18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
1
The impact of rotation on the flow through inclined rectangular channels
Leo R.M. Maas
Royal Netherlands Institute for Sea Research (NIOZ)
PO Box 59, 1790 AB Texel, the Netherlands
maas@nioz.nl
and
Institute for Marine and Atmospheric research Utrecht (IMAU)
Utrecht University
Abtract :
Flows in nature can turn unstable and generate waves that, depending on circumstances, retard or
accelerate flows. The importance of rotation on this process is studied by pumping fluid through a
rectangular container on a rotating platform, and by measuring the cross channel pressure difference as
well as flow rate, given an applied pump rate. As the flow passes the (rigid-lid) container, instabilities
develop, leading to inertial waves. Depending on a lateral tilt of this container, these waves may or may
not be focused onto a wave attractor, which may impact the through-flow.
R?sum? :
When possible, write the abstract of the paper in French (150 words maximum)
Key-words : inertial waves; wave attractor; fluid experiments
1 Introduction
When a fluid is pumped through a container with a given pressure head, is rotation going
to aid the through-flow, or obstruct it? One naively expects that rotation leads to additional drag
on the fluid motion. Not only there is energy needed to support the secondary circulation set-up
by Ekman fluxes in horizontal boundary layers, but also does the sheared through-flow turn
unstable on the pressure side of the channel, the instabilities sucking up energy. With increasing
rotation rate, all of this should lead to an increased pressure drop in the down-channel direction,
which, in other words, is sensed as increased friction (M?rtensson et al 2002). Remarkably
though, there are cases when the pressure drop does not seem to grow with increasing rotation
rate (Dobner 1959). This suggests that the answer to our question is more subtle, and that, in
fact, rotation may perhaps facilitate an increasing flow rate. The reason for this might lie in the
fate of instabilities. In a three-dimensional, non-rotating fluid, instability will bring energy to
the smallest scale through a process of nonlinear interactions, where energy is degraded into
heat. Rotation, however, endows a fluid with elastic properties, supporting inertial waves that
quickly propagate throughout the fluid, up to the largest scale available, that of the container to
which the waves are confined. Thus, when instability (turbulence) manifests itself in the form
of inertial waves this may completely change the cascading of energy; it is no longer necessarily
through nonlinear interaction but might as well proceed through the organization brought about
by multiply-reflecting inertial waves. It thus becomes of interest to consider the fate of these
inertial waves. Do they spread ergodically through the container, thus eventually again loosing
their energy due to internal friction and upon reflection at boundaries? Or, do the waves
organize themselves, e.g. in forming eigenmodes - spatially standing waves? It appears, perhaps
somewhat surprisingly, that the shape of the boundary may play an important role here, as it
turns out to be important whether the container shape is breaking the symmetry imposed by
rotation, or not. When it does (when one or more container side is inclined relative to the
rotation axis or its perpendicular plane) wave energy appears to collect on a limit cycle, called
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
2
wave attractor (see review in Maas 2005). This is a well-defined orbit, that is determined by
geometry and the ratio of wave frequency, ?, to inertial frequency, 2?, where ? is the frame's
rotation rate (see Fig. 1). It is on approach of this wave attractor that the inertial waves mix fluid
that is stratified in angular momentum, which, as a result sets up a (cyclonic) mean flow (Maas
2001) that might actually amplify also the through flow.
Fig 1. Example of a wave attractor, the rectangular shape visible in the streamfunction
field, of focused inertial waves that occurs in a tilted rectangular channel. The tilted rotation
axis indicates we are here looking at in the tilted reference frame.
The aim of this contribution is to present a number of fluid experiments that were
performed to check this hypothesis. Preliminary results will be reported here. A more
comprehensive discussion awaits further analysis.
2 Experiments
2.1 Experimental set-up
Fig. 2 shows the (fully enclosed) rectangular container (10 x10 x20 cm3) as viewed from
the top. It is given in its horizontal position. It is filled with homogeneous water. Via connecting
tubes of 0.5 cm width, this is pumped through the tank by the green pump, in the direction
Fig. 2 : topview of container, pump (green) and
propellor vane, all on turn table
1
2
3
4 6
5
q x
y
?
x
?
