Numerical simulation of forced convection over a periodic series of ...
International Journal of Heat and Fluid Flow 32 (2011) 1014–1023
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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 2010 Received 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 periodic series of shallow cavities. Finite-volume simulations are carried out on structured orthogonal curvilinear 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 geometry 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 because of the opacity of these fluids and the care required in handling them. In the present work, convective heat transfer in laminar conditions is investigated numerically for a Prandtl number Pr = 0.025, which is representative of liquid lead–bismuth; results are compared 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 reattachment are observed also for low Reynolds number, steady conditions. 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. ⇑ 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). 0142-727X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2011.05.009
Simulations are carried out for ten values of the Reynolds number in the laminar regime. Two different thermal boundary conditions imposed at the channel walls are considered, namely uniform temperature and constant heat flux. The flow over forward or backward steps in ducts or in freestream 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 transfer 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 governed 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 computations. In particular, the latter concluded that the similarity analysis by Chapman was somewhat inadequate to treat the problem completely and to derive general heat transfer correlations. They proposed 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 reported a limiting value of 10 for the length-to-height aspect ratio of the cavity, which separates ‘‘closed’’ flows (i.e. flows where reattachment occurs inside the cavity) from ‘‘open flows’’ for which
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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 cavities, encompassing the sensitivity to various parameters, such as freestream velocity, turbulent kinetic energy, aspect ratio and Reynolds number. Unfortunately all these works (Sarohia, 1977; Sinha et al., 1982; Zdanski et al., 2003) do not take into account heat transfer phenomena. In this context, the numerical work by Kondoh et al. (1993) is of primary concern. They investigated laminar heat transfer downstream 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 considered 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 considered 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 couple of orders of magnitude larger than the cases investigated.
2. Computational domain A three-dimensional domain, periodic in the streamwise direction 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-
flow H y x L
δ
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
Lz/d
Nx Ny Nz
10 10 5
20 20 10
5 5 5
1 1 1
132 58 9 263 115 9 133 115 9
y
2
2
1.5
1.5
y
1
1 0.5
0.5
0
0 4
5
6
x
7
4
5
6
7
x
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.
ity. As for AR = 10 reattachment occurs at the cavity bottom for Rem 6 1470 but not for Rem P 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 reported 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 direction was checked by comparing friction factor and Nusselt numbers calculated 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 1.0 106 (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 considerations. Details of the 132 58 and the 263 115 computational meshes in the x 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 Rem P 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 velocity us = (bd/q)1/2 for velocities and tref = d/us for time. In the definition of the friction velocity, b is the constant pressure drop imposed in the x direction along one periodicity, divided by the periodic length L
Pðx; y; zÞ Pðx þ L; y; zÞ ¼b L
ð1Þ
where the bar denotes time averaging. The pressure field P is subdivided into a linear and an unsteady, periodic contributions
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Pðx; y; z; tÞ ¼ bx þ pðx; y; z; tÞ
ð2Þ
Periodic boundary conditions are assigned to the pressure fluctuation p in the streamwise direction. From a time-averaged momentum balance, b equals
b¼
2hsw;x i Hav
ð3Þ
where Hav is the average channel height (Hav = H + d/2) and the angular brackets indicate a spatial average. The conservation equations for mass and momentum in dimensionless form are
ru¼0 @u 1 r2 u þ b þ r ðuuÞ ¼ rp þ @t Res
ð5Þ
ð8Þ
The energy balance leading to (8) does not take into account the effects of axial conduction: this is consistent with the linear temperature change in uniform cross section ducts and with the uniform c in streamwise periodic ducts. Nevertheless we wish to highlight that Eq. (8) loses its validity as soon as a different boundary condition is considered. The temperature field is made non dimensional by the reference quantity
Ts ¼
2qw d
qcus Hav
ð9Þ
and the non dimensional equation for / becomes
3.2. Energy equation For a physically realistic description of the heat transfer between 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 particular fluid/solid wall pair at once, with assigned thermophysical properties. In this study we consider two different thermal boundary conditions and therefore we calculate and discuss two separate temperature 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 temperature 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
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 domains with assigned heat flux at the walls is an extension of the one previously described (Stalio and Nobile, 2003) for ducts of uniform 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 temperature 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 cx
2qw Hav qcum
ð4Þ
where b is the unit vector in x direction since in the non-dimensional 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.
