PHYSICAL REVIEW E 90, 023015 (2014)
Natural convection in a fluid layer periodically heated from above M. Z. Hossain* and J. M. Floryan Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 (Received 13 May 2014; published 25 August 2014) Natural convection in a horizontal layer subject to periodic heating from above has been studied. It is shown that the primary convection leads to the cooling of the bulk of the fluid below the mean temperature of the upper wall. The secondary convection may lead either to longitudinal rolls, transverse rolls, or oblique rolls. The global flow properties (e.g., the average Nusselt number for the primary convection and the critical conditions for the secondary convection) are identical to those of the layer heated from below. However, the flow and temperature patterns exhibit phase shifts in the horizontal directions. DOI: 10.1103/PhysRevE.90.023015
PACS number(s): 47.20.−k, 44.25.+f
I. INTRODUCTION
It is well known that the buoyancy-driven instability of the conductive state of equilibrium appears only when heating is applied from below, i.e., Rayleigh-B´enard (RB) convection [1–3]. Laminar RB convection has been the origin of much research on nonlinear dynamics and pattern formation [4]. Recent efforts have been focused on the explanation of the roles played by various boundary imperfections [5,6]. A large enough heating intensity leads to turbulent RB convection which has many practical applications in the area of heat transfer [7,8]. Current efforts are focused on the use of boundary imperfections with an appropriate spatial pattern in order to increase the relevant Nusselt number [9]. It is less well known that the instability may occur when the fluid is heated from above (anticonvection) as long as the system contains at least one interface [10–13]. Anticonvection may occur in fluids with arbitrary properties when combined with heat release and extraction at the interface [14–17]; the current knowledge is summarized in [18,19]. Patterned heating represents another aspect of buoyancydriven convection. The heating represents an external forcing and convection represents a forced response which occurs regardless of the intensity of the heating and can have either a laminar or turbulent form. This structured convection is referred to as the primary convection. Its spatial pattern, which is locked in with the pattern of heating, affects contaminant transport in urban environments as local heating rates are determined by the dissimilar thermal properties of roofs, streets, and parks. It affects rural environments where local circulation is driven by variations in the heating rates of forests, fields, and lakes. It is known in geological applications that a system of fractures, leads, and polynyas in sea ice leads to convection in both the ocean and atmosphere with small leads of the order of several meters being more efficient than larger ones of the order of several hundred meters [20]. The insulating effect of continents on the mantle convection within the Earth represents a spatially structured convection with heating from above [21]. Further examples include systems of localized fires, systems of computer chips, thermal patterning in microfluidic devices, and so on. It has been demonstrated
that convection concentrates in the vicinity of the heated wall for large wave number heating while it penetrates into the complete flow domain for smaller wave number heating [6]. In the laminar case, the observable structure of convection may differ from that imposed by the heating pattern as the primary convection may undergo transition to secondary states. The onset conditions for the secondary convection are dictated by the interplay between two instability mechanisms, i.e., the RB mechanism and the spatial parametric resonance. A detailed description of the system response when the lower plate is exposed to a simple sinusoidal heating is given in [6]. Depending on the heating wave number, the secondary state may have the form of longitudinal rolls (rolls parallel to the primary rolls), transverse rolls (rolls orthogonal to the primary rolls), or oblique rolls, and the intensity of heating required to generate such states strongly depends on the heating wave number. It is not known how the fluid may respond when the same heating is applied at the upper plate. Less is known about the patterned turbulent convection but the available results suggest that significant intensification of convection can be expected when an appropriate pattern is used [12]. The present work is focused on the analysis of natural convection driven by periodic heating applied from above. The heating amplitude is expressed in terms of the periodic Rayleigh number Rap and its spatial distribution is characterized by the heating wave number α. Section II describes the problem formulation for the primary stationary convection and for its linear stability. Section III provides a description of the primary and secondary convections and draws an analogy with convection driven by the same heating applied at the lower wall. Section IV provides a short summary of the main conclusions.
y,v
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[email protected]
1539-3755/2014/90(2)/023015(7)
g
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1 2
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FIG. 1. Fluid layer with sinusoidal heating applied at the upper wall. 023015-1
©2014 American Physical Society
M. Z. HOSSAIN AND J. M. FLORYAN
PHYSICAL REVIEW E 90, 023015 (2014)
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FIG. 2. The flow topology and the isotherms for Rap = 400, α = 5 for the upper (a) and lower (b) heatings. Thick (thin) lines identify streamlines (isotherms); solid (dashed) lines identify positive (negative) temperature.
