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1 This article was downloaded by: [The University Of Melbourne Libraries] On: 18 October 2012, At: 21:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Turbulence Publication details, including instructions for authors and subscription information: The topology of skin friction and surface vorticity fields in wall-bounded flows M. S. Chong a, J. P. Monty a, C. Chin a & I. Marusic a a Department of Mechanical Engineering, The University of Melbourne, Victoria, 3010, Australia Version of record first published: 14 May To cite this article: M. S. Chong, J. P. Monty, C. Chin & I. Marusic (2012): The topology of skin friction and surface vorticity fields in wall-bounded flows, Journal of Turbulence, 13, N6 To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Turbulence Vol. 13, No. 6, 2012, 1 10 The topology of skin friction and surface vorticity fields in wall-bounded flows M.S. Chong, J.P. Monty, C. Chin and I. Marusic Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia (Received 4 December 2011; final version received 27 January 2012) In previous studies, the three invariants (P, Q and R) of the velocity gradient tensor have been widely used to investigate turbulent flow structures. For incompressible flows, the first invariant P is zero and the topology of turbulent flow structures can be investigated in terms of the second and third invariants, Q and R, respectively. However, all these three invariants are zero at a no-slip wall and can no longer be used to identify and study structures at the surface in any wall-bounded flow. An alternative scheme is presented here for the classification of critical points at a no-slip wall; the skin friction vector field at the wall is given by the wall normal gradients of the streamwise and spanwise velocity components; at a critical point, these gradients are simultaneously zero. The flow close to critical points in the surface skin friction field can be described by a no-slip Taylor series expansion and the topology of the critical point in the skin friction field is defined by the three invariants (P, Q and R) of the no-slip tensor. Like the invariants of the velocity gradient tensor, the no-slip tensor invariants can be easily computed and these invariants provide a methodology for studying the structure of turbulence at the surface of a no-slip wall in any wall-bounded flow. Keywords: turbulence; wall-bounded flows; surface skin friction fields; surface vorticity fields; local solutions of Navier Stokes equations 1. Introduction A critical point is a point in a flow field where the velocity components u 1, u 2 and u 3 (in any arbitrary x 1, x 2 and x 3 directions, respectively) are simultaneously zero and the streamline slopes are indeterminate. Close to the critical point, the velocity field can be described by the linear terms of a Taylor-series expansion. Two types of critical points can be described: one is the free-slip critical point, which is located away from a no-slip wall, and the other is the no-slip critical point, which occurs on a no-slip wall. Examples of critical points are the centres of vortical structures in wakes behind bluff bodies and stagnation/reattachment points in separated flows (more examples can be found in [1]). Corresponding author. min@unimelb.edu.au ISSN: online only C 2012 Taylor & Francis

3 2 M.S. Chong et al Free-slip critical points For free-slip critical points, the velocity u i in x i space is given by. u 1 x 1 dx 1 /dt A 11 A 12 A 13 x 1 u 2 =. x 2 = dx 2 /dt = A 21 A 22 A 23 x 2. u 3 x 3 dx 3 /dt A 31 A 32 A 33 x 3 or. x i = dx i dt where A ij = u i / x j is the velocity gradient tensor. = A ij x j, (1) 1.2. The invariants of the velocity gradient tensor For an observer moving in a non-rotating frame of reference with any point in a flow field, the flow surrounding the point is described in terms of the nine components of A ij.the characteristic equation of A ij is λ 3 i + Pλ2 i + Qλ i + R = 0, (2) where λ i is the eigenvalue and P, Q and R are the tensor invariants that are defined as P = (A 11 + A 22 + A 33 ), Q = A 11 A 22 A 12 A 21 + A 11 A 33 A 13 A 31 + A 22 A 33 A 23 A 32 and R = A 11 A 23 A 32 A 11 A 22 A 33 + A 12 A 21 A 33 A 12 A 23 A 31 A 13 A 21 A 32 + A 13 A 22 A 31. (3) The characteristic Equation (2) can have (1) all real roots that are distinct, (2) all real roots where at least two roots are equal or (3) one real root and a conjugate pair of complex roots. In the P Q R space, the surface that divides the real solutions from the complex solutions is given by 27R 2 + (4P 3 18PQ)R + (4Q 3 P 2 Q 2 ) = 0. (4) For an incompressible flow, the first invariant P is zero and hence all free-slip critical points can be described by the second and third invariants, i.e. Q and R. Details of the classification of three-dimensional flow fields using these invariants can be found in [2]. The invariants of the velocity gradient tensor have been widely used to study turbulent flow structures in order to extract information regarding the scales, kinematics and dynamics of these structures (see [3], p. 268 and [4]). These invariants can also be used to study turbulent structures using data from Direct Numerical Simulations (DNS), for example in the DNS of homogeneous isotropic turbulence by Ooi et al. [5] and in wall-bounded shear flows by Chong et al. [6], del Alamo et al. [7] and Wu et al. [8].

