Papers in Physics, vol. 4, art. 040002 (2012) Received: 29 August 2011, Accepted: 29 February 2012 Edited by: J-P. Hulin Licence: Creative Commons Attribution 3.0 DOI: http://dx.doi.org/10.4279/PIP.040002 www.papersinphysics.org ISSN 1852-4249 Beltrami flow structure in a diffuser. Quasi-cylindrical approximation Rafael González,1, 2∗ Ricardo Page,3 Andrés S. Sartarelli1 We determine the flow structure in an axisymmetric diffuser or expansion region connecting two cylindrical pipes when the inlet flow is a solid body rotation with a uniform axial flow of speeds Ω and U, respectively. A quasi-cylindrical approximation is made in order to solve the steady Euler equation, mainly the Bragg–Hawthorne equation. As in our previous work on the cylindrical region downstream [R González et al., Phys. Fluids 20, 24106 (2008); R. González et al., Phys. Fluids 22, 74102 (2010), R González et al., J. Phys.: Conf. Ser. 296, 012024 (2011)], the steady flow in the transition region shows a Beltrami flow structure. The Beltrami flow is defined as a field vB that satisfies ωB = ∇× vB = γvB, with γ = constant. We say that the flow has a Beltrami flow structure when it can be put in the form v = Uez + Ωreθ + vB, being U and Ω constants, i.e it is the superposition of a solid body rotation and translation with a Beltrami one. Therefore, those findings about flow stability hold. The quasi-cylindrical solutions do not branch off and the results do not depend on the chosen transition profile in view of the boundary conditions considered. By comparing this with our earliest work, we relate the critical Rossby number ϑcs (stagnation) to the corresponding one at the fold ϑcf [J. D. Buntine et al., Proc. R. Soc. Lond. A 449, 139 (1995)]. I. Introduction We have recently conducted studies on the for- mation of Kelvin waves and some of their fea- tures when an axisymmetric Rankine flow expe- riences a soft expansion between two cylindrical pipes [1, 2]. One of the significant characteristics of this phenomenon is that the downstream flow ∗E-mail: rgonzale@ungs.edu.ar 1 Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Gutierrez 1150, 1613 Los Polvorines, Pcia de Buenos Aires, Argentina. 2 Departamento de F́ısica FCEyN, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina . 3 Instituto de Ciencias, Universidad Nacional de General Sarmiento, Gutierrez 1150, 1613 Los Polvorines, Pcia de Buenos Aires, Argentina. shows a Rankine flow superposing a Beltrami flow (Beltrami flow structure [4])). Yet, upstream and downstream cylindrical geometries were considered without taking into account the flow in the expan- sion. This work considered that the base upstream flow, formed by a vortex core surrounded by a po- tential flow, would have the same Beltrami struc- ture at the expansion and downstream. Neverthe- less, the flow at the expansion was not determined. However, it has been seen that this flow is only possible when no reversed flow is present and if its parameters do not take the values where a vortex breakdown appears [6–8]. The starting point in the study of the expansion flow is an axysimmetric steady state resulting from the Bragg–Hawthorne equation [7, 9–11] for both the vortex breakdown and the formation of waves. Therefore, the solu- tion behavior, whether it branches off or shows a possible stagnation point on the axis, will be deter- 040002-1 Papers in Physics, vol. 4, art. 040002 (2012) / R González et al. minant to delimit both phenomena. Our previous research focused on the formation of Kelvin waves with a Beltrami flow structure downstream [1–3], when the upstream flow was a Rankine one. This present investigation considers only a solid body rotation flow with uniform axial flow at the inlet. As a first step in the study of the flow at the expansion, we only study the rotational flow. However, comparisons with our previous work [1] will be drawn. The aim of this present work is to obtain the steady flow structure at the expansion, consider- ing a quasi-cylindrical approximation when the in- let flow is a solid body rotation with uniform axial flow of speeds Ω and U, respectively. If a is the ra- dius of the cylindrical region upstream, a relevant parameter is the Rossby number ϑ = U Ωa . Thus, we would like to determine how this flow depends on the Rossby number, on the geometrical param- eters of the expansion and on the critical values of the parameters. We focus on finding the param- eter values for which a stagnation point emerges on the axis, or for