where u = (u,v,w), p, θ stand for the velocity, pressure, and temperature, respectively; j denotes the unit vector in the y direction; and Rap = gh3 Ta /νκ is the periodic Rayleigh number. The reader may note the difference between the periodic Rayleigh number Rap and the classical Rayleigh number which is based on the difference between the mean temperatures of both walls. The fluid has been assumed to be incompressible, Newtonian, with thermal conductivity k, specific heat per unit mass c, thermal diffusivity κ = k/ρc, kinematic viscosity ν, thermal expansion coefficient , and the Boussinesq approximation of density ρ variations. The half spacing between the plates h has been used as the length scale, Uv = ν/ h as the velocity scale, Pv = ρUv2 as the pressure scale, and Tv = Ta ν/κ as the temperature scale where Tv /Ta = Pr. Here Pr stands for the Prandtl number. System (2)–(4) is supplemented with homogeneous boundary conditions for the velocity and the temperature boundary conditions (1) are adjusted using the temperature scale Tv . The steady two-dimensional version of (2)–(4) describes the primary convection. The unknowns are repre sented as Fourier expansions ψ(x,y) = n=+∞ ϕ (n) (y)einαx n=−∞ (n) inαx and θ (x,y) = n=+∞ , where ψ stands for the n=−∞ φ (y)e stream function defined in the usual manner; the relevant solution has been determined numerically using the method
II. PROBLEM FORMULATION AND NUMERICAL SOLUTION
The problem formulation follows that given in [6] for a fluid heated from below. Consider a slot between two parallel plane plates extending to ±∞ in the x and z directions and placed at a distance 2h apart from each other with gravitational acceleration g acting in the negative y direction (Fig. 1). The lower plate is isothermal while the upper plate has x-periodic temperature variations, i.e., θ (x,h,z,t) = 12 Ta cos(αx), θ (x, − h,z,t) = 0.
(1)
In the above, θ = T − Tref denotes the relative temperature, T stands for the absolute temperature, and Tref denotes the reference temperature (temperature of the lower plate). Convection is described by the field equations of the form (2)
∂θ + (u · ∇)θ = Pr−1 (∇ 2 θ ), ∂t
(3)
∇ · u = 0,
(4)
3
5
(a)
5
10
yc= -1+2 10
15
-1
20
101
102 Rap 103
101
max
100 ) =45 10
) =72
10-1
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-1
10-1
Nuav
102 Rap 103 10 68 5 4 =3
104
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1 23
10
(c)
15
10-4 20
10-1
Nuav ) =45
10-2
) =72
10-3
-3 5 Rap 10 =10
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>
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=3 4 5 6 8 10
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10 1 10
>
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>
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-3
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yc
∂u + (u · ∇)u = −∇p + ∇ 2 u + Rap θ j, ∂t
10-4
1 23 -3 5 Rap 10 =10
10-3 5
10
15
10-5 20
(d)
FIG. 3. Variations of the location of the roll center yc as a function of α (a) and as a function of Rap (b), variations of the maximum of the stream function as a function of α and Rap (c) and variations of Nusselt number Nu as function of α and Rap (d). Solid (dotted) lines correspond to heating applied at the upper (lower [6]) wall in these figures. 023015-2
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FIG. 4. The mean temperature component for Rap = 400. Solid (dashed) lines correspond to the upper (lower [6]) heating.
described in [22]. The net heat transfer is expressed in terms of the Nusselt number, Nu = − Pr dφ (0) /dy|y=−1 . The stability analysis of the primary convection follows the method described in [6,23]. Small arbitrary three-dimensional disturbances of the form ud (x,y,z,t) = [gud (x,y),gvd (x,y),gwd (x,y)]ei[δx+βz−σ t] + c.c., θd (x,y,z,t) = gθd (x,y)ei[δx+βz−σ t] + c.c.
(5)
are imposed upon the stationary solution where ud = (ud ,vd ,wd ) is the disturbance velocity vector, θ d is the temperature disturbance, (δ, β) denotes the disturbance wave numbers in the (x, z) directions, respectively; the exponent σ is complex (σ = σr + iσi ) and its imaginary and real parts describe the rate of growth and the frequency of disturbances, respectively; and c.c. stands for the complex conjugate. The disturbance amplitudes are functions of x and y, and are periodic in x. The disturbance equations are linearized and the resulting system is posed as an eigenvalue problem for σ . The eigenvalues have been determined numerically using the method described in [6]. III. DISCUSSION OF RESULTS
The discussion will make frequent references to convection driven by the same heating but applied at the lower plate. Similarity with such convection can be demonstrated by (i) reversing the direction of gravity, i.e., Rap → −Rap , and (ii) changing the sign of the temperature, i.e., θ → −θ . The latter condition implies a change of the sign of the temperature imposed at the lower plate and results in a shift of the temperature field by half of a cycle in the x direction. All of the detailed results presented here are given for Pr = 0.71. Figure 2 illustrates the topology of the primary convection and temperature fields. The motion is driven by the cold (heavier) fluid concentrating around the cold spots at the upper plate and then descending. This fluid is turned sideways as it approaches the lower plate, then it is drawn upwards below the hot spots and pulled sideways as it approaches the upper plate resulting in the formation of pairs of counter-rotating rolls. The same figure displays the flow topology for the lower
4
6
8
10
FIG. 5. Variations of the critical periodic Rayleigh number Rap,cr as a function of α. Curves 1, 2, 3 correspond to the longitudinal, transverse, and oblique rolls, respectively. See [6] for derivation of the form of the asymptote of Rap,cr for α → ∞.
