Proceedings of CHT-08
ICHMT International Symposium on Advances in Computational Heat Transfer
May 11-16, 2008, Marrakech, Morocco
CHT-08-002
SOME SWIRLING-FLOW CHALLENGES FOR TURBULENT CFD
Tim Craft, Hector Iacovides, Brian Launder‡ and Athanasios Zacharos
School of Mechanical, Aerospace and Civil Engineering
The University of Manchester, PO Box 88, Manchester M60 1QD, UK
‡
Correspondence author. Fax +44 161 306 3723 Email: [email protected]
ABSTRACT The paper examines some of the continuing challenges, within a RANS framework,
of computing turbulent swirling flows such as are encountered in industry and the environment. The
principal focus is on modelling turbulent transport processes but serious problems also arise in
handling numerical issues, too. Recent researches of two of these types of flow by the authors and
their colleagues in the Turbulence Mechanics Group at Manchester are examined; namely, the
confined flow within a rotor-stator disc cavity and the trailing wing-tip vortex. The former flow,
while geometrically axisymmetric, has been found to create multiple rotating vortices necessitating a
three-dimensional time-dependent analysis and which leads to different heat transfer patterns than
are obtained from an axisymmetric, stationary analysis. The wing-tip vortex is extremely sensitive to
the choice of turbulence model and only a second-moment closure that complies with the constraints
of two-component turbulence has been found capable of handling both the flow over the wing and
the wake vortex. Moreover, because of the large distances downstream of the aircraft to which, for
practical cases, computations need to be carried, the numerical strategy is brought into question.
Finally, arising from these two test cases, outline remarks are made about a swirling flow that poses
one of the major computational challenges of the 21st Century.
INTRODUCTION
Thirty-five years have elapsed since Bradshaw [1973] re-awakened the world to the great sensitivity
of turbulent transport processes to streamline curvature, an extreme case of which is that of swirling
flows. Indeed, the much earlier work of Taylor [1923] on the instabilities arising in the flow in the
space between coaxial cylinders where one or both may be rotating had already brought out clearly
the susceptibility of the (laminar) flow to forming large scale vortices, enhancing transport rates,
when the inner cylinder was rotating and the outer cylinder fixed (i.e. where the angular momentum
decreased with increasing radius). That analogous effects should carry over to the turbulent flow
regime was only to be expected. Indeed, examination of the production terms in the transport
equations for the Reynolds stresses strikingly brings out the sensitivity of the stresses to the straining
produced by even weak amounts of streamline curvature, as may be inferred from the generation
terms in the stress transport equations. Typically, a 1% curvature strain produces for a boundary
layer a 10% or greater effect on the turbulent stresses [Bradshaw 1973].
Swirling flows or, more generally, flows with closed secondary circulations, arise in many practical
problems in aeronautical and mechanical engineering. The eddy viscosity approach usually adopted
in CFD computations of such flows takes the local turbulent stresses directly proportional to the
local strains, a practice which rules out any special sensitivity to curvature. It is therefore hardly
surprising that, in many cases, the accuracy of prediction is low. Moreover, numerical issues very
commonly arise at the same time, adding a further layer of complexity.
The simultaneous presence of both physical and numerical issues certainly arises in the present
contribution, which principally examines two very different flows relevant to aviation. First, the flow
in rotor-stator disc cavities is addressed, a geometric configuration which has until recently been
handled by steady flow analyses but where, following an experimental flow visualization study
[Czarny et al. 2002], a time-dependent CFD exploration has now been undertaken. The second flow
considered, the development of the shed wing-tip vortex, poses a different set of challenges; for most
of the popular models for the Reynolds stresses lead to a rapid dispersion of the vortex within the
first wingspan of downstream development. In practice, however, the vortex retains its identity for
several kilometres downstream of the wing and, as is well known, that can pose serious dangers for
following aircraft. For the CFD computationalist there are thus two problems: to model the mixing
of the shear flow correctly and secondly to devise a numerical scheme that permits such large
downstream distances to be mapped in circumstances where, as will be shown, the usual threedimensional parabolic approach [Patankar & Spalding, 1972] on which the CFD world has so widely
relied over the decades, does not provide a sufficiently accurate route for determining the evolution
of the flow.
