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The topology of skin friction andsurface vorticity fields in wall-boundedflowsM. S. Chong a , J. P. Monty a , C. Chin a & I. Marusic aa Department of Mechanical Engineering, The University ofMelbourne, Victoria, 3010, Australia

Version of record first published: 14 May 2012.

To cite this article: M. S. Chong, J. P. Monty, C. Chin & I. Marusic (2012): The topology of skinfriction and surface vorticity fields in wall-bounded flows, Journal of Turbulence, 13, N6

To link to this article: http://dx.doi.org/10.1080/14685248.2012.663093

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Journal of TurbulenceVol. 13, No. 6, 2012, 1–10

The topology of skin friction and surface vorticity fieldsin 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 tensorhave been widely used to investigate turbulent flow structures. For incompressibleflows, the first invariant P is zero and the topology of turbulent flow structures canbe 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 beused to identify and study structures at the surface in any wall-bounded flow. Analternative scheme is presented here for the classification of critical points at a no-slipwall; the skin friction vector field at the wall is given by the wall normal gradients ofthe streamwise and spanwise velocity components; at a critical point, these gradientsare simultaneously zero. The flow close to critical points in the surface skin frictionfield can be described by a no-slip Taylor series expansion and the topology of thecritical 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-sliptensor invariants can be easily computed and these invariants provide a methodology forstudying the structure of turbulence at the surface of a no-slip wall in any wall-boundedflow.

Keywords: turbulence; wall-bounded flows; surface skin friction fields; surfacevorticity fields; local solutions of Navier–Stokes equations

1. Introduction

A critical point is a point in a flow field where the velocity components u1, u2 and u3

(in any arbitrary x1, x2 and x3 directions, respectively) are simultaneously zero and thestreamline slopes are indeterminate. Close to the critical point, the velocity field can bedescribed by the linear terms of a Taylor-series expansion. Two types of critical pointscan be described: one is the free-slip critical point, which is located away from a no-slipwall, and the other is the no-slip critical point, which occurs on a no-slip wall. Exam-ples of critical points are the centres of vortical structures in wakes behind bluff bod-ies and stagnation/reattachment points in separated flows (more examples can be foundin [1]).

∗Corresponding author. Email: min@unimelb.edu.au

ISSN: 1468-5248 online onlyC© 2012 Taylor & Francis

http://dx.doi.org/10.1080/14685248.2011.663093http://www.tandfonline.com

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1.1. Free-slip critical points

For free-slip critical points, the velocity ui in xi space is given by⎡⎣u1

u2

u3

⎤⎦ =

⎡⎣

.x1.

x2.

x3

⎤⎦ =

⎡⎣dx1/dt

dx2/dt

dx3/dt

⎤⎦ =

⎡⎣A11 A12 A13

A21 A22 A23

A31 A32 A33

⎤⎦

⎡⎣x1

x2

x3

⎤⎦

or

.xi = dxi

dt= Aijxj , (1)

where Aij = ∂ui/∂xj is the velocity gradient tensor.

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 Aij . Thecharacteristic equation of Aij is

λ3i + Pλ2

i + Qλi + R = 0, (2)

where λi is the eigenvalue and P , Q and R are the tensor invariants that are defined as

P = −(A11 + A22 + A33),

Q = A11A22 − A12A21 + A11A33 − A13A31

+ A22A33 − A23A32 and

R = A11A23A32 − A11A22A33 + A12A21A33

− A12A23A31 − A13A21A32 + A13A22A31. (3)

The characteristic Equation (2) can have (1) all real roots that are distinct, (2) all real rootswhere at least two roots are equal or (3) one real root and a conjugate pair of complexroots. In the P − Q − R space, the surface that divides the real solutions from the complexsolutions is given by

27R2 + (4P 3 − 18PQ)R + (4Q3 − P 2Q2) = 0. (4)

For an incompressible flow, the first invariant P is zero and hence all free-slip criticalpoints can be described by the second and third invariants, i.e. Q and R. Details of theclassification 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 flowstructures in order to extract information regarding the scales, kinematics and dynamicsof these structures (see [3], p. 268 and [4]). These invariants can also be used to studyturbulent structures using data from Direct Numerical Simulations (DNS), for example inthe DNS of homogeneous isotropic turbulence by Ooi et al. [5] and in wall-bounded shearflows by Chong et al. [6], del Alamo et al. [7] and Wu et al. [8].

