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Fluid–structure interaction analysis of mixed convection heat transfer in a lid-driven cavity with a flexible bottom wall
International Journal of Heat and Mass Transfer 54 (2011) 3826–3836
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Fluid–structure interaction analysis of mixed convection heat transfer in a lid-driven cavity with a flexible bottom wall Abdalla Al-Amiri a, Khalil Khanafer b,⇑ a b
Mechanical Engineering Department, United Arab Emirates University, United Arab Emirates Vascular Mechanics Lab, Biomedical Engineering Department and Section of Vascular Surgery, University of Michigan, Ann Arbor, MI 48109, USA
a r t i c l e
i n f o
Article history: Received 19 January 2011 Received in revised form 21 April 2011 Accepted 21 April 2011 Available online 18 May 2011 Keywords: Elasticity Fluid–structure interaction Lid-driven cavity Mixed convection
a b s t r a c t A numerical investigation of steady laminar mixed convection heat transfer in a lid driven cavity with a flexible bottom surface is analyzed. A stable thermal stratification configuration was considered by imposing a vertical temperature gradient while the vertical walls were considered to be insulated. In addition, the transport equations were solved using a finite element formulation based on the Galerkin method of weighted residuals. In essence, a fully coupled fluid–structure interaction (FSI) analysis was utilized in this investigation. Moreover, the fluid domain is described by an Arbitrary-Lagrangian– Eulerian (ALE) formulation that is fully coupled to the structure domain. Comparisons of streamlines, isotherms, bottom wall displacement and average Nusselt number were made between rigid and flexible bottom walls. The results of this investigation revealed that the elasticity of the bottom wall surface plays a significant role on the heat transfer enhancement. Furthermore, the contribution of the forced convection heat transfer to that offered by natural convection heat transfer has a profound effect on the behavior of the flexible wall as well as the momentum and energy transport processes within the cavity. This investigation paves the road for future research studies to consider flexible walls when augmentation of heat transfer is sought. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Augmentation of flow and heat transfer in geometries with irregular surfaces is a topic of fundamental importance. This interest stems from its significance in many engineering and industrial applications such as flat-plate solar collectors, flat plate condensers in refrigerators [1], micro-electronic devices, cooling system of micro-electronic devices, and cooling of electrical components [2]. Further, all the studies on mixed convection flow and heat transfer in geometries assuming regular and irregular surfaces have been restricted to rigid walls [1–14]. For example, Al-Amiri et al. [2] conducted a study to analyze mixed convection heat transfer in a lid-driven cavity with a sinusoidal wavy bottom surface. Their results had illustrated that the average Nusselt number would increase with an increase in both the amplitude of the wavy surface and Reynolds number. Moreover, optimum heat transfer was achieved when the wavy surface was designated with two undulations while subjected to low Richardson numbers. Khanafer et al. [7] carried out a numerical study on natural convection heat transfer inside a porous cavity with a sinusoidal vertical wavy wall.
⇑ Corresponding author. Tel.: +1 (734) 763 5240; fax: +1 (734) 647 4834. E-mail address: [email protected] (K. Khanafer). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.04.047
Their results showed that the amplitude of the wavy surface and the number of undulations affect heat transfer characteristics inside the cavity. Owing to the difficulties associated with machining geometries with wavy surfaces; the authors propose in this investigation the use of flexible surfaces with known elasticity to enhance heat transfer characteristics as the fluid motion will cause the deformation of the flexible solid structure. The authors have proposed a liddriven cavity to demonstrate their case owing to its fundamental nature. The scenario of a flexible wall disturbing a fluid motion, which involves the coupling of fluid mechanics and structural mechanics, is referred to in the literature as the fluid–structure interaction (FSI). FSI approach has received a great attention in recent years and its importance is of a growing interest in mechanical, aerospace and biomedical engineering applications [15–18]. In an FSI situation, the stresses and deformations of a given structure are computed simultaneously with the fluid flow and heat transfer variables that surrounds the structure. The lid-driven cavity with a flexible bottom surface was used by several authors as a benchmark for validating their respective numerical codes for fluid– structure interaction problems [19–23]. It is worth noting that the energy transport was ignored in these cited studies as the prime motivation has been centered around proper modeling of the fluid motion in the presence of a flexible structure.
