Khalifa University, Nuclear Engineering, P.O. Box 127788, Abu Dhabi, U.A.E.

Turbulence, Heat and Mass Transfer 7

2012 Begell House, Inc.

DNS of Buoyancy Driven Flow Inside a Horizontal

Coaxial Cylinder

Imama Zaidi 1, Yacine Addad

2, Dominique Laurence

1

1 The University of Manchester, School of Mechanical, Aerospace and Civil Eng., M60 1QD,

U.K. (Imama.Zaidi@postgrad.manchester.ac.uk, Dominique.Laurence@manchester.ac.uk ) 2

(Yacine.addad@kustar.ac.ae)

Abstract – A direct numerical simulation of the buoyancy driven turbulent flow inside a horizontal annular

cavity has been carried out at higher Rayleigh number, Ra = 1.18x109, and the cylinders ratio of 4.87. Kinetic

energy budgets have been calculated to verify the accuracy of the CCM+ unstructured finite volume code on

polyhedral cells in Direct Numerical Simulation (DNS) mode. Comparison of DNS results with wall resolved

URANS models shows that the later models are able to capture the general flow features but fail to predict the

large unsteadiness and high turbulence levels in the plume. However local heat transfer rates along the inner

and outer cylinder walls are on average of acceptable accuracy for engineering purposes.

1. Introduction

Natural convection in annular cavities bounded by co-axial, horizontal cylinders has many

practical applications ranging from nuclear reactors, underground cables, to thermal storage

systems. Complex flow patterns inside the annuli and most challenging features for

simulation of this flow is the coexistence of turbulent and stagnation regions along with the

large recirculation and a wide range of timescales, from boundary layer heat transfer to global

cavity thermal equilibrium. Majority of the earlier studies were dedicated to the lower

Rayleigh numbers (Ra), onset of the laminar and transitional flows. Earlier experimental and

numerical work carried out by Kuehn et al.[1] and Padilla et al. [2] has shown that at Ra=104

a thermal plume appears above the inner cylinder, thinning with increasing Ra. To examine

the turbulent flow regime, Bishop [3] performed an experimental study for Ra ranging from

8x106 to 2x10

9 for a ratio of the outer to inner cylinder diameters of 3.36. It has been claimed

that this range of Ra was well above the transitional flow range. Later, McLeod et al.[4]

extended the previous study with the annuli of diameter ratio 4.85. These two studies

provided a baseline of experimental data for the heat transfer within the annular cavity which

was later on used for comparing computational results. Miki et al.[5] performed an LES of

this flow for Ra ranging from 2.51x106 to 1.18x10

9 with different values of Smagorinsky

model constant (Cs), but with Cs=0, he was unable to obtain fully developed flow in the

cavity at Ra=109. Additionally, the results showed a dependence on the value of Cs. They

concluded that a lower value of Cs (compared to the common value Cs=0.1) had to be used

along with a fine grid to take into account the viscous effects introduced by the boundary

layer at the inner and outer cylinder surfaces. In this THMT conference series, Addad et al.[6]

successfully performed an LES with the STAR-CD code for this coaxial cavity at Ra =

1.18×109 and 2.38×10

10 using Cs = 0.04. A comparison with RANS models showed rather

large differences for temperature distribution, and it has been concluded that additional

detailed data was needed to draw definite conclusions.

Turbulence, Heat and Mass Transfer 7

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To the Author’s knowledge, no DNS has been performed for Ra109 since the common

structured polar meshes would be refined on the inner cylinder instead of the outer one where

fully turbulent flow occurs. Furthermore, the absence of a detailed reference data and recent

demonstrations of unstructured finite-volume codes can be used for direct numerical

simulations (Fig. 1 shows CCM+ and polyhedral cells results for channel flow DNS) has

motivated the present numerical study using the CCM+ code (CD-Adapco) to provide

detailed and additional reference data for the verification and validation (V&V) of the existing

RANS turbulence models.

2. Case Description:

The cavity bounded by inner and outer cylinder is filled with incompressible Newtonian fluid.

Different isothermal boundary conditions are set on the inner (hot) and the outer (cold)

cylinders. A buoyancy plume flow forms on top of the hotter inner cylinder, then impinges

and later cools down in a wall-jet boundary layer flow along the outer cold cylinder. Due to

the small temperature difference the Boussinesq assumption is applied, i.e. thermal density

variation is only accounted for in the buoyancy force while fluid properties are set constant in

all other terms. The diameters of inner and outer cylinder are Di and Do. The dimensionless

parameters considered in this case are 91.18 10Ra , / 4.87o iD D and Pr 0.688 .