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
3
indicated by the arrow, at a rate q [l/min]. Here, q=.4572 Vq +.0639, where Vq (varying from
0.25-7.0V) is the applied pump voltage. Pressure differences are measured between pressure
holes, indicated by numbers 1-6. Flow rate is measured by a propeller vane (H?ntzsch, Fa
40/10), the device indicated on the left.
The platform is set into rotation by a KMF WD251 electromotor (Electro ABI) at an
angular velocity ?? [rad/s]. Here, ?=1.004V?-0.2942, where V? (varying from 0.25-7.0V) is
the voltage applied to rotate the turntable.
Differential pressure, e.g between pressure holes 5 and 6, is measured with the LPM5480
sensor (Druck), which can measure up to ?200 Pa, with a precision of ?0.08 Pa.
Automatic control of applied pump and rotation rate over prescribed measurement and
adjustment periods, Tm and Ta respectively, as well as measurement (at 4 Hz rate) of differential
pressure and flow rate, is made by means of software package LabView. Typically a scan is
made over the full indicated pump and rotation voltage ranges. We usually chose Ta=50 s and
Tm=150 s, meaning that at fixed pump rate, the rotation rate is kept fixed over 200s, of which
the final 150 s are used to measure differential pressure and flow speed. Then, the rotation rate
is increased by an increment (0.2 or 0.25V) and this is repeated until maximum rotation rate is
obtained. Subsequently, the flow rate is increased (in similar increments), and the rotation rate is
incrementally decreased until the lowest rotation rate is reached again, where the flow rate is
increased again, and this whole procedure is repeated a number of times.
2.2 Results
When the container is horizontal, we measure a time-average pressure difference as given
in figure 3. The table is rotating anticlockwise. We thus expect that the Coriolis force deflects
the through flow to the right, so that (see figure 2) the pressure at hole 6 should be increased
relative to that at hole 5. As the pressure at hole 5 minus that at 6 is presented in figure 3, this is
indeed observed in the triangular area to the right of the straight solid line; that is, for relatively
strong rotation. In this region, we see that the pressure difference is nearly constant along
hyperbola (curved solid lines), betraying the geostrophic equilibrium which the lateral pressure
gradient apparently satisfies: dp/dy= 2 ? u?. Here ? is the density of the fluid. The straight line
itself is close to a line for which the Rossby number, Ro= u/2 ? L, is constant, where L is a
length scale to be discussed below. The part above the straight line consists of a positive 'ridge'
(red), and an anomalous negative pressure difference, perhaps a brief return to geostrophy. The
interruption of the latter region at u=4mm/s is not an artefact. It is found repeatedly in this
region and betrays the presence of multiple equilibria, which are reached depending on whether
the rotation rate is incrementally increased or decreased. Physically, it is caused by the presence
of an eddy. After spin-up to a new rotation rate, the container is usually filled with two vortices:
one, big cyclonic one, that fills almost the entire container, and a smaller anticyclonic vortex,
which sits near the entrance of the container, on the low-pressure side (in this case, below hole
5, on the left of the through-flow), which seems to cause the anomaly.
The aforementioned length scale L is usually taken as a scale associated with the geometry.
The frequency spectrum of the differential pressure signal, however, suggest this scale to be
much smaller though. The spectrum shows significant variance usually only below 2 ?,
indicative of inertial waves. For low pump voltages this is present in particular near ?=?,
probably due to a slight misallignment of the table's rotation axis with gravity. For higher pump
rates, this spreads out. However, below the straight line, the energetic peaks all stay below the
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
4
Fig. 3 Observed time-averaged differential pressure between holes 5 and 6 [Pa], measured
by varying rotation rate ? and pump rate. The latter is here expressed as the through flow
u=q/A, obtained from pump rate q divided by cross-sectional area A, that would result in the
absence of rotation, and which is linearly related to the applied pump voltage.
inertial frequency, 2 ?. Only above the straight line, the energy no longer seems to be confined
to the inertial wave frequency band and energy is also found in bands above 2 ?. This strongly
suggests that physically the straight line should be interpreted as Ro=1, which separates a
strongly-rotating 'hyperbolic' area (Ro<1), where the variance ('turbulence') is taking shape in
the form of a 'sea of inertial waves' (Tritton 1978), from a weakly-rotating 'elliptic' area (Ro>1),
where variance appears as a spectrally broad-band phenomenon, as turbulence in non-rotating
fluids. However, this interpretation suggests the length scale L present in Ro, to be set by the
diameter of the feeding tube or perhaps by the Ekman boundary layer thickness.