On the basis of experience with these two moderately complex types of turbulent flow, we finally
consider the challenges posed by what must, in the field of swirling flows, be one of the major
computational challenges of the present century: the growth of hurricanes and the development of
strategies for promoting their decay.
THE ROTOR-STATOR DISC CAVITY
The flow here considered is that created within the enclosed space formed by two coaxial discs, of
radius R and distance s apart, and an enclosing cylindrical shroud. One of the discs and the shroud
are at rest while the other disc rotates at an angular velocity Ω. The configuration is an idealization
of that occurring in a gas turbine between a stage of a turbine or compressor rotor disc and the
adjacent stator, though the central shaft of the actual gas turbine is not present. There have been
numerous experimental and computational studies of this flow configuration in the last three decades
of the 20th Century which all adopted the view that the flow was essentially axisymmetric and steady.
However, Owen [2000] arising from the difficulties his group had had in correctly predicting the
flow in certain enclosed rotor-disc configurations queried this assumption. Thereafter, Czarny et
al.[2002] undertook flow visualization studies for the rotor-stator disc cavity and found that,
depending upon disc spacing and rotor speed, from two to five distinct large-scale vortices could be
present in the core region (i.e. outside the intense shear layers next to the two discs) rotating at about
half the speed of the disc, Fig. 1.
Figure 1: Dye-trace visualization of organized structures in rotating disc cavity, Czarny et al [2002]
Clearly, any attempt to mimic numerically that behaviour would require a fully 3-dimensional timedependent simulation. Large-eddy simulation (LES) is being increasingly used to simulate such
naturally induced time dependence (e.g. Severac et al. [2006]). However, our aim has been to keep
computational time to a minimum; thus, the modelling strategy adopted has been that of an unsteady
RANS computation rather than LES.
Our first attempt adopted a “low-Reynolds-number” k-ε model of turbulence (that is, a model that
could be used in a code extending computations through the sublayers up to the wall itself), Gant
[2004]. This strategy required a very fine mesh across the sublayers because the turbulence
properties and the source terms in the turbulence conservation equations - especially in that for the
dissipation rate, ε - undergo such steep variations across these very thin but very influential regions.
The reason it had seemed desirable to resolve this zone in detail was that close to the discs, a further
set of much finer-grained spiral vortices had been apparent in the experiments of Czarny et al, Fig 2a.
Indeed, this feature, known as Ekman or Bödewadt spirals (depending on whether they form
adjacent to the rotating or stationary discs) has been recorded over the years in visualization images
of numerous rotating disc experiments and direct numerical simulations [e.g. Crespo del Arco et al.
2005]. Figure 2b shows the numerical results from Gant’s computations for an aspect ratio h/D =
0.0975 at a spin Reynolds number ΩRs/ν = 1.16 x 105 on a 100 x 150 x 141 grid (circumferential x
radial x axial). The figure shows a slice in a plane parallel to the disc close to the stationary surface.
The spiral near-wall structures are, indeed, clearly evident; but further from the wall in the core
region of the disc there was no trace of the larger-scale organized structures we sought.
(a) Experiments near rotor disc.
(b) URANS computations near stator by Gant
[2004] with low-Re k-ε model.
Figure 2: Fine scale near-wall spiral structures in rotating disc flows.
This failure prompted a reappraisal of strategy. The Introduction explicitly highlighted the
inadequacy of the eddy viscosity hypothesis in mimicking the behaviour of turbulent flows where
streamline curvature was important. Perhaps, therefore, the modelling level should be raised.
However, Hanjalić and Kenjeres [2002], in one of a series of papers on this theme, have reported
that an unsteady RANS computation of Rayleigh-Bénard convection, using a simple eddy viscosity
model, captured the large–scale, unsteady thermal fingers characteristic of this thermally-driven flow
very successfully. Effectively, the chaotically unsteady, resolved motion was itself accounting for
the bulk of the momentum and heat transport associated with turbulent mixing and the turbulence
model thus had rather little effect. So, we concluded that upgrading the modelling of turbulence may
not represent the most important area for improvement.