@T þ r ðu TÞ ¼ ar2 T @t
c¼
ð7Þ
the average temperature slope c is evaluated by an energy balance as
@/ u 1 ¼ r2 / þ u r/ þ @t um Res Pr
ð10Þ
Boundary conditions applied at the walls to the non dimensional, periodic variable / are:
@/ Hav ¼ Res Pr ar @ g w 2
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 conditions so that a streamwise periodic variable can be calculated instead of the actual temperature field. Since the most common normalizations require the knowledge of the bulk temperature at every step, normalization is usually performed through an iterative procedure. The technique employed in this study instead directly solves the transport equation of the periodic variable h
@h @h þ r ðuhÞ ¼ ar2 h þ ða k2L þ u kL Þh 2 a kL @t @x
ð12Þ
where the normalized temperature h is defined as
hðx; y; z; tÞ ¼
Tðx; y; z; tÞ ekL x
ð13Þ
An energy balance is used for the evaluation of the space averaged temperature decay rate kL thus closing the system of equations. Effects of axial diffusion, which are significant at low Péclet numbers, are included in the equation for kL as well as in (12) and are therefore accounted for in the solution. The recovery of the actual temperature field is finally performed through Eq. (13). The interested reader is referred to the work by Stalio and Piller (2007) for a thorough 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 projection scheme is employed for the segregated solution of the pressure field and the three velocity components.
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3.4. Non dimensional parameters
Nuqw ðxÞ ¼
The Reynolds number, the friction factor and the Nusselt number are defined as:
Rem 2
Qs
m
;
f
DP 2Hav ; L 12 qu2av
Nu
2Hav h k
ð14Þ
where 2Hav is always used as the reference length, Qs is the timeaveraged flow rate per unit spanwise width of the channel and uav Qs/Hav. In the numerical code and in terms of non dimensional quantities, Rem and f are evaluated by
Rem ¼ 2Res Q s ;
f ¼
4Hav u2av
ð15Þ
The Nusselt number for the imposed heat flux temperature field is evaluated from:
Nuqw ¼
H2av Res Pr hT b i hT w i
ð16Þ
where Tb 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 temperature is evaluated from
NuT w ¼
2Hav @hTi hT b i T w @ g w
ð17Þ
A local Nusselt number can be defined from each of the two expressions (16) and (17) providing global values.
H2av Res Pr ; Tb Tw
NuT w ðxÞ ¼
2Hav @T T b T w @ g w
ð18Þ
From Eq. (18) two different Nu (x) functions can be evaluated and compared to assess the heat transfer performance of specific portions 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 displayed 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 already from Rem = 98.4. Between Rem = 98.4 and Rem = 841 the circulation region increases in size until changing its shape between Rem = 841 and 1130, where the separation bubble elongates to reach and surpass half the cavity length. Starting from the same regime 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.
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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
xs
Dc/Df
1 2 3 4 5 6 7 8 9 10
24.9 98.4 218 383 591 841 1130 1470 1840 2260
5.4 5.7 6.1 6.5 8.0 9.8 11.5 12.6 – –
14.7 14.7 14.7 14.7 14.5 14.4 14.3 14.1 – –
0.901 0.875 0.836 0.791 0.746 0.705 0.671 0.643 0.622 0.606
1
10
and separation points at the different regimes are provided in Table 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 Rem = 2260. Influence of the channel geometry over friction drag is even more apparent when comparing the friction drag on the flat wall to the one evaluated on the wall with steps. Results are given in Table 2, where Dc/Df 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 decreases to as low as Dc/Df = 0.606, for Rem = 2260. Due to the presence of progressively larger regions of flow reversals and, in general, due to smaller velocities close to the cavity floor, Dc/Df becomes 33% smaller between Rem = 24.9 and Rem = 2260. These results 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
0
10
f −1
10
−2
10
1
10
10
2
10
3
4
10
Re Fig. 4. Friction factor as a function of the Reynolds number. Results for the periodic channel 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.