heating; the motion is driven by the lighter fluid rising above the hot spots and then turning sideways as it approaches the upper plate, forming an identical system of counter-rotating vortex pairs. There is remarkable up-down symmetry of the flow fields for both types of heating. The bulk of the fluid is cooled below the mean plate temperature for heating at the upper plate while its temperature rises above the mean plate temperature when the lower plate is heated. The temperature field exhibits an up-down symmetry combined with the phase shift of λα /2 in the x direction. The global properties of the primary convection are not affected by the location of the heating, e.g., convection penetrates the entire depth of the fluid layer for α = 0(1) but concentrates closer to the heated wall when α increases as illustrated by the variations of the distances between the roll centers and the heated plate [Figs. 3(a) and 3(b)]. Furthermore, the intensity of convection [Fig. 3(c)] and the Nusselt numbers [Fig. 3(d)] decrease when α increases but are the same for both types of heating for the same Rap and α. When α is large enough, a conduction layer forms away from the heated plate regardless of whether the upper or lower plates are exposed to the heating (see Figs. 2–4). The net heat transfer between the plates is driven by the mean component of the 2
2 cr
1.6
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cr
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1.6
=1.56 cr =1.56
cr
1.2 0.8
0.8 3
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0.4 0
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0 10
FIG. 6. Variations of the critical wave numbers δcr and βcr as functions of α. Curves 1, 2, 3 correspond to the longitudinal, transverse, and oblique rolls, respectively.
023015-3
M. Z. HOSSAIN AND J. M. FLORYAN
PHYSICAL REVIEW E 90, 023015 (2014)
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FIG. 9. The disturbance flow field at y = 0 for the upper (a) and lower (b) heatings for the transverse rolls for α = 5, β = 1.62, and Rap = 3326. Path lines are illustrated using solid lines. 023015-4
NATURAL CONVECTION IN A FLUID LAYER . . .
PHYSICAL REVIEW E 90, 023015 (2014)
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FIG. 10. The disturbance isotherms for the upper (a) and lower (b) heatings for conditions from Fig. 9. Solid (dotted) lines identify the positive (negative) temperatures.
temperature field whose distribution is illustrated in Fig. 4. The mean temperature’s rise in the interior of the fluid layer for the lower heating and its decrease for the upper heating are clearly seen. Nevertheless, the average Nusselt numbers are the same in both cases; the heat flow is driven by the bulk temperature decrease for the upper heating and the bulk temperature increase for the lower heating. The appearance of a potentially unstable vertical temperature gradient is clearly seen for a sufficiently large α with a thin boundary layer forming at the heated plate (see inset in Fig. 4). When the heating is applied at the lower wall, the mean fluid temperature rises across the boundary layer and the bulk of the fluid sees the edge of the boundary layer as a hot plate, which leads to a situation similar to the classical Rayleigh-B´enard convection; the resulting temperature gradient is able to drive secondary motions, as shown in [6]. When the heating is applied at the upper plate, the mean temperature decreases across the boundary layer and, thus, the bulk of the fluid sees the edge of the boundary layer as a cold plate, also resulting in a potentially unstable vertical temperature gradient. Two instability mechanisms are activated by the heating. The RB mechanism requires the presence of a negative mean vertical temperature gradient and the formation of such a
gradient has been discussed above. The spatial resonance mechanism is associated with the x-periodic heating. The RB mechanism is expected to dominate for large α when the conductive layer forms away from the heated plate and the spatial parametric resonance is expected to play a dominant role for α = 0(1) [6]. The similarity between the linear stability problems for the upper and lower heatings can be demonstrated by changing the sign of gravitational acceleration and the sign of the temperature in the disturbance equations, similarly as in the case of the primary convection. This similarity has been confirmed by numerical solutions of the eigenvalues problem. Variations of the critical periodic Rayleigh number Rap,cr as a function of α are displayed in Fig. 5 and the forms of the critical curves are the same as for the lower heating; transverse rolls are dominant for large α, longitudinal rolls for α ࣈ 4, and oblique rolls for α < 4. Variations of the critical wave numbers (Fig. 6) demonstrate the same qualitative features of the disturbance patterns as found in the case of the lower heating [6]. The longitudinal rolls form two types of patterns. When α < 4.2 the disturbance wave number is locked with the primary roll wave number as 2δcr = α. There is no direct relation between both wave numbers for larger α and commensurable and noncommensurable states may exist [6]. The topologies
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FIG. 11. The disturbance flow field at y = 0 for the upper (a) and lower (b) heatings for the oblique rolls for α = 2.7, β = 0.8, δ = 1.35, and Rap = 2411. 023015-5
M. Z. HOSSAIN AND J. M. FLORYAN
PHYSICAL REVIEW E 90, 023015 (2014)
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FIG. 12. The disturbance isotherms for the upper (a) and lower (b) heatings. Conditions are the same as in Fig. 11. Solid (dotted) lines identify the positive (negative) temperatures.