The other possible culprit was the numerical solving strategy itself. Resolving the Ekman spirals
(and the corresponding Bödewadt layer near the stationary disc) might, it was felt, have required so
many of the 141 nodes distributed between the discs that the central core, occupying some 90% of
the gap width, was inadequately resolved. If, however, these fine-scale, near-wall spirals were not
causative in the appearance of the much larger scale structures shown in Fig 1, perhaps a coarser
scale of treatment would suffice, enabling a better resolution of the core flow. In fact, Craft et al.
[2002] have proposed an analytical wall-function (AWF) which has been successfully applied to a
wide range of non-equilibrium turbulent flows. In a nutshell, the scheme assumes a linear increase of
turbulent viscosity from zero at the edge of the viscous sublayer, the thickness of which (yv*) varies
according to the change in shear stress across the viscous region, Fig 3(a). This prescription was
sufficiently simple that the mean velocity and temperature variation could be integrated analytically
and the results packaged into a wall function which enabled the near-wall node to be placed outside
the viscosity affected sublayer. Since no especially fine grid was needed for the sublayer an
appreciably greater density of grid points could thus be deployed in the interior.
While the above scheme provided a powerful way of reducing the storage demands of the near-wall
layer, it required extension for use on the present problem since, within the immediate near-wall
sublayer of a spinning disc, the velocity vector undergoes strong skewing, a feature not taken into
account in Craft et al. [2002]. Accounting for sublayer skewing is, however, readily accomplished.
Figure 3(b) shows the immediate near-wall region close to a stationary surface over which the AWF
is to be applied. The standard AWF approach is used to obtain friction levels, etc. in the direction of
the resultant velocity at node P and the secondary velocity is obtained by solving a simplified
momentum equation in the direction tangential to the resultant velocity at P with the boundary
conditions that, at node P and the wall, the secondary velocity is zero. This approach has proved
highly effective. For example, when used with a linear k-ε model, Fig 4(a) clearly shows the streaks
(a) Assumed variation of turbulent viscosity.
(b) The approach adopted for skewed near-wall
motion.
Figure 3: Analytical Wall Functions
(a) Analytical Wall Functions
U/ΩR
(b) Log-Law Wall Functions.
Figure 4: Comparison of predicted structures close to the stationary wall
in the Bödewadt layer captured for a disc-cavity with s/R = 0.08. The property shown is the flow
towards the stationary disc. The bulk radial inflow near the stationary disc is slightly away from the
surface (negative values of axial velocity, U) but, because of the spiral vortices generated, there are
filaments where the flow is towards the wall and these appear as red bands in the coloured plots.
(The appearance is similar to that shown in Fig 2 for a low-Re model requiring an order of
magnitude more computing time.) However, the feature is missed if a standard log-law wall function
is adopted and no provision for sublayer skewing incorporated, Fig. 4(b).
For s/R = 0.127, the smaller of the ratios examined by Czarny et al., the experiment showed that two
large vortices, forming an S-shape, were present over a range of spin Reynolds numbers, rotating at
about half the speed of the disc. This two-vortex structure was indeed found in the computations,
Fig.5. Here the property shown in the computed contours is the normalized turbulence level, k½/ΩR
as these contours most closely matched the dye trace in appearance. However, the same S-shaped
structure near the stationary wall is also evident from the contours of axial velocity, Figure 6(b). Of
even greater interest, however, in this figure is the structure near the rotating disc where for r/R > 0.5
Ekman spirals are clearly apparent, Fig. 6(a). It is also noted that, for calculations with the usual log-
(a) Experimental dye trace, Czarny et al.
[2002]
(b) Computations of k½/ Ω R.
Figure 5: S-shaped vortex pair for s/R = 0.127
(a) x/s = 0.05
(b) x/s = 0.95
Figure 6: Differing structures near stationary and rotating discs s/R = 0.127. Contours
of axial velocity.
law wall function (not shown), neither the Ekman spiral nor the large-scale S-shaped vortices were
present.