3. Numerical Procedure:

In the present study, the unstructured finite volume code (CCM+, v.5.04) is used for the DNS.

As highlighted above, the polyhedral cells feature in this case is ideal, since the active flow

and fine-scale turbulence is limited to the narrow plume and cooling along the outer wall

boundary layers in the top half of the cavity. Hence a Cartesian polar mesh (as used for classic

DNS codes) would be very wasteful. A very gradually refined adaptive polyhedral cells mesh

has been used (Fig. 2 and 3). The Kolmogorov length scale, 3 1/4( / ) distribution was

estimated from a precursor RANS calculation, then local mesh refinement for the DNS was

chosen such that: 2.5 , 2.5 , 5x y z , using here local Cartesian mesh notations

for a better understanding. The mesh is polyhedral in the 2D plane and extruded in

homogeneous direction, resulting in a total of 6 million cells. Bounded central differencing

scheme (BCD) is employed for the spatial discretization [7] to enhance stability, whilst for

time discretization, the implicit three level second order scheme has been used. This

simulation was left to run for 120 seconds (487 Lref/Vref), starting statistical sampling from

70 sec when probes seemed to exhibit a self similar turbulent signal (no mean drift).

Prior to this cavity simulation, a DNS mode validation test case for the code and schemes

has been performed by simulating the well known turbulent channel flow at Re 395 [8].

This case also served to test and validate the user-coded Java scripts to compute the turbulent

energy budgets; see Fig. 1 showing the mean flow profiles and budget of turbulent kinetic

energy (TKE). DNS was run by selecting LES mode in CCM+ and setting Cs=0 (LES subgrid

scale model set to zero). Note that dissipation is recomputed here from its definition rather

than as a balance of all other terms, hence matching the reference DNS dissipation profile will

demonstrate that even Kolmogorov scales are fairly well resolved. Turbulent transport is only

in qualitative agreement, possibly due to insufficient time averaging as it is related to larger

scales. Nevertheless, sum of terms in the budget is very nearly zero.

I. Zaidi et al.

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Table 1: Dimensionless parameters definitions

0.5ref o iL D D ref refV g TL

1T

3 2

refGr g TL v Pr /v 3 2Pr refRa g TL v

ln( / )( / )i i o iNu R R R dT dn

ln( / )( / )o o o iNu R R R dT dn * ( ) / ( )i o iT T T T T

* ( ) / ( )i o iR R R R R

* / refV V V

2. . .* ( . . .) / ( )refT K E T K E V

4. Cavity Flow Results:

The Iso Q contours, coloured by temperature, in Fig. 4 show highly turbulent activity in the

plume and outer cylinder boundary layer whereas the inner cylinder boundary layer remains

laminar. Compared to Addad et al.’s earlier LES, even smaller structures are captured on this

finer mesh. In good agreement with previous experiments and numerical simulations, lower

region of the cavity is stagnant below the inner cylinder. The lower value of Q highlights the

large structures which are observed to penetrate further down along the outer cylinder wall,

while in the upper part the turbulent activity extends far outside the plume and boundary

layers. Fig. 6 represents an instantaneous snap-shot of temperature distribution. Animations

corresponding to Fig. 6 shown that the plume at 0 angle undergoes a drastic change in both;

the laminar plume length and angle of its’ inclination with the vertical line. The latter

oscillatory motion closely matches the flow structure postulated by McLeod et al.[4] and is

canceled out in the long time-averaged distribution (Fig. 5). Nevertheless, this low frequency

motion motivated the use of Unsteady RANS (URANS) approach in the present study

compared to the steady calculations reported in previous numerical work.

Fig. 7 and 8 illustrate the qualitative comparison made of the predicted flow patterns from

URANS and DNS. Both eddy-viscosity type models under-predict turbulence levels by more

than two times the new reference DNS data, hence the choice of different colour scale for the

DNS results. Furthermore, the hot plume is observed to be much thinner in the URANS

predictions. This might be linked back to the oscillatory motion being largely damped by

these two models tested herein.

