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24074 Manuscript This article is downloaded from http://researchoutput.csu.edu.au It is the paper published as: Author: Z. Li and G. Mallinson Title: Dual Stream Function Visualization of flows Fields Dependent on Two Variables Journal: Computing and Visualization in Science ISSN: 1432-9360 1433-0369 Year: 2006 Volume: 9 Issue: 1 Pages: 33-41 Abstract: The construction of streamlines is one of the most common methods for visualising fluid motion. Streamlines can be computed from the intersection of two nonparallel stream surfaces, which are iso-surfaces of dual stream functions. Stream surfaces are also useful to isolate part of the flow domain for detailed study. This paper introduces a technique for calculating dual stream functions for momentum fields that are defined analytically and depend on only two variables. For axi-symmetric flows, one of the dual stream functions is the well-known Stokes stream function. The analysis reduces the problem from the solution of partial differential equations to the solution of two ordinary differential equations. Example applications include the Moffat [17] vortex bubble, for which new solutions are presented. DOI/URL: http://dx.doi.org/10.1007/s00791-006-0015-z http://researchoutput.csu.edu.au/R/-?func=dbin-jump-full&object_id=24074&local_base=GEN01-CSU01 Author Address: [email protected] CRO Number: 24074 Dual stream function visualization of flows fields dependent on two variables ZHENQUAN LI¹ AND GORDON MALLINSON² ¹Department of Mathematics and Computing Science, University of the South Pacific, Suva, Fiji Island Email: [email protected] ²Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand Email: [email protected] Abstract The construction of streamlines is one of the most common methods for visualising fluid motion. Streamlines can be computed from the intersection of two nonparallel stream surfaces, which are iso-surfaces of dual stream functions. Stream surfaces are also useful to isolate part of the flow domain for detailed study. This paper introduces a technique for calculating dual stream functions for momentum fields that are defined analytically and depend on only two variables. For axi-symmetric flows, one of the dual stream functions is the well-known Stokes stream function. The analysis reduces the problem from the solution of partial differential equations to the solution of two ordinary differential equations. Example applications include the Moffat [17] vortex bubble, for which new solutions are presented. 1 Introduction The construction of the lines of a vector field is a useful aid towards understanding its structure. For a steady state fluid flow, the lines of the velocity or momentum field are called streamlines. For the case of a vector field V = (v1 , v2 , v3 ), in Cartesian space, (x1 , x2 , x3 ), a streamline is defined by the solution of: dx dx dx 1 = 2 = 3 v1 v2 v3 if vi ≠ 0 (i = 1, 2 , 3), or by introducing an integration variable such as time, t, dX = V . (1) dt The integration of (1) is usually solved by a numerical process that involves interpolation schemes for the velocity field and integration along the streamline from a starting point [3, 7, 8, and 20]. The computation can be a complicated and computationally expensive process [2, 11, and 16]. 1 Stream surfaces are surfaces across which there is no flow and can be more useful than streamlines because they permit regions of the flow domain to be isolated for detailed study. They can be used to visualise the structure of a complicated three-dimensional flow field, more quickly that relying on streamlines to delineate flow regions. Streamlines are contained within stream surfaces and can be generated by finding the intersections of appropriately chosen stream surfaces. Mass conservation is a key issue associated with construction of streamlines and stream surfaces [15], particularly when the velocity field is interpolated from computational data. These issues can be resolved by interpolating suitable vector potentials or stream functions for the velocity field rather than the velocity itself [16, 10, 11]. Yih [21] proved that, for a steady compressible fluid, there exist two stream functions f and g related to the momentum vector by ρV = ∇f × ∇g (2) where ρ is the fluid density. The momentum field described by (2) obeys the steady state law of mass conservation, as can be seen by taking the divergence of (2): ∇.()ρV = ∇(∇f × ∇g) = ∇g.()∇ × ∇f − ∇f .