Numerical Simulation of Forced Convection Over a Periodic Series 2011

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Numerical simulation of forced convection over a periodic series

of rectangular cavities at low Prandtl number

E. Stalio ⇑, D. Angeli, G.S. Barozzi

Dipartimento di Ingegneria Meccanica e Civile, Università degli Studi di Modena e Reggio Emilia, Via Vignolese 905/B, 41125 Modena, Italy

a r t i c l e i n f o

Article history:

Received 28 May 2010Received in revised form 13 April 2011

Accepted 17 May 2011

Available online 21 June 2011

Keywords:

Laminar forced convection

Periodic channel

Cavity

Liquid metal

Low Prandtl

a b s t r a c t

Convective heat transfer in laminar conditions is studied numerically for a Prandtl number Pr = 0.025,

representative of liquid lead–bismuth eutectic (LBE). The geometry investigated is a channel with a peri-

odic series of shallow cavities. Finite-volume simulations are carried out on structured orthogonal curvi-

linear grids, for ten values of the Reynolds number based on the hydraulic diameter between Rem = 24.9

and Rem = 2260. Flow separation and reattachment are observed also at very low Reynolds numbers and

wall friction is found to be remarkably unequal at the two walls. In almost all cases investigated, heat

transfer rates are smaller than the corresponding flat channel values. Low-Prandtl number heat transfer

rates, investigated by comparison with Pr = 0.71 results, are large only for uniform wall temperature and

very low Re. Influence of flow separation on local heat transfer rates is discussed, together with the effect

of different thermal boundary conditions. Dependency of heat transfer performance on the cavity geom-

etry is also considered.

Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction

Periodic, corrugated geometries are very common in heat

exchangers and separated flow is sought in the cooling passages

of heat transfer devices. Liquid lead–bismuth eutectic (LBE) is a

candidate coolant for sub-critical nuclear reactors, but reliable

physical models of convective heat transfer in liquid metals are

still missing. A detailed knowledge of velocity and thermal fields

in separated flow conditions are needed for model development

and a necessary starting point for this process is the understanding

of the laminar regime. Temperature and velocity fields in liquid

metals are almost impossible to obtain through experiments be-

cause of the opacity of these fluids and the care required in han-

dling them.

In the present work, convective heat transfer in laminar condi-

tions is investigated numerically for a Prandtl number Pr = 0.025,which is representative of liquid lead–bismuth; results are com-

pared to the Pr = 0.71 case. The geometry selected is a periodic

channel with forward and backward facing steps giving place to

a periodic series of shallow cavities, where separation and reat-

tachment are observed also for low Reynolds number, steady con-

ditions. The ratio between length and depth of the cavities, which

defines their aspect ratio, is AR = 10 but the AR = 5 case has also been

considered for assessing the dependency of results upon geometry.

Simulations are carried out for ten values of the Reynolds number

in the laminar regime. Two different thermal boundary conditionsimposed at the channel walls are considered, namely uniform tem-

perature and constant heat flux.

The flow over forward or backward steps in ducts or in free-

stream represents a classical benchmark for the study of turbulent

fluid flow and heat transfer, and the amount of literature on the

case is undoubtedly huge, see for example the paper by Avancha

and Pletcher (2002). However, very few works considered the

influence of separation and reattachment on convective heat trans-

fer in flows of low-Prandtl number fluids.

A similarity solution for laminar flows was carried out first by

Chapman (1956), who assumed heat transfer to be completely gov-

erned by the shear layer. The problem was tackled later by Aung

(1983) by means of interferometric techniques, and by Bhatti and

Aung (1984) who performed a set of finite-difference computa-tions. In particular, the latter concluded that the similarity analysis

by Chapman was somewhat inadequate to treat the problem com-

pletely and to derive general heat transfer correlations. They pro-

posed a correlation for the average Nusselt number over the

cavity, valid for laminar and transitional values of the Reynolds

number, and for a considerable range of aspect ratios.

A very complete experimental study of the laminar flow over

cavities in freestream was carried out by Sinha et al. (1982) by

means of smoke visualization and hot wire anemometry. They re-

ported a limiting value of 10 for the length-to-height aspect ratio of

the cavity, which separates ‘‘closed’’ flows (i.e. flows where reat-

tachment occurs inside the cavity) from ‘‘open flows’’ for which

0142-727X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.ijheatfluidflow.2011.05.009

⇑ Corresponding author. Tel.: +39 059 2056144; fax: +39 059 2056126.

E-mail addresses: [email protected] (E. Stalio), [email protected]

(D. Angeli), [email protected] (G.S. Barozzi).