Finally, the flow regime with s/R = 0.195 is examined, a configuration where Czarny et al. reported a
range of different vortex patterns as the disc rotation rate was changed. In this case, during the
progress of the calculation, a successive change of structure with time was found, Figure 7. After
seven revolutions a 3-lobed structure developed which was maintained until more than 20
revolutions but which reorganized to a two lobe structure by 40 revolutions. However, this two-cell
configuration still continued to evolve until 70 revolutions (the maximum that resources and time
permitted). A clearer view of this successive evolution is provided by the static pressure contours on
a plane close to the rotating disc in Fig. 8. The breakdown from three to two lobes is seen to come
about from two diminished lobes breaking away, joining together and then reattaching to the primary
vortex to form two equal partners. That position does not last, however, and by 70 revolutions one of
the two lobes appears to be close to disappearing. Thus, it appears quite likely that the final state
from these simulations will be axisymmetric.
______________________
Figure 7: Evolution of flow structures near the stationary disc for s/R=0.195, Re = 0.9 x 106.
Contours of axial velocity after (a) 20 revolutions; (b) 40 revolutions; (c) 70 revolutions.
Figure 8: Time evolution of static pressure for s/R = 0.195; Re = 0.9 x 106
From this part of the present contribution two important, positive conclusions have emerged: firstly,
that a relatively straightforward extension of the economical AWF wall-function strategy enables
flows where the velocity vector undergoes strong skewing across the viscosity-affected sublayer to
be computed; and, secondly, that this strategy allows many of the time-dependent organized-flow
features actually recorded in the rotor-stator disc cavity to be replicated.
The overall picture is not entirely clear, however. As Fig. 8 has shown, the computed results are by
no means in final agreement with the observed behaviour even though some interesting features
present in the experiments have been captured. But what if, instead of an eddy viscosity model, a
second-moment transport closure had been used instead, where the Reynolds stresses were obtained
from transport equations for each component? Wouldn’t that have achieved better results – perhaps
ones where the unsteadiness persisted permanently, as it did in the experiments? In fact, such
computations have been made [Zacharos 2008] and the result was that the flow developed rapidly to
a steady form with none of the interesting multiple vortex patterns reported above. The reason for
this unexpected behaviour is still being sought, may possibly be resolved by the time of the oral
presentation and, if it is not, leaves an interesting challenge for the future.
WINGTIP TRAILING VORTICES
The pressure difference between the upper (suction) and lower (pressure) surfaces of an aircraft wing
causes air in the boundary layer beneath the wing to “leak” around the tip thus creating a streamwise
vortex which is shed from the trailing edge of the wing, Fig. 9. As is well known, these vortices have
a remarkable persistence and can pose a serious threat to following light aircraft that may get caught
up in the intense swirling wake. The fact that the vortex does “persist” implies that the usual mixing
and dispersion process gets blocked, a feature that can only be reproduced with an eddy viscosity
model by introducing ad hoc spin-related modifications. A second implication of the persistence of
Figure 9: Streamlines in the vicinity of the
wingtip. TCL computations
Figure 10: Axial Velocity at Vortex Centre:
∆ Experiments, Chow et al.[1997];
____ TCL model, Craft et al. [1996a];
__ . __ Non-linear k-ε model, Craft et al.
[1996b];
.. __ .. Linear k-ε model.
the vortex is that one needs to be able to compute its behaviour very far downstream from the
aircraft that created it.