For the discussion of heat transfer phenomena in the cavities, the periodic variables / and h will be used instead of the true temperature fields as different profiles and isolines can be more easily compared. Moreover their y derivatives coincide with the y derivatives of the corresponding true temperatures, see Eqs. (7) and (13). Fig. 5 displays the isolines of the periodic variables / for isoflux conditions and h for isothermal walls for three different Reynolds 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 in steady, laminar regime is discussed in closer detail in the following, first through the analysis of the Nusselt number behavior as a function of the Reynolds number, and secondly using plots of local Nusselt number for the different cases investigated.
Fig. 5. Contours of / and h for Rem = 24.9, 591, 2260 and Pr = 0.025.
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9
14
8.5
12
8 10
Nu 7.5
Nu 8
7 6
6.5 6
0
500
1000
1500
2000
2500
4
0
5
10
15
20
x
Re 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.
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 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 equally heated from the walls.
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). 6
4
y 2
0 4
6
8
10
12
14
16
x 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.
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 comparison with the flat channel values it appears that, within this Reynolds number range, the presence of the cavity lowers the overall 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 laminar, 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 Tb Tw, where Tw is the wall temperature within the recirculating region, must be large (see Fig. 5b, d and f). When a large temperature difference is locally
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 diffusion in a fairly uniform temperature region; as a consequence heat transfer across the recirculating bubble is poor, we can conclude that the heat flux entering from the portion of wall bounded by the dividing streamline must be small. This mechanism becomes more 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 Tw overcomes the insulating effect of the cavity vortex 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 prescribed qw 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 surface), 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 channel provide a closer look to heat transfer performances of the cavities 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 concluded 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 Nusselt numbers in the cavity as well as downstream the forward step.
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14
12
10
Nu 8
6
4
0
5
10
15
20
x 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. 9 displays the same results for the prescribed wall temperature case. A symmetric behavior of the Nusselt number is not observed 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 allowing 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 single cavities 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 prescribed wall heat flux (Fig. 8) and prescribed wall temperature conditions (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 conditions of 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 number 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 indicate 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 increasing 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 observed that @h/@y calculated on the cavity floor is larger for Pr = 0.025 thus leading locally to a higher heat transfer rate. Diffusion 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 downstream the forward step, heat transfer in the higher Prandtl number 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 imposed qw cavities is smaller for Pr = 0.025 than for Pr = 0.71. Moreover, 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
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, 2.5 and jvhj = 0.5, 1.5, 2.5, 3.5.
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20
20
Nu 10
Nu 10
0
0
5
10
15
20
0
0
5
x
20
Nu 10
Nu 10
0
5
10
15
20
15
20
x
20
0
10
15
20
0
0
5
x
10
x
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.
15
15
Nu 10
Nu 10
5
0
5
10
15
20
5
0
5
x
15
Nu 10
Nu 10
0
5
10
15
20
15
20
x
15
5
10
15
20
x
5
0
5
10
x
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.
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 prescribed Tw 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 always found 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 laminar 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 coefficients 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 the lower wall of the channel.
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20
20
Nu 10
Nu 10
0 0
2
4
6
8
10
0 0
2
4
x
20
20
Nu 10
Nu 10
0 0
2
4
6
8
10
6
8
10
x
6
8
10
0 0
2
x
4
x
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.
9 8.5 8
Nu 7.5 7 6.5 6
0
500
1000
1500
2000
2500
Re 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.
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 corresponding Pr = 0.71 cases have been investigated for comparison purposes. It is shown that flow separation and recirculation past the backward step occurs even at very low Reynolds number values. For increasing Re the recirculating bubble progressively expands and finally 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 compared to the flat channel values. In laminar flow conditions and for isothermal walls, the presence 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 (prescribed Tw 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 transfer 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 reattachment 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 Programme 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.
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References Aung, W., 1983. An interferometric investigation of separated forced convection in laminar flow past cavities. ASME J. Heat Transfer 105 (8), 505–512. Avancha, R.V.R., Pletcher, R.H., 2002. Large eddy simulation of the turbulent flow past a backward-facing step with heat transfer and property variations. Int. J. Heat Fluid Flow 23, 601–614. Bhatti, A., Aung, W., 1984. Finite difference analysis of laminar separated forced convection in cavities. ASME J. Heat Transfer 106 (2), 49–54. Chapman, D., 1956. A Theoretical Analysis of Heat Transfer in Regions of Separated Flow. Technical Report TN 3792, NACA. Kondoh, T., Nagano, Y., Tsuji, T., 1993. Computational study of laminar heat transfer downstream of a backward-facing step. Int. J. Heat Mass Transfer 36 (3), 577– 591. Metzger, D.E., Bunker, R.S., Chyu, M.K., 1989. Cavity heat transfer on a transverse grooved wall in a narrow flow channel. ASME J. Heat Transfer 111, 73–79.