4 Journal of Turbulence No-slip critical points In wall-bounded flows, there is no slip at the wall, i.e. u 1 = u 2 = u 3 = 0 when x 3 = 0, where x 3 is the wall-normal direction (hereafter, x 1 and x 2 are the streamwise and spanwise components, respectively). If the origin of a Taylor series expansion is located at any arbitrary point on the no-slip wall, i.e. at x 3 = 0, the linear terms of the expansion are given by u A 13 x 1 u 2 = 0 0 A 23 x 2, (5) u 3 where A 13 = u 1 / x 3 and A 23 = u 2 / x 3. Even for finite A 13 and A 23, all three invariants (P, Q and R) of the above tensor are zero at the wall and hence they cannot be used to map the topology of flow patterns on a no-slip surface. To classify critical points on a no-slip wall, the Taylor series expansion at any point on the wall (i.e. at x 3 = 0) can be written to include higher order terms and at the same time satisfy the no-slip boundary condition. This expansion is given by If o x i is defined as u 1 = A 13 x 3 + (A 11 x 1 + A 12 x 2 + A 13 x 3 ) x 3, u 2 = A 23 x 3 + (A 21 x 1 + A 22 x 2 + A 23 x 3 ) x 3, u 3 = 0 + (A 31 x 1 + A 32 x 2 + A 33 x 3 ) x 3. (6) x 3 o x i = dx i dτ, (7) where dτ = x 3 dt, the no-slip velocity field can be expressed as u 1 = x o 1 = dx 1 x 3 dτ = A 13 + (A 11 x 1 + A 12 x 2 + A 13 x 3 ), u 2 = x o 2 = dx 2 x 3 dτ = A 23 + (A 21 x 1 + A 22 x 2 + A 23 x 3 ), u 3 = x o 3 = dx 3 x 3 dτ = 0 + (A 31x 1 + A 32 x 2 + A 33 x 3 ). (8) A critical point occurs on the wall when A 13 = A 23 = 0 and at this critical point o x 1 dx 1 /dτ A 11 A 12 A 13 x 1 o x 2 = dx 2 /dτ = A 21 A 22 A 23 x 2 (9) o x dx 3 /dτ A 31 A 32 A 33 x 3 3 or o x i = dx i dτ = A ij x j. (10)