Experiments on the initial development of the vortex reported by Chow et al [1997] have formed the
subject of a detailed computational study by Craft et al. [2006]. The computations using a range of
turbulence models were carried out in two parts using the 3D elliptic solver, STREAM, Lien &
Leschziner[1994]. First, the flow approaching and over the wing was computed with a nonorthogonal mesh with the grid extending from 1.7 chord-lengths upstream to 0.7 chord-lengths
downstream from the trailing edge. This whole-domain computation employed 4.5 million cells in
26 blocks. Downstream from the wing, however, a second computation was made using a mesh
designed to track the shed vortex better. This second computation employed 4.8 million cells on a
purely Cartesian mesh clustered around the vortex using computed dependent variable values from
the first to provide the upstream boundary conditions. A compact summary of the results is conveyed
by Fig. 10 which plots the streamwise development of the axial velocity at the centre of the vortex
(i.e. where the static pressure is a minimum). As the vortex wraps up and intensifies the pressure
falls at the centre of the vortex and thus the axial velocity increases (to more than 1.7 times the
undisturbed velocity). A fall in axial velocity downstream from the trailing edge is indicative of
vortex decay (since the increase in pressure at the vortex centre associated with this decay retards the
axial velocity). In fact, the experimental data show only a very slow decay, a feature that indicates
hardly any mixing in the core of the vortex and one which contrasts strongly with the behaviour
predicted by the k-ε linear eddy viscosity model, EVM. This is a generic weakness with models that
assume the turbulent stresses to be directly proportional to the mean strain. In fact, the non-linear
EVM of Craft et al. [1996b] does only moderately better: the cubic stress-deformation relationship
for this scheme correctly mimics the diminished mixing in the vortex core but produces excessive
transport and decay beyond the core where the angular momentum decreases with radius. The TCL
model (a 2nd moment closure that is designed to satisfy kinematic constraints in the two-component
Figure 11. Comparison of Measured and Computed Swirl Velocities at 0.68 chords downstream.
limit where turbulent fluctuations lie in a plane), Craft et al.[1995a], does dramatically better,
however. As with the failure of linear EVMs, the success of the second-moment closure can also be
said to be generic since the much simpler “basic” model, Launder et al.[1975], not shown, also did
much better than the linear and non-linear EVM’s (though not as well as the TCL scheme). A more
complete impression of the performance of different modelling levels is provided by the contours of
velocity perpendicular to the vortex axis in Fig. 11. Again, the success of the TCL scheme in
mimicking the measured flow and the serious failure of the other models is clearly brought out.
With the actual vortex decay being so slow, one needs to be able to track its development far
downstream. No really far-wake data are available but the EU C-Wake Project has provided data up
to 10 wingspans (b) downstream. A photograph of the wind tunnel and test model appears in Fig 12:
the composite wing comprised a NACA 4412 main profile at zero incidence with a NACA 0012 flap
at an incidence of 20°. Recent computations of the vortex development in this flow are here reported
for the first time with the calculation starting 1.25b downstream of the model craft using measured
data to furnish or derive boundary conditions. Three computations were carried out using in all cases
the TCL turbulence model: the first presumed axial symmetry of the vortex and thus employed a 2D
fully elliptic solver; the second adopted a 3D elliptic computation of the actual wake while the third
employed a 3D parabolic treatment. Figure 13 compares the development of the axisymmetric
computation with experiment. A weakly expanding grid both in the streamwise and radial directions
was used in this case with internode spacing at the upstream plane in both radial and axial directions
being 5x10-4 b within the vortex core. The experiment shows little variation in peak swirl velocity
throughout the development, a feature that the computed results reproduce closely.
The 3D computations for this case are compared with experiments at the last cross-sectional plane at
Figure 12: Model and Instrumentation for C-Wake Experiments
10b downstream of the wake generator. Figure 14 shows the horizontal component of mean velocity
on this plane. One sees that the 3D elliptic results display reasonable agreement with the
experimental data whereas the 3D parabolic computation exhibits a more diffused and,
correspondingly, much reduced strength of vortex.
This strikingly different behaviour initially surprised us as essentially the same grid was employed in
both computations. Moreover, we had earlier computed the 3-dimensional wall jet with both an
elliptic and a parabolic scheme [Craft & Launder 2001] and found only minor differences between
the two sets of results (which could be attributed to neglected terms containing streamwise gradients
in the turbulence equations). We concluded, nevertheless, that the source of the different computed
behaviours in the swirling wake must lie with the inherent approximations in producing the parabolic
marching scheme.