of the disturbance velocity fields for the locked-in case are displayed in Fig. 7. For the upper heating, the hot spots at the upper wall define the borders between the secondary rolls while for the lower heating the cold spots at the lower plate play this role. This results in the up-down symmetry and the phase shift of the secondary rolls by λα /2 in the x direction when the upper heating is replaced by the lower heating. The corresponding temperature disturbance fields are displayed in Fig. 8 and their pattern exhibits a similar up-down symmetry and a phase shift in the x direction. Disturbance patterns for the nonlocked case exhibit similar qualitative properties for the upper heating as described in [6] for the lower heating but details are not shown due to the existence of a large variety of possible responses. The pattern of the transverse rolls does not exhibit any direct correlation with the pattern of the primary convection (Fig. 6) [6]. The change from the upper to the lower heating results in the up-down symmetry of the disturbance fields, a phase shift of the velocity field by λα /2 in the x direction (Fig. 9), and a phase shift of the temperature field by λα /2 in the x direction and by λβ /2 in the z direction (Fig. 10). The oblique rolls also do not exhibit direct correlation with the pattern of the external heating (Fig. 6). A change from the upper to the lower heating results in the up-down symmetry of the disturbance fields, a phase shift of the disturbance flow
[1] H. B´enard, Rev. Gen. Sci. Pures Appl. 11, 1261 (1900). [2] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford, 1961). [3] J. W. S. Rayleigh, Philos. Mag. 32, 529 (1916). [4] E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid Mech. 32, 709 (2000). [5] G. Freund, W. Pesch, and W. Zimmermann, J. Fluid Mech. 673, 318 (2011). [6] M. Z. Hossain and J. M. Floryan, J. Fluid Mech. 733, 33 (2013). [7] G. Ahlers, S. Grossmann, and D. Lohse, Rev. Mod. Phys. 81, 503 (2009). [8] D. Lohse and K. Q. Xia, Annu. Rev. Fluid Mech. 42, 335 (2010). [9] O. Shiskina and C. Wagner, J. Fluid Mech. 686, 568 (2011). [10] P. Welander, Tellus 16, 349 (1964).
and temperature fields by λα /2 in the x direction and by λβ /4 in the z direction (Figs. 11 and 12). IV. CONCLUSIONS
Natural convection in a fluid layer periodically heated from above has been analyzed with detailed results presented for the Prandtl number Pr = 0.71. The global properties of convection, including the primary and secondary states, are the same as for convection driven by the same heating applied at the lower wall. The patterns of the primary convection are also the same if one accounts for the up-down symmetry. The patterns of the secondary convection display the same updown symmetry combined with phase shifts in the horizontal directions; the phase shifts depend on whether the longitudinal rolls, the transverse rolls, or the oblique rolls dominate at the onset. ACKNOWLEDGMENTS
This work has been carried out with support from the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to thank Dr. Savory for proofreading the manuscript.
[11] G. Z. Gershuni and E. M. Zhukhovitskii, Fluid Dyn. 15, 816 (1980). [12] O. V. Perestenko and L. K. H. Ingel, Izv. Akad. Nauk SSSR Fiz. Atmosf. Okeana 27, 408 (1991). [13] O.V. Perestenko and L. K. H. Ingel, J. Fluid Mech. 287, 1 (1995). [14] A. A. Nepomnyashchy and I. B. Simanovskii, Fluid Dyn. 25, 340 (1990). [15] A. A. Nepomnyashchy, I. B. Simanovskii, and L. M. Braveman, Phys. Fluids 12, 1129 (2000). [16] A. A. Nepomnyashchy and I. B. Simanovskii, Eur. J. Mech., B: Fluids 20, 75 (2001). [17] A. A. Nepomnyashchy, I. B. Simanovskii, and J. C. Legros, Interfacial Convection in Multilayer Systems (Springer, New York, 2006).
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PHYSICAL REVIEW E 90, 023015 (2014)
[18] T. Boeck, A. Nepomnyashchy, and I. Simanovskii, Fluid Dyn. Mater. Process. 4, 11 (2008). [19] D. Merkt and M. Bestehorn, Fluid Dyn. Res. 44, 031413 (2012). [20] S. Marcq and J.Weiss, Cryosphere 6, 143 (2012).
[21] A. Lenardic, L. Moresi, A. M. Jellinek, and M. Manga, Earth Planet. Sci. Lett. 234, 317 (2005). [22] M. Z. Hossain and J. M. Floryan, J. Heat Transfer 135, 022503 (2013). [23] J. M. Floryan, J. Fluid Mech. 335, 29 (1997).