International Journal of Heat and Fluid Flow 32 (2011) 1014–1023

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the circulation inside the cavity is hydrodynamically isolated, and

flow reattaches past the forward step (Sarohia, 1977). Building on

the work of Sinha et al. (1982), Zdanski et al. (2003) presented a

numerical study of laminar and turbulent flows over shallow cav-

ities, encompassing the sensitivity to various parameters, such as

freestream velocity, turbulent kinetic energy, aspect ratio and Rey-

nolds number. Unfortunately all these works (Sarohia, 1977; Sinha

et al., 1982; Zdanski et al., 2003) do not take into account heattransfer phenomena.

In this context, the numerical work by Kondoh et al. (1993) is of

primary concern. They investigated laminar heat transfer down-

stream of a backward facing step for a wide range of Prandtl values

and different channel expansion ratios, and concluded that the

dependence on Pr of the global and local heat transfer rates is

strong. In particular, it is found that for Pr < 0.1 diffusive effects

tend to smoothen the downstream profiles of the Nusselt number.

For higher Prandtl values, the peak heat transfer rate – which does

not locate necessary at the point of flow reattachment – is seen to

increase with powers of the Prandtl number. Also Metzger et al.

(1989) investigated the flow over a cavity, with reference to heat

transfer problems in the clearance gaps of turbine blades, but the

aspect ratio investigated are different, Reynolds numbers are much

larger, the fluid considered is air (Pr % 0.71) and the domain con-

sidered is not periodic.

The purpose of this study is to describe the main features of

laminar flow and low-Prandtl number thermal fields over a series

of periodic cavities, with particular focus on flow separation, and to

examine the influence of flow features on the local and global heat

transfer rates. This should provide a basis for the understanding of

convection phenomena also in turbulent regime and can be consid-

ered as a first step toward the development of physical models for

turbulent convection at low-Prandtl numbers. The ultimate goal of

this research is to devise reliable turbulent heat transfer models for

the design of accelerator driven sub-critical system (ADS) cooled

by lead bismuth eutectic, where the Reynolds numbers are a cou-

ple of orders of magnitude larger than the cases investigated.

2. Computational domain

A three-dimensional domain, periodic in the streamwise direc-

tion and homogeneous in the spanwise direction is considered in

this study. A longitudinal view of the domain geometry is shown

in Fig. 1, together with the coordinate system, whose origin is set

halfway between the forward and the backward steps, at the same

height of the cavity bottom. The size of the domain and the number

of grid points for all the simulations are given in Table 1, where the

reference length-scale d corresponds to the step height. Three

dimensionality would not be necessary for the investigation of

the laminar flow and heat transfer in a two dimensional geometry

while it has been introduced here because it is a basic feature of the numerical code used.

The periodic length L/d = 20 is equally subdivided between the

narrow channel section and the expansion. The aspect ratio of

the geometry investigated AR = L/(2d) = 10 identifies a shallow cav-

ity. As for AR = 10 reattachment occurs at the cavity bottom for

Rem 6 1470 but not for RemP 1840, the cavity considered in this

study falls within the closed cavities in the lower range of Reynolds

numbers simulated, but is to be considered an open cavity for

Rem % 1700 and larger, see (Sarohia, 1977). The decision to select

a high AR value is motivated by the interest in the heat transfer

characteristics of the flow reattachment region when the shear

layer reattaches to the cavity floor, see the discussion by Kondoh

et al. (1993). Besides AR = 10, results for the AR = 5 case are also re-

ported in the present study for assessing the dependency of results

upon the cavity geometry.

Simulations are performed over one periodicity L along the x

direction after which the flow and temperature patterns repeat

themselves. The periodic assumption in streamwise directionwas checked by comparing friction factor and Nusselt numbers cal-

culated over a single cavity of length L against values calculated

over a domain of length 2L, including two cavities. Results of this

preliminary test for Rem = 591 show that the relative error for f

and Nu is about 1.0 Â 10À6 (0.0001 %).

Results presented in this study are obtained on the fine mesh of

Table 1 while the coarse mesh is used only for grid refinement con-

siderations. Details of the 132 Â 58 and the 263 Â 115 computa-

tional meshes in the x À y plane are illustrated in Fig. 2. The

circulating region calculated on the fine mesh at Rem = 98.4 is

superimposed to the coarse mesh picture in order to show that

the relevant flow structures for RemP 98.4 are well discretized

by both grids.