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Nomenclature cp €s d
specific heat local acceleration of the solid region
df
displacement vector of the fluid domain
ds
displacement vector of the solid domain
E
Young’s Modulus
B ff B fs
fluid body force per unit volume
g Gr h H Nu Pr Re Si T u
gravitational acceleration Grashof number, gbðT H T C ÞH3 =m2 thickness of the flexible bottom wall side length of the cavity Nusselt number Prandtl number, m=a Reynolds number, U 0 H=m interface of the fluid and solid domains temperature fluid velocity vector
ug U0 x, y Xb, Yb
moving coordinate velocity sliding lid velocity Cartesian coordinates dimensionless displacements
Greek symbols a thermal diffusivity, k=ðqf cp Þ b volumetric expansion coefficient m kinematic viscosity # Poisson ratio h dimensionless temperature, ðT T C Þ=ðT H T C Þ q density r stress tensor
solid externally applied body force vector
Subscript f fluid domain s solid domain
Table 1 Comparison of the average Nusselt number at the top wall between various studies at Gr = 102. Re
Present FEM
100 2.02 400 4.05 1000 6.45
Ref. [32] FDM
Ref. [33] FVM
Ref. [34]
Ref. [35] FVM
Ref. [36] FDM
Ref. [37] FVM
1.94 3.84 6.33
2.01 3.91 6.33
1.99 3.88 6.35
NA 4.05 6.55
2.03 4.02 6.48
2.01 3.91 6.33
Table 2 Comparison of the average Nusselt number at the top wall between various studies at Gr = 104 and 106. Re
Gr = 104
Gr = 106
Present
Ref. [32]
Ref. [35]
Present
Ref. [32]
Ref. [35]
100 400 1000
1.38 3.76 6.56
1.34 3.62 6.29
– 3.82 6.50
1.02 1.17 1.72
1.02 1.22 1.77
– 1.17 1.81
⁄ FEM: finite element method; FDM: finite difference method; FVM: finite volume method.
Fig. 1. Schematic diagram of the physical model.
There are many studies reported in the literature on investigating the effect of moving boundaries on the flow and heat transfer characteristics [24–28]. These studies involved boundaries that
move with a prescribed motion (e.g., displacement, pressure, or velocity). For example, Khaled and Vafai [24] studied the effects of both external squeezing and internal pressure pulsations on
0.12 Present [20] Bathe and Zhang [36]
Y-Displacement (m)
0.06
0.00
-0.06
-0.12
-0.18 0
10
20
30
40
50
60
70
Time (s) Fig. 2. Comparison of the y-displacement between the present results and that of Bathe and Zhang [20] in the absence of heat transfer.
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Fig. 3. Comparison of the streamlines between rigid and flexible bottom wall at various Re (Gr = 102, E = 25,000 N/m2).
flow and heat transfer inside non-isothermal and incompressible thin film that has a small thickness compared to its length. The lower plate of the thin film was fixed while the vertical motion of the upper plate was considered to assume a sinusoidal behavior when the thin film gap is not charged with the working fluid. Nakamura et al. [26] investigated numerically the influence of the wall oscillation on the heat transfer characteristics of a two dimensional channel. In their study, the moving boundary problem was transformed into a fixed boundary problem using the coordinate transformation method. Based on the literature survey conducted by the authors and up to their knowledge, there has not been any attention given to investigate the momentum and energy transport processes in liddriven cavities while incorporating flexible walls. Therefore, this fundamental investigation focuses on studying the effect of flexible bottom wall on the heat transfer characteristics for various pertinent parameters such as Reynolds number, elasticity of the bottom wall and Grashof number.
2. Mathematical formulation Consider a two-dimensional square cavity with the physical dimensions shown in Fig. 1. The cavity side length is considered to be H while the thickness of the flexible bottom wall was set to be 0.002 of the side length. The working fluid is assumed to be an incompressible Newtonian fluid with a Prandtl number of 0.71 that is operating under steady-state laminar heat transfer regime. Furthermore, the thermophysical properties of the working fluid are taken to be constant except for the density variation, which is modeled according to the Boussinesq approximation. The mechanically induced lid motion (top wall) is assumed to move from left to right at a constant speed Uo and is maintained at a higher temperature TH than the flexible bottom wall TC. Such an imposed temperature condition will render a thermal stratification in the cavity. Meanwhile, the two vertical walls are assumed rigid and are subjected to an insulated boundary condition. Finally, the conduction through the flexible wall is assumed to be negligible.
