CALCULATION OF THE SEPARATION POINT FOR THE TURBULENT FLOW IN PLANE DIFFUSERS UDC 532

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1 FACTA UNIVERSITATIS Series: Mechanics, Atomatic Control and Robotics Vol.3, N o 15, 003, pp CALCULATION OF THE SEPARATION POINT FOR THE TURBULENT FLOW IN PLANE DIFFUSERS UDC 53 Mile Vjičić 1, Cvetko Crnojević 1 Universit of Serb Sarajevo, Repblic of Serbska Universit of Belgrade, Faclt of Mechanical Engineering 7. marta 80, 1110 Belgrade, Serbia and Montenegro ccrnojevic@mas.bg.ac. Abstract. This std examines the trblent flow in plane-wall diffsers. For calclation we se the eqations of trblent bondar laer in integral form, adjsted for internal flow, and for closing the sstem of eqations we se trblent viscosit model based on the mixing length. Velocit profile in ever cross-section of the diffser is approximated b a sixth-order polnomial, while the coefficients of the polnomial depend on three form parameters. B this transformation sstem of governing eqations is redced to three ordinar differential eqations for form parameters, which is solved nmericall. The obtained reslts show that the performance, position of the separation point and other flow characteristics of diffsers depend on the angle and Renolds nmber. 1. INTRODUCTION A diffser, as an element where the stream cross-section changes from inlet to otlet, either plane or axismetrical, has a great importance in man practical engineering applications. The flow strctre in the diffser transforms into the velocit profile with stream separation (stall), which is defined where the vale of shear stress on the wall is eqal to zero, then after this, the cross section stream starts to separate from the channel wall. The problem of stall is ver old bt ver important. In the diffser several regime flows can exist, which depend on the geometr and Renolds nmber, and which are defined b Kline's diagram [1], [] and [3] for trblent flow (obtained b Kline's work and its Stanford grop) and his appendix for laminar flow [4]. In the Kline's diagram (see Fig. 3) for the trblent flow in the plane, or conical diffsers, which is obtained experimentall, for different regimes exist: (a) no appreciable stall, (b) transitor stall, (c) fll stall on one wall with detachment close to the inlet, and (d) jet flow with fll stall on all walls. In man practical cases for plane diffsers designed with good geometrical parameters it is shown (see ref. [3]) that the optimal geometr of diffsers corresponds the lengths x s / o = 5 to 15, and the area ration s / o to 4 (see Fig. 1 and Fig. 3). Received April 0, 003

2 100 M. VUJIČIĆ, C. CRNOJEVIĆ The problem which is examined in this paper is incompressible stationar trblent flow in a plane diffser, which is solved b integral method of flow developed in the references [5] and [6]. This method is based in the integral eqations of the incompressible trblent bondar laer adopted for the problem of interior flow in diffsers. In this pattern it is necessar to make an approximation of the velocit profile, and in this paper we se an approximation of the velocit profile b the sixth-order polnomial based in the edd trblent viscosit. With this approximation we obtained the sstem of three nonlinear ordinar differential eqations for form parameters, which are solved b classical nmerical method of the Rnge-Ktte. Finall, we obtained the soltions for the flow in a plane diffser for different vales of the Renolds nmber and the diffser angle, starting with qasi-developed velocit profile in the inlet cross section p to the downstream cross section in which trblent flow separates from the wall. B sing the reslts of or calclation for the separation flow in diffser we obtained an addition to the Kline's diagram, which show that the position of separation point depend the angle of diffser and Renolds nmber.. PROBLEM STATEMENT AND THE GOVERING EQUATIONS Two dimensional axismetric stationar trblent flow in the diffser is shown in Fig. 1. Geometr of the diffser is defined b its half-width (0) = o, half-angle θ and b the downstream changes of half-width (. The trblent flow in the diffser is the one which takes place between the inlet cross section, defined b the maximm velocit in the axis: eo = e (0) and b the average velocit m (0), and which is transformed downstream into the velocit profiles (x,), the axis velocit e ( and the average velocit m (, and the separation cross section x s in which the flow separation begins and in which the shear stress on the wall τ w (x s ) is zero. V eo θ x e x S m(0) m( Fig. 1. Flow throgh a diffser Two-dimensional incompressible trblent flow is described b Renolds's eqations and eqation of continit, which are after transformation (see [5] and [6] ) written in the integral form: d ( η S S τ =0 w ' e τw ν 1) = ( ) e e ρe e ' e 3 e e 0 e e ρe v x, (1) d 3 ν τ 3 = ( ) ( ) d ()

