A fluid mechanic’s analysis of the teacup singularity

The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.

In 1926 Einstein published a short paper explaining the meandering of rivers [1].He famously began the paper by discussing the secondary flow generated in a stirred tea cup -the flow now widely known to be responsible for the collection of tea leaves at the center of a stirred cup of tea.In 2014, Luo and Hou presented detailed numerical evidence of a finite-time singularity at the boundary of a rotating, incompressible, inviscid flow [2,3].
The key to generating this singularity is the teacup effect.The present work is not aimed at proving the existence of a singularity for this flow, nor is it aimed at generating more highly resolved numerical evidence for the singularity than already exists.Rather, I assume that the flow simulated by Luo and Hou genuinely develops a singularity in finite time.My goal is to understand, from a fluid-mechanics perspective, why.
The flow under investigation is depicted in Fig. 1.The system is initialized with a pure azimuthal flow (swirl) having a sinusoidal dependence on the axial coordinate z.A pressure field is instantaneously generated to provide the radially inward force necessary to keep fluid parcels moving along circular paths.This results in high pressure near the cylinder wall where the circulation is largest (z = ±L/4) and low pressure where this is no azimuthal flow (z = 0 and z = ±L/2).Necessarily, then, there is a vertical variation in the pressure and this drives a secondary meridional flow.This is the teacup effect -the portion of the fluid just from z = 0 to z = L/4 corresponds to a cup of tea.(In an actual cup of tea, the variation in swirl with z is due to a boundary layer at the bottom of the cup.)

Mathematical preliminaries
The fluid flow is governed by the incompressible Euler equations where u is the fluid velocity and p is the pressure.Without loss of generality the fluid has unit density.We work naturally in cylindrical coordinates (r, θ, z).The flow is axisymmetric (independent of θ), but has swirl (u θ = 0 in general).Hence the velocity has components u(r, z, t) = u r (r, z, t)ê r + u θ (r, z, t)ê θ + u z (r, z, t)ê z , where êr , êθ , and êz are standard basis vectors for cylindrical coordinates.The flow takes place inside an axially periodic cylinder of period L = 1/6 and radius 1.The boundary condition at the cylinder wall is The initial condition employed by Luo and Hou, and reproduced here, is a pure swirl This initial condition possesses symmetries that are preserved under evolution of (1).The most important is centro symmetry about z = 0 The full set of symmetry planes is z j = jL/4, j = 0, ±1, ±2; u r is even and u z is odd about all planes; u θ is odd about planes z o , z ±2 and is even about planes z ±1 .The pressure p is even about all four planes.
Extensive analysis of finely resolved numerical simulations indicates that starting from the above initial condition, the flow evolves to form a singularity on the critical ring, r = 1, z = 0, at time T 0.0035056 [2,3].In the present work, simulations are well resolved to t = 0.0032.
To be conservative, the flow is analyzed at the early time of t = 0.0031.I rely heavily on the studies of Luo and Hou (hereafter referred to as LH), to know that the flow at t = 0.0031 is indicative of the flow all the way to t = 0.003505, extremely close to the singularity time.To be clear, the simulations presented here are not aimed at numerically establishing a singularity (LH have already done this), but instead at understanding the mechanisms at work, and for this they are fully adequate.

Pressure preliminaries
Pressure is the only stress acting within an inviscid fluid and it is the only means to provide force to, and thereby accelerate, the flow.The role of pressure is seen by taking the divergence of (1a) Given a solenoidal field u, in general ∇ • (u • ∇u) = 0, meaning that nonlinearity generates dilatation or compression.The pressure stress accelerates fluid exactly so as to counterbalance this effect, and it does so simultaneously everywhere.The initial flow [(2)] is solenoidal.
From (3), the relationship between pressure and velocity to maintain this is the Poisson equation This is not the full story, however.The flow of interest is wall bounded and this puts a condition on the stress field within the fluid.The initial flow satisfies (1c).From the êr component of the momentum equation at the wall, this will be maintained as long as Thus pressure is determined by the pressure Poisson equation together with its boundary condition where these expressions define the source term S and the boundary term b.As long as p satisfies (4), the flow evolving under (1a) will remain incompressible and confined within the cylinder.A primary focus of this work is distinguishing stresses associated with the incompressibility constraint from those associated with fluid confinement.

