APPH 4200 Physics of Fluids

Rotating Fluid FlowOctober 6, 2009

1.!! Hydrostatics of a Rotating Water Bucket (again)2.! Bath Tub Vortex3.! Ch. 5: Problem Solving

1

Key Definitions & Concepts• Ω ≡ ∇ × U

• Cylindrical coordinates (Appendix B)

• Two simple cases of circulation: rigid rotation and “line vortex” (irrotational)

• When Ω = 0 (irrotational) then flow is “additive” linear and Bernoulli’s Principle applies everywhere

• Circulation is Γ ≡ "dA⋅Ω, and Γ is conserved in (“moves” along with) an inviscid fluid. (Kelvin’s Theorem)

2

What is the Surface of Rotating Bucket?

3

Solid Body Rotation (Special Case of Rotational Flow)

4

“Bernoulli-like” Formula for Solid-Body Rotation (Condition for force balance in rotating frame)

5

Bath Tub Vortex

The free surface of a time-independent bathtub vortex in a rotating cylindrical container with a drain-hole at the bottom. (a) 6 rpm, (b) 12 rpm, and (c) 18 rpm. The water in the central region is spinning fast and the depth of the surface dip increases when the rotational velocity of the container is increased.

6

Anatomy of a BathtubVortex

A. Andersen,1,2,* T. Bohr,1 B. Stenum,2 J. Juul Rasmussen,2 and B. Lautrup3

1The Technical University of Denmark, Department of Physics, DK-2800 Kgs. Lyngby, Denmark2Risø National Laboratory, Optics and Fluid Dynamics Department, DK-4000 Roskilde, Denmark

3The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark(Received 11 March 2003; published 5 September 2003)

We present experiments and theory for the ‘‘bathtub vortex,’’ which forms when a fluid drains out ofa rotating cylindrical container through a small drain hole. The fast down-flow is found to be confinedto a narrow and rapidly rotating ‘‘drainpipe’’ from the free surface down to the drain hole. Surroundingthis drainpipe is a region with slow upward flow generated by the Ekman layer at the bottom of thecontainer. This flow structure leads us to a theoretical model similar to one obtained earlier byLundgren [J. Fluid Mech. 155, 381 (1985)], but here including surface tension and Ekman upwelling,comparing favorably with our measurements. At the tip of the needlelike surface depression, weobserve a bubble-forming instability at high rotation rates.

DOI: 10.1103/PhysRevLett.91.104502 PACS numbers: 47.32.–y, 47.45.Gx

The generation of strongly localized vorticity is a fas-cinating and complicating ingredient of a broad variety offluid flows ranging from vortex shedding at solid surfaces(such as paddles, sand ripples, or insect wings) over flowsthrough turbines to large-scale tornadoes [1]. In general,these flows are poorly understood, since the interplaybetween fast axial motion and intense, localized vorticityleads to difficult mathematical problems outside the com-fortable realm of classical subjects such as potential flowor standard boundary layer theory.

One of the most well-known examples of such flows isthe so-called ‘‘bathtub vortex,’’ which forms when waterdrains out of a container. The strong, localized deforma-tion of the free surface makes the vortex beautifullyvisible, and has made the bathtub vortex the prototype‘‘vortex.’’ This popularity is in stark contrast to theattention which the phenomenon receives in the literature.The few classic papers about it either neglect the axialflow [2] or consider the problem without a free surface [3].Similarly, textbooks very seldom mention the bathtubvortex, and if they do [4,5] the flow is modeled withinpotential theory (with the inclusion of an ad hoc viscouscore [4]) without incorporating the axial motion. In thevortex core the axial velocity can be high, an essentialingredient of the strong ‘‘swirl’’ [6] which makes the flowso fascinating. Our aim in this Letter is to provide basicunderstanding of the stationary bathtub vortex: the flowstructure, the shape of the free surface, and the interde-pendence of important characteristics such as the size ofthe central surface depression, the rate of the out-flow,and the rotation rate in the vortex core.