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Fig. 4. Comparison of the isotherms between rigid and flexible bottom wall at various Re (Gr = 102, E = 25,000 N/m2).
2.1. FSI analysis
7
ru¼0
ð1Þ
qf ðu ug Þ ru ¼ r rf þ qf f Bf
ð2Þ
u rT ¼ ar2 T
ð3Þ
6
Flexible Wall Average Nu
An Arbitrary Lagrangian–Eulerian formulation was employed to describe the fluid motion in the FSI model. The governing equations for the fluid domain are the continuity, Navier–Stokes equations, and energy equation which are described as:
5
Rigid Wall 4
3
B
where a = k/(qfcp) is the thermal diffusivity, f f the body force per unit volume, qf the fluid density, rf the fluid stress tensor, T the temperature, u the fluid velocity vector, ug the moving coordinate velocity, and (u ug) the relative velocity of the fluid with respect to the moving coordinate velocity. The governing equation for the
2 0
250
500
750
1000
Re Fig. 5. Comparison of the average Nusselt number between rigid and flexible bottom wall for various Reynolds number (Gr = 102, E = 25,000 N/m2).
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Fig. 6. Effect of varying the elasticity of the bottom wall surface on the streamlines and isotherms (Re = 100, Gr = 102).
solid domain of the FSI model can be described by the following elastodynamics equation:
2.4
qs d€ s ¼ r rs þ f Bs Average Nu
ð4Þ
€ s represents the local acceleration of the solid region where d B € _ ðds ¼ ug Þ; f s the externally applied body force vector at time t, qs the density of the bottom wall, and rs the solid stress tensor, and. In the current investigation, the physical properties of the flexible bottom wall were assumed to be constant and homogeneous. The default values chosen for the flexible wall were as follows: a density qs = 500 kg/m3, a Young’s Modulus E = 25,000 N/m2 and a Poisson’s ratio # = 0.45.
2.3
Flexible Wall 2.2
2.1
Rigid Wall
2.2. Boundary conditions 2 0
20000
40000
60000
80000
100000
E (N/m2) Fig. 7. Effect of varying the elasticity of the bottom wall on the average Nusselt number (Re = 100, Gr = 102).
The following boundary conditions are specified for the fluid domain:
u ¼ v ¼ 0;
T ¼ TC
y ¼ 0;
06x6H
ð5aÞ
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Fig. 8. Effect of varying the elasticity of the bottom wall on the displacement (Re = 100, Gr = 102).
u ¼ Uo ; u¼v ¼
v ¼ 0;
T ¼ TH
@T ¼ 0 x ¼ 0; H; @x
y ¼ H;
06x6H
06y6H
ð5bÞ
ð5cÞ
The final set of boundary conditions is applied to the FSI interfaces such that the conditions of displacement compatibility and traction equilibrium along the structure–fluid interfaces must be satisfied. These conditions can be expressed mathematically as follows [29]: Displacement compatibility:
d f ¼ ds
ðx; tÞ 2 Si
ð6Þ
3. Numerical scheme A finite element formulation based on the Galerkin method was utilized to solve the governing equations of fluid–structure interaction model in a lid-driven cavity with a flexible bottom surface using ADINA software subject to the boundary conditions described above. A variable grid-size system was employed to capture the rapid changes in the dependent variables especially in the vicinity of the walls where the major gradients occur inside the boundary layer. As such, a non-uniform mesh of 120 120 nodes was used for the solution of the Navier–Stokes equations [30]. Further, the bottom surface was modeled using iso-parametric beam elements [31].
Traction equilibrium: 4. Code validation
rf ¼ rs ðx; tÞ 2 Si
ð7Þ
where df and ds are the displacements, rf and rs the tractions of the fluid and solid, respectively, and Si the interface of the fluid and solid domains.
The validation of the current algorithm is carried out in 2-fold. First, the present numerical solution using ADINA software is verified against the average Nusselt number calculated at the top wall of the cavity for rigid bottom surface and the results were found to
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Fig. 9. Effect of varying Reynolds number on the displacement (Gr = 103, E = 25 103 N/m2).
be in excellent agreement between the present results and others reported in the literature as shown in Tables 1 and 2, respectively. The second fold of validating the accuracy of the present numerical scheme examines the predictions of the vertical displacement at mid-section of the flexible bottom wall in a driven cavity problem. In this regard, the comparison between the present results against those reported by Bathe et al. [29] in the absence of heat transfer as shown in Fig. 2. The comparison manifests a striking agreement between both results as the maximum relative difference between the two models does not exceed 7%.
average Nusselt number and fluid displacement in the subsequent sections. In addition, the implications of varying the Reynolds number Re = U0H/m and Grashof number Gr ¼ gbðT H T C ÞH3 =m2 on the transport phenomena will be investigated. In essence, the Reynolds number is considered to vary between 100 and 1000, Grashof number between 100 and 1000, and the elasticity of the bottom wall is varied in the range of 103 and 105 N/m2.
