell surface; D, diffusion coefficient; f, calculated variable; G, compressed air flow rate, kg/s; n, rotational speed of impeller, rpm; Q, source term of equation; SΦ, source term; SΦg, flow generation; SΦa, flow annihilation; Vi, volume of a cell; v, velocity vector; Δt, time step of calculation; ΓΦ, coefficient of diffusion exchange; πk, degrees of pressure increase; Φ, arbitrary dependent variable; Φ , time-averaged dependent variable; Φ′, pulsating value of dependent variable.

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