We study a stationary bathtub vortex in a rotatingcontainer with a small drain hole at the bottom. Thecontainer is filled to a constant level with water, and theout-flow is driven by gravity. The free surface shapedepends strongly on the rate of the out-flow and less onthe rotation rate of the container. In the experiment, wefix the size of the drain hole (and thus approximately the

flow rate) and vary the rotation rate (angular velocity) !.Pictures of the observed free surface profiles are shown inFig. 1. The surface depression increases with !, and at18 rpm the surface dip has a narrow, needlelike shape andis rotating very rapidly (around 104 rpm). At higher val-ues of the rotation rate, the tip is no longer stable: Airbubbles detach from it and are dragged down by thesurrounding flow as shown in Fig. 2. When ! is increasedfurther, the frequency of bubble shedding increases untilthe air-filled core extends all the way down through thedrain hole.

FIG. 1. The bathtub vortex in a cylindrical container with adrain hole at the center of the bottom (not shown). The fluid inthe central region is rotating rapidly, and the depth of thefree surface dip depends on the rotation rate of the container.Picture (a) was taken at 6 rpm, (b) at 12 rpm, and (c) at 18 rpm.The drain hole radius was 0.1 cm and the unperturbed fluiddepth 10.9 cm. The rates of the out-flow were (a) 3:62 cm3 s!1,(b) 3:54 cm3 s!1, and (c) 3:16 cm3 s!1. The photographs weretaken through the square reservoir tank surrounding the cy-lindrical tank, and in each photograph the area is4:0 cm "horizontally# $ 7:2 cm "vertically# and the lower edgeof the frame is 4.2 cm above the drain hole. The structure abovethe surface is due to reflections. The water is recirculatedthrough a rotationally symmetric inlet at the bottom of theouter cylindrical boundary.

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The flow structure is surprisingly complex. Figure 3(a)shows an overlay of two flow visualizations made byadding fluorescent dye at the surface (red) and at thebottom inlet (green), respectively. The control parameterscorrespond to Fig. 1(b). The central down-flow region(red) below the free surface dip is strongly localized,and the downward velocities are of the order of 1 m s!1.This region appears similar to a weakly conical drainpipewith a radius comparable to that of the drain hole. Ifthis corresponds to the ‘‘esophagus’’ of the vortex, the‘‘mouth’’ would appear to be the surface region above thedrain, but in the stationary flow only a small fraction ofthe flow rate through the drain hole actually comes fromthis region. Nearly all of the flow rate is provided by thethin Ekman layer at the bottom of the container. In thisboundary layer, the fluid spirals inward. Close to the drainhole, a large part of it is diverted upward and the rest goesdirectly down the drain. We note that a similar up-flowwas observed by Lewellen [3] close to the drain hole in arotating flow without a free surface. Figure 3(b) summa-rizes the flow structure of the bathtub vortex: The flow1 cm away from the axis is upward (with velocities of theorder of 1 cm s!1), and most of the fluid passing outthrough the drain hole has thus taken a contorted pathin through the bottom Ekman layer, up through the next-to-central upwelling region, and finally down through therapidly rotating ‘‘porous’’ central drainpipe.

In the bulk of the container the flow is essentially twodimensional and well described as a central line vortexsuperimposed on the rigid rotation of the container.This means that the radial velocity u and the verticalvelocity w both vanish, whereas the azimuthal velocity

v " vB#r$ " ! r% "=r, where " is the line vortexstrength. Large deviations from this are observed onlynear the boundaries and in the central region. Because ofthe mismatch between the rigidly rotating bottom and the"=r term in vB, an Ekman boundary layer is formed, withthe thickness ! "

!!!!!!!!!!

"=!p

& 1 mm, where " is the kine-matic viscosity. By standard linear theory [7], one finds,

FIG. 2. The tip of the surface depression is unstable when therotation rate of the container is high. An air bubble has justdetached from the tip and is dragged down the drain by thesurrounding flow.

FIG. 3 (color). (a) An overlay of two flow visualizations madeby adding fluorescent dye at the surface (red) and at the bottominlet (green), respectively. The red dye is flowing in a thinsurface layer to the central vortex core where it is diverteddownwards to the drain hole. The free surface dip is seen abovethe central red region (the red pattern above the surface is dueto reflections). The green dye spirals inward in the thin Ekmanlayer at the bottom, then upwards around the central drainpipe,and finally down the drain. Only a thin vertical cross sectionthrough the axis of symmetry is illuminated and thereby thespiraling upward motion gives rise to the layered structure ofthe green dye. (b) Sketch of the flow structure (not to scale).