Overview of the singularity
Figure 2A shows the pressure field and meridional flow near the cylinder wall.Only one quarter of the axial period is shown; the behavior over the full period follows from symmetry.
We see the teacup effect: high pressure is generated from the rotating fluid near z = L/4 and this generates a vertical pressure gradient driving fluid near the cylinder wall downward.A secondary local pressure maximum forms on the critical ring to provide the stress necessary to bend (accelerate) the downward velocity to a radially inward velocity.In the vicinity of the critical ring the meridional flow is a saddle.Figure 2B shows an enlargement of Fig 2A near the critical ring and Fig. 2C shows the vorticity magnitude |ω| in this region.Here the vorticity is dominated by the radial component ω r , which is just the axial shear of the swirl velocity ω r = −∂ z u θ .See Fig. 1.As time evolves, the axial flow along the cylinder wall advects the swirl towards the critical ring, where a singularity develops in a nearly, but not exactly, self-similar way [2][3][4][5].
The pressure field shown in Fig. 2 is similar to that reported by LH at t = 0.003505, very close to the singularity time T 0.0035056.(See Ref. [3], but note that its Fig. 17 has a distorted aspect ratio.)LH note that the pressure maximum on the critical ring means that there is locally an adverse axial pressure gradient that decelerates flow on the cylinder wall.However, this does not mean that the pressure maximum inhibits the singularity.On the contrary, a pressure maximum like that in Fig. 2B will drive a singularity.This fact is central to this work.
Consider the velocity-gradient dynamics on the critical ring.Differentiating velocity gives the velocity-gradient tensor ∇u and differentiating the pressure gradient gives the pressure Hessian ∇(∇p).Symmetries dictate that on the critical ring the only non-zero derivatives entering these are where | c means evaluated on the critical ring.We will refer to Q and P as pressure curvatures.Straightforward differentiation of (1a) gives By incompressibility on the critical ring: V + W = 0. Thus V can be eliminated, giving the velocity-gradient dynamics These equations are exact, and while they are not closed ((5c) is insufficient to determine From Fig. 2B we see that both pressure curvatures, Q and P , are negative (a pressure maximum occurs on the critical ring), but that they are not equal.The radial curvature is larger in magnitude than axial curvature, that is |Q| > |P |.To understand the importance of this, suppose that for t ≥ t 0 , (More precisely, we need inf t≥t 0 a > 1.) Using ( 6) to eliminate Q from (5c) gives P = −2W 2 /(a 2 + 1), which can then be used to eliminate P from (5a).The velocity-gradient equations ( 5) then become where γ = (a 2 + 1)/(a 2 − 1) < ∞.
The flow at t 0 is assumed to be axially contracting: W (t 0 ) < 0. Without loss of generality we redefine the origin of time so that t 0 = 0. Sacrificing generality here for simplicity, we take a > 1 to be constant.The solution to Eqs. ( 7) is then just where T = −γ/W (0) > 0 is the singularity time and Ω 0 = Ω(0).These are the known divergences as t → T [2,3].In particular, Ω = ω ∞ diverges with exponent −γ.All other divergences associated with the singularity follow immediately from invariances of the Euler equations and the value of γ.We know from LH that γ 2.46, corresponding to a 1.54.
The corresponding ratio of length scales is indicated in Fig. 2B.The contours do not exactly manifest this ratio of scales, in part because contours are a finite distance from the critical ring and in part because the flow is a finite distance from the singularity.From the data at t = 0.031, Q/P 1.62.
The fundamental point is the following.Incompressibility locks radial expansion and axial contraction together such that it is not the signs of Q and P that are important for singularity formation; it is their mismatch.A persistent mismatch in pressure curvatures on the critical ring will drive the flow to a singularity.Of interest here is |Q| > |P |.The pressure contours in Fig. 2B are the signature of this simple mechanism.One can deduce from the results of LH that a mismatch of approximately the same amount is still in effect as close to the singularity time as they could resolve (Fig. 17 of Ref. [3]).The remainder of the paper addresses why this happens in the teacup flow.