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Papoutsakis, E., Ramkrishna, D., Lim, H.C., 1980. The extended Graetz problem with prescribed wall flux. AIChE J. 26 (5), 779–787. Patankar, S.V., Liu, C.H., Sparrow, E.M., 1977. Fully developed flow and heat transfer in ducts having streamwise-periodic variations of cross-sectional area. ASME J. Heat Transfer 99, 180–186. Sarohia, V., 1977. Experimental investigation of oscillations in flows over shallow cavities. AIAA J. 15 (7), 984–991. Sinha, S., Gupta, A., Oberai, M., 1982. Laminar separating flow over backsteps and cavities part II: cavities. AIAA J. 20, 370–375. Stalio, E., Nobile, E., 2003. Direct numerical simulation of heat transfer over riblets. Int. J. Heat Fluid Flow 24, 356–371. Stalio, E., Piller, M., 2007. Direct numerical simulation of heat transfer in converging–diverging wavy channels. ASME J. Heat Transfer 129 (7), 769–777. Zdanski, P., Ortega, M., G.C.R., N., Fico, J., 2003. Numerical study of the flow over shallow cavities. Comput. Fluids 32, 953–974.
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
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 2010 Received 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 periodic series of shallow cavities. Finite-volume simulations are carried out on structured orthogonal curvilinear 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 geometry 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 because of the opacity of these fluids and the care required in handling them. In the present work, convective heat transfer in laminar conditions is investigated numerically for a Prandtl number Pr = 0.025, which is representative of liquid lead–bismuth; results are compared 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 reattachment are observed also for low Reynolds number, steady conditions. 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. ⇑ 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). 0142-727X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2011.05.009
Simulations are carried out for ten values of the Reynolds number in the laminar regime. Two different thermal boundary conditions imposed at the channel walls are considered, namely uniform temperature and constant heat flux. The flow over forward or backward steps in ducts or in freestream 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 transfer 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 governed 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 computations. In particular, the latter concluded that the similarity analysis by Chapman was somewhat inadequate to treat the problem completely and to derive general heat transfer correlations. They proposed 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 reported a limiting value of 10 for the length-to-height aspect ratio of the cavity, which separates ‘‘closed’’ flows (i.e. flows where reattachment occurs inside the cavity) from ‘‘open flows’’ for which
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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 cavities, encompassing the sensitivity to various parameters, such as freestream velocity, turbulent kinetic energy, aspect ratio and Reynolds number. Unfortunately all these works (Sarohia, 1977; Sinha et al., 1982; Zdanski et al., 2003) do not take into account heat transfer phenomena. In this context, the numerical work by Kondoh et al. (1993) is of primary concern. They investigated laminar heat transfer downstream 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 considered 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 considered 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 couple of orders of magnitude larger than the cases investigated.
2. Computational domain A three-dimensional domain, periodic in the streamwise direction 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-
flow H y x L
δ
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
Lz/d
Nx Ny Nz
10 10 5
20 20 10
5 5 5
1 1 1
132 58 9 263 115 9 133 115 9
y
2
2
1.5
1.5
y
1
1 0.5
0.5
0
0 4
5
6
x
7
4
5
6
7
x
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.