which the solution of the Bragg– Hawthorne equation branches off. We take them as the conditions for the vortex breakdown to develop. First, this paper presents the inlet flow and the corresponding Bragg–Hawthorne equation written for the transition together with the boundary con- ditions in section II. Second, it works on the quasi-cylindrical approximation for the Bragg– Hawthorne equation and its solution is developed in section III. Third, results and discussions are of- fered in section IV together with a comparison with our previous work [1]. Finally, conclusions are pre- sented in section V. II. The Bragg–Hawthorne equation We assume an upstream flow in a pipe of radius a as an inlet flow in an axisymmetric expansion of length L connecting to another pipe with radius b, b > a. The inlet flow filling the pipe consists of a solid body rotation of speed Ω with a uniform axial flow of speed U: v = Uez + Ωreθ, (1) U and Ω being constants. The equilibrium flow in the whole region is determined by the steady Euler equation which can be written as the Bragg– Hawthorne equation [10] ∂2ψ ∂z2 + r ∂ ∂r ( 1 r ∂ψ ∂r ) + r2 ∂H ∂ψ + C ∂C ∂ψ = 0, (2) where ψ is the defined stream function vr = − 1 r ∂ψ ∂z , vz = 1 r ∂ψ ∂r , (3) and H(ψ),C(ψ) are the total head and the circula- tion, respectively H(ψ) = 1 2 (v2r + v 2 θ + v 2 z) + p ρ , C(ψ) = rvθ. (4) To solve Eq. (2), the boundary conditions must be established. These consist of giving the inlet flow, of being both the centerline and the bound- ary wall, streamlines, and of being the axial veloc- ity positive (vz > 0). For the upstream flow, the stream function is ψ = 1 2 Ur2, and H(ψ),C(ψ) are given by H(ψ) = 1 2 U2 + Ωγψ, C(ψ) = γψ, (5) γ = 2U Ω being the eigenvalue of the flow with Bel- trami structure [3]. Thus, by considering the inlet flow, Eqs. (5) are valid for the whole region. The second condition regarding the streamlines implies the following relations ψ(r = 0,z) = 0, ψ(r = σ(z),z) = 1 2 Ua2, 0 ≤ z ≤ L (6) where r = σ(z) gives the axisymmetric profile of the pipe expansion. Deducing from Eq. (6), the boundary conditions are determined by the inlet flow. Additionally, curved profiles are considered, so ∂ψ ∂z (r,z = L) = 0, 0 ≤ r ≤ b. (7) 040002-2 Papers in Physics, vol. 4, art. 040002 (2012) / R González et al. III. Quasi-cylindrical approximation If we consider that ∂ 2ψ ∂z2 = 0, the solutions to Eqs. (2) and (5) for the cylindrical regions are given by [10] ψ = 1 2 Ur2 + ArJ1[γr], (8) where A is a constant. The quasi-cylindrical ap- proximation consists of taking the dependence of A(z) on z but with the condition ∂ 2ψ ∂z2 ≈ 0 com- pared with the remaining terms of (2). The ampli- tude A(z) is then obtained by imposing the bound- ary conditions (6) which depend on the wall profile r = σ(z), giving A(z) = 1 2 U ( a2 −σ2(z) ) σ(z)J1[γσ(z)] . (9) By using the dimensionless quantities r̃ = r a , z̃ = z a , ṽ = v U the stream function in the quasi- cylindrical approximation can be written as ψ̃ = 1 2 r̃2 + Ã(z̃)r̃J1[ 2 ϑ r̃], Ã(z̃) = 1 2 ( 1 − σ̃2(z̃) ) σ̃(z̃)J1[ 2 ϑ σ̃(z̃)] , (10) where ϑ = U Ωa is the Rossby number. Hence the velocity field becomes ṽr(r̃, z̃) = −à ′ (z̃)J1[ 2 ϑ r̃] (11) ṽθ(r̃, z̃) = 1 ϑ r̃ + 2 ϑ Ã(z̃)J1[ 2 ϑ r̃] (12) ṽz(r̃, z̃) = 1 + 2 ϑ Ã(z̃)J0[ 2 ϑ r̃], (13) where à ′ (z̃) = dÃ(z̃)/dz̃. Finally, it is necessary to give the wall profile σ̃(z) to completely determine the flow. Two kinds of profiles were seen: i- conical profile σ̃(z̃) = 1 + ( η − 1 L̃ ) z̃, 0 ≤ z̃ ≤ L̃ and η = b a . (14) ii- curved profile σ̃(z̃) = 1 + η 2 − ( η − 1 2 ) cos ( πz̃ L̃ ) , 0 ≤ z̃ ≤ L̃. (15) The latter meets the boundary condition (7) as well. Therefore, Eqs. (11-15) together with the boundary conditions (6,7) allow to determine the flow structure for both the conical and curved wall profile. IV. Results and discussion We note that the flow keeps a Beltrami flow struc- ture in the quasi-cylindrical approximation. Effec- tively, giving (11-13) ṽr(r̃, z̃) = ṽBr(r̃, z̃) (16) ṽθ(r̃, z̃) = 1 ϑ r̃ + ṽBθ(r̃, z̃) (17) ṽz(r̃, z̃) = 1 + ṽBz(r̃, z̃), (18) it is easy to see that under this approximation ∇× vB(r̃, z̃) = 2 ϑ vB(r̃, z̃) and so, the whole flow is the sum of a solid body rotation flow with a uniform axial flow plus a Beltrami flow, given the latter in a system with uniform translation velocity U = 1.