Figure 13: C-Wake Swirl Velocity at Initial, Intermediate and Downstream Planes
(a) x/b = 1.25; (b) x/b = 5.0; (c) x/b = 10.0
(a)
(b)
(c)
Figure 14: Horizontal Component of Swirl Velocity at x/b = 10.
(a) Elliptic Computation; (b) Experiment; (c) Parabolic Computation.
One area of difference is that, with the classical 3D parabolic solver, Patankar and Spalding [1972],
while the pressure variations in the cross-sectional plane of the vortex are computed via the SIMPLE
scheme, a uniform streamwise pressure gradient is applied (in the present case, zero). Though this
may be adequate in the great majority of ducted flows, it may not be for the swirling wake. In the 3D
wall jet noted above, the velocities in the cross-sectional plane, while significant, were at least an
order of magnitude less than in the stream direction. In the present flow, however, the crosssectional-plane velocities are several times larger and the flow-induced pressure variations
correspondingly higher. Yet, it does not appear that differences in handling pressure between the
elliptic and parabolic solvers could produce the type of difference shown in Fig 14. Figure 13 shows,
for example, that there are only very minor changes in swirl velocity over the measured development
of the shear flow. Thus by implication the streamwise pressure gradients even with the elliptic solver
will have been unimportant and could thus not have been responsible for the differences between
Figs 14a and 14c
The more likely explanation would thus seem to lie with the handling of convection. With the
elliptic solver the non-dispersive UMIST scheme (essentially a bounded QUICK scheme), Lien &
Leschziner [1994] was applied in all directions. With the parabolic scheme, because velocity data are
not available at the downstream plane, a simple upwind scheme is employed. The susceptibility of
this discretization strategy to numerical diffusion when flow-to-grid skewness is substantial is well
known. The numerical error is proportional to sin2φ where φ is the angle between the flow vector
and the normal to the control-volume face. In most parabolic flows, the in-plane velocities are so
weak that the flow-to-grid skewness is small enough for such errors to be negligible. Here, as noted
above, far more significant in-plane velocities exist and these appear likely to have caused the
unphysical diffusion.
In summary, this swirling wake study has confirmed that the TCL closure, already successfully
applied to many flows near walls, also performs far better than simpler competitors in modelling this
swirling free flow at distances up to ten wingspans behind the aircraft. However, in attempting to use
the classical parabolic approach to reduce the cost of computing the shear flow far downstream, we
have found the existing scheme to suffer from diffusion errors linked with the persistence of high
swirl. This needs to be rectified before CFD can be economically applied to what is a serious
practical problem.
HURRICANE MODELLING – THE NEXT GRAND CHALLENGE?
Nowadays one hears from all sides that RANS approaches to computing practically interesting
turbulent flows are on their way out. To be sure, it is acknowledged, RANS schemes will be used for
another decade or two, but, as an area for serious research, funding for such approaches will come to
an end not too long hence. To close, therefore, it seemed appropriate to look forward to the day
when what must be the supreme challenge in turbulent swirling flows can be reliably tackled via
CFD using RANS modelling. The flow in question is the hurricane: from its genesis through to its
emergence as a full-blown tropical storm. Or perhaps, one may even learn from CFD the strategy
that needs to be adopted to prevent the hurricane in its early stages ever growing to maturity. This
aspiration is given added impact by the report that the duration and intensity of hurricanes has
increased markedly over the past 30 years, probably associated with rising sea-water temperatures
[Emanuel 2005].
(a)
(b)
Figure 15: Illustrative Computational Images of Hurricane (a) Potential Temperature (b)
Swirl Velocity in Hurricane, Emanuel [2008]
At present, most computations of hurricanes, at least those that the general public hears about,
involve the tracking of tropical storms using a Cartesian grid where, in the horizontal plane,
internode distances are measured in kilometres. Many will remember the tracking of Hurricane
Katrina as it approached landfall somewhere near New Orleans. While such meteorological
computations, however imperfect, are an essential element in alerting residents, it is not the way to
learn fundamentals about hurricane behaviour! Much of the physical character of a hurricane is
better explored assuming an axisymmetric flow and, with such a simplification, radial spacing
between nodes can be reduced to ten metres or so – even less during the early stages when one
wishes to explore the issue of why Hurricane A dies out before it becomes dangerous while
Hurricane B goes on to become a monster. Thus, what is proposed is that engineering RANS
software should be applied to the analysis of a hurricane in much the way that turbomachinery and
IC-engine flows have been examined for the past two decades. It should, of course, be emphasized
that atmospheric physicists have been working on developing such computer models for many years.