3 Calclation of the Separation Point for the Trblent Flow in Plane Diffsers 1003 where (1) represents momentm eqation and () represents mechanical energ eqation, sbscript e represents the vale in the axis of diffser, and where total sheer stress τ is identified as: τ ρ ν ν = ( t ), (3) variable ' e represents the derivative ' e = de /, and the variables: 1, and 3 - displacement thickness, momentm thickness and energ thickness respectivel, are defined in the classical wa: 1 = ( 1 ) d, = ( 1 )d, 0 e 3 = [ 1 ( ) ] d (4) 0 e e 0 e e For closing the phsical-mathematical model which describes trblent flow we se for edd viscosit the mixing length model where νt = l d / dη, where mixing length l(η) is defined b expression [6]: l 3 η η η η = 0, (5) where κ=0,4 is the Prandtl's constant. With the aim of solving eqations (1)-() we predicted that velocit profile is the sixth-order polnomial: 4 6 ( x, ) = a( b( c( d(. (6) The velocit profile has to be smmetrical with relation to the axis of the channel and for this reason we onl applied even nmbers in the power ratios, and a(, b(, c( and d( denote the coefficient of the polnomial. Analsis of trblent flow shows that it is sefl to introdce: a coordinate measred positive from the wall: η =, friction velocit *( = τw ( / ρ and dimensionless variables: * η =, = = η, η * =, * =. (7) * ν ν ν If we se velocit profile (6) and satisf bondar conditions as: = 0, ( x,0) = e (, / = 0 ; =, ( x, ) = 0, v( x, ) = 0 ; V! = d = const. ; 0 we will determine polnomial coefficients as: ax ( )= q. Re. e, b( = λ 70q 5.5 Re λ, c( =, q. Re. d( = λ, (8) 7 in which: Re = e / ν is the Renolds nmber, and parametrical forms: 4

4 1004 M. VUJIČIĆ, C. CRNOJEVIĆ e e' ν e' e λ( = = 3 * ν, q ( x )= e. (9) If we se formla (9) we find a relationship between the parametrical forms: 1 3 q' = ( λ q '). (10) q Velocit profile (6) is defined b formla (8) as a fnction of parametrical forms, as = ( λ, q, ). After a hge mathematical process, which is dictated b eqations (3) and (4), and b formla (7), a sstem of three simple differential eqations are obtained: dλ dq = f ( x, λ, q, ) ; = gx (, λ, q, ) ; d = hx (, λ, q, ) (11) The fnctions which depend on the longitdinal coordinate and parametrical forms: f ( x, λ, q, ), gx (, λ, q, ) and h( x, λ, q, ), and which are given in the reference [5]. Initial conditions of the parametrical forms according to ref. [5] are: λ(0) 0, (1) bt which is ver near zero, and: Re q ( 0) = (1 3,75 C f (0) / ), (13) Re ( 0) = C f (0) /. (14) Finall, the sstem of differential eqations (11) and initial conditions (1) to (14) is solved b the application of the Rnge-Ktte nmerical method. Nmerical calclations are stopped in the downstream cross section in which the flow separates from the wall of the diffser, that is when on a distance x = xs shear stress becomes τw( x s ) = NUMERICAL RESULTS AND DISCUSSION In order to have concrete nmerical reslts we have to define the geometr of the diffser and the vale of the Renolds nmber at the diffser inlet. In this paper the geometr of the diffser (see Fig. 1) is defined as a straight walled slope of half-angle θ, and the cross section change is defined b the linear fnction: ( = 0 tgθ x. In Fig. velocit profiles are shown in a diffser with the half-angle of 15 and the Renolds nmber:, in the inlet cross section (x/x s = 0) and in the downstream cross section (x/x s = 1) in which the flow separation occrs. In this diagram velocit is normalized with the maximm velocit in the corresponding cross section, and the transversed coordinate is normalized with the corresponding half-width of the channel. The inlet velocit profile (for x = 0) is different from the fll developed trblent velocit profile taking place between parallel plates pstream from the diffser. This difference originates in the slight transformation of the fll developed flow immediatel in front of the diffser. Velocit profiles in the separation cross section (x/x s = 1) for