Meridional and swirl pressures
We exploit the linearity of the Poisson equation ( 4) to separate pressure into contributions from distinct effects.To begin, the source is written: S = S 2D + S swirl , where S 2D depends only on the (2D) meridional flow (u r , u z ) and S swirl depends only on the swirl u θ .(See Materials and Methods.)Thus p = p 2D + p swirl , where These pressures are plotted in Fig.Let The saddle structure of p swirl is evident.
maximum implies that Q 2D P 2D < 0, while for the saddle swirl pressure Q swirl < 0 < P swirl .
In fact, Graphically this can be seen by adding the corresponding pressure slices (blue to blue and red to red), in Figs.3C and 3D.We will now address in more depth the two key features responsible for the pressure mismatch.) Such a saddle flow is to be anticipated [6][7][8] and the associated approximate symmetry of p 2D is not particularly surprising.
However, it is the pressure curvatures on the critical ring that matter for singularity formation.Figure 4 shows second derivatives of p 2D along slices at the midplane, z = 0, and at the cylinder wall, r = 1.The general agreement between the two curves is a manifestation of the near symmetry of p 2D .However, the curves behave differently approaching the critical ring.Necessarily ∂ 2 z p 2D is even about z = 0, since p 2D is.There is no such constraint on ∂ 2 r p 2D about r = 1.Hence, although the slices in Fig. 3C appear nearly identical approaching the critical ring, they are not.The significant observation is that Q 2D < P 2D < 0. While this ordering does not seem a priori obvious, it appears to be a natural consequence of the conditions at the wall and symmetry plane.

Swirl pressure
The swirl stress p swirl both maintains incompressibility of the flow and confines the fluid within the cylinder wall.Fully decoupling these two effects is not achievable, but we can partially separate them via the decomposition p swirl = p a + p b + p c , where The signs of the curvatures Q a < 0 < P a can be understood in two ways.As expected, the axially varying pressure is larger away from the critical ring (Fig. 5D) where the swirl is also larger.Hence P a > 0, and since ∇ 2 p a | c = 0, Q a < 0. We can also consider the radial dependence of p a .Since We have seen (Fig. 5C,E) that ∂ r p c | c > 0, hence ∂ r p a | c < 0. (Note that a negative gradient ∂ r p a | r=1 < 0 is required to contain a "negative density" fluid within the cylinder.)Now The pressure p a is the essence of the teacup effect near the critical ring -axial variation of the swirl at the cylinder wall necessitates the confining stress p a , whose derivative ∂ z p a then produces axial force toward the critical ring.At the critical ring, its opposite-signed curvatures, Q a < 0 < P a , arise naturally and are at the heart of the pressure mismatch driving the blowup.(Recall (10).)The meridional stress p 2D is a more passive player.In response to the incoming axial flow generated by the teacup effect, p 2D develops a local maximum with approximate rotational symmetry in the region around the critical ring.The symmetry is only approximate (Fig. 4) and the meridional pressure curvatures satisfy Q 2D < P 2D < 0.
This ordering is significant since it gives |Q 2D | > |P 2D |, meaning the asymmetry in p 2D does not act against the pressure mismatch generated by the swirl.It acts to enhance it.Momentarily we will exploit this by making the symmetric approximation knowing that this approximation is safe, in the sense that if the flow develops a singularity with this approximation, then it will certainly develop one in the actual asymmetric case.

One-dimensional models and closure
There is a rich literature on one-dimensional modeling of singularities in inviscid flow.
See [9] for a recent summary.For the teacup flow, LH propose the model [2, 3] with the identifications ω(z) ∼ ω θ | r=1 , θ(z) ∼ u 2 θ | r=1 , and u(z) ∼ u z | r=1 .(We abuse notation, by conflicting with usage elsewhere in the paper and by not strictly distinguishing between model quantities and their full-flow counterparts.)Eqs. ( 12) are closed by determining u from ω via the Hilbert transform This is natural from a vorticity-formulation viewpoint.The model captures very well features of the teacup flow and exhibits a finite-time singularity [9].
One can ask -what about the Hilbert transform of θ? From ( 4) and (11a) we have that (We have used linearity of H and H( b ) = H(const) = 0.The final equality is by definition.)Thus the model variable θ is equivalent to the axial gradient of p a .This is important because for any model to capture the correct singularity mechanism, it must capture p a .The LH model does.This also helps explain why the model can so successfully capture the singularity using only variables on the cylinder wall.
This suggests a different approach to closure -working in a primitive-variable formulation and obtaining pressure by Hilbert transform.This approach appears to be inferior to the LH model and will not be pursued here, except as it provides insight into the velocity-gradient blowup.We can express Eqs.(5) in terms of quantities only at the cylinder wall under two assumptions: that the meridional curvatures are equal, Q 2D = P 2D , and that the curvature P b is negligible.With the first assumption, (5c) gives 2P 2D = −2W 2 .Using this to eliminate W 2 in (5a) and neglecting P b gives Ẇ = −P a .Then the curvature P a can be obtain by Hilbert transform as . With these approximations, the velocity-gradient dynamics on the critical ring become These equations make explicit the vital role of the pressure p a and the global character of the blowup problem through dependence on the swirl along the boundary, θ = u 2 θ | r=1 .As the flow evolves, the contracting axial velocity transports swirl towards the critical ring, while also increasing the velocity gradient on the critical ring.This will produce blowup if H(∂ z θ)(0) ∼ W 2 .More quantitatively, blowup will occur with the known exponents ( (7) and following), if approaching the singularity time This establishes a relationship for singularity formation involving the diverging gradient W on the critical ring and the (global) gradient ∂ z θ along the cylinder wall.By writing the Hilbert transform in integral form and plotting the integrand using data from simulations, one can observe numerically that θ evolves along the cylinder wall so that the left-hand-side of ( 14) approaches a finite value as the system approaches the singularity.This, however, is not a new result; it is a direct consequence of the known nearly self-similar collapse at the singularity [2,3].The more important objective is to find a first-principles derivation that would explain why the left-hand-side of ( 14) approaches a finite value, and thereby explain the selection of the exponent γ.I have been unsuccessful in this.Hou and Lui [10] were able to make progress on the selection problem by replacing the Hilbert transform with a simpler closure [7].Perhaps that approach could be employed here, but this must await future work.