For example, Emanuel [2008] has provides interesting qualitative behaviour of a model hurricane,
Fig 15. However, the present authors take the view that by drawing on and extending the wide range
of modelling devised within engineering CFD schemes, such software could be beneficially applied
to this multi-faceted, supremely complex class of flows. For the research programme to have a
chance of success the contributors must, naturally, include a wide diversity of experience: upperatmosphere physicists, oceanographers, civil engineering hydrodynamicists as well as the
mechanical/aerospace engineering CFD community addressed in the earlier parts of this paper.
Of course, the boundary conditions pose many problems and numerous modular researches will be
needed along the way to establish best practices. For example, computations will need to extend
from the top of the stratosphere to a depth of 100 metres (or should it be more?) beneath the sea
surface. It is essential to include the sea in the computation not least because the bringing of cooler
water from the deep to the surface can be the means of calming the hurricane. Indeed, for the sea
water, the rotating air-sea interface is rather like the rotor disc considered earlier: the rotation draws
sea water up to the surface and pumps it radially outward. Indeed, the air-sea interface provides one
of the major areas where research is needed to parameterize the input. A crucial relation to establish
is that between near-ground wind speed and wave structure. A wall-function-type approach will
clearly be needed. In this connection it is noted that Suga et al. [2006] have recently extended the
AWF scheme to study rough walls. The air-sea interface is, of course, much more complex than a
rough rigid wall but at least that provides a starting point. Another vital physical issue to address is
that connecting the rate of evaporation of tropical sea water (the sustaining enthalpy source from the
sea) which is clearly dependent upon the sprays formed which in turn will depend on the wave
height and wind shear. It may be that the best way of getting a first form for many such sub-models
would be by way of experiment! Regarding the upper boundary, this would need to extend to a
height of some 20km where very fine ice crystals form which are strongly affected by electrical
charges. Finally, in computing the convective heat and mass transfer budgets for the water content in
the air in solid or liquid phases, a link must be provided between the size of the drops and their
history as their size is crucial to their flow dynamics. Indeed, an aggregation of tropical thunderstorms is seen as instrumental in initiating the formation of certain types of hurricane, Hart [2006].
That is surely an area where CFD fuel-spray specialists can provide invaluable input.
The above paragraphs have done no more than provide a brief, superficial impression of the
challenges and prospects for hurricane modelling via RANS CFD which, nevertheless, within the
next twenty years could and, indeed, needs to become a reality.
SUMMARIZING REMARK
The paper has focused on the complex character of turbulent swirling flows and explored CFD
modelling strategies within a RANS framework that have brought success in accounting for some of
the flow features while noting several aspects that are not yet fully explained or satisfactorily
handled. It is finally argued that it would seem fruitful to apply the types of CFD software employed
in engineering to the larger scale swirling flow processes encountered in the environment, most
notably to modelling the birth, development and termination of hurricanes.
ACKNOWLEDGEMENTS
This contribution is dedicated to Brian Spalding on his recently attained 85th anniversary. It is our
hope that the mixture of physical and numerical challenges found in the problems discussed above
will be to his taste and that, perhaps, he may be stimulated to devise simple and characteristically
imaginative remedies to the problems we encountered.
We are pleased to acknowledge the contributions of Drs S. E. Gant, A. V. Gerasimov and C. M.
Robinson to the research presented above. AZ acknowledges support during his PhD from the UK
Engineering & Physical Sciences Research Council through Grant No. GR/R23657/01 The wingtip
vortex research was sponsored by the European Union under grant MDAW-UMI-207-N007.
Authors’ names are listed in alphabetical order.
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