Application of the B-Determining Equations Method to One Problem of Free Turbulence

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1 Smmetr, Integrabilit and eometr: Methods and Applications SIMA , 073, 10 pages Application of the B-Determining quations Method to One Problem of Free Turbulence Oleg V. KAPTSOV and Alexe V. SCHMIDT Institute of Computational Modeling SB RAS, Akademgorodok, Krasnoarsk, , Russia -mail: kaptsov@icm.krasn.ru, schmidt@icm.krasn.ru Received Ma 17, 2012, in final form October 04, 2012; Published online October 16, Abstract. A three-dimensional model of the far turbulent wake behind a self-propelled bod in a passivel stratified medium is considered. The model is reduced to a sstem of ordinar differential equations b a similarit reduction and the B-determining equations method. The sstem of ordinar differential equations satisfing natural boundar conditions is solved numericall. The solutions obtained here are in close agreement with experimental data. Ke words: turbulence; far turbulent wake; B-determining equations method 2010 Mathematics Subject Classification: 76M60; 76F60 1 Introduction Most flows occurring in nature and engineering practice are turbulent see, e.g., [10, 20, 22]. Semiempirical models of turbulence are widel used in the modeling of turbulent flows [16, 21, 27]. However, there are onl a few analtical results on such models see, e.g., [2, 11]. The far turbulent wake behind an axismmetric bod in a stratified medium is an example of a free shear flow. Sufficientl complete experimental data on the dnamics of turbulent wakes generated b moving bodies in stratified fluids were obtained b Lin and Pao [17]. The far turbulent wake behind an axismmetric towed bod in a linearl stratified medium was numericall simulated in [9]. Chernkh et al. [6] carried out the numerical simulation of the dnamics of turbulent wakes in a stable stratified medium based on hierarch of second order closure models. Calculation results obtained in [6, 9] are in close agreement with experimental data [17]. Similarit solutions for several turbulence models were constructed in [7, 13, 14, 15]. In the current paper we consider three-dimensional semiempirical model of the far turbulent wake behind an axismmetrical self-propelled bod in a passivel stratified medium see [6, 4, 25] and the references therein. This paper is organized as follows. In Section 3 we determine the most general continuous classical smmetr group of the model and obtain the similarit reduction of the model. In Section 4 we use the B-determining equations BDs method [1, 12] to transform the reduced sstem into a sstem of ordinar differential equations ODs. In the last section, we consider a boundar value problem for the sstem of ODs. We use the modified shooting method and the asmptotic expansion of the solution in the vicinit of the singular point to solve this problem. Finall, computational results are given. This paper is a contribution to the Special Issue eometrical Methods in Mathematical Phsics. The full collection is available at