ity. As for AR = 10 reattachment occurs at the cavity bottom for Rem 6 1470 but not for Rem P 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 reported 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 direction was checked by comparing friction factor and Nusselt numbers calculated 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 1.0 106 (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 considerations. Details of the 132 58 and the 263 115 computational meshes in the x 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 Rem P 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 velocity us = (bd/q)1/2 for velocities and tref = d/us for time. In the definition of the friction velocity, b is the constant pressure drop imposed in the x direction along one periodicity, divided by the periodic length L
Pðx; y; zÞ Pðx þ L; y; zÞ ¼b L
ð1Þ
where the bar denotes time averaging. The pressure field P is subdivided into a linear and an unsteady, periodic contributions
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Pðx; y; z; tÞ ¼ bx þ pðx; y; z; tÞ
ð2Þ
Periodic boundary conditions are assigned to the pressure fluctuation p in the streamwise direction. From a time-averaged momentum balance, b equals
b¼
2hsw;x i Hav
ð3Þ
where Hav is the average channel height (Hav = H + d/2) and the angular brackets indicate a spatial average. The conservation equations for mass and momentum in dimensionless form are
ru¼0 @u 1 r2 u þ b þ r ðuuÞ ¼ rp þ @t Res
ð5Þ
ð8Þ
The energy balance leading to (8) does not take into account the effects of axial conduction: this is consistent with the linear temperature change in uniform cross section ducts and with the uniform c in streamwise periodic ducts. Nevertheless we wish to highlight that Eq. (8) loses its validity as soon as a different boundary condition is considered. The temperature field is made non dimensional by the reference quantity
Ts ¼
2qw d
qcus Hav
ð9Þ
and the non dimensional equation for / becomes
3.2. Energy equation For a physically realistic description of the heat transfer between 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 particular fluid/solid wall pair at once, with assigned thermophysical properties. In this study we consider two different thermal boundary conditions and therefore we calculate and discuss two separate temperature 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 temperature 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
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 domains with assigned heat flux at the walls is an extension of the one previously described (Stalio and Nobile, 2003) for ducts of uniform 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 temperature 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 cx
2qw Hav qcum
ð4Þ
where b is the unit vector in x direction since in the non-dimensional 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.
@T þ r ðu TÞ ¼ ar2 T @t
c¼
ð7Þ
the average temperature slope c is evaluated by an energy balance as
@/ u 1 ¼ r2 / þ u r/ þ @t um Res Pr
ð10Þ
Boundary conditions applied at the walls to the non dimensional, periodic variable / are:
@/ Hav ¼ Res Pr ar @ g w 2
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 conditions so that a streamwise periodic variable can be calculated instead of the actual temperature field. Since the most common normalizations require the knowledge of the bulk temperature at every step, normalization is usually performed through an iterative procedure. The technique employed in this study instead directly solves the transport equation of the periodic variable h
@h @h þ r ðuhÞ ¼ ar2 h þ ða k2L þ u kL Þh 2 a kL @t @x
ð12Þ
where the normalized temperature h is defined as
hðx; y; z; tÞ ¼
Tðx; y; z; tÞ ekL x
ð13Þ
An energy balance is used for the evaluation of the space averaged temperature decay rate kL thus closing the system of equations. Effects of axial diffusion, which are significant at low Péclet numbers, are included in the equation for kL as well as in (12) and are therefore accounted for in the solution. The recovery of the actual temperature field is finally performed through Eq. (13). The interested reader is referred to the work by Stalio and Piller (2007) for a thorough 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 projection scheme is employed for the segregated solution of the pressure field and the three velocity components.
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3.4. Non dimensional parameters
Nuqw ðxÞ ¼
The Reynolds number, the friction factor and the Nusselt number are defined as:
Rem 2
Qs
m
;
f
DP 2Hav ; L 12 qu2av
Nu
2Hav h k
ð14Þ
where 2Hav is always used as the reference length, Qs is the timeaveraged flow rate per unit spanwise width of the channel and uav Qs/Hav. In the numerical code and in terms of non dimensional quantities, Rem and f are evaluated by
Rem ¼ 2Res Q s ;
f ¼
4Hav u2av
ð15Þ
The Nusselt number for the imposed heat flux temperature field is evaluated from:
Nuqw ¼
H2av Res Pr hT b i hT w i
ð16Þ
where Tb 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 temperature is evaluated from
NuT w ¼
2Hav @hTi hT b i T w @ g w
ð17Þ
A local Nusselt number can be defined from each of the two expressions (16) and (17) providing global values.
H2av Res Pr ; Tb Tw
NuT w ðxÞ ¼
2Hav @T T b T w @ g w
ð18Þ
From Eq. (18) two different Nu (x) functions can be evaluated and compared to assess the heat transfer performance of specific portions 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 displayed 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 already from Rem = 98.4. Between Rem = 98.4 and Rem = 841 the circulation region increases in size until changing its shape between Rem = 841 and 1130, where the separation bubble elongates to reach and surpass half the cavity length. Starting from the same regime 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.