Fig. 9, streamlines of velocity coloured by temperature show a pair of elliptical

recirculation bubbles in the upper region, entrained by only the top half of the plume, as it

becomes fully turbulent, whereas there is a weak horizontal entrainment of the flow in the

plume from stratified lower temperature region, connecting to the lower point of the inner

cylinder. These features consistent with the velocity magnitude map, Fig. 8 where it can be

also observed that the URANS predictions for these recirculation patterns are rather poor.

Fig. 10 shows terms of the TKE budget. Turbulence is mainly produced on either sides of

the plume. Buoyancy production is not symmetrical indicating that probably further averaging

is needed. It is positive at the centre then negative on the sides of the plume as turbulence

works against gravity to entrain upwards cooler fluid coming from the sides, as seen on

streamlines in Fig. 9. Total viscous dissipation is nicely symmetrical as this is made up of

small-scale eddies which come in very large numbers even in short statistical samples. More

dissipation occuring along the outer cylinder boundary layer is present but hard to see on this

colour scale. In order to confirm the validity of the DNS performed, Fig. 11 gives the ratio of

the Kolmogrov length scale to the current grid spacing, on the plume axis, i.e. at 00. This

figure shows that the maximum ratio of grid spacing to the Kolmogrov scale does not exceed

5 for the whole domain and for natural convection problems this ratio has been safely used[9].

Turbulence, Heat and Mass Transfer 7

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Note that near the walls there is a prism layer hence cells are very fine in the boundary layer.

Therefore, we can claim that our grid is fine enough to resolve all length scales and the

precursor RANS simulation, even if qualitative, has been very helpful for this purpose, in

opposition to classic DNS benchmarks for which mesh generation guidelines are

well-established . Fig. 12 & 13 illustrate instantaneous velocity history and the TKE spectral

density curve at mid height of the plume axis at 00. First peak in the curve at 0.35 frequency

corresponds to the plume large scale instability with a period of 3 (s) also very visible in Fig.

4 which is showing a quasi sinusoidal signal superimposed with smaller amplitude turbulence.

The spectrum follows the classical high-frequency slope of -5/3 which shows that the

turbulence is already well developed.

The Nusselt number (Nu) distribution is the principle objective of many engineering CFD

studies, but is not available from the experiments, hence the main motivation for the present

DNS. Fig. 14 provides Nu on the inner cylinder, a clear laminar boundary-layer growth is

observed along the inner cylinder wall followed by a sharp decrease around the 10 angle due

to the presence of a small separation bubble (the flow direction is from 180 at the bottom to

0-10 at the top where the plume is generated). Both low-Re RANS models should perform

well in this laminar flow region (if this was an isolated cylinder). The underestimation of Nu

predicted by the k sst is due to the global flow pattern and average temperature in the

cavity. Indeed temperature profiles shown later (Fig. 15) will confirm that the

k sst overestimates the bulk cavity temperature (hence the slower buoyancy-induced

laminar boundary layer). The switch from under- to over-estimation between 70 and 20 angle

may be due to early turbulent eddy viscosity activation.

On the outer cold cylinder the plume behaves like an impingement jet developing into a

wall jet, although here the DNS Nu profile is flatter, i.e. with a lower and wider peak. This is

due to the plume’s axial offset instability. The inertia of the wall jet allows it to overshoot the

stratification level down to 120 angle, as seen previously on Fig. 6.

Both 2 f and k sst model fail to predict the heat transfer in the plume

impingement region i.e. they produce a high and narrow peak around to 0-30 more similar

to a steady impinging jet on a flat plate for which they are known to produce accurate results.

The 2 f model’s over-prediction at the stagnation point is more severe and is most likely due

to the higher velocity of the impinging plume discussed below (rather than to the somewhat

downgraded code-friendly version used here, instead of Durbin’s original model – but space

does not allow a RANS modeller specialist’s discussion here). The 2 f and

k sst respectively only slightly under- and over-shoot the wall jet penetration into the

stratified stagnant flow region around 120 angle. On average and through integration and

error compensation, the predictions of the global heat transfer prediction from inner to outer

cylinder should be quite satisfactory for engineering purposes.

Fig. 15 now shows comparisons with the available experimental temperature profiles. The

DNS is in fairly good agreement at all angles, particularly along the plume axis and as

concerns the bulk temperature in the cavity. It does differ however significantly in the outer

cylinder wall jet region where the DNS shows a much thicker thermal boundary layer. Along

the plume axis both RANS models show a significantly slower temperature decrease,

consistent with a severe over-estimation of the plume axial velocity (Fig. 16). On profiles at

60 and 90 angles the k sst is seen to significantly overestimate the bulk temperature in

the cavity.

