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Similarity Transformations for the Two-Dimensional, Unsteady, Stream-Function Equation

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Similarity Transformations for the Two-Dimensional, Unsteady, Stream-Function Equation J. Fluid Meek. (1978), vol. 85, part 2, pp. 257-271 257 Printed in Great Britain Similarity transformations for the two-dimensional, unsteady, stream-function equation By BRIAN J. CANTWELL Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena (Received 2 March 1977 and in revised form 12 July 1977) The methods described by Bluman & Cole (1974) are used to derive the infinitesimals of the general invariance group of the unsteady, two-dimensional, stream-function equation for the case where the kinematic viscosity v is equal to a constant and the case where v = 0. The infinitesimals in each case involve ten independent parameters, seven of which appear explicitly and three of which are contained implicitly in three arbitrary functions of time. The various finite groups and similarity transformations which may be derived from the infinitesimals are discussed through examples. Two of the arbitrary functions of time are non-trivial and represent invariance of the stream­ function equation under a transformation to a co-ordinate system which moves in a non-uniform irrotational fashion. A general similarity form is derived for which the equations dxfdt = u(x, y, t) and dyfdt = v(x, y, t) for the particle paths may be reduced to an autonomous system. This form is general enough to suggest the hypothesis that, under certain restrictions, the entrainment processes of unsteady flows dominated by two-dimensional large-scale motions may be :displayed diagrammatically on a phase-plane plot of particle trajectories. 1. Introduction It has been known for some time that the process of Reynolds averaging hides from view many of the salient features of turbulent flow. There is now strong evidence which suggests that a wide variety of turbulent shear flows, once thought to consist of un­ differentiated masses of randomly moving, vorticity-bearing fluid elements, are actually dominated by large-scale, often two-dimensional, unsteady motions which are not random. The most notable example is the experiment by Brown & Roshko (1974), who discovered large eddies in a fully developed mixing layer. The effect of this work has been to motivate a renewed interest in the general problem of the unsteady flow of an incompressible fluid in two dimensions. There are several corollaries to the coherent-eddy picture of turbulent shear flow. To begin with, the problem becomes much more complicated since any description of the flow must include the fundamental unsteadiness. Furthermore, since instantaneous velocity fluctuations in the x andy directions may be of the same order, the traditional boundary-layer approach must be replaced by an examination of the complete equa­ tions of motion. On the other hand, since the eddies are convected by the main flow, one might hope to simplify things a bit by moving in a co-ordinate system whose origin was somehow fixed to an eddy. Lastly the message which comes through in the 258 B. J. Cantwell photographs of the mixing layer by Brown & Roshko (1974), the photographs of the wake by Zdravkovich (1969), photographs of the early region of jets and many other pictures of unsteady fluid flows is the message of symmetry. The existence of an array of vortex structures, given certain boundary conditions and the requirement of continuity, imposes certain compatibility conditions on the flow. Entrainment of free-stream fluid occurs as a natural consequence of the induced motions of the large eddies and the compatibility conditions are manifested in terms of points of fluid accumulation. For example, when the velocity field in the near wake of a circular cylinder is viewed from a frame of reference moving with a vortex (Cantwell197 5) the result is a slowly varying pattern of centres and saddles. This paper is concerned, in one way or another, with each of the above corollaries. To begin with, there is a growing number of experimenters, sometimes called eddy chasers,t who are trying to map large eddy motions in space as well as time; see, for example, Browand (1975, mixing-layer measurements), Fiedler (1975, mixing-layer measurements), Cantwell (1975, cylinder near-wake measurements), Cantwell, Coles & Dimotakis (1977, turbulent-spot measurements), Coles & Barker (1975, turbulent­ spot measurements), Wygnanski, Sokolov & Friedman (1976, turbulent-spot measure­ ments) and Turner (1964, turbulent spherical-vortex model). The eddy chasers face the practical problem of collapsing onto a two-dimensional plot measurements or com­ putations made in a three-dimensional space; there is a need for finding similarity transformations which reduce the number of independent variables in the problem. Furthermore, the velocity field, which is the quantity one is usually concerned with, depends on the motion of a moving frame