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Journal of Turbulence 3

1.3. No-slip critical points

In wall-bounded flows, there is no slip at the wall, i.e. u1 = u2 = u3 = 0 when x3 = 0,where x3 is the wall-normal direction (hereafter, x1 and x2 are the streamwise and spanwisecomponents, respectively). If the origin of a Taylor series expansion is located at anyarbitrary point on the no-slip wall, i.e. at x3 = 0, the linear terms of the expansion are givenby

⎡⎢⎣

u1

u2

u3

⎤⎥⎦ =

⎡⎢⎣

0 0 A13

0 0 A23

0 0 0

⎤⎥⎦

⎡⎢⎣

x1

x2

x3

⎤⎥⎦ , (5)

where A13 = ∂u1/∂x3 and A23 = ∂u2/∂x3. Even for finite A13 and A23, all three invariants(P , Q and R) of the above tensor are zero at the wall and hence they cannot be used to mapthe 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 onthe wall (i.e. at x3 = 0) can be written to include higher order terms and at the same timesatisfy the no-slip boundary condition. This expansion is given by

u1 = A13x3 + (A11x1 + A12x2 + A13x3) x3,

u2 = A23x3 + (A21x1 + A22x2 + A23x3) x3,

u3 = 0 + (A31x1 + A32x2 + A33x3) x3. (6)

Ifoxi is defined as

oxi = dxi

dτ, (7)

where dτ = x3dt , the ‘no-slip velocity’ field can be expressed as

u1

x3= o

x1= dx1

dτ= A13 + (A11x1 + A12x2 + A13x3) ,

u2

x3= o

x2= dx2

dτ= A23 + (A21x1 + A22x2 + A23x3) ,

u3

x3= o

x3= dx3

dτ= 0 + (A31x1 + A32x2 + A33x3) . (8)

A critical point occurs on the wall when A13 = A23 = 0 and at this critical point

⎡⎢⎢⎣

ox1ox2ox3

⎤⎥⎥⎦ =

⎡⎢⎣

dx1/dτ

dx2/dτ

dx3/dτ

⎤⎥⎦ =

⎡⎢⎣A11 A12 A13

A21 A22 A23

A31 A32 A33

⎤⎥⎦

⎡⎢⎣

x1

x2

x3

⎤⎥⎦ (9)

or

oxi = dxi

dτ= Aij xj . (10)

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4 M.S. Chong et al.

Aij will be referred to as the ‘no-slip tensor’. The above vector field can be integrated withτ to generate surface streamlines (limiting streamlines or skin friction lines).

1.4. 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

Aij =

⎡⎢⎣A11 A12 A13

A21 A22 A23

0 0 −1

2(A11 + A22)

⎤⎥⎦ . (11)

By substitution of the above expansion into the Navier–Stokes equation, it can be shownthat A11, A12, A21 and A22 are related to the vorticity gradients and A13 and A23 arerelated to pressure gradients. Like the velocity gradient tensor, the no-slip tensor has threeinvariants, P , Q and R. For incompressible flow, the first invariant P is no longer zero andis given by

P = −1

2(A11 + A22) . (12)

The second and third invariants, Q and R, respectively, are given by

Q = −A12A21 −[

1

2(A2

11 + A222)

], (13)

R = (A11A22 − A12A21)

[1

2(A11 + A22)

]. (14)

It can also be shown that the relationship between the three invariants of the no-slip tensoris given by

2P3 + 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 frictionfields and the surface vorticity field at the wall in any wall-bounded flows, especially usingdata from DNS. Wall shear stress or skin friction is defined as the wall normal derivativeof 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 ofthe near-wall velocity vector when projected onto the plane is in the same direction asthe wall shear stress vector if the first plane is sufficiently close to the wall. Thus, skinfriction lines (also known as limiting streamlines at the no-slip wall) can be obtained byintegrating the streamwise velocity u1 and spanwise velocity u2 in the first plane from thewall. It is generally assumed that this first plane is very close to the wall. How close thisplane has to be to produce sufficiently accurate skin friction lines is an open question. Inmost computations of wall-bounded flows, the first plane is located at x+

3 << 1, where

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Journal of Turbulence 5

Figure 1. Skin friction lines at the wall using data from DNS of channel flow at Reτ = 934, see [9]for details of computations. This figure shows a small region (for x+

1 × x+2 plane ≈ 600 × 600) of

the entire channel wall in the computational domain (23, 474 × 8, 803 in viscous units). The size ofthe blue square is 100 × 100 wall units.

the wall normal distance is given in viscous wall units, i.e. x+3 = x3uτ/ν, where ν is the

kinematic viscosity and uτ is the friction velocity. Figure 1 shows skin friction lines in astreamwise–spanwise planer surface at the no-slip wall using data from the DNS of channelflow made available by del Alamo et al. [9]. The Reynolds number is Reτ = huτ/ν = 934,where h is the channel half-height. In the computation by del Alamo et al. [9], the firstplane from the wall is at x+

3 = 0.0313. The skin friction lines shown in Figure 1 are for arelatively small region of the flow, i.e. for x+

1 × x+2 plane ≈ 600 × 600; the size of the entire

streamwise–spanwise channel wall, in viscous units, is 23, 474 × 8, 803. Skin friction linesof many such small random regions indicate that Figure 1 is typical of a general turbulentskin friction field, consisting of bifurcation lines that are associated with surface streaklines 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 inthe skin friction field? This question is best addressed by considering the surface vorticityfield, 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 areorthogonal to the surface skin friction lines, see [6]. It can be shown that

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6 M.S. Chong et al.

Figure 2. Skin friction lines (black) shown in figure with surface vortex lines (red) . The surfacevortex lines are orthogonal to the skin friction lines. No critical points are detected in both the skinfriction 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 vorticityfield,

2. if the critical point in the skin friction field is a saddle, the critical point in surfacevorticity field is also a saddle and

3. if the critical point in the skin friction field is a node/focus, the critical point in thesurface vorticity field is also a node/focus.