ẑ and uniform rigid rotation velocity V = 1 ϑ r̃θ̂. Given the flow field and its structure, the param- eters are considered by evaluating the behavior of ṽz(r̃, z̃0) with z̃0 = L̃ i.e., taken at outlet, and with L̃ = 1. In order to do so, a wall profile is selected (14 or 15) and three different values of the expan- sion parameter are taken, mainly η1 = 1.1, η2 = 1.2 and η3 = 1.3. The first step is to analyze the flow dependence on the Rossby number. In Fig. 1, the contour flows corresponding to the conical and curved pro- files for η1 = 1.1, ϑ1 = 0.695 are shown. Graph- ics in Fig. 2 represent the same configuration but for ϑ = 0.68 < ϑ1. The broken lines represent points for which ṽz = 0. Inflow and recirculation are present but it is not a real flow because the model fails when considering inflow. It can be seen 040002-3 Papers in Physics, vol. 4, art. 040002 (2012) / R González et al. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z r 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z r Figure 1: Contour flow in the transition region for conical and curved profiles for η1 = 1.1, ϑ1 = 0.695. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z r 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 z r Figure 2: Contour flow in the transition region for conical and curved profiles for η1 = 1.1, ϑ1 = 0.68. The broken lines represent points with ṽz = 0. that for ϑ1 = 0.695, ṽz = 0 at the outlet, on the axis. For the Rossby numbers with ϑ ≥ ϑc, the azimuthal flow vorticity is negative (ωφ < 0), re- sulting in an increase in the axial velocity with the radius, and so having a minimum on the axis where the stagnation point appears [6]. Therefore, the critical Rossby number can be defined ϑc as the value where ṽz is zero at the outlet on the axis i.e., where the flow shows a stagnation point. This is the necessary condition to produce a vortex break- down [6]. We find the same critical Rossby number for both wall profiles and so we will not treat them separately from now on. The critical Rosssby val- ues for η2 = 1.2 and η3 = 1.3 are ϑ2 = 0.869 and ϑ3 = 1.052, respectively. Given the previous analysis, the second step is to show the behavior of ṽz on the axis at the outlet as a function of ϑ for each η in order to study the ex- 040002-4 Papers in Physics, vol. 4, art. 040002 (2012) / R González et al. istence of folds in the Rossby number-continuation parameter (equivalent to the swirl parameter in [5,7,11]); indeed, we have seen that ṽz has the min- imum on the axis. Besides, when using Eq. (13) when r = 0, it is easy to see that ṽz decreases with z and so it reaches the minimum at the outlet being ṽz ≥ 0. In Fig. 3, the radial dependence of ṽz is plotted at the outlet for η1,η2,η3 and its variation with ϑ when it is slightly shifted from ϑ1. In Fig. 4, it can be seen that the minimum of ṽz on the axis increases with ϑ so there is no fold of ṽzmin as defined by Buntine and Saffman in a similar ap- proximation [5]. Η1 <Η2 <Η3 Η1Η3 H a L 0.5 1 v ~ z 1 r ~ 1 r ~ 3 r ~ J1 0.95 J1 1.2 J1 H b L 0.5 1 v ~ z 1 r ~ 1 r ~ Figure 3: (a) ṽz at the outlet as a function of r for η1,η2,η3 and the corresponding critical Rossby numbers ϑ1,ϑ2,ϑ3. (b) ṽz at the outlet as a function of r for ϑ1 and for values of ϑ slightly shifted from ϑ1 . In each case, the minimum of ṽz is reached on the axis. The dependence of the results on L is analyzed. It can be seen that when z = L in Eqs. (14) and (15), σ̃(L̃) = η is obtained. By replacing this in Eq. (13) for z = L and r = 0 it gives ṽzmin = 1 + ( 1 −η2 ) ϑηJ1[ 2 ϑ η] , (19) and so ϑc is obtained as a function of η by solving the last equation when ṽzmin = 0, as shown in Fig. 5. This result seems to be surprising, but it is not so if it is considered as derived from the quasi-cylindrical approximation: the dependence of 0.5 1.5J1 J2 J3 0.0 0.2 0.4 0.6 Rossby number J A x ia l v e lo c it y v~ z Η1 Η2 Η3 Figure 4: ṽz at the outlet on the axis as a function of the Rossby number ϑ for η1 = 1.1,η2 = 1.2, η3 = 1.3. Here ϑ1 = 0.695,ϑ2 = 0.869 and ϑ3 = 1.052 correspond to stagnation points. the flow on z is obtained through the boundary conditions expressed by Eq. (6). At the same time, these boundary conditions depend on the inlet flow and on the parameter η. This explains the fact that the same results, for both conical and curved profiles, have been obtained and that the condition given by Eq. (7) at the outlet has not influenced them. 1.2 1.4 1.6 1.8 2.0 0.5 1.0 1.5 2.0 2.5 Expansion Η C ri ti c a l R o s s b y J c Figure 5: Critical Rossby number ϑc as a function of η. Differences with Batchelor’s seminal work should be marked [10]. Mainly, he works in cylindrical ge- ometry and does not consider the dependence