5. Results and discussion
Figs. 3 and 4 depict the streamlines and isotherms for both the FSI model and the rigid model for various Reynolds number values. The results show an augmentation in heat transfer when the flexible bottom wall is invoked. As can be seen from the presented results, the shape of the flexible bottom wall is found to depend significantly on the employed Reynolds number value. At high Reynolds number (Re = 1000), the shape of the bottom wall of the cavity approaches a sinusoidal wave pattern. Moreover, Fig. 5 demonstrates that the flexible bottom wall has a profound effect
As mentioned earlier, the main objective of the present investigation is to explore the sensitivity of the elasticity behavior of the flexible bottom wall on the momentum and energy transport in a square lid-driven cavity, which is operating under a steady-state laminar mixed convection heat transfer regime. The transport phenomena are presented by demonstrating and analyzing the streamlines, isotherms [dimensionless temperature; h = (T TC)/(TH TC)],
5.1. Comparison of isotherms, streamlines, and Nusselt number between FSI and rigid models
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Fig. 10. Effect of varying Reynolds number on the displacement of the bottom wall (Gr = 104 and E = 25 103 N/m2).
on the average Nusselt number. As such, flexible bottom wall exhibits higher Nusselt numbers than rigid wall model for various Reynolds numbers. It is worth noting that the maximum relative difference in Nusselt number prediction between both models is 9.4% and occurs at Re = 100. As the Reynolds number increases, it is interesting to note that the Nusselt number difference between both considered models decreases. This is likely attributed to the behavior of the flexible bottom wall with increasing Re. The ‘parabolic’ nature of the flexible wall at Re 6 400 augments heat transfer in as it facilitate the motion of the cold fluid layers from the bottom towards the center of the cavity. Such an augmentation becomes less pronounced when Re = 1000 (i.e., presents a dominant forced convection mechanism as Gr/Re2 1) where the flexible wall assumes a sinusoidal pattern. The sinusoidal pattern brings about an adverse effect which depreciates the contribution of the flexible wall in augmenting the energy transport. Hence, the magnitude of Nusselt number enhancement at higher employed Reynolds number values tends to decrease when compared to the rigid model. This suggests that heat transfer enhancement can be achieved using flexible walls rather than incorporating, for example, irregular geometries.
5.2. Effect of varying the Young’s Modulus of the bottom wall on streamlines, isotherms and Nusselt numbers Fig. 6 shows the effect of varying the elasticity of the bottom wall on the streamlines and isotherms when using Re = 100 and Gr = 102. The variation of the elasticity in the considered range in this investigation appears to have less dramatic impact on the shape of the flexible wall as compared to the considered range of the Reynolds number. In addition, it can be seen from this figure that the bottom surface bulges significantly inward at a small value of elasticity, i.e., E = 1000 N/m2. As the elasticity of the bottom wall increases, its stiffness will consequently increase. As a result, less momentum and thermal penetrations into the cavity are manifested. Furthermore, the impact of elasticity on the average Nusselt number is appraised as depicted in Fig. 7. As can be shown from the figure, the average Nusselt number decreases and asymptotically approaches the value of the rigid model case as the elasticity of the bottom wall increases. It is worth noting that the relative difference in the Nusselt number predictions between both considered models was found to be 15.3% and 4.5% at E = 103 and 105, respectively.
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Fig. 11. Effect of varying Reynolds number on the streamlines and isotherms (Gr = 103, E = 25 103 N/m2).