Discussion
The discovery of the teacup singularity by Luo and Hou [2,3] has significantly advanced our understanding of finite-time singularities in the Euler equations.However, the blowup takes place on the cylinder wall and not in the flow interior.The present analysis provides a direct connection between the singularity and flow confinement.The pressure stress at the heart of the teacup effect is present solely to confine the rotating fluid within the cylinder; it plays no role in maintaining incompressibility.This stress forces a mismatch of pressure curvatures on the critical ring -a mismatch that persists as the axial flow advects swirl toward the critical ring, driving the blowup of velocity gradients.Not only does the stress from the teacup effect arise from flow confinement, but this field also has a local minimum with second derivatives of opposite sign.Fundamentally, such a minimum can only occur at a boundary.For better or worse, no modification of the mechanism can move the singularity to the flow interior.
There is an important connection between this mechanism and recent popular models for singularity formation [2,3,7].These models involve two variables, vorticity and square swirl on the cylinder wall.The Hilbert (or similar) transform of the vorticity is used to obtain the velocity.The Hilbert transform of the square swirl is, uniquely, the axial gradient of the confining pressure at the core of the singularity mechanism.
There are many future directions suggested by this work.Pressure could possibly provide physical insight into the role of the boundary in the rapid growth of vorticity gradients shown by Kiselev and Šverák [6].One could simulate a cylindrical configuration with a no-penetration condition at z = 0 to impart more symmetry to the saddle in the vicinity of the critical ring (achieving something similar in spirit to [8]).Translating these result to the Boussinesq system should be straightforward [9].The selection mechanism for the exponent γ remains open as does the role of pressure in other configurations, such as anti-parallel vortices [11].Finally, it should be possible to develop precise theorems along the lines of Chae et al. [12] to address the specific pressure fields observed in the teacup flow.This could possibly lead to a new line of attack on proof of a singularity in the Euler equations.

Euler equations for axisymmetric flow with swirl
The Euler equations in component form are where û = (u r , u z ) and ∇ = (∂ r , ∂ z ).
Taking the divergence of the nonlinear terms gives the source term S on the right-handside of the pressure Poisson equation The first and third terms are independent of the swirl velocity u θ , while the middle term depends only on u θ .This leads us to define Thus the pressure Poisson equation, with boundary condition, is The Euler equations have been simulated in the vorticity-streamfunction formulation as given by Eqs.(2) in [2].The essential difference between the simulations here and those of LH [2,3] is that here a fixed computation grid is used.A Fourier pseudospectral representation is used in z with dealiasing given by Hou and Li [13].A Chebychev grid is used in r with no dealiasing.Fourth-order Runge-Kutta time stepping is used with an adaptive time step such that the CFL number is less than 0.2.Exploiting the separation in the Fourier representation, the Poisson problem for the streamfunction is solved directly.Solving similar Poisson problems, pressure fields are computed in a post-processing step.
For all results reported the computation grid has 769 radial points for r ∈ [0, 1] and 2048 axial points for z ∈ [0, L/4).At time t = 0.003 simulations produce a vorticity maximum ω ∞ = 90846.6,agreeing to at least 5 digits of precision with the value ω ∞ = 90847 reported by LH [3].The flow is resolved until t = 0.0032, which is sufficient for our purposes.Recall, (13), that P a is determined from the Hilbert transform of ∂ z u 2 θ (z), evaluated at zero.From this, as stated in (14), a condition for blowup is that (15) approach a finite limit as the flow evolves.Using the integral representation of the Hilbert transform, this can be written as The coordinate ξ is the unique rescaling of z such that the integrand h(ξ) has value 1 at ξ = 0.   Points (circles) are used to show the last resolved time in the simulations, t = 0.0032.