Figure 2 shows the same skin friction lines (black lines) as in Figure 1 with the surfacevorticity lines (shown in red). In the DNS of wall-bounded flows, the flow is assumed tobe 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 suggeststhat the surface vorticity field (and hence the skin friction field) at the no-slip wall can bemapped onto the surface of a toroid. This conjecture is important since the Poincare–Hopfindex theorem or the Poincare–Bendixson1 theorem (see [10], [11] or [12]) states that thetopology of a smooth vector field on a toroidal surface is such that the number of nodes

1The Poincare–Hopf index theorem states that the topology of a smooth vector field on a surface isgiven by

∑(Nodes) − ∑

(Saddles) = 2 − 2g, where g is the genus of the surface.

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Journal of Turbulence 7

Figure 3. Skin friction lines (black) and surface vortex lines (red) at the wall, using same data asfor 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 Poincare index of –1 and a node/focus has a Poincare index of+1. Hence, if the skin friction field for the entire no-slip wall is similar to that shownin Figure 1, where no critical points can be seen, then the Poincare–Hopf index theoremis satisfied. Hence, it was initially felt that there are probably no critical points on thesurface of a no-slip wall, since a single critical point (a single saddle or a single node)will violate the Poincare–Hopf theorem for the topology of the flow field on a toroidalsurface.

However, contrary to expectations, there are critical points in the skin friction field atthe wall. Figure 3 shows a close-up (x+

1 × x+2 plane ≈ 150 × 150) 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 frictionand surface vorticity fields shown in Figure 3 is that these critical points consist of saddle–node pairs so that the Poincare index around the pair of critical points is such that it is zeroaround a closed circuit enclosing the two critical points. Figure 4 shows another region

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8 M.S. Chong et al.

Figure 4. Skin friction lines (black) and surface vortex lines (red) at the wall showing complex setof critical points – 3 saddles and 3 nodes/foci.

using data from the same computation as in Figure 1. Even for such complex surface flowpatterns, the number of nodes equals the number of saddles, satisfying the Poincare–Hopfindex theorem for a toroidal surface.

3. Conclusion

There is an increasing reliance on DNS of wall-bounded flows (pipes, channels and bound-ary layers) at low to moderate Reynold’s number to try understand the physics of turbulentstructures near the wall. It has been conjectured in this paper that the surface vorticity fieldcan be mapped onto toroidal surface such that vortex lines at the surface are closed. IfFigures 3 and 4 are typical of all regions with critical points at the wall, then the numberof nodes/foci equals the number of saddles, hence satisfying the Poincare–Hopf index the-orem, confirming the conjecture postulated in this paper. It should be noted that regionswith critical points are ‘rare’; roughly 120 such regions covering an estimated 0.6% of theentire wall have been identified in the data from [9], and the topology of all these regionssatisfy the Poincare–Hopf index theorem locally.

Since the finding of these critical points in the channel flow simulation of [9], similarcritical points have also been found on the surface using data from the DNS of a pipe flowby Chin et al. [13] and also from the recent spatial computation of a zero pressure gradientturbulent boundary layer by Wu et al. [8]. Recently, Lenears et al. [14] found negative

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Journal of Turbulence 9

streamwise velocities near the wall in turbulent flows using the DNS data, and stated thatthis 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 noflow reversals; however, this paper has shown, it is possible to have critical points on thesurface of wall-bounded flows, indicating that in some regions at the wall, complex three-dimensional structures exist. These three-dimensional ‘turbulent’ structures will cause flowreversals near the wall. The invariants of the no-slip tensor provide an analytical frameworkand methodology to study the ‘footprint’ of turbulent structures near the wall. However, itis yet to be established conclusively that critical points exist in all DNS of wall-boundedflows and whether they are necessary for the generation of ‘turbulence’ at the wall. Inan incompressible flow, vorticity is only generated at the boundary (see [11]) and it isconjectured that these critical points must be associated with the generation and transportof vorticity and, therefore, turbulence ‘at the wall’. What is even more important to establishis how to ‘define’ a smooth wall in experimental investigations and whether critical pointssimilar to that in wall-bounded simulations exist. This is significant, since having criticalpoints 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.

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10 M.S. Chong et al.

and many anemometry techniques, such as hot-wire anemometry, do not generally allowfor measurement of negative velocities.

An example of how one of the invariants of the no-slip tensor can be used to studystructures at a no-slip wall is shown in Figure 5, where the first invariant P is plottedwith the skin friction and surface vorticity lines. At points where P = 0, the flow is two-dimensional. 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 expansionwith the origin at the no-slip wall can be used to study flow structures at the wall. Thetopology of the skin friction and surface vorticity fields shows that there may be three-dimensional structures at the wall, and an example showing the use of the first invariant Pis shown in Figure 5. The complete study of all three invariants, showing the full potentialof the technique, has yet to be carried out and is beyond the scope of this paper.

AcknowledgementsThe 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 GrantDP1093585.

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