3. Governing equations

3.1. Momentum equations

The momentum equations are set in dimensionless form using d

as the reference quantity for spatial coordinates, the friction veloc-

ity us = (bd/q)1/2 for velocities and t ref = d/us for time. In the defini-

tion of the friction velocity, b is the constant pressure drop

imposed in the x direction along one periodicity, divided by the

periodic length L

P ð x; y; z Þ À P ð x þ L; y; z Þ

L¼ b ð1Þ

where the bar denotes time averaging. The pressure field P is subdi-vided into a linear and an unsteady, periodic contributions

flow

Lδ

H

x

y

Fig. 1. Periodic geometry of the problem, longitudinal view.

Table 1

Domain dimensions and number of grid points for the fine and grid points around the

AR = 10 cavity; domain and mesh size for AR = 5.

AR L/d H /d L z /d N x Â N y Â N z

10 20 5 1 132 Â 58 Â 9

10 20 5 1 263 Â 115 Â 9

5 10 5 1 133 Â 115 Â 9

4 5 6 70

0.5

1

1.5

2

x

y

4 5 6 70

0.5

1

1.5

2

x

y

Fig. 2. Details of coarse and fine orthogonal meshes around the back step.

Streamlines in the circulation region at Rem = 98.4 are superimposed to the coarse

mesh picture.

E. Stalio et al. / International Journal of Heat and Fluid Flow 32 (2011) 1014–1023 1015



P ð x; y; z ; t Þ ¼ Àb x þ pð x; y; z ; t Þ ð2Þ

Periodic boundary conditions are assigned to the pressure fluc-

tuation p in the streamwise direction. From a time-averaged

momentum balance, b equals

b ¼2hsw; xi

H avð3Þ

where H av is the average channel height (H av = H + d/2) and theangular brackets indicate a spatial average. The conservation equa-

tions for mass and momentum in dimensionless form are

r Á u ¼ 0 ð4Þ

@ u

@ t þ r Á ðuuÞ ¼ Àr p þ

1

Resr2

u þ b ð5Þ

where b is the unit vector in x direction since in the non-dimen-

sional form, b = 1. Res is the friction Reynolds number, Res = usd/m.

No-slip boundary conditions are enforced at the walls, periodicity

is set in the homogeneous spanwise direction and in the streamwise

direction.

3.2. Energy equation

For a physically realistic description of the heat transfer be-

tween a fluid and a solid wall, the choice of boundary conditions

to be applied on the temperature field is a key parameter to be

carefully considered. While the only way to correctly represent

actual thermal boundary conditions is to include conjugate heat

transfer effects, such a model, besides introducing additional

complexity and demanding more computer resources, does not

provide easily scalable results, but can only manage one particu-

lar fluid/solid wall pair at once, with assigned thermophysical

properties.

In this study we consider two different thermal boundary con-

ditions and therefore we calculate and discuss two separate tem-

perature fields. They are obtained by setting uniform wall

temperature and imposed wall heat flux conditions respectively.This is in order to represent the two limiting cases of the physical

boundary conditions. Since buoyancy is neglected and the temper-

ature does not influence the flow, the two temperature fields are

computed together for the same velocity field solution. For both

conditions at the wall, the following energy equation with no heat

sources nor sinks is numerically solved

@ T

@ t þ r Á ðu T Þ ¼ ar2T ð6Þ

where thermophysical properties are assumed to remain constant

and viscous dissipation is neglected.

3.2.1. Imposed heat flux

The algorithm for solving Eq. (6) in streamwise periodic do-mains with assigned heat flux at the walls is an extension of the

one previously described (Stalio and Nobile, 2003) for ducts of uni-

form cross section. Using the assumption of fully developed flow

and heat transfer, a periodic variable / can be computed instead

of the temperature field. As the fluid temperature change becomes

linear in fully developed conditions (Papoutsakis et al., 1980), then

this can be extended to streamwise periodic ducts as soon as tem-

perature differences are evaluated over a periodic length (Patankar

et al., 1977). The ratio between the time-averaged temperature

drop and the domain length c DT =L, is independent of x and a

normalized temperature variable / is defined by

T ¼ / þ c x ð7Þ

the average temperature slope c is evaluated by an energy balanceas

c ¼2qw

H avqcum

ð8Þ

The energy balance leading to (8) does not take into account the ef-

fects of axial conduction: this is consistent with the linear temper-

ature change in uniform cross section ducts and with the uniform cin streamwise periodic ducts. Nevertheless we wish to highlight

that Eq. (8) loses its validity as soon as a different boundary condi-

tion is considered.The temperature field is made non dimensional by the reference

quantity

T s ¼2qwd

qcusH avð9Þ

and the non dimensional equation for / becomes

@ /

@ t þ u Á r/ þ

u

um

¼1

ResPrr2

/ ð10Þ

Boundary conditions applied at the walls to the non dimensional,

periodic variable / are:

@ /

@ g

w

¼ ÀResPr

H av2 ar ð11Þ

where ar is the ratio between the wall surface projected in the

streamwise direction and the actual wall surface. In this way the

heat flux imposed on the two projected surfaces is the same and

the fluid is equally heated from the top wall and the wall with steps.