023015-7
Natural convection in a fluid layer periodically heated from above M. Z. Hossain* and J. M. Floryan Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 (Received 13 May 2014; published 25 August 2014) Natural convection in a horizontal layer subject to periodic heating from above has been studied. It is shown that the primary convection leads to the cooling of the bulk of the fluid below the mean temperature of the upper wall. The secondary convection may lead either to longitudinal rolls, transverse rolls, or oblique rolls. The global flow properties (e.g., the average Nusselt number for the primary convection and the critical conditions for the secondary convection) are identical to those of the layer heated from below. However, the flow and temperature patterns exhibit phase shifts in the horizontal directions. DOI: 10.1103/PhysRevE.90.023015
PACS number(s): 47.20.−k, 44.25.+f
I. INTRODUCTION
It is well known that the buoyancy-driven instability of the conductive state of equilibrium appears only when heating is applied from below, i.e., Rayleigh-B´enard (RB) convection [1–3]. Laminar RB convection has been the origin of much research on nonlinear dynamics and pattern formation [4]. Recent efforts have been focused on the explanation of the roles played by various boundary imperfections [5,6]. A large enough heating intensity leads to turbulent RB convection which has many practical applications in the area of heat transfer [7,8]. Current efforts are focused on the use of boundary imperfections with an appropriate spatial pattern in order to increase the relevant Nusselt number [9]. It is less well known that the instability may occur when the fluid is heated from above (anticonvection) as long as the system contains at least one interface [10–13]. Anticonvection may occur in fluids with arbitrary properties when combined with heat release and extraction at the interface [14–17]; the current knowledge is summarized in [18,19]. Patterned heating represents another aspect of buoyancydriven convection. The heating represents an external forcing and convection represents a forced response which occurs regardless of the intensity of the heating and can have either a laminar or turbulent form. This structured convection is referred to as the primary convection. Its spatial pattern, which is locked in with the pattern of heating, affects contaminant transport in urban environments as local heating rates are determined by the dissimilar thermal properties of roofs, streets, and parks. It affects rural environments where local circulation is driven by variations in the heating rates of forests, fields, and lakes. It is known in geological applications that a system of fractures, leads, and polynyas in sea ice leads to convection in both the ocean and atmosphere with small leads of the order of several meters being more efficient than larger ones of the order of several hundred meters [20]. The insulating effect of continents on the mantle convection within the Earth represents a spatially structured convection with heating from above [21]. Further examples include systems of localized fires, systems of computer chips, thermal patterning in microfluidic devices, and so on. It has been demonstrated
that convection concentrates in the vicinity of the heated wall for large wave number heating while it penetrates into the complete flow domain for smaller wave number heating [6]. In the laminar case, the observable structure of convection may differ from that imposed by the heating pattern as the primary convection may undergo transition to secondary states. The onset conditions for the secondary convection are dictated by the interplay between two instability mechanisms, i.e., the RB mechanism and the spatial parametric resonance. A detailed description of the system response when the lower plate is exposed to a simple sinusoidal heating is given in [6]. Depending on the heating wave number, the secondary state may have the form of longitudinal rolls (rolls parallel to the primary rolls), transverse rolls (rolls orthogonal to the primary rolls), or oblique rolls, and the intensity of heating required to generate such states strongly depends on the heating wave number. It is not known how the fluid may respond when the same heating is applied at the upper plate. Less is known about the patterned turbulent convection but the available results suggest that significant intensification of convection can be expected when an appropriate pattern is used [12]. The present work is focused on the analysis of natural convection driven by periodic heating applied from above. The heating amplitude is expressed in terms of the periodic Rayleigh number Rap and its spatial distribution is characterized by the heating wave number α. Section II describes the problem formulation for the primary stationary convection and for its linear stability. Section III provides a description of the primary and secondary convections and draws an analogy with convection driven by the same heating applied at the lower wall. Section IV provides a short summary of the main conclusions.
y,v
x,u
[email protected]
1539-3755/2014/90(2)/023015(7)
g
U
x
1 2
Ta cos x L
z,w
*
+h
x
0
-h
FIG. 1. Fluid layer with sinusoidal heating applied at the upper wall. 023015-1
©2014 American Physical Society
M. Z. HOSSAIN AND J. M. FLORYAN
PHYSICAL REVIEW E 90, 023015 (2014)
1
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0.1
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-1 0
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1 -0 . 3 -0 .
0.3
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0.3 0.6
0 02 -0 .
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x/
(a)
(b)
FIG. 2. The flow topology and the isotherms for Rap = 400, α = 5 for the upper (a) and lower (b) heatings. Thick (thin) lines identify streamlines (isotherms); solid (dashed) lines identify positive (negative) temperature.
where u = (u,v,w), p, θ stand for the velocity, pressure, and temperature, respectively; j denotes the unit vector in the y direction; and Rap = gh3 Ta /νκ is the periodic Rayleigh number. The reader may note the difference between the periodic Rayleigh number Rap and the classical Rayleigh number which is based on the difference between the mean temperatures of both walls. The fluid has been assumed to be incompressible, Newtonian, with thermal conductivity k, specific heat per unit mass c, thermal diffusivity κ = k/ρc, kinematic viscosity ν, thermal expansion coefficient , and the Boussinesq approximation of density ρ variations. The half spacing between the plates h has been used as the length scale, Uv = ν/ h as the velocity scale, Pv = ρUv2 as the pressure scale, and Tv = Ta ν/κ as the temperature scale where Tv /Ta = Pr. Here Pr stands for the Prandtl number. System (2)–(4) is supplemented with homogeneous boundary conditions for the velocity and the temperature boundary conditions (1) are adjusted using the temperature scale Tv . The steady two-dimensional version of (2)–(4) describes the primary convection. The unknowns are repre sented as Fourier expansions ψ(x,y) = n=+∞ ϕ (n) (y)einαx n=−∞ (n) inαx and θ (x,y) = n=+∞ , where ψ stands for the n=−∞ φ (y)e stream function defined in the usual manner; the relevant solution has been determined numerically using the method
II. PROBLEM FORMULATION AND NUMERICAL SOLUTION
The problem formulation follows that given in [6] for a fluid heated from below. Consider a slot between two parallel plane plates extending to ±∞ in the x and z directions and placed at a distance 2h apart from each other with gravitational acceleration g acting in the negative y direction (Fig. 1). The lower plate is isothermal while the upper plate has x-periodic temperature variations, i.e., θ (x,h,z,t) = 12 Ta cos(αx), θ (x, − h,z,t) = 0.