5 4 M.S. Chong et al. A ij will be referred to as the no-slip tensor. The above vector field can be integrated with τ to generate surface streamlines (limiting streamlines or skin friction lines) The invariant of the no-slip tensor To satisfy boundary conditions on a no-slip surface (and assuming incompressible flow), the no-slip tensor is given by A 11 A 12 A 13 A ij = A 21 A 22 A (11) 2 (A 11 + A 22 ) By substitution of the above expansion into the Navier Stokes equation, it can be shown that A 11, A 12, A 21 and A 22 are related to the vorticity gradients and A 13 and A 23 are related to pressure gradients. Like the velocity gradient tensor, the no-slip tensor has three invariants,p, Q and R. For incompressible flow, the first invariant P is no longer zero and is given by P = 1 2 (A 11 + A 22 ). (12) The second and third invariants, Q and R, respectively, are given by [ ] 1 Q = A 12 A 21 2 (A A2 22 ), (13) [ ] 1 R = (A 11 A 22 A 12 A 21 ) 2 (A 11 + A 22 ). (14) It can also be shown that the relationship between the three invariants of the no-slip tensor is given by 2P 3 + PQ + R = 0. (15) 2. Surface skin friction and surface vorticity fields The no-slip tensor and its invariants can be used to investigate the topology of skin friction fields and the surface vorticity field at the wall in any wall-bounded flows, especially using data from DNS. Wall shear stress or skin friction is defined as the wall normal derivative of the velocity vector at the wall. In simulations of wall-bounded flow over a smooth wall, the first plane away from the wall is usually parallel to the wall. Hence, the direction of the near-wall velocity vector when projected onto the plane is in the same direction as the wall shear stress vector if the first plane is sufficiently close to the wall. Thus, skin friction lines (also known as limiting streamlines at the no-slip wall) can be obtained by integrating the streamwise velocity u 1 and spanwise velocity u 2 in the first plane from the wall. It is generally assumed that this first plane is very close to the wall. How close this plane has to be to produce sufficiently accurate skin friction lines is an open question. In most computations of wall-bounded flows, the first plane is located at x + 3 << 1, where

6 Journal of Turbulence 5 Figure 1. Skin friction lines at the wall using data from DNS of channel flow at Re τ = 934, see [9] for details of computations. This figure shows a small region (for x + 1 x+ 2 plane ) of the entire channel wall in the computational domain (23, 474 8, 803 in viscous units). The size of the blue square is wall units. the wall normal distance is given in viscous wall units, i.e. x + 3 = x 3u τ /ν, where ν is the kinematic viscosity and u τ is the friction velocity. Figure 1 shows skin friction lines in a streamwise spanwise planer surface at the no-slip wall using data from the DNS of channel flow made available by del Alamo et al. [9]. The Reynolds number is Re τ = hu τ /ν = 934, where h is the channel half-height. In the computation by del Alamo et al. [9], the first plane from the wall is at x + 3 = The skin friction lines shown in Figure 1 are for a relatively small region of the flow, i.e. for x + 1 x+ 2 plane ; the size of the entire streamwise spanwise channel wall, in viscous units, is 23, 474 8, 803. Skin friction lines of many such small random regions indicate that Figure 1 is typical of a general turbulent skin friction field, consisting of bifurcation lines that are associated with surface streak lines and are signatures of streamwise vortical motions (or streamwise eddying motions) at the wall. The major question this paper seeks to address is as follows: Are there critical points in the skin friction field? This question is best addressed by considering the surface vorticity field, since Helmholtz s second theorem states that a vortex filament cannot end in a fluid (it must extend to the boundaries of a fluid or form a closed path). Surface vortex lines are orthogonal to the surface skin friction lines, see [6]. It can be shown that

7 6 M.S. Chong et al. Figure 2. Skin friction lines (black) shown in figure with surface vortex lines (red). The surface vortex lines are orthogonal to the skin friction lines. No critical points are detected in both the skin friction and surface vorticity fields in this region of the flow field. 1. critical points in the skin friction field are also critical points in the surface vorticity field, 2. if the critical point in the skin friction field is a saddle, the critical point in surface vorticity field is also a saddle and 3. if the critical point in the skin friction field is a node/focus, the critical point in the surface vorticity field is also a node/focus. Figure 2 shows the same skin friction lines (black lines) as in Figure 1 with the surface vorticity lines (shown in red). In the DNS of wall-bounded flows, the flow is assumed to be periodic in the spanwise direction and the outflow is re-cycled and fed back as inflow. With regard to surface vortex lines, a consideration of Helmholtz s second theorem suggests that the surface vorticity field (and hence the skin friction field) at the no-slip wall can be mapped onto the surface of a toroid. This conjecture is important since the Poincaré Hopf index theorem or the Poincaré Bendixson 1 theorem (see [10], [11] or [12]) states that the topology of a smooth vector field on a toroidal surface is such that the number of nodes 1 The Poincaré Hopf index theorem states that the topology of a smooth vector field on a surface is given by (Nodes) (Saddles) = 2 2g, where g is the genus of the surface.