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Figure 4 shows a comparison of the measured surfaceprofile and azimuthal velocities at 12 rpm [see Figs. 1(b)and 3(a)] and the corresponding numerical solution ofLundgren’s equations with and without surface tension.The numerical solution with surface tension agrees wellwith the measurements, whereas the model without sur-face tension overestimates the depth of the central surfacedepression by 70%. Difficulties in obtaining reliable ex-perimental data for the velocity field and numerical in-stabilities encountered with Eqs. (3) and (4) at highrotation rates have kept us from making a similar com-parison at 18 rpm.

The bubble-forming instability (Fig. 2) is induced bysurface tension. The capillary instability of a fluid col-

umn [13] does not change qualitatively when it is rotating[14]. We therefore speculate that the critical value of !where the instability sets in occurs when the downwarddrag becomes large enough to overcome the upward buoy-ancy force and drag the bubbles formed by the capillaryinstability along with the flow. Using values for the dragon a spherical bubble at high Reynolds number [15], weestimate the critical value of the vertical velocity asWC ! ga2="9!#, where a is the radius of the bubble.With a bubble radius of a $ 0:5 mm, we find WC !25 cm s%1, which is in the measured range of w. We donot, however, have measurements of w immediately be-low the surface tip for ! around 22 rpm when the tipbecomes unstable, and a precise test of the criterion forthe tip instability remains to be carried out.

We thank Jens Eggers and Vachtang Putkaradze fordrawing our attention to Ref. [12] and Tom Lundgrenfor helpful suggestions. We are thankful to MortenErnebjerg for help with measurements of the free surfacedepression, and to Mark Schram Christensen, MortenNørgaard Nielsen, and Jacob Richter for inspirationthrough a series of preparatory experiments. We acknowl-edge support from the Danish Natural Science ResearchCouncil Grant No. 24-56880.

*Present address: Cornell University, Dept. of Theoreticaland Applied Mech., Ithaca, NY 14853, USA.

[1] H. J. Lugt, Vortex Flow in Nature and Technology(Krieger, Malabar, Florida, 1995).

[2] H. A. Einstein and H. Li, in Proceedings of the HeatTransfer and Fluid Mechanics Institute, StanfordUniversity, 1951, p. 33.

[3] W. S. Lewellen, J. Fluid Mech. 14, 420 (1962).[4] H. Lamb, Hydrodynamics (Cambridge University Press,

Cambridge, U.K., 1932).[5] R. P. Feynman, R. B. Leighton, and M. Sands, The

Feynman Lectures on Physics (Addison-Wesley,Reading, 1964).

[6] G. K. Batchelor, An Introduction to Fluid Dynamics(Cambridge University Press, Cambridge, U.K., 1967).

[7] D. J. Acheson, Elementary Fluid Dynamics (OxfordUniversity Press, Oxford, 1990).

[8] M. Mory and N. Yurchenko, Eur. J. Mech. B 12, 729(1993).

[9] A. Andersen, Ph.D. thesis, The Technical University ofDenmark, 2002).

[10] J. M. Owen, J. R. Pincombe, and R. H. Rogers, J. FluidMech. 155, 233 (1985).

[11] A. Andersen, B. Lautrup, and T. Bohr, J. Fluid Mech.487, 81 (2003).

[12] T. S. Lundgren, J. Fluid Mech. 155, 381 (1985).[13] P. G. Drazin and W. H. Reid, Hydrodynamic Stability

(Cambridge University Press, Cambridge, U.K., 1981).[14] L. M. Hocking, Mathematika 7, 1 (1960).[15] J. Magnaudet and I. Eames, Annu. Rev. Fluid Mech. 32,

659 (2000).

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FIG. 4 (color). Numerical solution of Lundgren’s equationswith surface tension (blue) and without surface tension (green)compared with experimental results (red) at 12 rpm. (a) Thefree surface and (b) the azimuthal velocity in the corotatingreference frame. We used g $ 981 cm s%2, " $ 1:0 g cm%3,! $ 0:0089 cm2 s%1, # $ 65 g s%2, and d $ 0:35 cm. To ob-tain the best possible agreement with the measurements, we setR $ 0:16 cm and Q $ 1:79 cm3 s%1 in the solution with sur-face tension, and R $ 0:17 cm and Q $ 1:76 cm3 s%1 in thesolution without surface tension. We measured # in the experi-ment, and the values of R, d, and Q are based on the best fit toour measured vertical velocities 1 cm above the bottom.