(1)
In the above, θ = T − Tref denotes the relative temperature, T stands for the absolute temperature, and Tref denotes the reference temperature (temperature of the lower plate). Convection is described by the field equations of the form (2)
∂θ + (u · ∇)θ = Pr−1 (∇ 2 θ ), ∂t
(3)
∇ · u = 0,
(4)
3
5
(a)
5
10
yc= -1+2 10
15
-1
20
101
102 Rap 103
101
max
100 ) =45 10
) =72
10-1
-0.5 -1 0 10
100
-1
10-1
Nuav
102 Rap 103 10 68 5 4 =3
104
10-2
(b)
1 23
10
(c)
15
10-4 20
10-1
Nuav ) =45
10-2
) =72
10-3
-3 5 Rap 10 =10
5
10-2
104
>
-1
3
100
=3 4 5 6 8 10
4
10 1 10
>
2
-0.5
0
max
2 3 10 Rap 10 10 68 5 =3 4
>
2
0.5
1
10-2
>
1
yc
10
>
0
5
-1
0
>
0.5
yc=1-2 Rap 10-3=10
10 1 10
1
10
-3
>
1
>
yc
∂u + (u · ∇)u = −∇p + ∇ 2 u + Rap θ j, ∂t
10-4
1 23 -3 5 Rap 10 =10
10-3 5
10
15
10-5 20
(d)
FIG. 3. Variations of the location of the roll center yc as a function of α (a) and as a function of Rap (b), variations of the maximum of the stream function as a function of α and Rap (c) and variations of Nusselt number Nu as function of α and Rap (d). Solid (dotted) lines correspond to heating applied at the upper (lower [6]) wall in these figures. 023015-2
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PHYSICAL REVIEW E 90, 023015 (2014) 20000 15000
1
2
Unstable
10000
Rap,cr 3
5000
Rap,cr=233.8
1.5
Stable 2
FIG. 4. The mean temperature component for Rap = 400. Solid (dashed) lines correspond to the upper (lower [6]) heating.
described in [22]. The net heat transfer is expressed in terms of the Nusselt number, Nu = − Pr dφ (0) /dy|y=−1 . The stability analysis of the primary convection follows the method described in [6,23]. Small arbitrary three-dimensional disturbances of the form ud (x,y,z,t) = [gud (x,y),gvd (x,y),gwd (x,y)]ei[δx+βz−σ t] + c.c., θd (x,y,z,t) = gθd (x,y)ei[δx+βz−σ t] + c.c.
(5)
are imposed upon the stationary solution where ud = (ud ,vd ,wd ) is the disturbance velocity vector, θ d is the temperature disturbance, (δ, β) denotes the disturbance wave numbers in the (x, z) directions, respectively; the exponent σ is complex (σ = σr + iσi ) and its imaginary and real parts describe the rate of growth and the frequency of disturbances, respectively; and c.c. stands for the complex conjugate. The disturbance amplitudes are functions of x and y, and are periodic in x. The disturbance equations are linearized and the resulting system is posed as an eigenvalue problem for σ . The eigenvalues have been determined numerically using the method described in [6]. III. DISCUSSION OF RESULTS
The discussion will make frequent references to convection driven by the same heating but applied at the lower plate. Similarity with such convection can be demonstrated by (i) reversing the direction of gravity, i.e., Rap → −Rap , and (ii) changing the sign of the temperature, i.e., θ → −θ . The latter condition implies a change of the sign of the temperature imposed at the lower plate and results in a shift of the temperature field by half of a cycle in the x direction. All of the detailed results presented here are given for Pr = 0.71. Figure 2 illustrates the topology of the primary convection and temperature fields. The motion is driven by the cold (heavier) fluid concentrating around the cold spots at the upper plate and then descending. This fluid is turned sideways as it approaches the lower plate, then it is drawn upwards below the hot spots and pulled sideways as it approaches the upper plate resulting in the formation of pairs of counter-rotating rolls. The same figure displays the flow topology for the lower
4
6
8
10
FIG. 5. Variations of the critical periodic Rayleigh number Rap,cr as a function of α. Curves 1, 2, 3 correspond to the longitudinal, transverse, and oblique rolls, respectively. See [6] for derivation of the form of the asymptote of Rap,cr for α → ∞.
