Wall-bounded laminar sink flows

Transcription

1 Aeta Meehanica 125, (1997) ACTA MECHANICA 9 Springer-Verlag 1997 Wall-bounded laminar sink flows S. Haas and W. Schneider, Vienna, Austria Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of his 60th birthday (Received April 14, 1997) Summary. The laminar flow near an infinite plane wall perpendicular to a line sink of constant strength is investigated in the limit of large Reynolds numbers. Self-similarity requires that fluid is issuing from the boundary layer. The inviscid flow outside the boundary layer is governed by the Euler equations. A one-parametric set of solutions to the Euler equations with appropriate boundary conditions is given. Uniqueness of the iuviscid flow solution is obtained from matching with the boundary layer expansion. The solution of the boundary-layer equations is given both in closed form and numerically. It is found that at the edge of the boundary layer the vorticity decays algebraically. I Introduction Most laminar boundary layers exhibit (positive) entrainment, i.e. irrotational fluid elements, originating from an outer potential flow, enter the boundary layer, where vorticity is generated by viscous forces. There are, however, boundary layers that deviate from this common feature. A well-documented example is the boundary layer at a plane wall in two-dimensional sink flow (cf. [1, p. 116]). Owing to the self-similarity of that flow, the outer edge of the boundary layer coincides with a potential-flow streamline, i.e. entrainment is zero. Though the lack of entrainment might appear rather unusual, the main properties of boundary layers are retained in that case: The boundary-layer solution can be found by matching outer and inner expansions in an hierarchical order, i.e. beginning with an outer potential-flow solution that is independent of the inner boundary-layer solution, to be determined subsequently. A more peculiar example is the boundary layer at a plane wall perpendicular to the axis of a line sink of constant strength, cf. Fig. 1. If the wall extends to infinity, the flow is self-similar. While the velocity increases proportional to the inverse of the distance from the sink, the boundary-layer thickness decreases as the distance itself. Thus the mass flow rate in the boundary layer decreases in main flow direction, and entrainment is negative. The consequences are severe. First of all, the streamlines in the inviscid outer flow ("core region") originate from the boundary layer, where the flow is rotational. Thus an a priori assumption of irrotational inviscid flow, usually based on Helmholtz's vortex theorem, is not justified in this case. Secondly the hierarchical order of classical boundary-layer theory breaks down, as the inviscid outer flow depends on the boundary-layer flow, and vice versa. Further peculiarities of this flow will become apparent from the solution to be given below. The present paper is concerned with laminar flow in the limit of large Reynolds numbers, i.e. Re = Q/v ~ oo, where 2nQ is the constant strength of the line sink, cf. Fig. 1, and v is the constant

2 212 S. Haas and W. Schneider kinematic viscosity of the incompressible fluid. More general solutions, in particular for varying sink strength, are given in [2], where the problem of heat transfer is also addressed. Related problems associated with turbulent flow are considered in [3] and [4]. 2 Core region In the limit of large Reynolds numbers the flow in the core layer is inviscid. As the flow is not necessarily irrotational, the Euler equations are to be solved in the core layer. According to [5] the Euler equations can be written as 9( 1 ~) ~ ( 1 ~)=r2sinof(~k) ' (1) ~r sino & +~ r 2sinO~-O where 0 is the streamfunction and f(0) is an arbitrary function. Assuming a similarity solution of the form 0 = Qrf(O) gives J(O) = - A0-3 (A = const). (3) Introducing x = 1 - cos O, (4) Eq. (1) becomes d2f dx 2 + AF- ~ = 0. (5) The boundary conditions are (2) F(0) = 1 and F(1) = 0. If A = 0, the solution is the potential-flow solution F=l-x. This solution is unique for given sink strength. If A r 0 one obtains the solution F = 1/( / (1 - (6) (7) (8) ~aminar boundary [ayer I / /' / / -) / / ' ~ ---> Q = eonst Fig. 1. Flow field, coordinate systems, and potential-flow streamlines