FIG. 1 :
FIG.1:The teacup flow in a cylinder, periodic in the axial direction.The primary azimuthal flow (swirl) generates an axial variation in the pressure.This produces a secondary meridional flow that in turn drives azimuthal flow along the cylinder wall towards the critical ring.The shear of this azimuthal flow generates intense vorticity on the critical ring, ultimately leading to a singularity and a breakdown of the Euler equations.Note that by symmetry a second critical ring (not indicated) exists at z = L/2, which by periodicity is also at z = −L/2.In the actual configuration studied, the height L is only one sixth of the radius.

3 BFIG. 2 :
FIG. 2: The teacup flow at t = 0.0031.(A) The pressure field (color) and meridional-flow streamlines (black) near the cylinder wall for 0 ≤ z ≤ L/4.The behavior over a full axial period follows from symmetry.The surfaces z = 0, z = L/4, and r = 1 are flow invariant.High pressure form near the outer wall in the vicinity of z = L/4 where the swirl is largest and this drives meridional downward flow.A secondary pressure maximum exists on the critical ring to divert the incoming flow.(B) Enlargement near the critical ring.The length ratio 1.54-to-1 associated with exponent γ is indicated (see text).(C) Magnitude of vorticity |ω| near the critical ring.A contour plot of just the radial component |ω r | is nearly identical.The color bar in A is used for all plots in the paper; the values of Low and High vary.For pressure only the difference is relevant: (A) High -Low = 275, (B) High -Low = 23.(C) High = 1.54 × 10 5 , Low = 1.2 × 10 4 .

Q
and P separately), they are extremely useful in examining what transpires in singularity formation.(5b) is commonly referred to as vortex stretching.For this flow, Ω = − ω r | c is the absolute vorticity maximum [2, 3], so Ω = ω ∞ .(5c) is the pressure Poisson equation on the critical ring.

3 .
Contours of p 2D are nearly circular arcs indicating approximate rotational symmetry about a pressure maximum on the critical ring.Contours of p swirl are those of a saddle with the expected high pressure along the cylinder wall where the swirl is largest.The pressure slices in Figs.3C and 3D further demonstrate the near symmetry and the saddle.The core cause for the mismatch in pressure curvatures, |Q| > |P |, is immediately evident.

FIG. 3 :
FIG. 3: Pressure components from (A) the meridional (2D) flow and (B) the swirl.The contours of p 2D are nearly circular arcs centered on the critical ring, while p swirl is a saddle with high pressure along the cylinder wall.Colors are given by the color bar in Fig. 2A where in (A) High-Low = 20.2 and in (B) High-Low = 7.9.Only differences in pressure are relevant.(C) Slices of p 2D at the midplane, z = 0, as a function of r (red), and at the cylinder wall, r = 1, as function of z (blue).The z coordinate is oriented to align the slices with the critical ring on the right.The vertical bar indicate a pressure difference of 5.The near symmetry of p 2D is evident.(D) Same as (C) for p swirl .
11c) where • denotes axial mean and tilde denotes axial fluctuations.(See Materials and Methods.)p a contributes to confining the flow, p b contributes to maintaining incompressibility, and p c contributes to both.These are plotted in Fig. 5.We also decompose the pressure curvatures, Q swirl = Q a + Q b + Q c and P swirl = P a + P b + P c , with the obvious meanings.The most significant finding is best seen in Fig. 5D.Near the critical ring, the axial variation of p swirl is almost exclusively dictated by the component p a .As we will see, p a is the only component that has curvatures with the signs Q a < 0 < P a needed to generate the pressure mismatch that drives the singularity.The component p b has very weak variation near the critical ring; the green curve in Fig. 5D is nearly flat.By definition p c does not vary with z and hence P c = Q c = 0. We begin the discussion with p c , since it is easy to interpret physically.p c (r) is the axially-independent pressure generated by the swirl flow u 2 θ (r)ê θ , whose speed at each r is the axial r.m.s. of u θ .Although p c contributes nothing to the pressure curvature (P c = 0), it is by far the dominant component of the swirl stress (Fig. 5E); −∇p c (r) is the radiallyinward force curving each circular streamline of the r.m.s.swirl flow, both maintaining incompressibility and confining the fluid.The important stress p a is more difficult to interpret physically.It does nothing to maintain incompressibility.There is no physical flow that has pressure field p a , since b takes on negative values and no value of u 2 θ is negative.In fact, on the critical ring: b c = b| c − b | c = − b | c < 0. (One could think of a fluctuating component of fluid in this region as having "negative density".)