5 Calclation of the Separation Point for the Trblent Flow in Plane Diffsers 1005 conditions stated in Fig. 1, as well as for other conditions, agrees well with the experimental date stated in [7] and [8]. / e x/x s =0 x/x s =1 θ= Fig.. Velocit profile / For a possible application of the obtained reslts on the flow separation the velocit profile in the cross section x/x s = 1 is important onl. B variations of the Renolds nmber and the different angle, the whole spectrm of different separation regimes of the flow is obtained, shown in Fig. 3, and compared with Kline's diagram, where n = x s / o is the characteristic parameter. It is seen from this figre that most of the calclated flow regimes belong to the region of transitional flow in the Kline's diagram, as well as that the position of separation point depends on the Renolds nmber and the diffser angle in sch a wa that for a fixed Renolds nmber the separation is enhanced with the increase of the angle. Also, it is seen that for a fixed diffser angle the separation is enhanced with the increase of the Renolds nmber. 100 θ [ ] Jet flow Hsteresis zone Fll stall 100 θ [ ] optimm condition [3] Transitor stall s / o = s / o = Transitor stall 1 Present work: Re=10000 Re=50000 No stall 1 n 10 Fig. 3. Dependence of the separation point on the parameters: Re, θ and n. Dash lines correspond a Kline's diagram and fll lines correspond at or reslts. Present work: Re=10000 Re=50000 No stall n 100 Fig. 4. Comparison of or reslts and the optimal regime flow in plane diffser [3]. In Fig. 4 we show the position of or reslts within the optimal region [3], which is defined b s / o = p to s / o = 4, and n between 5 and 15. It is seen that optimal parameters can be achieved for relativel small vales of both the Renolds nmber and the diffser angle. The presented reslts can be sed for the choice of optimal dimensions of plane diffsers in the case in which no flow separation occrs.

6 1006 M. VUJIČIĆ, C. CRNOJEVIĆ 4. CONCLUSION For the calclation of trblent flow in plane diffsers in this paper we se the method of integral eqations of the bondar laer theor, adopted for the problem of interior flow. The reslts obtained b or calclations contain the development of velocit profiles from the inlet cross section of the diffser p to the separation crosssection. The reslts obtained show that the position of the separation point is a fnction of the Renolds nmber and the diffser angle, and their changes are more intense in the diffsers with greater angles. For the fixed vale of the Renolds nmber the position of the separation cross section is postponed for smaller diffser angles. Or reslts of calclations can b se for design or choice of the optimal geometrical and flow characteristics of plane diffsers. Acknowledgment. This work is spported b the Ministr of Sciences, Technolog and Development of the Repblic Serbia, Project No 138. REFERENCES 1. Kline S. J., On the Natre of Stall, Transactions of the ASME, Jornal of Basic Engineering, Vol. 1, Series D, No 3, pp , Fox R. W., Kline S. J., Flow Regimes in Crved Sbsonic Diffsers, Jornal of Basic Engineering, Vol. 84, pp , Johnston P.J., Review: Diffser design and performance analsis b a nified integral method Transactions of the ASME, Jornal of Flids Engineering, Vol. 10, pp. 6-18, Crnojević C.: Détermination d point de décollement d'n écolement visqex incompres-sible dans les diffsers bidimensionnels. 11ème Congrès Français de Mécaniqe, Villeneve d'ascq, p , Tome II, Septembre Vjičić M.: Comptation of trblent flow in plane diffsers. Masters Thesis, Faclt of Mechanical Eng., Belgrade, Vjičić M., Crnojević C., Djordjević V.: Calclation of the Trblent Flow in a Plane Diffser b sing the Integral Method (the paper sbmited in the FME Transactions). 7. Rakesh K. Singh, Ram S. Azad, Asmptotic Velocit Defect Profile in a Incipient-Separating Axismmetric Flow, Jornal AAIA, Vol. 33, No 1, pp , Stieglmeier M., Tropea C., Weiser N., Nitsche W.: Experimental Investigation of the Flow Throgh Axismmetric Expansions, Transactions of the ASME, Jornal of Flid Engineering, Vol. 111, pp , PRORAČUN TAČKE ODVAJANJA PRI TURBULENTNOM STRUJANJU U RAVANSKIM DIFUZORIMA Mile Vjičić, Cvetko Crnojević U rad se pročava trblentno strjanje ravanskim difzorima. Za proračn se koriste jednačine trblentnog graničnog sloja prilagodjene za ntrašnja strjanja, i to njihov integralni oblik. Za zatvaranje sistema jednačina koristi se model trblentne viskoznosti baziran na ptanji mešanja. Profil brzina poprečnom presek je aproksimiran polinomom šestog stepena, pri čem se pokazje da koeficijenti polinoma zavise od tri parametra oblika. Sa ovom pretpostavkom polazni sistem jednačina se transformiše na tri obične diferencijalne jednačine koje s rešene nmerički. Dobijeni rezltati proračna pokazj da performanse, položaj tačke odvajanja i drge strjne karakteristike difzora zavise od Rejnoldsovog broja i gla širenja difzora. Rezltatima proračna koji se odnose na tačk odvajanja strje izvršena je dopna Kline-ovog dijagram mogćih režima strjanja ravanskim difzorima.

Calculation of the Turbulent Flow in a Plane Diffuser by using the Integral Method

Calculation of the Turbulent Flow in a Plane Diffuser by using the Integral Method Calclation of the Trblent Flow in a Plane Diffser by sing the Integral Method Mile Vji~i} Assistant University of Serb Sarajevo Repblic of Serbska Cvetko Crnojevi} Associate professor University of Belgrade