of reference and therefore may lead to con­ flicting or erroneous conclusions when used to argue the disposition of the large eddies; there is a need for understanding the relationship between the frame of reference of a moving observer and the unsteady flow being observed. Finally, the entrainment process, which is an essential part of literally all unconfined turbulent shear flows, can properly be described only in terms of particle paths, the computation of which may present formidable analytical difficulties even if the unsteady velocity field is known exactly (Hama 1962). We may hypothesize that approximating the flow field by a similarity form might prove helpful in this regard even if the similarity were not exact, but could be applied only over a limited space, of the order of one or two eddy lengths, and/or for a limited period of time, comparable to the lifetime of an eddy. To see how this might work, imagine an observer moving in a co-ordinate system with origin attached to the centre of an eddy defined, say, by a peak in the vorticity field or a minimum in the pressure field. In such a system the flow, locally, would be quasi­ steady; the eddy would be growing in size and, for times small compared with its lifetime, might remain relatively undistorted by neighbouring eddies. If one could fit a similarity form to the stream function over this limited region of the flow, then the problem of computing particle paths might be considerably simplified. The possibilities of this approach to the entrainment problem will be addressed in the latter part of this paper. First, the problem of proper frames of reference and the symmetry properties of two-dimensional unsteady flows is approached through the theory of Lie groups. The t Invariably the eddy chaser is a cartoonist whose sketches of the whorls and convolutions of vortex flows are at once informative because they show how the flow must go and frustrating because they defy definition. This paper describes a possible escape from this dilemma. Two-dimensional, unsteady, stream-function equation 259 methods described by _Bluman & Cole (1974) are used to derive the infinitesimals of the general invariance group of the stream-function equation for the case where the kinematic viscosity vis equal to a constant and the case where v = 0 (Euler's equation). The infinitesimals in each case involve ten independent parameters, seven of which appear explicitly and three of which are contained implicitly in three arbitrary once­ differentiable functions of time. The various finite groups implied by the infinitesimals are discussed. Two of the arbitrary functions of time contained in the infinitesimals are non-trivial and represent invariance of the stream-function equation under a trans­ formation to a co-ordinate system which moves in a non-uniform irrotational fashion. The infinitesimals are used to derive a few of the possible similarity transformations which the equation will admit, and for one rather general case it is shown that the equations for the unsteady particle paths may be reduced to an autonomous system. The reader who is unfamiliar with group theory may wish to refer to the introductory works of Birkhoff (1950), Hansen (1964) and especially Cohen (1911). A thorough treatment of group methods using infinitesimal transformations may be found in Ovsjannikov (1962) and a comparison of methods using finite and infinitesimal transformations may be found in Woodard & Ames (1971). Ames (1972) contains a survey of group methods, along with a valuable bibliography on the subject. Trans­ formation techniques have been applied to related problems by Strumpf ( 1964) and Rosen & Ullrich (1973). 2. The problem The motion of an incompressible constant-property fluid in two dimensions is described by the stream-function equation V2ift+ifv V2ifx-ifx V2ifv = vV4if (V2 = fJ21fJx2+o21fJy2), (1) where the stream function if is defined by u = oif1 ay and v = - oif1 ax in order to satisfy the continuity equation identically. We wish to compute the various trans­ formation groups which leave the functional form of ( 1) invariant. The infinitesimals derived from these groups may be used to find similarity transformations of the form if= g(x, y, t, O(x, y, t), rp(x, y, t), F(O, rp)), (2) which, when substituted into (1), lead to a new partial differential equation for Fin terms of() and rp, and thus to a reduction in the number of independent variables from three to two. The application of such transformations to specific cases requires that the boundary curves and the functions of 1/f specified on them satisfy the same conditions of in­ variance. However, the effort in this paper will be restricted to the determination of similarity transformations, not similarity solutions. 3. The invariance condition Consider one-parameter Lie groups of the form x' = X(x, y, t, if; e), y' = Y(x, y, t, if; e),} (3) t' = T(x, y, t, if; e), 1/f' = 'Y(x, y, t, 1/f; e). The surface F(x, y, t, if) = 0 (4) 260 B. J.
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