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Turbulent large-scale structures in natural convection vertical channel flow
International Journal of Heat and Mass Transfer 53 (2010) 4168–4175
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Turbulent large-scale structures in natural convection vertical channel flow J. Pallares *, A. Vernet, J.A. Ferre, F.X. Grau Department of Mechanical Engineering, University Rovira i Virgili, Av. Països Catalans, 26, 43007 Tarragona, Spain
a r t i c l e
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Article history: Received 11 December 2009 Accepted 11 March 2010 Available online 11 June 2010 Keywords: Turbulent natural convection Vertical channel Heat transfer rate Direct numerical simulation Conditional sampling Large-scale structures
a b s t r a c t We analyzed a database of a direct numerical simulation of natural convection in a vertical channel. The flow is driven by a constant temperature difference imposed at the walls (Ra = 5.4 105, Pr = 0.7). The averaged flow and turbulent statistics are in good agreement with previous direct numerical simulations reported in the literature. Contrary to forced convection flows, the fluctuations of the heat transfer rate are uncorrelated with the fluctuations of the wall shear stress, which exhibit a symmetric probability density function. At the low Rayleigh number considered, the large-scale structures, which consist mainly in two counter-rotating vortices, with sizes comparable to the separation of the walls, are responsible for the extreme fluctuations of the wall heat transfer rate. The occurrence and the averaged topology of these structures have been determined using a conditional sampling technique. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The fluid flow driven by turbulent natural convection in a vertical channel is of practical and fundamental importance. This flow has implications for problems as the heating and cooling of building spaces, the insulation properties of double paned windows or the cooling of electronic components. From a fundamental point of view, this flow configuration is suitable for studying the production and sustainment of turbulence by buoyancy and by shear, generated as a consequence of buoyancy. Direct numerical simulations (DNS) of this flow are reported and analyzed by, Philips [1], Boudjemadi et al. [2], Versteegh and Nieuwstadt [3,4] and Tsujimoto et al. [5]. The determination and analyses of the averaged momentum and wall heat transfer rate generated by turbulent flows provide quantitative information useful for design purposes and to understand the overall interaction of the flow with the wall. The study of the statistics and the dynamics of the instantaneous fluctuations of the heat transfer rate can be considered a step forward in the understanding of this interaction. Particularly, it is interesting to detect and to describe the dynamics of the near wall flow structures since they are responsible for the instantaneous fluctuations of the wall transfer rates. Knowing how these structures evolve and interact with the wall provide information that can be used to develop efficient techniques to manipulate the flow in order to obtain a desired modification of the wall transfer rates. This is the foundation of some control techniques recently developed that
* Corresponding author. Tel.: +34 977 559 682; fax: +34 977 559 691. E-mail address: [email protected] (J. Pallares). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.05.039
are oriented to reduce the skin friction in turbulent boundary layers [6]. Contrary to turbulent forced convection flows in channels or natural convection flows in horizontal layers (Rayleigh–Bénard convection), the analyses of the instantaneous flow structures in natural convection flows in vertical channels are scarce in the literature and, to the authors’ knowledge, their role in the production of turbulence and in the fluctuations of the wall transfer rates has not been reported. In this study, we analyze a database of a DNS of turbulent natural convection in a vertical channel to extract the topology of the flow structures that produce extreme fluctuations of wall heat transfer rate. The physical and the mathematical models are described in Section 2, the conditional sampling technique used is outlined in Section 3 and results are presented in Section 4.
2. Physical and mathematical models Fig. 1 shows the coordinate system and the computational domain of the vertical channel. The natural convection flow, driven by the temperature difference imposed at the walls of the channel, is assumed to be hydrodynamically and thermally fully developed. The two walls of the channel located at y = d/2 and y = d/2 are rigid, smooth and they are kept at constant but different temperatures. All physical properties of the fluid, with a Prandtl number (Pr = m/a) of 0.7, are considered constant except for the linear variation of the density with temperature which is considered only in the buoyancy term according to the Boussinesq approximation. The viscous dissipation and the radiation heat transfer are neglected.
J. Pallares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4168–4175
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Nomenclature d g h k L Nu p Pr q00 Ra T t u, v, w x, y, z
channel width (m) gravitational acceleration (m s2) convective heat transfer coefficient (W m2 K1) thermal conductivity (W m1 K1) length (m) Nusselt number, Nu = hd/k pressure (N m2) Prandtl number, Pr = m/a heat flux density (W m2) Rayleigh number, Ra = gbDNd3/(ma) temperature (K) time (s) Cartesian velocity components (m s1) Cartesian coordinates (m)
d D k2
l m h
q s
Superscripts and subscripts c cold h hot o reference w wall 0 fluctuation hi time-averaged
Greek letters a thermal diffusion coefficient (m s2) b thermal coefficient of expansion (K1)
d
12.5d
6.3d
Hot wall
Cold wall
Th
Tc
g
zx y
Fig. 1. Physical model and coordinate system.