2 2 O.V. Kaptsov and A.V. Schmidt 2 Model The following three-dimensional semiempirical model of turbulence was constructed in [4, 6, 25] to calculate characteristics of the far turbulent wake behind an axismmetric self-propelled bod in a passivel stratified medium: e x = C e 2 e e ε + C e 2 e e ε, ε 1 ε x = C e 2 ε ε ε + C e 2 ε ε ε C ε 2 ε2 e, 2 ρ 1 x = C e 2 ρ 1 ρ ε + C e 2 ρ 1 ρ ε C e 2 ρ ε, 3 ρ 2 x = C e 2 ρ 2 1ρ + ε C e 2 ρ 2 e 2 ρ 1 1ρ + 2C ρ ε ε e 2 ρ1 2 ρ 2 ε + 2C ρ 1 C T, 4 ε e where e is the turbulent kinetic energ, ε is the kinetic energ dissipation rate, ρ 1 is the averaged densit defect, and ρ 2 is the densit fluctuation variance. All the unknown functions depend on x,, and z. The quantities C e = 0.136, C ε = C e /δ, δ = 1.3, C ε2 = 1.92, C ρ = 0.208, C 1ρ = 0.087, C T = 1.25 are generall accepted empirical constant [8, 21]. is the free stream velocit. The marching variable x in the equations 1 4 acts as time. This model is based on the three-dimensional parabolized sstem of averaged Navier Stokes equations in the Oberbeck Boussinesq approximation see [5, 26] U d x + V U d + W U d = u v + u w, 5 V x + V V + W V = 1 ρ 0 p 1 v2 v w, 6 W x + V W + W W = 1 p 1 ρ 0 v w w2 g ρ 1, ρ 0 7 ρ 1 x + V ρ 1 + W ρ 1 + W ρ s = v ρ w ρ, 8 V + W = U d x, 9 where U d = U is the defect of the averaged longitudinal velocit component; U, V and W are the mean flow velocit component along x-, - and z-axes, respectivel; p 1 is the deviation from the hdrostatic pressure due to stratification ρ s z; g is the gravit acceleration; ρ 1 is the averaged densit defect: ρ 1 = ρ ρ s ; ρ s = ρ s z is the undisturbed fluid densit assumed to be linear: ρ s z = ρ 0 1 az, a > 0 is a constant; the prime indicates the pulsating components; indicates averaging. In [9, 26] the Renolds stress tensor components u i u j, the turbulent flows u i ρ, and the densit fluctuation variance ρ 2 are defined b the algebraic relations [21]. Since the flow in the far turbulent wake is considered, these relations are simplified as follows U d = K U d, 10 u v = 1 c 2 e v 2 c 1 ε 1 c 2 e w 2 1 c 31 c 2T g e 2 u w c = 1T ρ 0 ε w ρ c 1 ε 1 1 c 3 g e 2 ρ c 1 c 1T ρ 0 ε 2 2 U d = K z U d, 11

3 Application of the BDs Method to One Problem of Free Turbulence 3 v 2 = 2 3 e 1 1 c 2 P c 1 ε 1 c 2 c 1 w 2 = 2 3 e 1 1 c 2 P c 1 ε + 21 c 2 c 1 ρ 2 = 2 c T e ε w ρ ρ u ρ = 1 e c 1T ε v ρ = 1 e c 1T ε v2 ρ w ρ = P = = e c 1T ε ε ε, 12, 13, 14 u w ρ + 1 c 2T w ρ U, 15 ρ = K ρ w 2 ρ + 1 c 2T g ρ 2 ρ 0 e w 2 c 1T ε c 2T c 1T c 2T u v U d + u w U d, 16 ρ = K ρ ρz, 17 g e 2 ρ ρ 0 ε 2, 18 = 1 ρ 0 w ρ g. 19 Differential transport equations [21] are used in [4, 6, 25] to determine the kinetic turbulent energ e, the kinetic energ dissipation rate ε, and the shear Renolds stress v w : e x + V e + W e = K e e + K e ez + P + ε, 20 ε x + V ε + W ε = K ε ε + K ε εz + c ε ε1 e P + c ε 2 ε2 e, 21 v w + V v w + W v w = x K v w e + K v w ez + c 2 1 v 2 W + w2 V + 1 c 3 g ρ 0 v ρ c 1 ε e v w. 22 The turbulent viscosit coefficients in these equations are K e = K, K ez = K z, K ε = K e /δ, K εz = K ez /δ. The model 1 4 is an analogue of the equations 5 22 for the diffusion approximation in a homogeneous fluid V = 0, W = 0, and g = 0. In what follows, we assume that the free stream velocit equals unit. 3 Similarit solution The infinitesimal smmetr group [18, 19] of the model 1 4 is spanned b the eight vector fields X 1 = x, X 2 =, X 3 =, X 4 = ρ 1, X 5 = z + + ρ 1, X 6 = + z + 2e e + 2ε ε + ρ 1 ρ ρ2 ρ 2, X 7 = x x 2e e 3ε ε, X 8 = ρ 1 z ρ ρ2 ρ 2.