8 Journal of Turbulence 7 Figure 3. Skin friction lines (black) and surface vortex lines (red) at the wall, using same data as for Figures 1 and 2. In this region of the surface, there are two critical points. S saddle, N node/ focus. minus the number of saddles is equal to zero, i.e. (Nodes) (Saddles) = 0. (16) Note that a saddle has a Poincaré index of 1 and a node/focus has a Poincaré indexof +1. Hence, if the skin friction field for the entire no-slip wall is similar to that shown in Figure 1, where no critical points can be seen, then the Poincaré Hopf index theorem is satisfied. Hence, it was initially felt that there are probably no critical points on the surface of a no-slip wall, since a single critical point (a single saddle or a single node) will violate the Poincaré Hopf theorem for the topology of the flow field on a toroidal surface. However, contrary to expectations, there are critical points in the skin friction field at the wall. Figure 3 shows a close-up (x + 1 x+ 2 plane ) from another area of a skin friction field using data from the same computation as that used to produce Figure 1. Critical points in the skin friction field are rare but what is more surprising in skin friction and surface vorticity fields shown in Figure 3 is that these critical points consist of saddle node pairs so that the Poincaré index around the pair of critical points is such that it is zero around a closed circuit enclosing the two critical points. Figure 4 shows another region

9 8 M.S. Chong et al. Figure 4. Skin friction lines (black) and surface vortex lines (red) at the wall showing complex set of critical points 3 saddles and 3 nodes/foci. using data from the same computation as in Figure 1. Even for such complex surface flow patterns, the number of nodes equals the number of saddles, satisfying the Poincaré Hopf index theorem for a toroidal surface. 3. Conclusion There is an increasing reliance on DNS of wall-bounded flows (pipes, channels and boundary layers) at low to moderate Reynold s number to try understand the physics of turbulent structures near the wall. It has been conjectured in this paper that the surface vorticity field can be mapped onto toroidal surface such that vortex lines at the surface are closed. If Figures 3 and 4 are typical of all regions with critical points at the wall, then the number of nodes/foci equals the number of saddles, hence satisfying the Poincaré Hopf index theorem, confirming the conjecture postulated in this paper. It should be noted that regions with critical points are rare ; roughly 120 such regions covering an estimated 0.6% of the entire wall have been identified in the data from [9], and the topology of all these regions satisfy the Poincaré Hopf index theorem locally. Since the finding of these critical points in the channel flow simulation of [9], similar critical points have also been found on the surface using data from the DNS of a pipe flow by Chin et al. [13] and also from the recent spatial computation of a zero pressure gradient turbulent boundary layer by Wu et al. [8]. Recently, Lenears et al. [14] found negative

10 Journal of Turbulence 9 streamwise velocities near the wall in turbulent flows using the DNS data, and stated that this contradicted the experimental work of Eckelmann [15], who stated that with certainty, there are no negative velocities near the wall. It is generally assumed that near the wall the flow is two-dimensional and there are no flow reversals; however, this paper has shown, it is possible to have critical points on the surface of wall-bounded flows, indicating that in some regions at the wall, complex threedimensional structures exist. These three-dimensional turbulent structures will cause flow reversals near the wall. The invariants of the no-slip tensor provide an analytical framework and methodology to study the footprint of turbulent structures near the wall. However, it is yet to be established conclusively that critical points exist in all DNS of wall-bounded flows and whether they are necessary for the generation of turbulence at the wall. In an incompressible flow, vorticity is only generated at the boundary (see [11]) and it is conjectured that these critical points must be associated with the generation and transport of vorticity and, therefore, turbulence at the wall. What is even more important to establish is how to define a smooth wall in experimental investigations and whether critical points similar to that in wall-bounded simulations exist. This is significant, since having critical points at the wall implies that there are regions near the wall with instantaneous flow reversal Figure 5. Colour contours of the first invariant P of the no-slip tensor plot with skin friction lines (black) and surface vortex lines (red). At points where P = 0, the flow is two-dimensional.