The non-dimensional continuity, Navier–Stokes and thermal energy equations are
@ui ¼ 0; @xi @ui @uj ui @p @ 2 ui ¼ þ Pr þ Ra Pr hdi1 þ @xi @t @xj @xj @xj
ð1Þ ð2Þ
and
@h @uj h @2h þ ¼ ; @t @xj @xj @xj
Kronecker’s delta increment second largest eigenvalue of the velocity gradient tensor (s1) dynamic viscosity (kg m1 s1) kinematic viscosity (m s2) non-dimensional temperature density (kg m3) wall shear stress (N m2)
ð3Þ
respectively. The scales used to obtain the non-dimensional variables are the channel width (d) and the thermal diffusion time (d2/a). The nondimensional temperature is defined as h = (T To)/(Th Tc) where Th and Tc are the temperatures of the hot and cold wall, respectively, To = (Th + Tc)/2 is the mean temperature and DT = Th Tc is the temperature increment. The last term on the right-hand side of Eq. (2) corresponds to the non-dimensional buoyancy acceleration along the, vertical, xdirection. In Eq. (2) Ra is the Rayleigh number, Ra = gbDTd3/(ma) which has been set to 5.4 105 in order to compare and validate the present simulation with the DNS of Versteegh and Nieuwstadt
[3,4]. Periodic boundary conditions for velocities and temperature are applied along the homogeneous x and z-directions. The nonslip condition is imposed at the walls. The thermal boundary conditions are h = 0.5 and h = 0.5 at the hot and cold wall, respectively. The governing transport equations (Eqs. (1)–(3)), together with the boundary conditions are solved numerically with the 3DINAMICS code. This second-order accuracy finite volume code uses central differencing of the diffusive and convective terms on a staggered grid and Crank-Nicolson and Adams-Bashforth schemes for the temporal discretization. The Poisson equation resulting from the coupling between the velocity and pressure fields is solved with an efficient parallel multigrid solver. More details can be found in Fabregat [7]. The code has been used successfully in DNS of forced [8], mixed [9] convection turbulent flows. The computational domain, with dimensions Lx = 12.5 d, Ly = d, Lz = 6.3 d, is divided into 121 100 121 grid nodes. They are uniformly distributed along the streamwise and spanwise homogeneous directions (Dx 0.1 d and Dz 0.05 d) in which periodic boundary conditions are imposed, while hyperbolic tangent distributions are used to stretch the nodes near the walls (Dymin 1.7 103 d and Dymax 2.2 102 d where the noslip condition is applied. The time step used to integrate numerically the transport equations is Dt = 5 105 d2/a. Fig. 2 shows the time averaged velocity and temperature profiles as well as the Reynolds stresses. It can be seen that the present predictions are in good agreement with the DNS of Versteegh and Nieuwstadt [4] performed with a second order finite volume code. These authors used the Richardson interpolation of the results obtained with two different grid resolutions (432 96 216 and 180 48 90) to increase the accuracy of the simulations. Although the effect of the smallest scales is not completely captured by our grid resolution (121 100 121), the agreement shown in Fig. 2 indicates that this effect does not affect considerably the primary statistics of the flow and the main characteristics of the large flow structures analyzed in the present study. 3. Conditional sampling technique The time series of the wall heat transfer rate are needed to detect the flow structures responsible of their extreme values. When the flow was statistically fully developed the time evolution of the instantaneous velocity, pressure and temperature fields in the
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instantaneous wall heat fluxes (q00 )0 , were computed assuming a linear variation of the instantaneous temperatures within the distance between the first near wall grid node located at y = 1.7 103 d from the wall. The conditional pattern recognition technique used is based on the detection of extreme values of the fluctuations of the local wall heat transfer rate in the instantaneous distributions of q00 on the walls of the channel. This technique has been previously applied to educe the flow structures responsible for large fluctuations of wall transfer rates in forced [8] and mixed [9] convection turbulent channel flows. Fig. 3 illustrates the main steps of the pattern recognition technique. As indicated in Step 1 of Fig. 3, the cross-correlation coefficients, between the instantaneous distributions of (q00 )0 and a template, are stored in a three-dimensional matrix (i.e. a twodimensional matrix for each time step). As an example, Fig. 4a shows an instantaneous distribution of the fluctuations of the heat transfer rate on the hot wall. It can be seen that the regions with large positive fluctuations, indicated in black, have different shapes and sizes and they coexist with regions of extreme negative fluctuations, indicated in white. On the hot wall these regions with extreme fluctuations appear, evolve and disappear while are advected, mainly, along the positive streamwise direction, but also along the spanwise direction. The template used to detect the large positive fluctuations is shown in Fig. 4b. It consists in a positive lobe with a diameter of, approximately, d. The continuous grey line contours in Fig. 4a enclose regions where the correlation between the instantaneous fluctuations of q00 and the template is larger than 0.7. The dashed grey line contours correspond to regions where the correlation attains extreme negative values. As sketched in Fig. 3, the time evolution of the spatial distributions of the fluctuations produces that the regions where the maxima of the correlation coefficients appear as elongated volumes in the 3D-matrix of the correlation coefficients according to the fact that the extreme values of the wall heat transfer rate move along the streamwise and spanwise directions. Within these volumes, which correspond to the detected events, only the region of the plane of time where the absolute maximum of correlation occurs is selected to obtain the ensemble average of the wall event (see Step 2 in Fig. 3). This procedure prevents the selection of the different stages of the same event at different times. The positions and the time at which the selected events occur are stored in a file (see Step 2 in Fig. 3). This information is used to obtain the conditional ensemble average of the flow near the wall when and where the selected wall events occur (see Step 3 in Fig. 3).
4. Results and discussion
Fig. 2. Time averaged (a) velocity and (b) temperature profiles. (c) Reynolds stresses. The lines correspond to DNS of Versteegh and Nieuwstadt [4].
computational domain during 0.5 non-dimensional time units are recorded. This period is about 300 times the integral time scale of the history of the local instantaneous heat flux density. The
The probability density functions (pdf) of the fluctuations of the wall heat transfer rate and of the fluctuations of the streamwise wall shear stress on the hot wall are shown in Fig. 5a. It can be seen that the negative fluctuations of q00 are more probable (62%) that the positive fluctuations (38%). About 8% of the fluctuations are larger than the averaged value ði:e Nu0 =hNui1Þ. The distribution of the fluctuations of the streamwise component of the wall shear stress on the hot wall has a high degree of symmetry. However, the relative large values of the positive tail of the pdf produce a skewness of 0.2, in agreement with the value reported by Tsujimoto et al. [5], and a kurtosis of 7. It is known that the application of flow control techniques to obtain modifications of the wall heat transfer rate of the turbulent forced convection in a plane channel affects directly the wall shear stress, because of the similarity between both wall transfer rates [10]. The joint probability function between the fluctuations of q00 and the fluctuations of the streamwise component of the wall
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Instantaneous distributions of (q”)’
Correlation coefficients 3D matrix
t+n Δt
. . .
Step 1
Cross-correlation
t+2 Δt
Template
t+ Δt
. . .
t
t
x z
Step 2
x z
Positions and time of each event
Step 3
event 1 x z t1 event 2 x z t1 event 3 x z t2
. . .
t1
Ensemble average of τ’, (q”)’, p’
. . .
t2
Ensemble average of the flow
y
x z
y
x z
Fig. 3. Steps of the conditional sampling procedure.