4 4 O.V. Kaptsov and A.V. Schmidt Available experimental data [10, 17] and numerical calculations [9, 20, 27] show that the flow in the far turbulent wake can be considered to be close to a self-similar flow. We therefore consider the linear combination of scaling vector fields X 6 and X 7 Z = x x + α + αz + 2α 1e e + 2α 3ε ε + α ρ 1 ρ 1 + 2α ρ2 ρ 2. The similarit variables associated with the infinitesimal generator Z are ξ = x α, η = z x α, e = x2α 2 ξ, η, ε = x 2α 3 ξ, η, ρ 1 = x α Hξ, η, ρ 2 = x 2α Rξ, η, where,, H, and R are arbitrar functions. According to phsical considerations the influence of gravit is neglected in this problem, the functions and must be of the form ξ, η = ξ 2 + η 2, ξ, η = ξ 2 + η We obtain the reduced sstem b introducing similarit variables. Using 23 and changing to polar coordinates ξ = r cos φ, η = r sin φ the reduced sstem becomes C e r + αr + 21 α = 0, 24 C ε C ρ 2 C 1ρ r + αr + 3 2α C ε2 2 H rr + 1 r 2 H φφ 2 + C ρ R rr + 1 r 2 R φφ 2 + 2C ρ + H 2 r + 1 r 2 H2 φ C ρ αr H r αh r = 0, 25 sin φ = 0, 26 r αr R r C T + 2α R 2 + 2C ρ 4C 2 cos φ ρ H r sin φ + H φ = 0, 27 r + C 1ρ where = r, = r, H = Hr, φ, and R = Rr, φ. Here, and throughout, subscripts denote derivatives, so H r = H/ r, etc. Lie s classical method do not provide solution of the reduced sstem agreed with experimental data. We therefore use the BDs method. 4 BDs method The concept of BDs of a sstem of partial differential equations PDs was introduced in [1, 12]. Consider a scalar PD Ω x, u, u 1, u 2,... = 0, 28 where x = x 1, x 2,..., x n denotes n independent variables, u denotes the dependent variable, and u k u k = x i1 x i2 x ik

5 Application of the BDs Method to One Problem of Free Turbulence 5 denotes the set of coordinates corresponding to all k-th order partial derivatives of u with respect to x. In the BDs method, an extension of the classical smmetr determining relations is made b incorporating an additional factor Bx, u, u 1, u 2,.... For a scalar PD 28, BD is h Ω u + α 1 D α h Ω u α + Bh Ω=0 = 0, 29 where h is a function of x, u, u 1, u 2,... ; D α = D α 1 x 1 Dx αn n, D xi is the total x i derivative; α is a multi-index. qualit 29 must hold for all solutions of 28. Now we use the BDs method to reduce 26 and 27 to some ODs. Consider more general equation than 26 H φφ + r 2 H rr + ArH r + BrH + Cr sin φ = 0, 30 where Ar, Br, and Cr are arbitrar functions. BD corresponding to 30 is D 2 φ h + r2 D 2 rh + b 1 r, φd r h + b 2 r, φh = Here, and throughout D φ, D r are the operators of total differentiation with respect to φ and r. The function h ma depend on r, φ, H and derivatives of H. The functions b 1 r, φ and b 2 r, φ are to be determined together with the function h. Note that if we let in 31 b 1 r, φ = Ar, b 2 r, φ = Br, we obtain the classical determining equation [18, 19]. For simplicit, we assume that a solution of 31 is independent of r and partial derivatives of H with respect to r h = H φφ + h 1 φ, H, H φ. 32 Substituting 32 into 31 gives a polnomial equation for derivatives of the fourth order. This polnomial must identicall vanish. We can express the derivatives H rrφφ, H φφφφ, H rφφ, H φφφ, and H φφ using 30. The coefficient of H rrr implies b 1 r, φ = Ar. As a result, the left side of 31 is a polnomial in H rr and H rφ. Collecting similar terms we obtain Hence 2 ArH r + BrH + Cr sin φ h 1Hφ H φ 2H φ h 1HHφ 2h 1φHφ + Br b 2 r, φ = 0, h 1Hφ H φ = 0, h 1HHφ = 0. h 1 φ, H, H φ = h 2 φh φ + h 3 φ, H, b 2 r, φ = Br 2h 2φ. Substituting the functions b 1, b 2, and h 1 into the left side of 31 we obtain the polnomial with respect to H r and H φ. This polnomial must identicall vanish. Collecting similar terms we have Thus, BrH + Cr sin φh 3H h 3φφ + 2h 2 Brh 3 + Crh 2 cos φ sin φ = 0, h 3HH = 0, 2h 3φH + h 2 2h 2h 2 = 0. h 3 φ, H = where h 4 is an arbitrar constant. 1 2 h h 2 + h 4 H, h 2 h cot φh h 4 = 0, 33