3.2.2. Uniform wall temperature

As for imposed heat flux, a normalization of the temperature

field is introduced also for simulating prescribed temperature con-

ditions so that a streamwise periodic variable can be calculated in-

stead of the actual temperature field.

Since the most common normalizations require the knowledge

of the bulk temperature at every step, normalization is usually per-

formed through an iterative procedure. The technique employed in

this study instead directly solves the transport equation of the

periodic variable h

@ h

@ t þ r Á ðuhÞ ¼ ar2

h þ ða k2L þ u kLÞh À 2 a kL@ h

@ xð12Þ

where the normalized temperature h is defined as

hð x; y; z ; t Þ ¼T ð x; y; z ; t Þ

eÀkL xð13Þ

An energy balance is used for the evaluation of the space averaged

temperature decay rate kL thus closing the system of equations. Ef-

fects of axial diffusion, which are significant at low Péclet numbers,

are included in the equation for kL as well as in (12) and are there-fore accounted for in the solution. The recovery of the actual tem-

perature field is finally performed through Eq. (13). The interested

reader is referred to the work by Stalio and Piller (2007) for a thor-

ough description of the method.

3.3. Discrete form of the equations

The second order finite-volume code used for the simulations

does not differ from the one used in former studies of the flow

and heat transfer over corrugated surfaces (Stalio and Nobile,

2003; Stalio and Piller, 2007), where the transport equation for

the three velocity components and the temperature fields are

solved using standard numerical techniques. A second order pro-

jection scheme is employed for the segregated solution of the pres-sure field and the three velocity components.

1016 E. Stalio et al. / International Journal of Heat and Fluid Flow 32 (2011) 1014–1023



3.4. Non dimensional parameters

The Reynolds number, the friction factor and the Nusselt num-

ber are defined as:

Rem 2Q sm

; f ÀDP

L

2H av1

2qu2

av

; Nu 2H av h

kð14Þ

where 2H av is always used as the reference length, Q s is the time-averaged flow rate per unit spanwise width of the channel and

uav Q s/H av. In the numerical code and in terms of non dimensional

quantities, Rem and f are evaluated by

Rem ¼ 2ResQ s; f ¼4H avu2av

ð15Þ

The Nusselt number for the imposed heat flux temperature field

is evaluated from:

Nuqw ¼H 2avResPr

hT bi À hT wið16Þ

where T b is the bulk temperature and the angular brackets indicate

that a space average on the computational domain is performed.

Nu for the temperature field with prescribed wall temperatureis evaluated from

NuT w ¼2H av

hT bi À T w

@ hT i

@ g

w

ð17Þ

A local Nusselt number can be defined from each of the two

expressions (16) and (17) providing global values.

Nuqwð xÞ ¼H 2avResPr

T b À T w; NuT w ð xÞ ¼

2H avT b À T w

@ T

@ g

w

ð18Þ

From Eq. (18) two different Nu ( x) functions can be evaluated and

compared to assess the heat transfer performance of specific por-

tions of the periodic channel.

4. Results

4.1. Fluid flow

In the range of Reynolds number values investigated

(Rem = 24.9–2260), the velocity and temperature fields finally

reach steady conditions. Steady-state fluid flow patterns are dis-

played in Fig. 3, different Reynolds numbers are characterized by

reattachment and separation occurring in different locations of

the cavity floor.

For Rem = 24.9 the oncoming flow touches the cavity floor with

almost no separation. Separated flow and the presence of a steady

vortex close to the backward step is instead distinctly observed al-

ready from Rem = 98.4. Between Rem = 98.4 and Rem = 841 the cir-

culation region increases in size until changing its shape betweenRem = 841 and 1130, where the separation bubble elongates to

reach and surpass half the cavity length. Starting from the same re-

gime range, a secondary circulation region close to the forward

step is observed. The open cavity regime is recorded from

Rem = 1840, when reattachment on the cavity floor does not occur

anymore and fluid particles of the cavity region are distinct from

those flowing above the cavity. Axial coordinates of reattachment

Fig. 3. Streamlines inside the cavity for all the Reynolds numbers investigated.




and separation points at the different regimes are provided in Ta-

ble 2, where they are indicated by xr and xs, respectively.