shear stress plotted in Fig. 5b shows that, contrary to the forced convection turbulent channel flow [8], these quantities do not show a high degree of correlation. This differential feature of the natural convection flow analyzed here with respect to the forced convection, suggests the possibility of the manipulation of the flow by means of a control technique to obtain a modification of heat transfer rate without the direct modification of the wall shear stress. Fig. 6 shows the streamwise (Fig. 6a) and spanwise (Fig. 6b) autocorrelations of the fluctuations of the heat transfer rate on the hot wall. Fig. 6a includes the decay of the streamwise autocorrelation at different time lags. It can be seen that the fluctuations have a typical size of order d along the streamwise and spanwise directions. The averaged velocity of the fluctuations along the streamwise direction, estimated from the displacement of the position of the maximum of the autocorrelation at different time lags is 140 a/d which is the averaged velocity of the flow at a distance of 0.016 d from the wall (see Fig. 2a). We applied the conditional sampling technique described in Section 3 to obtain the averaged flow structure responsible for the large positive fluctuations of the heat transfer rate on the hot and cold walls, as well as the averaged flow structure responsible for large negative fluctuations of the heat transfer rate on the hot and cold walls. Table 1 summarizes the number of the events detected on the hot and cold walls, and the number of events selected for averaging. As explained in Section 3, the selected events correspond to
the time and the location where the maximum correlation between the template and the detected event occurs. The finite number of samples used to obtain the ensemble-averaged distribution of the fluctuations of the heat transfer rate produces an estimated averaged error of the mean values of about 5%. As it will be shown later, the footprints of the flow structures on the walls have dimensions of about 3 d and 2 d along the streamwise (x) and spanwise (z) directions, respectively. Considering the dimensions of the walls of the computational domain (Lx = 12.5 d, Lz = 6.3 d) and the data of Table 1, the events detected associated with positive fluctuations of Nu contribute about 16% to the overall history of the wall heat transfer rate while the contribution of those producing large negative fluctuations is about 33%. The larger contributions of the negative fluctuations in comparison with those of the positive fluctuations agree with the larger probability of the negative fluctuations shown in Fig. 5a. The average duration of the detected events can be estimated from the ratio between the number of detected events and the number of selected events and the time step between two consecutive flow fields in the database (Dt = 104 d2/a i.e. average duration Dt [detected events/selected events]. According to this estimation, the events associated with positive fluctuations of q” last, on average, about 9 104 d2/a [or 0.6 (d/gbDT)1/2], while the average duration of the events associated with negative fluctuations is approximately 7 104 d2/a [or 0.4 (d/gbDT)1/2]. Note that these times correspond to a short streamwise displacement of the maximum of the autocorrelation shown in Fig. 6a. In fact, these times are about
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Fig. 4. (a) Instantaneous distribution of the fluctuations of the wall heat transfer rate on the hot wall. The white color corresponds to the extreme negative fluctuations, the grey, to the moderate positive and negative fluctuations and the black, to the large positive fluctuations. The continuous line contours indicate the regions where the correlation of the template, shown in (b) and the instantaneous distribution is larger than 0.7 while the dashed line contours indicate where the correlation is lower than 0.7. (b) Template used for the determination of the regions with large positive fluctuations.
two orders of magnitude smaller than the through-flow time, which is about 0.082 d2/a considering an averaged streamwise velocity in the channel half-width of 152 a/d (see Fig. 2a). The contribution to the overall turbulent kinetic energy of the flow of the events detected producing large positive fluctuations of the heat transfer rate ranges from 16% near the center of the channel to 20% near the wall. Fig. 7 shows the conditionally averaged flow structure responsible for the large positive fluctuations of the heat transfer rate on the hot wall. The isosurface corresponds to a negative value of the second largest eigenvalue of the velocity gradient tensor (k2). A quantity proposed by Jeong and Hussain [11] to detect the occurrence of vortex cores. It can be seen that the flow structure has a hairpin shaped topology. The legs of the hairpin are two counterrotating vortices that are inclined about 30° on the x–y plane and tilted about ±30° on the x–z plane. The vortices of the legs advect cold fluid towards the hot wall and they produce a maximum of the wall heat transfer rate in the region between them, as shown in Fig. 7b. The injection of cold fluid towards the hot wall caused by the legs of the hairpin vortices produces a descending flow near the hot wall, where the flow, on average is ascending (see Fig. 2a). The equivalent flow structure near the cold wall can be obtained by mirroring the flow structure near the hot wall, shown in Fig. 7, firstly, with respect to the plane x = 0 and secondly, with respect to the plane y = 0. The conditional averaged flow structure obtained from the events associated with negative fluctuations of the heat transfer rate on the hot wall is shown in Fig. 8. This flow structure consists
in two counter-rotating vortices that convect fluid from the region between them towards the external part of the vortices, as shown in Fig. 8c. The inclination of the vortices on the x–y plane is about 30° and the tilting angle, on the x–z plane is 20°. The vortices produce an ejection of fluid from the wall, which has associated the imprint depicted in Fig. 8b. It can be seen in Fig. 8 that the induced negative fluctuations of the heat transfer rate are located between the legs of the vortices, which accelerate hot fluid near the hot wall along the positive streamwise direction. The corresponding flow structure responsible for the extreme negative fluctuations of the heat transfer rate on the cold wall can be obtained by mirroring the flow structure near the hot wall, shown in Fig. 8, firstly, with respect to the plane x = 0 and secondly, with respect to the plane y = 0. The extensions of the imprints corresponding to the extreme fluctuations of the heat transfer rate, with approximate dimensions lx = d, lz = d, as shown in Figs. 7b and 8b, is in agreement with the distances at which the autocorrelations of Nu’ reach low values (see Fig. 6). As shown in Fig. 7b, the extreme values of the positive fluctuations of the wall heat transfer rate induced by the flow structure on the wall are about 1.5 times the value of the averaged heat transfer rate, indicating that these events contribute to the positive tail of the pdf of Nu’ shown in Fig. 5a. The scale of the contour legend of Fig. 7b is indicated by the right arrow in Fig. 5a. Correspondingly, the averaged extreme negative fluctuations detected by the conditional sampling (Fig. 8b) are about 0.6 times hNui and they contribute to the pdf in the range indicated by the left arrow in Fig. 5a, which correspond to the more probable fluctuations.
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1
(a)
Δt=0
R [ (q")' t ,(q")' t+Δt ]
0.8
Δt=5 10 d /α -4
0.6
0.4
2
Δt=10 d /α -3
2
Δt=2 10-3 d2/α
0.2
Δt=3 10 d /α -3
2
0
0
0.5
1
1.5
2
2.5
3
x 1
(b)
R [ (q")' ,(q")' ]
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
z Fig. 5. (a) Probability density functions of the fluctuations of the wall heat transfer rate and the fluctuations of the streamwise wall shear stress. The arrows indicate the ranges of the fluctuations of q00 shown in Figs. 7 and 8. (b) Joint probability density function of the fluctuations of the heat transfer rate and the streamwise wall shear stress.