(∇ × ∇g)= 0. The iso-surfaces of f and g are stream surfaces [21]. Because ∇f and ∇g are perpendicular to the stream surfaces that f and g represent, they are also perpendicular to the velocity field V and hence the cross product ∇f × ∇g is in the direction of the velocity field. Thus the intersection of the two stream surfaces represented by f and g in (2) is tangential everywhere to the velocity field V ; that is, the intersection is the streamline. Thus a potentially attractive method for constructing streamlines is to calculate f and g for a given velocity field and then use the intersections their iso-surfaces to construct the streamlines. Unfortunately, as simple as the above method might seem, the calculation of global dual stream functions for general three-dimensional flow fields is an unsolved problem since one or both of the stream functions can be multi-valued. Previous applications of dual stream functions [2, 6, 9] have been limited to non-recirculating flows where both stream functions were single valued. Examples include extrusions, parabolic flows in ducts and potential flows in diffusers. 2 Reztsov and Mallinson [19] demonstrated that dual stream functions could be applied to classes of recirculating or swirling flow by assigning to one of the stream surfaces a structural significance in that it defined closed regions within the flow. The primary stream function (as it will be called here) was single valued and the other function multi-valued. Their approach differs from conventional dual stream function methods in which the two functions have equal significance. This paper extends the concepts in [19] to velocity or momentum fields that can be reduced, by a coordinate transformation if necessary, to a dependence on two variables. This allows an easy identification of the structural stream function. The paper then considers the issue of calculating the second stream function and thence the streamlines. A novel method is presented whereby the process of finding the second stream function is reduced the solution of an ordinary differential equation. The process is illustrated for axi-symmetric flows with swirl that include a vortical bubble describe by Moffatt [17] for which the expressions for the second stream function are new results. 2 Mathematical preliminaries The reduction to a dependence on two variables will generally rely in the definition of an appropriate curvilinear coordinate system. In this section the description of the velocity field and the transformed forms of equation (1) in an arbitrary curvilinear system is presented. 2.1 Curvilinear coordinate system Consider a transformation from Cartesian coordinates (x1, x2 , x3 ) to a new set of coordinates denoted by (ξ1, ξ2, ξ3). The transformation equations are ξ1 = ξ1(x1, x2, x3) ξ2 = ξ2 ()x1, x2, x3 ξ3 = ξ3()x1, x2, x3 . The inverse transformation* is x1 = x1(ξ1, ξ2, ξ3) x2 = x2 ()ξ1, ξ2, ξ3 x3 = x3()ξ1, ξ2, ξ3 . * The inverse exists if and only if the Jacobian does not vanish, that is, when ∂x j / ∂ξi ≠ 0 . 3 A position vector from the origin to an arbitrary point in Cartesian coordinates ()x1, x2 , x3 , measured from the origin, is given by r = x1 i+ x2 j+ x3 k . In terms of the new coordinates, r = r(ξ1, ,ξ 2 ,ξ3 ). The differential dr can now be written ∂ r ∂ r ∂ r 3 ∂ r dr = dξ1 + dξ2 + dξ3 = ∑ dξi . ∂ξ1 ∂ξ2 ∂ξ3 i=1∂ξi We now denote the covariant basis by (eˆ1, eˆ 2 , eˆ3) as follows ∂ r ∂ r ∂ r eˆ1 ≡ , eˆ2 ≡ , eˆ3 ≡ . ∂ξ1 ∂ξ2 ∂ξ3 The unit vectors (e1, e2 , e3), tangent to the coordinate lines of the new coordinate system are eˆ eˆ eˆ e = i = i = i i = 1,2,3 i ˆ ei ∂r gi,i ∂ξi where gi, j denotes the metric tensor, 3 ∂xk ∂xk gi, j = ∑ . k=1 ∂ξi ∂ξ j The differential distance can now be written as 3 dr = ∑ dξieˆ i i=1 3 = ∑ dξi gi,i ei . i=1 2.2 Velocity field description in the curvilinear coordinate system The velocity field in the curvilinear coordinate system is given by 4 V = v1e1 + v2e2 + v3e3 = v1i + v2j + v3k d()x i + x j + x k = 1 2 3 dt dr = dt 3 dξi = ∑ gi,i ei . i=1 dt dξ (The components v are the physical components of V whereas the components i are the i dt contravariant components.) Comparison of the coefficients of ei leads to the following equations gi,i dξi gi,i dξi = vi or = dt i = 1, 2, 3. (3) dt vi For simplicity, we replace vi by vi (i = 1, 2, 3) in the rest of this paper so that it is understood that the components are physical components in the curvilinear system. Since the right hand sides of each of the equations in (3) are the same, and assuming that each of the scale factors gi,i is non- zero, we can write, dξ dξ dξ 1 = 2 = 3 . v1 g 2,2 g3,3 v2 g1,1 g3,3 v3 g1,1 g 2,2 Provided the density is not zero, the equivalent set of equations for the momentum field can be written as, dξ dξ dξ 1 = 2 = 3 . ρv1 g2,2g3,3 ρv2 g1,1g3,3 ρv3 g1,1g2,2 Without any loss of generality the scale factors gi,i can be replaced by hi so that, dξ dξ dξ 1 = 2 = 3 .
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