Comparison of the friction factor evaluated for the computed

velocity fields through Eq. (15) and the analytical result for the flat

channel in laminar conditions ( f = 96/Rem) is displayed in Fig. 4.

The friction factor keeps the same behavior as in a flat channel

while its value is increased up to the 23% for Re m = 2260. Influence

of the channel geometry over friction drag is even more apparentwhen comparing the friction drag on the flat wall to the one eval-

uated on the wall with steps. Results are given in Table 2, where

Dc /D f is the ratio between wall friction on the cavity side and the

flat side. As the circulation structure in the cavity becomes larger,

preventing the fresh fluid from reaching the wall, the drag ratio de-

creases to as low as Dc /D f = 0.606, for Rem = 2260. Due to the pres-

ence of progressively larger regions of flow reversals and, in

general, due to smaller velocities close to the cavity floor, Dc /D f be-

comes 33% smaller between Rem = 24.9 and Rem = 2260. These re-

sults together with the friction factor increase suggest that

friction on the flat wall is also influenced by the cavity on the lower

wall.

4.2. Heat transfer

For the discussion of heat transfer phenomena in the cavities,

the periodic variables / and h will be used instead of the true tem-

perature fields as different profiles and isolines can be more easily

compared. Moreover their y derivatives coincide with the y deriv-

atives of the corresponding true temperatures, see Eqs. (7) and

(13). Fig. 5 displays the isolines of the periodic variables / for iso-

flux conditions and h for isothermal walls for three different Rey-

nolds numbers and Pr = 0.025. Temperature fields are seen to be

greatly affected by boundary conditions. While in the isothermal

case and especially at low Re numbers, an area of strong heat

transfer can be identified at the forward step, for isoflux conditions

heat seems to be more evenly transferred across the channel.

The heat transfer efficiency of the periodic channel with steps insteady, laminar regime is discussed in closer detail in the follow-

ing, first through the analysis of the Nusselt number behavior as

a function of the Reynolds number, and secondly using plots of lo-

cal Nusselt number for the different cases investigated.

Table 2

Friction Reynolds number, Reynolds number of the averaged velocity, x coordinate of

the reattachment ( xr ) and the separation ( xs) points and friction drag ratios for

Rem = 24.9 to Rem = 2260. Reattachment and separation point positions are not

indicated for open cavity flow.

Res Rem xr x s Dc /D f

1 24.9 5.4 14.7 0.901

2 98.4 5.7 14.7 0.875

3 218 6.1 14.7 0.8364 383 6.5 14.7 0.791

5 591 8.0 14.5 0.746

6 841 9.8 14.4 0.705

7 1130 11.5 14.3 0.671

8 1470 12.6 14.1 0.643

9 1840 – – 0.622

10 2260 – – 0.606

101

102

103

104

10−2

10−1

100

101

Re

f

Fig. 4. Friction factor as a function of the Reynolds number. Results for the periodicchannel with cavities are indicated by round symbols, the solid line displays f = 96/

Rem for the flat channel. The x-mark correspond to the f = 0.1902 value evaluated for

Rem = 591 over the coarse grid and is displayed to provide confirmation that results

presented are grid-independent.

Fig. 5. Contours of / and h for Rem = 24.9, 591, 2260 and Pr = 0.025.




Fig. 6 displays the global Nusselt number behavior as a function

of Re, for the two different temperature fields, Eqs. (16) and (17),

and two Prandtl numbers representing air and LBE. From compar-

ison with the flat channel values it appears that, within this

Reynolds number range, the presence of the cavity lowers the over-

all heat transfer rates between the fluid and the wall. This is due to

the presence of a vortex in the region of the backward step. In lam-

inar, steady conditions, the recirculating bubble is bounded by a

steady dividing streamline between the edge of the backward step( x = 5, y = 1) and the reattachment point whose location depends

on Re as indicated in Table 2. There is no mass exchange between

the recirculating bubble and the core region. As a consequence,

heat can be exchanged through the dividing streamline only by

diffusion.

4.2.1. Prescribed heat flux

For prescribed heat flux at the wall, the same heat flux entering

from the portion of wall between the edge of the backward step

and the reattachment point must be exchanged by diffusion across

the dividing streamline. This requires a rather strong temperature

gradient in direction normal to the recirculating bubble boundary.

As a consequence the temperature difference T b À T w, where T w is

the wall temperature within the recirculating region, must be large(see Fig. 5b, d and f). When a large temperature difference is locally

required to sustain a given heat flux, it can be concluded that the

local heat transfer coefficient is low, see also the Nusselt number

expression, Eq. (16).