Tsujimoto et al. [5] analyzed the same flow, at the same Rayleigh and Prandtl numbers, using the proper orthogonal decomposition. Their analyses show that the reconstruction of the flow needs a relatively large number of modes. For example, the first ten modes for the reconstruction of the velocity field contain about 20% of the energy, being the contribution of each individual mode to the total energy between 1% and 3%. As a comparison, the first eigenfunction for fully developed turbulent plane channel flow is shown to account for 50% of the energy [12]. According to Tsujimoto et al. [5], for the turbulent natural convection flow in a vertical channel, the mode that makes the largest contribution to both the velocity and thermal fields corresponds to elongated streamwise counter-rotating vortices with a wavelength of about 2d along the spanwise direction and a wavelength of about 12d along the streamwise direction. This mode contributes about 2% to the total energy. As shown in Fig. 7b, the spanwise dimension of the flow structure responsible for the large positive fluctuations of the heat
Fig. 6. Spatial autocorrelation of the fluctuations of the wall heat transfer rate along the (a) streamwise direction and (b) along the spanwise direction.
Table 1 Number of events detected in the database and number of selected for the calculation of the conditional sampling average. The database contains 5000 times steps. The time increment between two consecutive flow fields is 104 d2/a. Hot wall
Cold wall
Positive fluctuations of Nu
Events detected Events selected
10936 1151
9813 1062
Negative fluctuations of Nu
Events detected Events selected
20585 2792
23032 3198
transfer rate agrees with the spanwise wavelength for the most energetic mode reported by Tsujimoto et al. [5]. However the conditional sampling technique applied in this study indicates that the streamwise dimension of the flow structure is about 3d, as shown in Fig. 7b and that the counter-rotating vortices are not perfectly aligned with the streamwise direction. Fig. 9 shows the cross-correlation between the fluctuations of the wall heat transfer rate on the hot and on the cold wall. It can be seen that the level of correlation is, in general, weak. This is in agreement with the dimension along the y-direction of the aver-
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Fig. 7. Conditionally averaged flow structure producing a large positive fluctuation of the heat transfer rate on the hot wall. The isosurface corresponds to a negative value of k2 [11]. (a) Lateral view. (b) Top view with the vector field of the fluctuations of the wall shear stress and the contours of the fluctuations of the wall heat transfer rate. The isosurface is depicted translucent to show the vector and contour fields underneath. (c) Three-dimensional view with some particle paths released from the line x = 1, y = 0 that follow the conditionally averaged flow field.
Fig. 8. Conditionally averaged flow structure producing an extreme negative fluctuation of the heat transfer rate on the hot wall. The isosurface corresponds to a negative value of k2 [11]. (a) Lateral view. (b) Top view with the vector field of the fluctuations of the wall shear stress and the contours of the fluctuations of the wall heat transfer rate. The isosurface is depicted translucent to show the vector and contour fields underneath. (c) Three-dimensional view with some particle paths released from the line x = 1, y = 0.2 that follow the conditionally averaged flow field.
aged flow structures shown in Figs. 7 and 8, which is about half of the distance between the walls. It can be seen in Fig. 9 that there is a region centered at x = 0, z = 0, (i.e. at zero displacement) with a minimum of correlation, and two regions symmetrically distributed, with respect to z = 0, with weak positive correlation. The cross-correlation indicates that some of the large-scale structures detected by the conditional sampling technique produce simultaneously fluctuations on both walls. In fact, the inspection of the individual events shows that about 13% of the events detected on the hot wall associated with large positive fluctuations of the heat transfer rate have weak negative fluctuations on the cold wall at the same x and z positions. In these cases, the vortex motions, that produces large positive fluctuations of q00 on the hot wall (Fig. 7c),
also induces positive fluctuations of the heat transfer rate on the cold wall at locations displaced by the position of the regions of positive correlation indicated in Fig. 9. 5. Conclusions We analyzed a database obtained from a direct numerical simulation of the turbulent natural convection flow in a vertical plane channel. It is shown that, contrary to the turbulent forced convection heat transfer in a plane channel flow, the fluctuations of the heat transfer rate are uncorrelated with the fluctuations of the streamwise component of the wall shear stress. This suggests the possibility that the individual modification of one of these two wall
J. Pallares et al. / International Journal of Heat and Mass Transfer 53 (2010) 4168–4175
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from the near-wall region towards the center of the channel. The duration of the events is about a half of the buoyancy time scale. The contributions to the overall flow history of the events detected are, respectively, about 16% and 33% for the events responsible for the large positive and extreme negative fluctuations of the wall heat transfer rate.
1
0.5 -0.04 -0.06
0 -0.08
Acknowledgements
-0.5
x
-0.02
The Spanish Ministry of Science of Technology and FEDER financially supported this study under project DPI2006-0477.
-1 0.03
References
0.03
-1.5
-2 0.02 0.02
-2.5
-3 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
z Fig. 9. Cross-correlation of the fluctuations of the wall heat transfer rate on the hot and cold walls.