6 6 O.V. Kaptsov and A.V. Schmidt Clearl, that the Riccati equation 33 has the partial solution h 2 = tan φ for h 4 = 1/2. Thus we find the solution of 31 h = H φφ + H φ tan φ. The corresponding differential constraint h = 0 has the general solution H = H 1 r sin φ + H 2 r, 34 where H 1 and H 2 are arbitrar functions. Next we use 34 and consider the equation 27 in more general form R φφ + r 2 R rr + KrR r + LrR + Mr sin 2 φ + Nr sin φ + P r = 0, 35 where Kr, Lr, Mr, Nr, and P r are arbitrar functions. The BDs method applied to the equation 35 gives rise to the following results: Nr = 0, b 1 r, φ = Kr, b 2 r, φ = Lr 8 sin 2 2φ, h = R φφ 2R φ cot 2φ. Integrating differential constraint h = 0 corresponding to the BD solution 34, we find R = R 1 r sin 2 φ + R 2 r, 36 where R 1 r and R 2 r are arbitrar functions. Using 34 and 36 we obtain the following corollar in terms of variables x,, and z. Corollar 1. The following expressions for the unknown functions e = x 2α z 2, ε = x 2α z 2, 37 x α ρ 1 = zh 2 + z 2, ρ 2 = z 2 R 2 + z 2 1 x α allow us to reduce the model 1 4 to a sstem of ODs. x α x α + x 2α R 2 + z 2 2 x α Indeed, substituting 37 and 38 into 1 4 we obtain α 1 + ατ = 0, 39 τ C e Cɛ 2α C ɛ2 τ ατ = 0, 40 2 H 2 H 2 3 τ α τ C ρ 2 2 τ 2 H 1 = 0, 41 2 R 1 2 R τ α τ C 1ρ 2 2 R 1 2 τ 4 + C T τ C 1ρ 3 H H = 0, 42 τ + 2 C ρ 2 H C 1ρ 2 R 2 2 R C ρ H C 1ρ where τ = ξ 2 + η τ α τ C 1ρ 2 R 2 C T C 1ρ + 2α R 1 38 = 0, 43

7 Application of the BDs Method to One Problem of Free Turbulence 7 a b c d Figure 1. Calculated profiles as ξ = 0: a normed profile of, b normed profile of, c profile of H, d normed profile of R. 5 Calculation results The sstem of ODs has to satisf the conditions = = H 1 = R 1 = R 2 = 0, τ = 0, 44 = = H 1 = R 1 = R 2 = 0, τ. 45 Conditions 44 take into account flow smmetr with respect to the OX axis. The boundar conditions 45 impl that all functions take zero values outside the turbulent wake. The sstem of ODs satisfing boundar conditions 44, 45 was solved numericall. Additional difficulties are caused b the fact that the coefficients of ODs have singularities. The problem was solved b the modified shooting method and asmptotic expansion of the solution in the vicinit of the singular point [3, 15] = c 1 τ a 10/7 + o τ a 10/7, = 30C ec 2 1 τ a 13/7 + o τ a 13/7, 7a 7C ρ H = τ a + o τ a, a7c ρ 10C e