11 10 M.S. Chong et al. and many anemometry techniques, such as hot-wire anemometry, do not generally allow for measurement of negative velocities. An example of how one of the invariants of the no-slip tensor can be used to study structures at a no-slip wall is shown in Figure 5, where the first invariant P is plotted with the skin friction and surface vorticity lines. At points where P = 0, the flow is twodimensional. It can be seen that, although a major portion of the flow is two-dimensional, the flow is three-dimensional in regions near critical points and in regions where there are kinks in the vortex lines. The aim of this paper is to show how a Taylor series expansion with the origin at the no-slip wall can be used to study flow structures at the wall. The topology of the skin friction and surface vorticity fields shows that there may be threedimensional structures at the wall, and an example showing the use of the first invariant P is shown in Figure 5. The complete study of all three invariants, showing the full potential of the technique, has yet to be carried out and is beyond the scope of this paper. Acknowledgements The authors would like to acknowledge the use of data from computations of wall bounded flows, especially the data from torroja.dmt.upm.es (described in [9]). This work is funded by ARC Grant DP References [1] A.E. Perry and M.S. Chong, A description of eddying motions and flow patterns using criticalpoint concepts, Ann. Rev. Fluid Mech. 19 (1987), pp [2] M.S. Chong, A.E. Perry, and B.J. Cantwell, A general classification of three-dimensiponal flow fields, Phys. Fluids A 4 (1990), pp [3] P.A. Davidson Turbulence, Oxford University Press, Oxford, UK, [4] G.E. Elsinga and I. Marusic, Evolution and lifetimes of flow topology in a turbulent boundary layer, Phys. Fluids 22 (2010), p [5] A. Ooi, M.S. Chong, and J. Soria, A study of the invariants associated with the velocity gradient tensor in homogeneous flows, J. Fluid Mech. 381 (1999), pp [6] M.S. Chong, J. Soria, A.E. Perry, J. Chacin, B.J. Cantwell, and Y. Na, Turbulence structures of wall-bounded shear flows found using DNS data, J. Fluid Mech. 357 (1998), pp [7] J.C. del Alamo, J. Jimenez, P. Zandonade, and R.D. Moser, Self-similar vortex clusters in the turbulent logarithmic region, J. Fluid Mech. 561 (2006), pp [8] X. Wu and P. Moin, Direct numerical simulation of turbulence in a nominally zero-pressuregradient flat-plate boundary layer, J. Fluid Mech. 630 (2009), pp [9] J.C. del Alamo, J. Jimenez, P. Zandonade, and R.D. Moser, Scaling of energy spectra in turbulent channels, J. Fluid Mech. 500 (2004), pp [10] J.C.R. Hunt, C.J. Abell, J.A. Peterka, and H. Woo, Kinematic studies of the flows around free or surface-mounted obstacles; applying topology to flow visualization, J. Fluid Mech. 86 (1987), pp [11] M.J. Lighthill, Attachment and separation in three-dimensional flow, inlaminar Boundary Layers, L. Rosenhead, ed., Oxford University Press, Oxford, UK, 1963, pp [12] G.C. Flegg, From Geometry to Topology, English Universities Press, London, [13] C. Chin, A.S.H. Ooi, I. Marusic, and H. Blackburn, The influence of pipe length on turbulence statistics computed from direct numerical simulation data, Phys. Fluids 22 (2010), pp [14] P. Lenears, Q. Li, G. Brethouwer, P. Schlatter, and R. Omis, Negative streamwise velocities and other rare events near the wall in turbulent flows, Proceedings of the Thirteenth European Turbulence Conference, Warszawa, Poland, [15] H. Eckelmann, The structure of the viscous sublayer and the adjacent wall region in a turbulent channel flow, J. Fluid Mech. 65 (1974), pp