3 Wall-bounded laminar sink flows = 0 (potential flow) =! 4 =1 0 z 10 Fig. 2. Streamlines according to the solutions of the Euler equations for various values of the vorticity parameter A (cylindrical coordinate system, arbitrary units of lengths) Thus, for given sink strength, there is a one-parametric set of solutions to the Euler equations, with the parameter A characterizing the vorticity of the flow field, cf. Fig. 2. The parameter A is to be determined from matching the inviscid core-layer solution to the viscous boundary layer attached to the wall Expanding the solution (8) of the Euler equations in the vicinity of the wall, and introducing a = (4A) TM, (9) one obtains for the velocity components in directions of increasing r and O, respectively, as 0 ~ hi2: vr = - a (10) 2r V(n/2) - O' vo = - a ]/(zc/2) - 0 (11) r IfA ~ 0 (a ~ 0), the Euler solutions become singular as the wall is approached. Thus matching to the wall boundary layer, as described below, is possible if, and only if, A = 0, i.e. the flow in the core region is irrotational. 3 Boundary layer Introducing the boundary-layer coordinate 0 = [(n/2)- O] ~/~ and the non-dimensional velocity component P'= V~ vor/q, differential equation the boundary-layer equations reduce to the ordinary d3f " _ dzv (d~'~ z ~ d~ + Vd- ~+ \~-~/ -1 =0. (12) Equation (12) is subject to the boundary and matching conditions V(O) = O, (13) d~(o) = 0 (1 4) and dp (Go) = - 1. (15)

4 214 S. Haas and W. Schneider Integrating Eq. (12) two times and using the transformation O = ~/~ - 5, (16) ~'(0) = ~-~ ( 1 z2 dz(2)),~ ] (17) d2~" 0 where 5 = d-~ (), gives Kummer's differential equation (1)d ) Y - +1 z=0. (18) x d2 2 ~ 4 \ 2 The solution of Eq. (18) can be expressed in terms of confluent hypergeometric functions M (cf. [6]) as z(2) = C,M ~-;~;2 + Cb2~M -~(C + 1); 2,2 (19) where C= 2\2 +1 ). (20) The constants Ca and Cb have to satisfy the relationship F1-C in order to allow matching of boundary layer and core region. Since dealing with the analytical solution is rather tedious, a numerical integration was performed. The following results were obtained: d217" 0 5 = ~ ( ) = ; (22) 2rw r c s=~q~- ~_- _~:_, (23) VRe VRe v~r/q I I I I I I I I I I I"-'~ ; = [(~/2)- O]Re 1/2 I I Fig. 3. Velocity profile in the boundary layer

5 Wall-bounded laminar sink flows 215 where cf is the local wall-friction coefficient, zw is the wall-shear stress, and Q is the density. The velocity profile is shown in Fig. 3. At the edge of the boundary layer the vorticity decays algebraically, i.e. r ~o, O(0-3) as 0 ~ oo. (24) QOO This is a contrast to the well-known exponential decay in the common case of a boundary layer with positive (or zero) entrainment, cf. [1]. 4 Conclusions The boundary layer considered in this paper has the peculiar property of negative entrainment. Despite the fact that fluid is issuing from the viscous boundary layer into the inviscid core region, the flow in the core region is irrotational. The vorticity decay at the edge of the boundary layer is algebraic, i.e. weaker than in the common case of positive (or zero) entrainment. Acknowledgement The authors are grateful to Prof. E. Terentev, Moscow, for valuable comments and dicussions regarding the solution of the boundary-layer equations. References [1] Schlichting, H., Gersten, K.: Grenzschicht-Theorie, 9. Aufl. Berlin: Springer [2] Haas, S.: Achsensymmetrische Quell- und Senkenstr6mungen. Dissertation, Technische Universitfit Wien [3] Haas, S., Schneider, W.: Asymptotic analysis of turbulent wall-bounded sink flows. ZAMM 73, T626-T628 (1993). [4] Haas, S., Schneider, W.: Axisymmetric turbulent sink flows. In: Asymptotic methods for turbulent shear flows at high Reynolds numbers (Gersten, K., ed.), pp IUTAM Symposium, Bochum, Germany, Dordrecht: Kluwer [5] Milne-Thomson, L. M.: Theoretical hydrodynamics, 5th ed. London: Macmillan [6] Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. New York: Dover Authors' address: S. Haas and W. Schneider, Technische Universitfit Wien, Institut ffir Str6mungslehre und Wfirmefibertragung, Wiedner Hauptstrage 7, A-1040 Vienna, Austria