FIG. 5 :
FIG. 5: Components of the swirl pressure: (A) p a , (B) p b , and (C) p c .Colors are given by the color bar in Fig. 2A where in (A) High-Low = 11, in (B) High-Low = 0.5, and in (C) High-Low = 14.Note that the range in (B) is much smaller than in (A) and (C).The curvature p a satisfies Q a < 0 < P a , even though p a has a local minimum on the critical ring.(D) Pressure slices at the cylinder wall, r = 1, with the z coordinate is oriented to agree with Fig. 3D.p a (orange), p b (green), and p swirl (blue points).p swirl is the same data as in Fig. 3D (blue curve), but here plotted as points at every fourth computational grid value so as to be visually distinguishable from p a .(E) Pressure as function of r at the midplane z = 0 over the full range of r.Here pressures values are aligned at r = 0. p a (orange), p b (green), and p c (purple).The inset shows enlargement near the critical ring.On the critical ring P a = ∂ 2 z p a c = 9.53 × 10 6 , P b = ∂ 2 z p b c = −0.84× 10 6 .

S
swirl and b are further decomposed into axial mean and fluctuating terms S swirl = S swirl + Sswirl , b = b + b,

FIG. 6 :
FIG. 6: (A) Approximate symmetry of the meridional flow near the critical ring.Contours of the Stokes streamfunction ψ(r, z) are shown in black.Also shown in dashed green are contours ofψ(1 − z, 1 − r).The two sets of contours are nearly identical.(B) Velocity profiles in along the cuts indicated by red and blue lines in A. The z coordinate is oriented to align the profiles.The red curve is u z (r, z = 3.9 × 10 −4 ) while the blue curve is u r (r = 1 − 3.9 × 10 −4 , z).Note that ∂ r u z | r=1 = 0 so that there is shear (vorticity) at the cylinder wall, r = 1.However, by symmetry ∂ z u r | z=0 = 0 and there is no shear (vorticity) at the midplane z = 0.

Figure 8 (
Figure 8(A) shows the time evolution of ∂ z u 2 θ over just a portion of the cylinder wall.

FIG. 7 :
FIG. 7: Time evolution of the axial velocity u z and the swirl velocity u θ along the cylinder wall.(A)u z and (B) u θ over the full cylinder length [−L/2, L/2], at times equally spaced from t = 0 to t = 0.003.Arrows at the top indicate the direction of the axial velocity, which also naturally orders the curves in time.(C) and (D) are the same as (A) and (B) except only over the region [−L/8, L/8] and for times equally space from t = 0.0025 to t = 0.0032.

Figure 8 ( 4
Figure 8(B)  shows the time evolution of the integrand h(ξ) for the same data as in Fig.8(A).The curves show convergence to a finite limit, thereby implying a finite-time blowup.Of course we already know from LH that the flow collapses to a singularity in a nearly, but not exactly, self-similar way[2][3][4][5].Hence this is not a new result, just a different way of looking at what is already known.The lack of exact self-similarity necessarily follows since the data is taken from simulations in an axially periodic cylinder and not an infinite cylinder.Therefore the integrands h(ξ) fundamentally cannot collapse because the (very weak) tails at large ξ cannot.This lack of exact self-similarity for this flow is well known[4,5].

FIG. 8 :
FIG. 8: (A) Time evolution of ∂ z u 2 θ along the wall.These profiles determine the pressure curvature of p a at the critical ring.(B) Plots of h versus ξ given by expressions Eqs.(16) for the same data as in (A).The bold black curve corresponds to the time t = 0.0031.Results in the main paper are all shown at this time.For reference, at t = 0.0031, ξ = 2 corresponds to z = 1.51 × 10 −3 .