4.2.2. Uniform wall temperature

For uniform wall temperature, the temperature field is fairly

uniform in the recirculating bubble region (see Fig. 5a, c, e and

Fig. 7) because of the presence of the uniform temperature side

walls and the circular motion of the same fluid particles within

the bubble. Heat across the dividing streamline is exchanged by dif-

fusion in a fairly uniform temperature region; as a consequence

heat transfer across the recirculating bubble is poor, we can con-

clude that the heat flux entering from the portion of wall bounded

by thedividing streamline must be small. This mechanism becomesmore important for increasing vortex size and strength and, from

the point of view of an heat exchanger designer, can be considered

an adverse advection effect . Only the Pr = 0.71 case with isothermal

walls, shows a slightly increasing (Nu, Re) curve for Re > 383, but

it is shown further in the text that this has to be ascribed to an even

more significant heat transfer augmentation in the region of the

boundary layer restart, downstream of the forward step.

A larger heat transfer rate as compared to the flat channel case

is observed in Fig. 6 only for Rem < 500, Pr = 0.025 and isothermal

walls. In this case, the increase in the convection surface area at

the prescribed T w overcomes the insulating effect of the cavity vor-

tex because the vortex is small and weak and conduction effects

are dominant. An increase in the Nusselt number is observed for

the uniform wall temperature case rather than in the prescribedqw results because of the choice to impose a larger heat flux on

the top wall with respect to the lower wall (which has a larger sur-

face), in order to ensure that an even amount of heat is added to

the fluid from the two walls.

Plots of the local Nusselt number on the lower wall of the chan-

nel provide a closer look to heat transfer performances of the cav-

ities at Pr = 0.025. The low heat transfer rate in the cavity region is

confirmed by Figs. 8 and 9, where Nu ( x) is displayed for different

Reynolds numbers in the steady laminar regime.

In Fig. 8 it is shown that at Rem = 24.9, Pr = 0.025 and imposed

heat flux heat is transported mostly by diffusion, as can be con-

cluded by observing the almost perfect symmetry of Nu ( x) about x = 10. An increase in Re produces a loss of symmetry indicating

that convection is now playing a role, and a shift to lower Nusseltnumbers in the cavity as well as downstream the forward step.

0 500 1000 1500 2000 25006

6.5

7

7.5

8

8.5

9

Re

Nu

Fig. 6. Nusselt number as a function of Rem for two Prandtl numbers and two

different boundary conditions of the temperature field. Square symbols indicate

results for Pr = 0.025, round symbols for Pr = 0.71. Solid lines are for imposed wall

flux, dashed lines for prescribed wall temperature. The x-marks correspond to

values calculated for Rem = 591 and Pr = 0.025 on the coarse grid. Nusselt numbers

for the flat channel in laminar conditions are indicated by horizontal lines for

comparison purposes, Nuqw ¼ 8:235 and NuT w ¼ 7:541.

4 6 8 10 12 14 16

0

2

4

6

x

y

Fig. 7. Profiles of the normalized temperature h in the recirculating region for

Rem = 2260. Solid lines indicate Pr = 0.025 results, dashed lines are for Pr = 0.71.

0 5 10 15 204

6

8

10

12

14

x

Nu

Fig. 8. Distribution of the local Nusselt number for the imposed heat flux case on

the lower wall, Pr = 0.025. Open circles indicate the Re m = 24.9 curve, triangles are

for Rem = 383, plus signs for Rem = 1130 and squares for Rem = 2260. The horizontal

solid line marks the value of Nusselt for a flat channel equally heated from the

walls.




Fig. 9 displays the same results for the prescribed wall temper-

ature case. A symmetric behavior of the Nusselt number is not ob-

served in this case, as with isothermal walls not even h is

symmetric, see Fig. 5. A peak Nusselt number is instead found at

the edge of the forward step, as well as at the corresponding edge

of the backward step. There both the velocity and temperature

boundary layers are very thin, see again Fig. 5a, c and e, moreover

in the same regions the v velocity component as well as the vertical

heat flux vh reach their maxima, as displayed in Fig. 10, thus allow-

ing for a very intense heat exchange with the solid wall.

A similar general behavior of Nu ( x) in single cavities of different

aspect ratios is reported by Metzger et al. (1989), where prescribed

wall temperature conditions are investigated experimentally,

although for higher Reynolds and using air as the working fluid.