In Fig. 16, at 0 the k sst exhibits a lower peak velocity. Both models then fail to

reproduce the plume axial velocity decay. Severe overestimation of the velocity in the region

I. Zaidi et al.

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of impingement on the outer cylinder is consistent with the Nu over-predictions. At 90 (Fig.

16), both RANS models predict the boundary layers surprisingly well, by chance as

agreement is poorer for 30 and 60.

Fig. 17 represents the turbulent kinetic energies in the annular cavity. For URANS models

TKE is the sum of resolved kinetic energy and modelled kinetic energy, but the former is 10%

for the k sst and negligible in the case of 2 f model. Both very severely underestimate

the onset of instability and turbulence in the early stage of the plume and this is likely the

origin of overall discrepancies. Fig. 18 illustrates the difference between the modeled and

resolved kinetic energy for the two models. It is clear that k sst predicts more instability

in the flow as compared to 2 f model. At other angels resolved part of TKE is almost zero.

Rms values are shown in Fig. 19. The origin of the plume is nearly a one dimensional

turbulence with high values of vrms and rms, i.e. a series of puffs before evolving to more

isotropic turbulence.

The balance of kinetic energy equation for buoyancy driven flow is:

t i ik U k P G VD (4.1)

with:

= = ( )

= / 2 = =

i k k i i i

k i i k k i k i

P u u U G g v pu

u u u D k u u

(4.2)

where P is shear production, G is buoyancy production turbulent transport, pressure

diffusion, VD viscous diffusion and is dissipation. Fig. 20 shows the turbulence kinetic

energy budgets at 00 and 60

0. At the top of the inner cylinder, i.e. at 0

0, due to recirculation

production shows a double peak, then buoyancy dominates, followed by convection. Near the

inner wall pressure diffusion dominates, balancing viscous dissipation and convection. On the

outer cylinder, the more classical wall jet budget is observed, with production occurring

mainly in the large free-shear layer side of the jet while it drops in the very narrow near wall

shear layer (see Fig. 16). Buoyancy contribution is also significant. At the plume centerline i.e.

at 00

balance of budgets is zero but near the cylinder it is not negligible, indicating that a

longer run is needed to reach sufficient statistical sampling. Additionally, animations show

tank sloshing phenomena in the lower stratified half of the cavity at an even lower frequency

than the plume instability.

5. Conclusions:

Buoyancy driven turbulent flow inside a horizontal annular cavity at high Rayleigh number,

free from inlet condition uncertainties, is a very apt but challenging test case for CFD. Thanks

to a precursor RANS simulation providing Kolmogorov scales, an optimal polyhedral cells

mesh for DNS using the commercial CCM+ code was generated, and checked a posterior

using the turbulent kinetic energy budget imbalance. The boundary layer flow on the inner hot

cylinder remains laminar, but the plume above it starts off with an unconventional quasi 1D

turbulence or series of hot puffs motion, later developing into a more classical buoyant jet

type flow which impinges on the cold outer cylinder summit, followed by a pair of curved hot

wall jets overshooting the stratified lower half of the cavity. URANS models fail to predict

the large scale oscillating motion of the plume, resulting in severe overestimation of the axial

velocity as a result of underestimating the turbulence level in the plume. The k sst model

overestimates the bulk temperature in the cavity, while the 2 f model and DNS more closely

Turbulence, Heat and Mass Transfer 7

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match the available sparse temperature measurements. The k sst ,and even more so the

2 f model, overestimate the maximum heat flux where the hot plume impinges on the cold

container cylinder summit, but overall, global, and in most region local, heat transfer rates

along the inner and outer cylinder walls are of acceptable accuracy for engineering purposes.

Further averaging seems to be needed to converge DNS statistics, possibly in connection with

low frequency tank sloshing motion in the stratified lower half of the cavity. The thermal

boundary layer width on the outer cylinder is visibly wider in the experiment as compared to

the DNS, but the latter is closer than the RANS model predictions.

The first author benefited from a PhD scholarship from the Higher Education Commission

of Pakistan on CFD for safety studies of industrial buildings, which remains a challenging

topic.

6. References:

[1] T. Kuehn and R. Goldstien, “An experimental study of natural convection heat transfer

in concentric and eccentric horizontal cylindrical annuli,” Journal of Heat transfer, vol.

100, pp. 635–640, 1978.