transfer rates, by means of an adequate control technique, would not affect directly the other. A conditional sampling technique has been applied to detect the flow structures responsible for the extreme fluctuations of the heat transfer rate. It has been found that the flow structures that produce large positive fluctuations of the wall heat transfer rate consist in two counter-rotating vortices that convect large temperature fluctuations from the center of the channel towards the wall. These strong inrushes generate instantaneous flow reversals near the walls (i.e. instantaneous descending flows near the hot wall and instantaneous ascending flows near the cold wall). The flow structures that produce large negative fluctuations of the wall heat transfer rate are more probable than those producing positive fluctuations. They are associated with a pair of counter-rotating vortices that convect flow
[1] J.R. Philips, Direct simulations of turbulent unstratified natural convection in a vertical slot for Pr = 0.71, Int. J. Heat Mass Transfer 39 (1996) 2485–2494. [2] R. Boudjemadi, V. Maupu, D. Laurence, P. Le Quéré, Budgets of turbulent stresses and fluxes in a vertical slot natural convection flow at Rayleigh Ra = 105 and 5.4 105, Int. J. Heat Fluid Flow 18 (1997) 70–79. [3] T.A.M. Versteegh, F.T.M. Nieuwstadt, Turbulent budgets of natural convection in an infinite, differentially heated, vertical channel, Int. J. Heat Fluid Flow 19 (1998) 135–149. [4] T.A.M. Versteegh, F.T.M. Nieuwstadt, A direct numerical simulation of natural convection between two infinite vertical differentially heated walls: scaling laws and wall functions, Int. J. Heat Mass Transfer 42 (1999) 3673–3693. [5] K. Tsujimoto, N. Ohya, T. Shakouchi, T. Ando, Analysis of turbulent structures of natural convection in vertical channel using DNS, in: K. Hanjalic, Y. Nagano, S. Jakirlic (Eds.), Turbulence, Heat and Mass Transfer, vol. 5, 2006. [6] N. Kasagi, Y. Suzuki, K. Fukagata, Microelectromechanical systems-based feedback control of turbulence for skin friction reduction, Ann. Rev. Fluid Mech. 41 (2009) 231–251. [7] A. Fabregat, Direct numerical simulation of turbulent dispersion of buoyant plumes in a pressure-driven channel flow, Ph.D. Thesis, Department of Mechanical Engineering, Universitat Rovira i Virgili; 2006. [8] J. Pallares, A. Vernet, J.A. Ferré, F.X. Grau, Eduction of near-wall flow structures responsible for large deviations of the momentum-heat transfer analogy and fluctuations of wall transfer rates in turbulent channel flow, Comput. Fluids 35 (2007) 1327–1334. [9] A. Fabregat, J. Pallares, A. Vernet, I. Cuesta, J.A. Ferré, F.X. Grau, Identification of near-wall flow structures producing large wall transfer rates in turbulent mixed convection channel flow, Comput. Fluids 39 (2010) 15–24. [10] K. Fukagata, K. Iwamoto, N. Kasagi, Novel turbulence control strategy for simultaneously achieving friction drag reduction and heat transfer augmentation, in: Proceedings of the 4th International Symposium on Turbulence and Shear Flow Phenomena, Williamsburg, Virginia, 2005, pp. 307–312. [11] J. Jeong, F. Hussain, On the identification of a vortex, J. Fluid Mech. 285 (1995) 69–80. [12] P. Moin, R.D. Moser, Characteristic-eddy decomposition of turbulence in a channel, J. Fluid Mech. 200 (1989) 471–509.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Turbulent large-scale structures in natural convection vertical channel flow J. Pallares *, A. Vernet, J.A. Ferre, F.X. Grau Department of Mechanical Engineering, University Rovira i Virgili, Av. Països Catalans, 26, 43007 Tarragona, Spain
a r t i c l e
i n f o
Article history: Received 11 December 2009 Accepted 11 March 2010 Available online 11 June 2010 Keywords: Turbulent natural convection Vertical channel Heat transfer rate Direct numerical simulation Conditional sampling Large-scale structures
a b s t r a c t We analyzed a database of a direct numerical simulation of natural convection in a vertical channel. The flow is driven by a constant temperature difference imposed at the walls (Ra = 5.4 105, Pr = 0.7). The averaged flow and turbulent statistics are in good agreement with previous direct numerical simulations reported in the literature. Contrary to forced convection flows, the fluctuations of the heat transfer rate are uncorrelated with the fluctuations of the wall shear stress, which exhibit a symmetric probability density function. At the low Rayleigh number considered, the large-scale structures, which consist mainly in two counter-rotating vortices, with sizes comparable to the separation of the walls, are responsible for the extreme fluctuations of the wall heat transfer rate. The occurrence and the averaged topology of these structures have been determined using a conditional sampling technique. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The fluid flow driven by turbulent natural convection in a vertical channel is of practical and fundamental importance. This flow has implications for problems as the heating and cooling of building spaces, the insulation properties of double paned windows or the cooling of electronic components. From a fundamental point of view, this flow configuration is suitable for studying the production and sustainment of turbulence by buoyancy and by shear, generated as a consequence of buoyancy. Direct numerical simulations (DNS) of this flow are reported and analyzed by, Philips [1], Boudjemadi et al. [2], Versteegh and Nieuwstadt [3,4] and Tsujimoto et al. [5]. The determination and analyses of the averaged momentum and wall heat transfer rate generated by turbulent flows provide quantitative information useful for design purposes and to understand the overall interaction of the flow with the wall. The study of the statistics and the dynamics of the instantaneous fluctuations of the heat transfer rate can be considered a step forward in the understanding of this interaction. Particularly, it is interesting to detect and to describe the dynamics of the near wall flow structures since they are responsible for the instantaneous fluctuations of the wall transfer rates. Knowing how these structures evolve and interact with the wall provide information that can be used to develop efficient techniques to manipulate the flow in order to obtain a desired modification of the wall transfer rates. This is the foundation of some control techniques recently developed that
* Corresponding author. Tel.: +34 977 559 682; fax: +34 977 559 691. E-mail address: [email protected] (J. Pallares). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.05.039
are oriented to reduce the skin friction in turbulent boundary layers [6]. Contrary to turbulent forced convection flows in channels or natural convection flows in horizontal layers (Rayleigh–Bénard convection), the analyses of the instantaneous flow structures in natural convection flows in vertical channels are scarce in the literature and, to the authors’ knowledge, their role in the production of turbulence and in the fluctuations of the wall transfer rates has not been reported. In this study, we analyze a database of a DNS of turbulent natural convection in a vertical channel to extract the topology of the flow structures that produce extreme fluctuations of wall heat transfer rate. The physical and the mathematical models are described in Section 2, the conditional sampling technique used is outlined in Section 3 and results are presented in Section 4.
2. Physical and mathematical models Fig. 1 shows the coordinate system and the computational domain of the vertical channel. The natural convection flow, driven by the temperature difference imposed at the walls of the channel, is assumed to be hydrodynamically and thermally fully developed. The two walls of the channel located at y = d/2 and y = d/2 are rigid, smooth and they are kept at constant but different temperatures. All physical properties of the fluid, with a Prandtl number (Pr = m/a) of 0.7, are considered constant except for the linear variation of the density with temperature which is considered only in the buoyancy term according to the Boussinesq approximation. The viscous dissipation and the radiation heat transfer are neglected.
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Nomenclature d g h k L Nu p Pr q00 Ra T t u, v, w x, y, z
channel width (m) gravitational acceleration (m s2) convective heat transfer coefficient (W m2 K1) thermal conductivity (W m1 K1) length (m) Nusselt number, Nu = hd/k pressure (N m2) Prandtl number, Pr = m/a heat flux density (W m2) Rayleigh number, Ra = gbDNd3/(ma) temperature (K) time (s) Cartesian velocity components (m s1) Cartesian coordinates (m)
d D k2
l m h
q s
Superscripts and subscripts c cold h hot o reference w wall 0 fluctuation hi time-averaged
Greek letters a thermal diffusion coefficient (m s2) b thermal coefficient of expansion (K1)
d
12.5d
6.3d
Hot wall
Cold wall
Th
Tc
g
zx y
Fig. 1. Physical model and coordinate system.