P H Y S I C A L R E V I E W L E T T E R S week ending5 SEPTEMBER 2003VOLUME 91, NUMBER 10

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7

Bath Tub Vortex (Example of irrotational flow)

8

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Chapter 5 Problems

• 5.2) Rankine vortex

• 5.4) “Vortex image” method

• 5.7) Mechanical energy of circulation

• 5.8) Vorticity dynamics

11

Problem 5.2162 Vorticity Dynamics

2. A tornado can be idealized as a Rankine vortex with a core of diameter 30 m.

The gauge pressure at a radius of 15 m is -2000 Njm2 (that is, the absolute pressure

is 2000 Njm2 below atmospheric). (a) Show that the circulation around any circuit

surrounding the core is 5485 m2 js. (Hint: Apply the Bernoull equation between

infinity and the edge of the core.) (b) Such a tornado is moving at a linear speed

of 25 mjs relative to the ground. Find the time required for the gauge pressure to

drop from -500 to -2000 Njm2. Neglect compressibilty effects and assume an ai

temperature of 25°C. (Note that the tornado causes a sudden decrease of the local

atmospheric pressure. The damage to structures is often caused by the resulting excess

pressure on the inside of the walls, which can cause a house to explode.)

3. The velocity field of a flow in cylindrical coordinates (R, cp, x) is

UR =0 Urp = aRx Ux =0

where a is a constant. (a) Show that the vorticity components are

úJR = -aR úJrp = 0 úJx = 2ax

(b) Verify that V 06) = O. (c) Sketch the streamlines and vortex lines in an Rx-plane.Show that the vortex lines are given by x R2 = constant.

4. Consider the flow in a 90° angle, confined by the walls () = 0 and () = 900.

Consider a vortex line passing through (x, y), and oriented parallel to the z-axis.

Show that the vortex path is given by

1 12 + 2 = constant.x y

(Hint: Convince yourself that we need three image vortices at points (-x, _y),

(-x, y) and (x, -y). What are their senses of rotation? The path lines are given

by dxjdt = U and dyjdt = v, where U and v are the velocity components at the

location of the vortex. Show that dyjdx = vju = _y3 jx3, an integratiop. of whichgives the result.)

5. Star with the equations of"1otion in the rotating coordinates, and prove

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14

Problem 5.4

162 Vorticity Dynamics

2. A tornado can be idealized as a Rankine vortex with a core of diameter 30 m.

The gauge pressure at a radius of 15 m is -2000 Njm2 (that is, the absolute pressure

is 2000 Njm2 below atmospheric). (a) Show that the circulation around any circuit

surrounding the core is 5485 m2 js. (Hint: Apply the Bernoull equation between

infinity and the edge of the core.) (b) Such a tornado is moving at a linear speed

of 25 mjs relative to the ground. Find the time required for the gauge pressure to

drop from -500 to -2000 Njm2. Neglect compressibilty effects and assume an ai

temperature of 25°C. (Note that the tornado causes a sudden decrease of the local

atmospheric pressure. The damage to structures is often caused by the resulting excess

pressure on the inside of the walls, which can cause a house to explode.)

3. The velocity field of a flow in cylindrical coordinates (R, cp, x) is

UR =0 Urp = aRx Ux =0

where a is a constant. (a) Show that the vorticity components are

úJR = -aR úJrp = 0 úJx = 2ax

(b) Verify that V 06) = O. (c) Sketch the streamlines and vortex lines in an Rx-plane.Show that the vortex lines are given by x R2 = constant.

4. Consider the flow in a 90° angle, confined by the walls () = 0 and () = 900.

Consider a vortex line passing through (x, y), and oriented parallel to the z-axis.

Show that the vortex path is given by

1 12 + 2 = constant.x y

(Hint: Convince yourself that we need three image vortices at points (-x, _y),

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by dxjdt = U and dyjdt = v, where U and v are the velocity components at the

location of the vortex. Show that dyjdx = vju = _y3 jx3, an integratiop. of whichgives the result.)