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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
xs
Dc/Df
1 2 3 4 5 6 7 8 9 10
24.9 98.4 218 383 591 841 1130 1470 1840 2260
5.4 5.7 6.1 6.5 8.0 9.8 11.5 12.6 – –
14.7 14.7 14.7 14.7 14.5 14.4 14.3 14.1 – –
0.901 0.875 0.836 0.791 0.746 0.705 0.671 0.643 0.622 0.606
1
10
and separation points at the different regimes are provided in Table 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 Rem = 2260. Influence of the channel geometry over friction drag is even more apparent when comparing the friction drag on the flat wall to the one evaluated on the wall with steps. Results are given in Table 2, where Dc/Df 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 decreases to as low as Dc/Df = 0.606, for Rem = 2260. Due to the presence of progressively larger regions of flow reversals and, in general, due to smaller velocities close to the cavity floor, Dc/Df becomes 33% smaller between Rem = 24.9 and Rem = 2260. These results 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
0
10
f −1
10
−2
10
1
10
10
2
10
3
4
10
Re Fig. 4. Friction factor as a function of the Reynolds number. Results for the periodic channel 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.
For the discussion of heat transfer phenomena in the cavities, the periodic variables / and h will be used instead of the true temperature fields as different profiles and isolines can be more easily compared. Moreover their y derivatives coincide with the y derivatives of the corresponding true temperatures, see Eqs. (7) and (13). Fig. 5 displays the isolines of the periodic variables / for isoflux conditions and h for isothermal walls for three different Reynolds 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 in steady, laminar regime is discussed in closer detail in the following, first through the analysis of the Nusselt number behavior as a function of the Reynolds number, and secondly using plots of local Nusselt number for the different cases investigated.
Fig. 5. Contours of / and h for Rem = 24.9, 591, 2260 and Pr = 0.025.
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9
14
8.5
12
8 10
Nu 7.5
Nu 8
7 6
6.5 6
0
500
1000
1500
2000
2500
4
0
5
10
15
20
x
Re 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.
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 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 equally heated from the walls.
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). 6
4
y 2
0 4
6
8
10
12
14
16
x 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.
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 comparison with the flat channel values it appears that, within this Reynolds number range, the presence of the cavity lowers the overall 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 laminar, 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 Tb Tw, where Tw is the wall temperature within the recirculating region, must be large (see Fig. 5b, d and f). When a large temperature difference is locally
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 diffusion in a fairly uniform temperature region; as a consequence heat transfer across the recirculating bubble is poor, we can conclude that the heat flux entering from the portion of wall bounded by the dividing streamline must be small. This mechanism becomes more 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 Tw overcomes the insulating effect of the cavity vortex 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 prescribed qw 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 surface), 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 channel provide a closer look to heat transfer performances of the cavities 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 concluded 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 Nusselt numbers in the cavity as well as downstream the forward step.
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x 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. 9 displays the same results for the prescribed wall temperature case. A symmetric behavior of the Nusselt number is not observed 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 allowing 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 single cavities 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 prescribed wall heat flux (Fig. 8) and prescribed wall temperature conditions (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 conditions of 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 number 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 indicate 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 increasing 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 observed that @h/@y calculated on the cavity floor is larger for Pr = 0.025 thus leading locally to a higher heat transfer rate. Diffusion 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 downstream the forward step, heat transfer in the higher Prandtl number 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 imposed qw cavities is smaller for Pr = 0.025 than for Pr = 0.71. Moreover, 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
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, 2.5 and jvhj = 0.5, 1.5, 2.5, 3.5.
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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.
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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.
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 prescribed Tw 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 always found 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 laminar 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 coefficients 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 the lower wall of the channel.
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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.
9 8.5 8
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Re 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.
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 corresponding Pr = 0.71 cases have been investigated for comparison purposes. It is shown that flow separation and recirculation past the backward step occurs even at very low Reynolds number values. For increasing Re the recirculating bubble progressively expands and finally 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 compared to the flat channel values. In laminar flow conditions and for isothermal walls, the presence 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 (prescribed Tw 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 transfer 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 reattachment 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 Programme 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.
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