Local heat transfer coefficient measured by Aung (1983) in singlecavities of smaller aspect ratios (AR = 1 and 4), for air flows in

laminar conditions, also share a comparable behavior.

The comparison between the local Nusselt number for pre-

scribed wall heat flux (Fig. 8) and prescribed wall temperature con-

ditions (Fig. 9) reveals a markedly different behavior in the regions

of the backward and forward facing steps. This can be explained by

a departure from the Reynolds analogy in the case of wall heat flux

conditions. At the edge of the forward step, where a peak of wall

friction is expected, the heat transfer is maximum only for the case

of imposed wall temperature,i.e.

for the same boundary conditionsof the velocity field.

In order to analyze the Prandtl number effect on heat transfer in

steady, separated flow conditions, flow patterns above the lower

wall are displayed in Fig. 11, together with the local Nusselt num-

ber for prescribed wall temperature and for the two different fluids

investigated. Two dashed vertical lines in correspondence of the xr

and xs points of Table 2 are also added to the plots, in order to indi-

cate the region of the cavity where the flow is attached. As already

noticed in the work by Kondoh et al. (1993) the peak heat transfer

location seems to be uncorrelated with the positions of attachment

and separation.

Fig. 11 shows that when heat transfer is adversely affected by

advection, i.e. in the cavities with isothermal walls and for increas-

ing Reynolds number, Nu is larger for the Pr = 0.025 fluid, because

in such conditions a larger temperature gradient is established at

the wall. The remark is confirmed by Fig. 7, where it can be ob-

served that @ h/@ y calculated on the cavity floor is larger for

Pr = 0.025 thus leading locally to a higher heat transfer rate. Diffu-

sion taking place in a very conductive fluid can locally overcome

adverse advection effects. Conversely, when advection effects are

beneficial, as in the case of the restart of the boundary layer down-

stream the forward step, heat transfer in the higher Prandtl num-

ber fluids can be more intense, at least for large enough Reynolds

numbers. This behavior is observed in Fig. 11c and d, i.e. for

Re > 383.

Local Nusselt number plots are quite different when evaluated

for imposed wall heat flux, as displayed in Fig. 12. As opposed to

the prescribed temperature case, the Nusselt number in the im-

posed qw cavities is smaller for Pr = 0.025 than for Pr = 0.71. More-over, in the region of the boundary layer restart and unlike the

isothermal walls case, Nu is larger for Pr = 0.025 than for air. The

0 5 10 15 204

6

8

10

12

14

x

Nu

Fig. 9. Distribution of the local Nusselt number for the prescribed wall temperature

case, Pr = 0.025 on the lower wall. Open circles indicate the Rem = 24.9 curve,

triangles are for Rem = 383, plus signs for Rem = 1130 and squares for Rem = 2260.

The horizontal solid line marks the value of Nusselt for a flat channel with

isothermal walls.

Fig. 10. Contours of the v velocity component and of the vertical advective heat fluxes vh for Rem = 24.9, 591 and 2260, Pr = 0.025 and prescribed wall temperature. Solid lines

indicate positive values, dashed lines negative values. Thin lines describe eight levels jvj = 0.1, 0.2, 0.3, 0.4 and jvhj = 0.1, 0.2, 0.3, 0.4 thick lines are for levels jvj = 0.5, 1.5, 2.5and jvhj = 0.5, 1.5, 2.5, 3.5.




different behavior of the isoflux case is to be ascribed to the fact

that when the temperature derivative is assigned at the wall, heat

transfer is less sensible to near-wall advection effects.

4.3. Aspect ratio effect

This section will discuss heat transfer over cavities of aspect

ratio AR = 5 in order to assess the dependency of results upon

geometry. The channel dimension and mesh size are indicated

in Table 1. Local Nusselt number plots for Pr = 0.025 and pre-

scribed T w of Fig. 13 qualitatively follow results for AR = 10. Two

heat transfer rate peaks can be observed for all the investigated

Reynolds numbers: the smaller one corresponds to the separation

point at the backward step edge while the larger peak is alwaysfound at the forward step, corresponding to the restart of the

boundary layer. Again there seem to be no identifiable effect of

the reattachment point location inside the cavity upon local heat

transfer.

Fig. 14 shows a decreasing behavior of Nu with Re in the lami-

nar regime. Nusselt numbers are always smaller than for AR = 10

and the heat transfer decrease is larger for lower Re and uniform

temperature walls. Focusing on the uniform temperature boundary

conditions, the reasons for a global heat transfer decrease over

smaller AR geometries are twofold. First, local heat transfer coeffi-

cients are considerably smaller for AR = 5 than for AR = 10 as can

be seen from the comparison between Fig. 9 to Nu ( x) in Fig. 13.