5. Star with the equations of"1otion in the rotating coordinates, and prove

Kelvin's circulation theorem

D-(ra) = 0Dt

where

ra= !(6)+2Q)odA

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17

Problem 5.7

;ci(y Dynamics

ameter 30 m.,lute pressureid any circuittion betweenlinear speed

e pressure toissume an air~ of the local

ulting excess.)

is

in Rx-plane.

nd () = 90°.) the z-axis.

s (-x, -y),es are given'nents at theop of which

, and prove

, Literature Cited 163

Assume that the flow is inviscid and barotropic and that the body forces are conser-vative. Explain the result physically.

6. Consider the interaction of two vortex rings of equal strength and similarsense of rotation. Argue that they go through each other, as described near the end ofSection 8.

7. A constant density irrotational flow in a rectangular torus has a circulationr and volumetrc flow rate Q. The inner radius is r¡, the outer radius is r2, and theheight is h. Compute the total kinetic energy of this flow in terms of only p, r, and Q.

8. Consider a cylindrical tank of radius R filled with a viscous fluid spinningsteadily about its axis with constant angular velocity Q. Assume that the flow is ina steady state. (a) Find fA 6) . dA where A is a horizontal plane surface through thefluid normal to the axis of rotation and bounded by the wall of the tank. (b) The tankthen stops spinning. Find again the value of fA 6) . dA.

9. In Figure 5.11, locate point G.

Literature Cited

Lighthil, M. J. (1986). An Informl Introduction to Theoretical Fluid Mechanics, Oxford, England: Claren-don Press.

Sommerfeld, A. (1964). Mechanics of Deformble Bodies, New York: Academic Press. (This book containsa goo discussion of the interaction of vortices.)

Supplemental ReadingBatchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.Pedlosky, J. (1987). Geophysical Fluid Dynamics, New York: Springer-Verlag. (Ths book discusses the

vorticity dynamcs in rotating coordinates, with application to geophysical systems.)Prandtl, L. and O. G. Tietjens (1934). Fundamentals of Hydro- and Aeromechanics, New York: Dover

Publications. (This book contans a goo discussion of the interaction of vortices.)

~,

"' ld-~. .) f~''',.

I.

18

Problem 5.7 ~~~

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~~~I

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19

Problem 5.8

;ci(y Dynamics

ameter 30 m.,lute pressureid any circuittion betweenlinear speed

e pressure toissume an air~ of the local

ulting excess.)

is

in Rx-plane.

nd () = 90°.) the z-axis.

s (-x, -y),es are given'nents at theop of which

, and prove

, Literature Cited 163

Assume that the flow is inviscid and barotropic and that the body forces are conser-vative. Explain the result physically.

6. Consider the interaction of two vortex rings of equal strength and similarsense of rotation. Argue that they go through each other, as described near the end ofSection 8.

7. A constant density irrotational flow in a rectangular torus has a circulationr and volumetrc flow rate Q. The inner radius is r¡, the outer radius is r2, and theheight is h. Compute the total kinetic energy of this flow in terms of only p, r, and Q.

8. Consider a cylindrical tank of radius R filled with a viscous fluid spinningsteadily about its axis with constant angular velocity Q. Assume that the flow is ina steady state. (a) Find fA 6) . dA where A is a horizontal plane surface through thefluid normal to the axis of rotation and bounded by the wall of the tank. (b) The tankthen stops spinning. Find again the value of fA 6) . dA.

9. In Figure 5.11, locate point G.

Literature Cited

Lighthil, M. J. (1986). An Informl Introduction to Theoretical Fluid Mechanics, Oxford, England: Claren-don Press.

Sommerfeld, A. (1964). Mechanics of Deformble Bodies, New York: Academic Press. (This book containsa goo discussion of the interaction of vortices.)

Supplemental ReadingBatchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.Pedlosky, J. (1987). Geophysical Fluid Dynamics, New York: Springer-Verlag. (Ths book discusses the

vorticity dynamcs in rotating coordinates, with application to geophysical systems.)Prandtl, L. and O. G. Tietjens (1934). Fundamentals of Hydro- and Aeromechanics, New York: Dover

Publications. (This book contans a goo discussion of the interaction of vortices.)

~,

"' ld-~. .) f~''',.

I.

20

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Problem 5.8

21

Summary• Two special examples of vorticity: solid-

body rotation (uniform) vorticity and the line vortex (zero vorticity away from origin)

• Sample problems in rotating fluid dynamics:

‣ Bathtub vortex

‣ Tornados, moving line vortices

‣ Mass flow

22