of the flow on z . We introduce this z dependence through the quasi-cylindrical approximation. This, there- fore, allows us to find the structure of the flow in the transition together with the Rossby critical number defined by considering this structure and by show- ing that the minimum of ṽz is reached at the outlet 040002-5 Papers in Physics, vol. 4, art. 040002 (2012) / R González et al. on the axis. Nevertheless, once the flow reaches the pipe downstream, the analysis coincides because, as shown, the problem depends on the inlet flow and on the parameter expansion η. This allows us to consider the issue of the vortex core that we have not considered at the inlet flow. As we know the structure of the flow in the downstream cylindri- cal region [1] and by assuming a quasi-cylindrical approximation for the vortex core in the transition region, the minimum of vcorez at the outlet on the axis is given by vcorezmin = 1 + ( 1 − η̂2 ) ϑ̂η̂J1[ 2 ϑ̂ η̂] , (20) where ϑ̂ = ϑ ι , η̂ = ξ ι and ξ and ι are the dimension- less radius of the core downstream and upstream, respectively. We note that η̂ is the expansion pa- rameter of the core. Hence Eqs. (19) and (20) have the same structure. In the present work, we have not found any fold in the Rossby number- continuation parameter of ṽz, as found in our pre- vious work [1] where the fold was associated with a critical Rossby number called ϑcf by Buntine and Saffman [5]. As we have already done, we define the Rossby critical number for which vcorezmin = 0 where there is a stagnation point, and we will call it ϑcs. In [1], for ι = 0.272 and pipe expansion parameters η1,η2,η3, we have found that ϑcf were 0.35, 0.44 and 0.53, respectively, while the core ex- pansion parameters η̂ were 1.25, 1.47 and 1.65, re- spectively. By replacing these values in Eq. (20) when vcorezmin is zero, we get the corresponding ϑ̂cs and then ϑcs for the vortex core. These are respectively 0.26, 0.38 and 0.49. That is to say that in all the cases we have ϑcs < ϑcf . Therefore, at the fold ṽz > 0. This coincides with the results found by Buntine and Saffman [5] in their analysis using a three-parameter family inlet flow. V. Conclusions The main conclusions drawn from the previous sec- tions are: 1. In the quasi-cylindrical approximation, the steady flow in the transition expansion region corresponding to a solid body rotation with uniform axial flow as inlet flow has the same Beltrami flow structure as in the pipe down- stream, which is compatible with the bound- ary conditions. Therefore, findings from our previous work on stability [1–3] can hold. 2. For fixed values of η and ϑ ≥ ϑc, ωφ < 0 and then ṽz in the transition region is an increasing function of r and a decreasing function of z reaching its the minimum on the axis at the outlet. 3. For fixed values of η, the minimum of ṽz on the axis is an increasing function of ϑ (Fig. 4), where the stagnation point corresponds to ϑc. 4. As a consequence, no branching off takes place for the solutions of Bragg–Hawthorne equa- tion. 5. The critical Rossby number ϑc corresponding to stagnation is an increasing function of η (Fig. 5). 6. The whole picture can be reached by putting together these results with those obtained in [1], where there is a branching owing to the boundary conditions at the frontier between the vortex and the irrotational flow. Moreover, since the results in [1] for the rotational flow depend on the inlet flow as well as on the ro- tational expansion parameter η̂ defined in Eq. (20), given a quasi-cylindrical approximation, it can be concluded that this expression is the minimum of vz in the core. Therefore, we can get the critical Rossby number ϑcs and com- pare it with that corresponding to the fold ϑcf . This present work verifies that ϑcs < ϑcf , in accordance with Buntine and Saffman’s results [5]. 7. In the quasi-cylindrical approximation, previ- ous results do not depend on the chosen profile. This can be explained by the boundary condi- tions chosen depending on the inlet flow and on the parameter expansion. Acknowledgements - We would like to thank Un- versidad Nacional de General Sarmiento for its sup- port for this work, and our colleague Gabriela Di 040002-6 Papers in Physics, vol. 4, art. 040002 (2012) / R González et al. Gesú for her advice on the English version of this paper. [1] R González, G Sarasúa, A Costa, Kelvin waves with helical Beltrami flow structure, Phys. Flu- ids 20, 24106 (2008). 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