[2] E. L. M. Padilla and A. Silveira-Neto, “Large-eddy simulation of transition to turbulence

in natural convection in a horizontal annular cavity,” International Journal of Heat and

Mass Transfer, vol. 51, no. 13–14, pp. 3656–3668, Jul. 2008.

[3] H. Bishop, “Heat transfer by natural convection of Helium between horizontal

isothermal concentric cylinders at cryogenic temperatures,” International Journal of

Heat and Mass Transfer, pp. 109–115, 1988.

[4] A. E. McLeod and E. H. Bishop, “Turbulent natural convection of gases in horizontal

cylindrical annuli at cryogenic temperatures,” International Journal of Heat and Mass

Transfer, vol. 32, no. 10, pp. 1967–1978, Oct. 1989.

[5] Y. Miki, K. Fukuda, and N. Taniguchi, “Large eddy simulation of turbulent natural

convection in concentric horizontal annuli,” International Journal of Heat and Fluid

Flow, vol. 14, no. 3, pp. 210–216, 1993.

[6] Y. Addad, D. Laurence, and M. Rabbit, “Turbulent natural convection between

horizontal coaxial cylinders,” presented at the Turbulence, Heat and Mass transfer,

Begell house, 2006, vol. 5, pp. 507–510.

[7] I. Afgan, Y. Kahil, S. Benhamadouche, and P. Sagaut, “Large eddy simulation of the

flow around single and two side-by-side cylinders at subcritical Reynolds numbers,”

Physics of Fluids, vol. 23, no. 7, p. 17, 2011.

[8] H. Kawamura, H. Abe, and Y. Matsuo, “DNS of turbulent heat transfer in channel flow

with respect to Reynolds and Prandtl number effects,” International Journal of Heat and

Fluid Flow, vol. 20, no. 3, pp. 196–207, Jun. 1999.

[9] R. P. J. Kunnen, B. J. Geurts, and H. J. H. Clercx, “Turbulence statistics and energy

budget in rotating Rayleigh–Bénard convection,” European Journal of Mechanics -

B/Fluids, vol. 28, no. 4, pp. 578–589, Jul. 2009.

I. Zaidi et al.

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Fig. 1: Channel flow DNS at Re 395 ; mean velocity and temperature (left); TKE budgets

(right), black symbols=ref. DNS, open symbols = polyhedral mesh STAR CCM+ DNS.

Fig. 2: Coaxial cylinder grid. Fig. 3: Near-wall refinement.

Fig. 4: Iso values of Q =1 coloured by mean T*

Fig. 5: DNS mean temp

Fig. 6: Instantaneous iso-temperatures illustrating transient behavior of the plume (left T= 84.1s,

center = 85.5s and right = 86.05s).

Turbulence, Heat and Mass Transfer 7

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Fig. 7: Iso values of mean TKE* (left =DNS, center = k sst model, right = 2 f model).

Fig. 8: Iso values of mean velocity (left =DNS, center = k sst model, right = 2 f model).

Fig. 9: Streamlines of mean velocity vectors coloured by temperature.

Fig. 10: Budgets of turbulent kinetic energy (a) shear production, (b) buoyancy production, (c)

dissipation and (d) turbulent transport.

Fig. 11: Ratio of / x . Fig. 12: Instantaneous velocity

at center of plume.

Fig. 13: Spectral density curve

at the center of plume.

(a)

(c) (d)

(b)

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Fig. 14: Nu number; left: inner hot cylinder, right: outer cold cylinder.

(a) (b)

(c) (d)

Fig. 15: Mean temperature at different angles (a) 0 , (b) 90 , (c) 30 , and (d) 60 ).

(a) (b)

Turbulence, Heat and Mass Transfer 7

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(c) (d)

Fig. 16: Mean vertical component of velocity((a) 0 , (b) 90 , (c) 30 and (d) 60 ).

(a) (b)

(c) (d)

Fig. 17: Turbulent kinetic energy((a) 0 , (b) 90 , (c) 30 and (d) 60 ).

k sst model 2 f model

Fig. 18: Comparison of resolved TKE with modeled TKE (red line resolved, black line modeled TKE).

I. Zaidi et al.

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(a) (b)

(c) (d)

Fig. 19: Rms values of u’, v’ w’ and t’ ((a) 0 , (b) 90 , (c) 30 and (d) 60 )

Fig. 20: Budgets of kinetic energy (left 0 and right 60 ).