8 8 O.V. Kaptsov and A.V. Schmidt a b c d Figure 2. Calculated functions: a the function / 0, b the function / 0, c the function H, d the function R/R 0. R 1 = 49C2 ρ72a + 1C ρ 20aC e 2a 2 7C ρ 10C e 2 5C e 7C ρ1 τ a2 + o τ a 2, R 2 = 7C ρ 25C e 7C 1ρ τ a2 + o τ a 2. The value of α is taken to be 0.23 in accordance with experimental data [9, 17]. The results for the problem solution are illustrated in Figs. 1 and 2. Fig. 1 shows the profiles of the functions / 0, / 0, H, and R/R 0 as ξ = 0, where subscript 0 denotes the axial value. The functions / 0, / 0, H, and R/R 0 are plotted in Fig. 2. The functions / 0 and / 0 are bellshaped and determine shapes of the normalized turbulent kinetic energ and the normalized kinetic energ dissipation rate respectivel. Similarl, H and R/R 0 determine shapes of the average densit defect and the normalized densit fluctuation varience respectivel. The function H0, η characterizing the degree of fluid mixing in the turbulent wake is given in Fig. 1c. The maximum value of this function equals This is in consistent with the present notions of incomplete fluid mixing in the wakes [23, 24]. In Fig. 3 adapted from [6] the normalized values of the turbulent energ along the wake axis e 1/2 0 / = ex, 0, 0/ are compared with experimental data [17], computational results [9]

9 Application of the BDs Method to One Problem of Free Turbulence 9 Figure 3. Axial values of the turbulent energ. and results of numerical simulation based on two semi-empirical turbulence models Model1 and Model2 in [4, 6]. The coordinate x is normalized b the bod diameter D. The results obtained here are in close agreement with Lin and Pao s experimental data. Acknowledgements The authors are grateful to Professor.. Chernkh for man helpful and stimulating discussions. The authors would like to thank unknown referees for valuable comments which corrected and improved the first version of this paper. This work was supported b the Russian Foundation for Basic Research project no and programme Leading scientific schools grant no. NSh References [1] Andreev V.K., Kaptsov O.V., Pukhnachov V.V., Rodionov A.A., Applications of group theoretical methods in hdrodnamics, Mathematics and its Applications, Vol. 450, Kluwer Academic Publishers, Dordrecht, [2] Barenblatt.I., alerkina N.L., Luneva M.V., volution of a turbulent burst, J. ng. Phs. Thermophs , [3] Cazalbou J.B., Spalart P.R., Bradshaw P., On the behavior of two-equation models at the edge of a turbulent region, Phs. Fluids , [4] Chashechkin Yu.D., Chernkh.., Voropaeva O.F., The propagation of a passive admixture from a local instantaneous source in a turbulent mixing zone, Int. J. Comp. Fluid Dn , [5] Chernkh.., Fedorova N.N., Moshkin N.P., Numerical simulation of turbulent wakes, Russian J. Theor. Appl. Mech , [6] Chernkh.., Fomina A.V., Moshkin N.P., Numerical models for turbulent wake dnamics behind a towed bod in a linearl stratified medium, Russian J. Numer. Anal. Math. Modelling , [7] fremov I.A., Kaptsov O.V., Chernkh.., Self-similar solutions of two problems of free turbulence, Mat. Model , in Russian. [8] ibson M.M., Launder B.., On the calculation of horizontal, turbulent, free shear flows under gravitational influence, J. Heat Transfer , [9] Hassid S., Collapse of turbulent wakes in stable stratified media, J. Hdronautics , [10] Hinze J.O., Turbulence: an introduction to its mechanism and theor, Mcraw-Hill Series in Mechanical ngineering, Mcraw-Hill Book Co., Inc., New York, 1959.

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