heating; the motion is driven by the lighter fluid rising above the hot spots and then turning sideways as it approaches the upper plate, forming an identical system of counter-rotating vortex pairs. There is remarkable up-down symmetry of the flow fields for both types of heating. The bulk of the fluid is cooled below the mean plate temperature for heating at the upper plate while its temperature rises above the mean plate temperature when the lower plate is heated. The temperature field exhibits an up-down symmetry combined with the phase shift of λα /2 in the x direction. The global properties of the primary convection are not affected by the location of the heating, e.g., convection penetrates the entire depth of the fluid layer for α = 0(1) but concentrates closer to the heated wall when α increases as illustrated by the variations of the distances between the roll centers and the heated plate [Figs. 3(a) and 3(b)]. Furthermore, the intensity of convection [Fig. 3(c)] and the Nusselt numbers [Fig. 3(d)] decrease when α increases but are the same for both types of heating for the same Rap and α. When α is large enough, a conduction layer forms away from the heated plate regardless of whether the upper or lower plates are exposed to the heating (see Figs. 2–4). The net heat transfer between the plates is driven by the mean component of the 2
2 cr
1.6
1
3
cr
1.2
1.6
=1.56 cr =1.56
cr
1.2 0.8
0.8 3
0.4
0.4 0
2
4
6
8
0 10
FIG. 6. Variations of the critical wave numbers δcr and βcr as functions of α. Curves 1, 2, 3 correspond to the longitudinal, transverse, and oblique rolls, respectively.
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M. Z. HOSSAIN AND J. M. FLORYAN
PHYSICAL REVIEW E 90, 023015 (2014)
x/
0.5
1
0 1
x/
0.5
1 0
0 1
-0.1
>
>
>
>
>
>
>
>
>
1
>
>
>
x/
>
>
>
>
0.5
>
0.5
>
z/
>
>
>
>
>
0 0
>
>
>
>
0.5
>
z/
1
>
1
0.5
x/
1
(b)
FIG. 9. The disturbance flow field at y = 0 for the upper (a) and lower (b) heatings for the transverse rolls for α = 5, β = 1.62, and Rap = 3326. Path lines are illustrated using solid lines. 023015-4
NATURAL CONVECTION IN A FLUID LAYER . . .
PHYSICAL REVIEW E 90, 023015 (2014)
1
1
z/
z/
0.5
0.5
0 0
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0 0
1
x/
0.5
(a)
1
x/
(b)
FIG. 10. The disturbance isotherms for the upper (a) and lower (b) heatings for conditions from Fig. 9. Solid (dotted) lines identify the positive (negative) temperatures.
temperature field whose distribution is illustrated in Fig. 4. The mean temperature’s rise in the interior of the fluid layer for the lower heating and its decrease for the upper heating are clearly seen. Nevertheless, the average Nusselt numbers are the same in both cases; the heat flow is driven by the bulk temperature decrease for the upper heating and the bulk temperature increase for the lower heating. The appearance of a potentially unstable vertical temperature gradient is clearly seen for a sufficiently large α with a thin boundary layer forming at the heated plate (see inset in Fig. 4). When the heating is applied at the lower wall, the mean fluid temperature rises across the boundary layer and the bulk of the fluid sees the edge of the boundary layer as a hot plate, which leads to a situation similar to the classical Rayleigh-B´enard convection; the resulting temperature gradient is able to drive secondary motions, as shown in [6]. When the heating is applied at the upper plate, the mean temperature decreases across the boundary layer and, thus, the bulk of the fluid sees the edge of the boundary layer as a cold plate, also resulting in a potentially unstable vertical temperature gradient. Two instability mechanisms are activated by the heating. The RB mechanism requires the presence of a negative mean vertical temperature gradient and the formation of such a
gradient has been discussed above. The spatial resonance mechanism is associated with the x-periodic heating. The RB mechanism is expected to dominate for large α when the conductive layer forms away from the heated plate and the spatial parametric resonance is expected to play a dominant role for α = 0(1) [6]. The similarity between the linear stability problems for the upper and lower heatings can be demonstrated by changing the sign of gravitational acceleration and the sign of the temperature in the disturbance equations, similarly as in the case of the primary convection. This similarity has been confirmed by numerical solutions of the eigenvalues problem. Variations of the critical periodic Rayleigh number Rap,cr as a function of α are displayed in Fig. 5 and the forms of the critical curves are the same as for the lower heating; transverse rolls are dominant for large α, longitudinal rolls for α ࣈ 4, and oblique rolls for α < 4. Variations of the critical wave numbers (Fig. 6) demonstrate the same qualitative features of the disturbance patterns as found in the case of the lower heating [6]. The longitudinal rolls form two types of patterns. When α < 4.2 the disturbance wave number is locked with the primary roll wave number as 2δcr = α. There is no direct relation between both wave numbers for larger α and commensurable and noncommensurable states may exist [6]. The topologies
>
>
>
>
>
>
>
>
>
> >
0.5
>
>
>
z/ >
0.5
>
>
>
>
>
z/
1
>
1
> > >
0 0
0.5
x/
1
(a)
0 0
0.5
x/
1
(b)
FIG. 11. The disturbance flow field at y = 0 for the upper (a) and lower (b) heatings for the oblique rolls for α = 2.7, β = 0.8, δ = 1.35, and Rap = 2411. 023015-5
M. Z. HOSSAIN AND J. M. FLORYAN
PHYSICAL REVIEW E 90, 023015 (2014)
1
1
z/
z/ 0.5
0 0
0.5
0.5
x/
0 0
1
(a)
0.5
x/
1
(b)
FIG. 12. The disturbance isotherms for the upper (a) and lower (b) heatings. Conditions are the same as in Fig. 11. Solid (dotted) lines identify the positive (negative) temperatures.