The non-dimensional continuity, Navier–Stokes and thermal energy equations are
@ui ¼ 0; @xi @ui @uj ui @p @ 2 ui ¼ þ Pr þ Ra Pr hdi1 þ @xi @t @xj @xj @xj
ð1Þ ð2Þ
and
@h @uj h @2h þ ¼ ; @t @xj @xj @xj
Kronecker’s delta increment second largest eigenvalue of the velocity gradient tensor (s1) dynamic viscosity (kg m1 s1) kinematic viscosity (m s2) non-dimensional temperature density (kg m3) wall shear stress (N m2)
ð3Þ
respectively. The scales used to obtain the non-dimensional variables are the channel width (d) and the thermal diffusion time (d2/a). The nondimensional temperature is defined as h = (T To)/(Th Tc) where Th and Tc are the temperatures of the hot and cold wall, respectively, To = (Th + Tc)/2 is the mean temperature and DT = Th Tc is the temperature increment. The last term on the right-hand side of Eq. (2) corresponds to the non-dimensional buoyancy acceleration along the, vertical, xdirection. In Eq. (2) Ra is the Rayleigh number, Ra = gbDTd3/(ma) which has been set to 5.4 105 in order to compare and validate the present simulation with the DNS of Versteegh and Nieuwstadt
[3,4]. Periodic boundary conditions for velocities and temperature are applied along the homogeneous x and z-directions. The nonslip condition is imposed at the walls. The thermal boundary conditions are h = 0.5 and h = 0.5 at the hot and cold wall, respectively. The governing transport equations (Eqs. (1)–(3)), together with the boundary conditions are solved numerically with the 3DINAMICS code. This second-order accuracy finite volume code uses central differencing of the diffusive and convective terms on a staggered grid and Crank-Nicolson and Adams-Bashforth schemes for the temporal discretization. The Poisson equation resulting from the coupling between the velocity and pressure fields is solved with an efficient parallel multigrid solver. More details can be found in Fabregat [7]. The code has been used successfully in DNS of forced [8], mixed [9] convection turbulent flows. The computational domain, with dimensions Lx = 12.5 d, Ly = d, Lz = 6.3 d, is divided into 121 100 121 grid nodes. They are uniformly distributed along the streamwise and spanwise homogeneous directions (Dx 0.1 d and Dz 0.05 d) in which periodic boundary conditions are imposed, while hyperbolic tangent distributions are used to stretch the nodes near the walls (Dymin 1.7 103 d and Dymax 2.2 102 d where the noslip condition is applied. The time step used to integrate numerically the transport equations is Dt = 5 105 d2/a. Fig. 2 shows the time averaged velocity and temperature profiles as well as the Reynolds stresses. It can be seen that the present predictions are in good agreement with the DNS of Versteegh and Nieuwstadt [4] performed with a second order finite volume code. These authors used the Richardson interpolation of the results obtained with two different grid resolutions (432 96 216 and 180 48 90) to increase the accuracy of the simulations. Although the effect of the smallest scales is not completely captured by our grid resolution (121 100 121), the agreement shown in Fig. 2 indicates that this effect does not affect considerably the primary statistics of the flow and the main characteristics of the large flow structures analyzed in the present study. 3. Conditional sampling technique The time series of the wall heat transfer rate are needed to detect the flow structures responsible of their extreme values. When the flow was statistically fully developed the time evolution of the instantaneous velocity, pressure and temperature fields in the
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instantaneous wall heat fluxes (q00 )0 , were computed assuming a linear variation of the instantaneous temperatures within the distance between the first near wall grid node located at y = 1.7 103 d from the wall. The conditional pattern recognition technique used is based on the detection of extreme values of the fluctuations of the local wall heat transfer rate in the instantaneous distributions of q00 on the walls of the channel. This technique has been previously applied to educe the flow structures responsible for large fluctuations of wall transfer rates in forced [8] and mixed [9] convection turbulent channel flows. Fig. 3 illustrates the main steps of the pattern recognition technique. As indicated in Step 1 of Fig. 3, the cross-correlation coefficients, between the instantaneous distributions of (q00 )0 and a template, are stored in a three-dimensional matrix (i.e. a twodimensional matrix for each time step). As an example, Fig. 4a shows an instantaneous distribution of the fluctuations of the heat transfer rate on the hot wall. It can be seen that the regions with large positive fluctuations, indicated in black, have different shapes and sizes and they coexist with regions of extreme negative fluctuations, indicated in white. On the hot wall these regions with extreme fluctuations appear, evolve and disappear while are advected, mainly, along the positive streamwise direction, but also along the spanwise direction. The template used to detect the large positive fluctuations is shown in Fig. 4b. It consists in a positive lobe with a diameter of, approximately, d. The continuous grey line contours in Fig. 4a enclose regions where the correlation between the instantaneous fluctuations of q00 and the template is larger than 0.7. The dashed grey line contours correspond to regions where the correlation attains extreme negative values. As sketched in Fig. 3, the time evolution of the spatial distributions of the fluctuations produces that the regions where the maxima of the correlation coefficients appear as elongated volumes in the 3D-matrix of the correlation coefficients according to the fact that the extreme values of the wall heat transfer rate move along the streamwise and spanwise directions. Within these volumes, which correspond to the detected events, only the region of the plane of time where the absolute maximum of correlation occurs is selected to obtain the ensemble average of the wall event (see Step 2 in Fig. 3). This procedure prevents the selection of the different stages of the same event at different times. The positions and the time at which the selected events occur are stored in a file (see Step 2 in Fig. 3). This information is used to obtain the conditional ensemble average of the flow near the wall when and where the selected wall events occur (see Step 3 in Fig. 3).
4. Results and discussion
Fig. 2. Time averaged (a) velocity and (b) temperature profiles. (c) Reynolds stresses. The lines correspond to DNS of Versteegh and Nieuwstadt [4].
computational domain during 0.5 non-dimensional time units are recorded. This period is about 300 times the integral time scale of the history of the local instantaneous heat flux density. The
The probability density functions (pdf) of the fluctuations of the wall heat transfer rate and of the fluctuations of the streamwise wall shear stress on the hot wall are shown in Fig. 5a. It can be seen that the negative fluctuations of q00 are more probable (62%) that the positive fluctuations (38%). About 8% of the fluctuations are larger than the averaged value ði:e Nu0 =hNui1Þ. The distribution of the fluctuations of the streamwise component of the wall shear stress on the hot wall has a high degree of symmetry. However, the relative large values of the positive tail of the pdf produce a skewness of 0.2, in agreement with the value reported by Tsujimoto et al. [5], and a kurtosis of 7. It is known that the application of flow control techniques to obtain modifications of the wall heat transfer rate of the turbulent forced convection in a plane channel affects directly the wall shear stress, because of the similarity between both wall transfer rates [10]. The joint probability function between the fluctuations of q00 and the fluctuations of the streamwise component of the wall
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Instantaneous distributions of (q”)’
Correlation coefficients 3D matrix
t+n Δt
. . .