Secondly, the local heat transfer coefficient at the side walls is well

below its averaged value but for cavities of smaller aspect ratio the

area of the two side walls have a larger share on the area of thelower wall of the channel.

0 5 10 15 200

10

20

x

Nu

0 5 10 15 200

10

20

x

Nu

0 5 10 15 200

10

20

x

Nu

0 5 10 15 200

10

20

x

Nu

Fig. 11. Streamlines and local Nusselt number of the lower wall for prescribed wall temperature conditions. Vertical dashed lines indicate attachment and separation points,

see Table 2. Squares are for Pr = 0.025, triangles for Pr = 0.71.

0 5 10 15 205

10

15

x

Nu

0 5 10 15 205

10

15

x

Nu

0 5 10 15 205

10

15

x

Nu

0 5 10 15 205

10

15

x

Nu

Fig. 12. Local Nusselt number of the lower wall for imposed heat flux conditions. Squares are for Pr = 0.025, triangles for Pr = 0.71.




5. Conclusions

Steady, laminar flow and heat transfer of liquid lead–bismuth

eutectic (Pr = 0.025) has been studied numerically in a periodic

series of cavities, for ten Reynolds number values, based on

the hydraulic diameter, ranging from Rem = 24.9 to Rem = 2260,

and for isothermal and isoflux boundary conditions. The corre-

sponding Pr = 0.71 cases have been investigated for comparison

purposes.

It is shown that flow separation and recirculation past the back-

ward step occurs even at very low Reynolds number values. For

increasing Re the recirculating bubble progressively expands andfinally encompasses the whole cavity leading to an open cavity

regime for Rem ’ 1700. The friction factor and the friction drag

balance between the two walls are considerably modified com-

pared to the flat channel values.

In laminar flow conditions and for isothermal walls, the pres-

ence of the cavity has a negative effect on heat transfer rates due

to the presence of a stable vortex downstream the backward step.

The insulating effect of the vortex increases with its strength, i.e.with the Reynolds number; this makes Nu a decreasing function

of Re in almost all cases. The same tendency is observed for

AR = 10 and AR = 5 but heat transfer rates are even smaller for

AR = 5. In the only case where the global Nu increases with Re (pre-

scribed T w at Pr = 0.71 and AR = 10), this is due to a stronger heat

transfer enhancement in the developing boundary layer area

downstream the cavity. Low Prandtl fluids show better heat trans-

fer characteristics only where advection effects are adverse like

across a stable recirculation bubble. When advection is beneficial

as in the restart of a boundary layer, higher Pr fluids display larger

heat transfer coefficients.

In the laminar regime of our investigation, local Nusselt number

results are quite different for the imposed heat flux case. When the

temperature derivative is prescribed at the wall, convection effects

are not as beneficial in the area of boundary layer restart and they

are not as adverse in the recirculating region in the cavity as for

isothermal walls.

In general, there seems to be no straightforward correlations

between the locations of the peak heat transfer rate and the reat-

tachment points at the cavity floor; instead Nusselt maxima for

prescribed wall temperature are always found at the edge of the

forward step, corresponding to the restart of the boundary layer.

Acknowledgments

Part of this work was performed during the NORDITA Pro-

gramme on ‘‘Turbulent Boundary Layers’’, April 2010 in Stockholm.

Thanks to Prof. Henrik Alfredsson and Dr. Philipp Schlatter for this.The authors thank Mr. Stefano Stalio for technical support.

0 2 4 6 8 100

10

20

x

Nu

0 2 4 6 8 100

10

20

x

Nu

0 2 4 6 8 100

10

20

x

Nu

0 2 4 6 8 100

10

20

x

Nu

Fig. 13. Streamlines and local Nusselt number of the lower wall for prescribed wall temperature conditions, Pr = 0.025 and AR = 5. Vertical dashed lines indicate reattachment

and separation points. The horizontal solid line marks the value of Nusselt for a flat channel with isothermal walls.

0 500 1000 1500 2000 25006

6.5

7

7.5

8

8.5

9

Re

Nu

Fig. 14. Nusselt number as a function of Rem for two different boundary conditions.

Results obtained in the AR = 5 cavity are indicated by x-marks, square symbols are

for AR = 10, as in Fig. 6. Solid lines are for imposed wall flux, dashed lines for

prescribed wall temperature. Nusselt numbers for the flat channel in laminar

conditions are indicated by horizontal lines for comparison.




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Numerical Simulation of Forced Convection Over a Periodic Series 2011