of the disturbance velocity fields for the locked-in case are displayed in Fig. 7. For the upper heating, the hot spots at the upper wall define the borders between the secondary rolls while for the lower heating the cold spots at the lower plate play this role. This results in the up-down symmetry and the phase shift of the secondary rolls by λα /2 in the x direction when the upper heating is replaced by the lower heating. The corresponding temperature disturbance fields are displayed in Fig. 8 and their pattern exhibits a similar up-down symmetry and a phase shift in the x direction. Disturbance patterns for the nonlocked case exhibit similar qualitative properties for the upper heating as described in [6] for the lower heating but details are not shown due to the existence of a large variety of possible responses. The pattern of the transverse rolls does not exhibit any direct correlation with the pattern of the primary convection (Fig. 6) [6]. The change from the upper to the lower heating results in the up-down symmetry of the disturbance fields, a phase shift of the velocity field by λα /2 in the x direction (Fig. 9), and a phase shift of the temperature field by λα /2 in the x direction and by λβ /2 in the z direction (Fig. 10). The oblique rolls also do not exhibit direct correlation with the pattern of the external heating (Fig. 6). A change from the upper to the lower heating results in the up-down symmetry of the disturbance fields, a phase shift of the disturbance flow
[1] H. B´enard, Rev. Gen. Sci. Pures Appl. 11, 1261 (1900). [2] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford, 1961). [3] J. W. S. Rayleigh, Philos. Mag. 32, 529 (1916). [4] E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid Mech. 32, 709 (2000). [5] G. Freund, W. Pesch, and W. Zimmermann, J. Fluid Mech. 673, 318 (2011). [6] M. Z. Hossain and J. M. Floryan, J. Fluid Mech. 733, 33 (2013). [7] G. Ahlers, S. Grossmann, and D. Lohse, Rev. Mod. Phys. 81, 503 (2009). [8] D. Lohse and K. Q. Xia, Annu. Rev. Fluid Mech. 42, 335 (2010). [9] O. Shiskina and C. Wagner, J. Fluid Mech. 686, 568 (2011). [10] P. Welander, Tellus 16, 349 (1964).
and temperature fields by λα /2 in the x direction and by λβ /4 in the z direction (Figs. 11 and 12). IV. CONCLUSIONS
Natural convection in a fluid layer periodically heated from above has been analyzed with detailed results presented for the Prandtl number Pr = 0.71. The global properties of convection, including the primary and secondary states, are the same as for convection driven by the same heating applied at the lower wall. The patterns of the primary convection are also the same if one accounts for the up-down symmetry. The patterns of the secondary convection display the same updown symmetry combined with phase shifts in the horizontal directions; the phase shifts depend on whether the longitudinal rolls, the transverse rolls, or the oblique rolls dominate at the onset. ACKNOWLEDGMENTS
This work has been carried out with support from the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to thank Dr. Savory for proofreading the manuscript.
[11] G. Z. Gershuni and E. M. Zhukhovitskii, Fluid Dyn. 15, 816 (1980). [12] O. V. Perestenko and L. K. H. Ingel, Izv. Akad. Nauk SSSR Fiz. Atmosf. Okeana 27, 408 (1991). [13] O.V. Perestenko and L. K. H. Ingel, J. Fluid Mech. 287, 1 (1995). [14] A. A. Nepomnyashchy and I. B. Simanovskii, Fluid Dyn. 25, 340 (1990). [15] A. A. Nepomnyashchy, I. B. Simanovskii, and L. M. Braveman, Phys. Fluids 12, 1129 (2000). [16] A. A. Nepomnyashchy and I. B. Simanovskii, Eur. J. Mech., B: Fluids 20, 75 (2001). [17] A. A. Nepomnyashchy, I. B. Simanovskii, and J. C. Legros, Interfacial Convection in Multilayer Systems (Springer, New York, 2006).
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PHYSICAL REVIEW E 90, 023015 (2014)
[18] T. Boeck, A. Nepomnyashchy, and I. Simanovskii, Fluid Dyn. Mater. Process. 4, 11 (2008). [19] D. Merkt and M. Bestehorn, Fluid Dyn. Res. 44, 031413 (2012). [20] S. Marcq and J.Weiss, Cryosphere 6, 143 (2012).
[21] A. Lenardic, L. Moresi, A. M. Jellinek, and M. Manga, Earth Planet. Sci. Lett. 234, 317 (2005). [22] M. Z. Hossain and J. M. Floryan, J. Heat Transfer 135, 022503 (2013). [23] J. M. Floryan, J. Fluid Mech. 335, 29 (1997).
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