Step 1
Cross-correlation
t+2 Δt
Template
t+ Δt
. . .
t
t
x z
Step 2
x z
Positions and time of each event
Step 3
event 1 x z t1 event 2 x z t1 event 3 x z t2
. . .
t1
Ensemble average of τ’, (q”)’, p’
. . .
t2
Ensemble average of the flow
y
x z
y
x z
Fig. 3. Steps of the conditional sampling procedure.
shear stress plotted in Fig. 5b shows that, contrary to the forced convection turbulent channel flow [8], these quantities do not show a high degree of correlation. This differential feature of the natural convection flow analyzed here with respect to the forced convection, suggests the possibility of the manipulation of the flow by means of a control technique to obtain a modification of heat transfer rate without the direct modification of the wall shear stress. Fig. 6 shows the streamwise (Fig. 6a) and spanwise (Fig. 6b) autocorrelations of the fluctuations of the heat transfer rate on the hot wall. Fig. 6a includes the decay of the streamwise autocorrelation at different time lags. It can be seen that the fluctuations have a typical size of order d along the streamwise and spanwise directions. The averaged velocity of the fluctuations along the streamwise direction, estimated from the displacement of the position of the maximum of the autocorrelation at different time lags is 140 a/d which is the averaged velocity of the flow at a distance of 0.016 d from the wall (see Fig. 2a). We applied the conditional sampling technique described in Section 3 to obtain the averaged flow structure responsible for the large positive fluctuations of the heat transfer rate on the hot and cold walls, as well as the averaged flow structure responsible for large negative fluctuations of the heat transfer rate on the hot and cold walls. Table 1 summarizes the number of the events detected on the hot and cold walls, and the number of events selected for averaging. As explained in Section 3, the selected events correspond to
the time and the location where the maximum correlation between the template and the detected event occurs. The finite number of samples used to obtain the ensemble-averaged distribution of the fluctuations of the heat transfer rate produces an estimated averaged error of the mean values of about 5%. As it will be shown later, the footprints of the flow structures on the walls have dimensions of about 3 d and 2 d along the streamwise (x) and spanwise (z) directions, respectively. Considering the dimensions of the walls of the computational domain (Lx = 12.5 d, Lz = 6.3 d) and the data of Table 1, the events detected associated with positive fluctuations of Nu contribute about 16% to the overall history of the wall heat transfer rate while the contribution of those producing large negative fluctuations is about 33%. The larger contributions of the negative fluctuations in comparison with those of the positive fluctuations agree with the larger probability of the negative fluctuations shown in Fig. 5a. The average duration of the detected events can be estimated from the ratio between the number of detected events and the number of selected events and the time step between two consecutive flow fields in the database (Dt = 104 d2/a i.e. average duration Dt [detected events/selected events]. According to this estimation, the events associated with positive fluctuations of q” last, on average, about 9 104 d2/a [or 0.6 (d/gbDT)1/2], while the average duration of the events associated with negative fluctuations is approximately 7 104 d2/a [or 0.4 (d/gbDT)1/2]. Note that these times correspond to a short streamwise displacement of the maximum of the autocorrelation shown in Fig. 6a. In fact, these times are about
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Fig. 4. (a) Instantaneous distribution of the fluctuations of the wall heat transfer rate on the hot wall. The white color corresponds to the extreme negative fluctuations, the grey, to the moderate positive and negative fluctuations and the black, to the large positive fluctuations. The continuous line contours indicate the regions where the correlation of the template, shown in (b) and the instantaneous distribution is larger than 0.7 while the dashed line contours indicate where the correlation is lower than 0.7. (b) Template used for the determination of the regions with large positive fluctuations.
two orders of magnitude smaller than the through-flow time, which is about 0.082 d2/a considering an averaged streamwise velocity in the channel half-width of 152 a/d (see Fig. 2a). The contribution to the overall turbulent kinetic energy of the flow of the events detected producing large positive fluctuations of the heat transfer rate ranges from 16% near the center of the channel to 20% near the wall. Fig. 7 shows the conditionally averaged flow structure responsible for the large positive fluctuations of the heat transfer rate on the hot wall. The isosurface corresponds to a negative value of the second largest eigenvalue of the velocity gradient tensor (k2). A quantity proposed by Jeong and Hussain [11] to detect the occurrence of vortex cores. It can be seen that the flow structure has a hairpin shaped topology. The legs of the hairpin are two counterrotating vortices that are inclined about 30° on the x–y plane and tilted about ±30° on the x–z plane. The vortices of the legs advect cold fluid towards the hot wall and they produce a maximum of the wall heat transfer rate in the region between them, as shown in Fig. 7b. The injection of cold fluid towards the hot wall caused by the legs of the hairpin vortices produces a descending flow near the hot wall, where the flow, on average is ascending (see Fig. 2a). The equivalent flow structure near the cold wall can be obtained by mirroring the flow structure near the hot wall, shown in Fig. 7, firstly, with respect to the plane x = 0 and secondly, with respect to the plane y = 0. The conditional averaged flow structure obtained from the events associated with negative fluctuations of the heat transfer rate on the hot wall is shown in Fig. 8. This flow structure consists
in two counter-rotating vortices that convect fluid from the region between them towards the external part of the vortices, as shown in Fig. 8c. The inclination of the vortices on the x–y plane is about 30° and the tilting angle, on the x–z plane is 20°. The vortices produce an ejection of fluid from the wall, which has associated the imprint depicted in Fig. 8b. It can be seen in Fig. 8 that the induced negative fluctuations of the heat transfer rate are located between the legs of the vortices, which accelerate hot fluid near the hot wall along the positive streamwise direction. The corresponding flow structure responsible for the extreme negative fluctuations of the heat transfer rate on the cold wall can be obtained by mirroring the flow structure near the hot wall, shown in Fig. 8, firstly, with respect to the plane x = 0 and secondly, with respect to the plane y = 0. The extensions of the imprints corresponding to the extreme fluctuations of the heat transfer rate, with approximate dimensions lx = d, lz = d, as shown in Figs. 7b and 8b, is in agreement with the distances at which the autocorrelations of Nu’ reach low values (see Fig. 6). As shown in Fig. 7b, the extreme values of the positive fluctuations of the wall heat transfer rate induced by the flow structure on the wall are about 1.5 times the value of the averaged heat transfer rate, indicating that these events contribute to the positive tail of the pdf of Nu’ shown in Fig. 5a. The scale of the contour legend of Fig. 7b is indicated by the right arrow in Fig. 5a. Correspondingly, the averaged extreme negative fluctuations detected by the conditional sampling (Fig. 8b) are about 0.6 times hNui and they contribute to the pdf in the range indicated by the left arrow in Fig. 5a, which correspond to the more probable fluctuations.