Concerning the property of most concern here, the through flow, let us look at observed
flows obtained from (1) the horizontal rotating tank compared to the non-rotating tank (Fig. 4a),
and (2) the tilted rotating tank compared to the horizontal (non-tilted) tank (Fig. 4b,c). In the
latter case the tank is tilted 10 and 20 degrees respectively.
Surprisingly, even in the non-tilted case (Fig. 4a), rotation seems to enhance the through-
flow, particularly in the rapidly rotating triangular (Ro<1) region of the parameter plane
spanned by the rotation rate (x-axis) and pump rate (y-axis), both expressed in terms of Volt.
?p? (Pa)
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
5
For Ro<1, a further enhancement is found over that already present when the tank is tilted
(Figs. 4b,c), with the exception of the region where the pump speed is small for which the
through flow is instead obstructed.
3 Conclusions
The results presented raise a number of surprising issues. The spectral distribution of
pressure perturbations tells us that the length scale that apparently sets the Rossby number,
separating the 3D turbulent regime from the inertial wave regime, is much smaller than the
length scale of the container. In the inertial wave regime, the average cross-channel pressure
difference is found to be in near-geostrophic balance (for Ro<<1). The perturbations in this area
take the form of inertial waves. For low pump rates, these are concentrated around certain
spectral peaks. For higher pump rates, the spectrum is broader (albeit always staying well below
the inertial frequency cut-off 2 ?). While the na?ve expectation, that rotation will inhibit the
through flow, is in general found to be true for Ro>1, this is surprisingly not true in the inertial
wave regime (Ro<1). Apart from the enhancement already found when the container is still in
its flat position (which is not understood well), there is a further enhancement observed when
the tank is tilted, especially when the pump rate is relatively large (and the spectrum rich). We
speculate that the latter enhancement is brought about by the organization of inertial waves. For
some of these frequencies they will be focused onto a simply-shaped wave attractor, where, as
was found in an earlier study (Maas 2001), this mixes background angular momentum leading
to the generation of a cyclonic mean flow. It is speculated that for the larger flow rates, the
broad-band spectrum encompasses those frequencies for which a wave attractor can be obtained
and thus a mean and through flow is driven, while for the low pump rates the few frequencies
observed are likely not inside an attractor window, so that they only lead to extra damping,
reducing the through-flow. Further analysis is needed to substantiate these speculations.
Fig. 4 (a) q(rot)-q (rest) : measured flow rate
[l/min] in rotating system, relative to flow rate
without rotation for horizontal (flat) tank. (b)
q(10,rot)-q(0,rot) : measured flow rate in tilted,
rotating frame (10 degrees tilt with horizontal)
compared to that in flat, rotating frame. (c) Same as
b, but for a 20 degrees tilt.
The colored spot near the origin of the graphs
(0.5,0.5) should be ignored.
18?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
6
4 Acknowledgements
I am grateful to Niels Smit and Maurits Kruijt for help with the experiments and to Stefan
Kopecz for providing Fig. 1.
References
Dobner, E. 1959 ? ?Uber den Str?omungswiderstand in einem rotierenden Kanal?, Dissertation,
Technische Hochschule Darmstadt.
Maas, L.R.M. 2001 Wave focusing and ensuing mean flow due to symmetry breaking in
rotating fluids J. Fluid Mech. 437, 13-28
Maas, L.R.M. 2005 Wave attractors: linear yet nonlinear. Int. J. Bifurc. & Chaos 15, 2757-2782
M?rtensson G. E., Gunnarsson J.; Johansson A. V.,Moberg H. 2002 Experimental investigation
of a rapidly rotating turbulent duct flow. Exp. fluids 33, 482-487
Tritton, D.J. 1978 Turbulence in rotating fluids. From: Rotating fluids, eds. P.H. Roberts and
A.M. Soward. Ac. Press., 105-138