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1
(a)
Δt=0
R [ (q")' t ,(q")' t+Δt ]
0.8
Δt=5 10 d /α -4
0.6
0.4
2
Δt=10 d /α -3
2
Δt=2 10-3 d2/α
0.2
Δt=3 10 d /α -3
2
0
0
0.5
1
1.5
2
2.5
3
x 1
(b)
R [ (q")' ,(q")' ]
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
z Fig. 5. (a) Probability density functions of the fluctuations of the wall heat transfer rate and the fluctuations of the streamwise wall shear stress. The arrows indicate the ranges of the fluctuations of q00 shown in Figs. 7 and 8. (b) Joint probability density function of the fluctuations of the heat transfer rate and the streamwise wall shear stress.
Tsujimoto et al. [5] analyzed the same flow, at the same Rayleigh and Prandtl numbers, using the proper orthogonal decomposition. Their analyses show that the reconstruction of the flow needs a relatively large number of modes. For example, the first ten modes for the reconstruction of the velocity field contain about 20% of the energy, being the contribution of each individual mode to the total energy between 1% and 3%. As a comparison, the first eigenfunction for fully developed turbulent plane channel flow is shown to account for 50% of the energy [12]. According to Tsujimoto et al. [5], for the turbulent natural convection flow in a vertical channel, the mode that makes the largest contribution to both the velocity and thermal fields corresponds to elongated streamwise counter-rotating vortices with a wavelength of about 2d along the spanwise direction and a wavelength of about 12d along the streamwise direction. This mode contributes about 2% to the total energy. As shown in Fig. 7b, the spanwise dimension of the flow structure responsible for the large positive fluctuations of the heat
Fig. 6. Spatial autocorrelation of the fluctuations of the wall heat transfer rate along the (a) streamwise direction and (b) along the spanwise direction.
Table 1 Number of events detected in the database and number of selected for the calculation of the conditional sampling average. The database contains 5000 times steps. The time increment between two consecutive flow fields is 104 d2/a. Hot wall
Cold wall
Positive fluctuations of Nu
Events detected Events selected
10936 1151
9813 1062
Negative fluctuations of Nu
Events detected Events selected
20585 2792
23032 3198
transfer rate agrees with the spanwise wavelength for the most energetic mode reported by Tsujimoto et al. [5]. However the conditional sampling technique applied in this study indicates that the streamwise dimension of the flow structure is about 3d, as shown in Fig. 7b and that the counter-rotating vortices are not perfectly aligned with the streamwise direction. Fig. 9 shows the cross-correlation between the fluctuations of the wall heat transfer rate on the hot and on the cold wall. It can be seen that the level of correlation is, in general, weak. This is in agreement with the dimension along the y-direction of the aver-
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Fig. 7. Conditionally averaged flow structure producing a large positive fluctuation of the heat transfer rate on the hot wall. The isosurface corresponds to a negative value of k2 [11]. (a) Lateral view. (b) Top view with the vector field of the fluctuations of the wall shear stress and the contours of the fluctuations of the wall heat transfer rate. The isosurface is depicted translucent to show the vector and contour fields underneath. (c) Three-dimensional view with some particle paths released from the line x = 1, y = 0 that follow the conditionally averaged flow field.
Fig. 8. Conditionally averaged flow structure producing an extreme negative fluctuation of the heat transfer rate on the hot wall. The isosurface corresponds to a negative value of k2 [11]. (a) Lateral view. (b) Top view with the vector field of the fluctuations of the wall shear stress and the contours of the fluctuations of the wall heat transfer rate. The isosurface is depicted translucent to show the vector and contour fields underneath. (c) Three-dimensional view with some particle paths released from the line x = 1, y = 0.2 that follow the conditionally averaged flow field.
aged flow structures shown in Figs. 7 and 8, which is about half of the distance between the walls. It can be seen in Fig. 9 that there is a region centered at x = 0, z = 0, (i.e. at zero displacement) with a minimum of correlation, and two regions symmetrically distributed, with respect to z = 0, with weak positive correlation. The cross-correlation indicates that some of the large-scale structures detected by the conditional sampling technique produce simultaneously fluctuations on both walls. In fact, the inspection of the individual events shows that about 13% of the events detected on the hot wall associated with large positive fluctuations of the heat transfer rate have weak negative fluctuations on the cold wall at the same x and z positions. In these cases, the vortex motions, that produces large positive fluctuations of q00 on the hot wall (Fig. 7c),
also induces positive fluctuations of the heat transfer rate on the cold wall at locations displaced by the position of the regions of positive correlation indicated in Fig. 9. 5. Conclusions We analyzed a database obtained from a direct numerical simulation of the turbulent natural convection flow in a vertical plane channel. It is shown that, contrary to the turbulent forced convection heat transfer in a plane channel flow, the fluctuations of the heat transfer rate are uncorrelated with the fluctuations of the streamwise component of the wall shear stress. This suggests the possibility that the individual modification of one of these two wall
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from the near-wall region towards the center of the channel. The duration of the events is about a half of the buoyancy time scale. The contributions to the overall flow history of the events detected are, respectively, about 16% and 33% for the events responsible for the large positive and extreme negative fluctuations of the wall heat transfer rate.
1
0.5 -0.04 -0.06
0 -0.08
Acknowledgements
-0.5
x
-0.02
The Spanish Ministry of Science of Technology and FEDER financially supported this study under project DPI2006-0477.
-1 0.03
References
0.03
-1.5
-2 0.02 0.02
-2.5
-3 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
z Fig. 9. Cross-correlation of the fluctuations of the wall heat transfer rate on the hot and cold walls.
transfer rates, by means of an adequate control technique, would not affect directly the other. A conditional sampling technique has been applied to detect the flow structures responsible for the extreme fluctuations of the heat transfer rate. It has been found that the flow structures that produce large positive fluctuations of the wall heat transfer rate consist in two counter-rotating vortices that convect large temperature fluctuations from the center of the channel towards the wall. These strong inrushes generate instantaneous flow reversals near the walls (i.e. instantaneous descending flows near the hot wall and instantaneous ascending flows near the cold wall). The flow structures that produce large negative fluctuations of the wall heat transfer rate are more probable than those producing positive fluctuations. They are associated with a pair of counter-rotating vortices that convect flow
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