5.3. Impact of the Young’s Modulus on the displacement of the bottom wall (Xb, Yb) The effect of manipulating the elasticity magnitude of the bottom wall is documented in terms of the incurring Xb wall and Yb displacement in the x- and y-directions, respectively when using Re = 100 and Gr = 102. Fig. 8 shows relatively large positive displacement of the bottom wall in the vertical direction when E = 25 103 N/m2. As the elasticity of the bottom wall increases, the vertical displacement Yb depreciates in value and consequently impedes its influence on the fluid layers in the vicinity of the bottom wall and, hence, retards heat transfer augmentation. It is worth noting that an increase of the elasticity of the flexible wall by 100-fold brings about a reduction in the vertical displacement by around 3-fold. 5.4. Effect of varying Reynolds and Grashof numbers on the streamlines, isotherms, displacement, and Nusselt number Figs. 9–13 display the effect of varying Reynolds number on the bottom wall displacement, streamlines, isotherms, and average Nusselt number predictions at relatively higher Grashof number
values, i.e., Gr = 103 and 104. The Young’s Modulus of elasticity was set equal to 25 103 N/m2 in all these figures. Giving that the Grashof number reflects the strength of the thermal buoyancy force; its value has a significant effect on the shape of the bottom wall and consequently on the heat transfer characteristics within the lid-driven cavity. This can be depicted from the results outlined in Figs. 9 and 10, respectively. This effect is more pronounced at low Reynolds number (Re = 100) when both the buoyancy contribution and the mechanical force of the sliding lid are equally important. At higher Reynolds numbers, the forced convection effect offered by the sliding mechanical lid overwhelms the natural convection sustained by the imposed Grashof number. Hence, the sliding lid tends to dampen the displacement of the flexible bottom wall. As can be seen from the results, the increase of Grashof number from 103 to 104 renders an appreciated increase in the vertical displacement by 1.71, 1.91 and 2.05 for assigned Reynolds number values of 100, 400 and 1000, respectively. Next, the streamlines and isotherms are presented for Grashof numbers of 103 and 104, respectively. The increase of Grashof number pushes the center of the main vortex closer to the sliding lid. Also, it confines the energy carried away from the sliding lid to
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Fig. 12. Effect of varying Reynolds number on the streamlines and isotherms (Gr = 104, E = 25 103 N/m2).
of the thermal currents to the vicinity of the bottom wall. Finally, Fig. 13 documents the appreciated increase in the average Nusselt number using the flexible wall as compared to the rigid model. Such an enhancement gap closes, however, as the Reynolds number value picks up.
8
6
Average Nu
Flexible Wall
6. Conclusions
4
Rigid Wall
2
0 0
200
400
600
800
1000
Re Fig. 13. Comparison of the average Nusselt number between flexible and rigid bottom wall for various Reynolds numbers (Gr = 104, E = 25 103 N/m2).
the upper half of the cavity. Such an effect becomes less pronounced when the Reynolds number increases. This is vivid from the enlargement of the main central vortex and the penetration
Flow and heat transfer in a lid-driven cavity was investigated numerically for various pertinent parameters using a fluid–structure interaction model. The bottom wall of the cavity was considered to be flexible whereas the vertical walls were taken to be rigid. In addition, the elasticity of the bottom wall was found to have a profound impact on the shape of the bottom wall and consequently on the heat transfer characteristics within the cavity. Moreover, the effect of increasing Grashof number was also found to significantly impact the flexible wall shape and displacement, especially at low Reynolds number. It was also noted that the increase of the Reynolds number brings about a acounter effect to that of the Grashof number. Hence, the considered values of the elasticity, Reynolds number and Grashof number will primarily depend on the applciation at hand. The current investigation invites
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futher research into the application of the fluid–structure interaction model as it offers promising opportunities for heat transfer augmentation. Engineering applications such as MEMS and electronic cooling are potential candidates for such applications. References [1] C.-K. Chen, C.-C. Wang, Transient analysis of forced convection along a wavy surface in micropolar fluids, J. Thermophys. Heat Transfer 14 (2000) 340–347. [2] A. Al-Amiri, K. Khanafer, J. Bull, I. Pop, Effect of sinusoidal wavy bottom surface on mixed convection heat transfer in a lid-driven cavity, Int. J. Heat Mass Transfer 50 (2007) 1771–1780. [3] J.L. Goldstein, E.M. Sparrow, Heat/mass transfer characteristics for flow in a corrugated wall channel, ASME J. Heat Transfer 99 (1977) 187–195. [4] J.E. O’Brien, E.M. Sparrow, Corrugated-duct heat transfer, pressure drop, and flow visualization, J. Heat Transfer 104 (1982) 410–416. [5] C.-C. Wang, C.-K. Chen, Forced convection in a wavy-wall channel, Int. J. Heat Mass Transfer 45 (2002) 2587–2595. [6] Abdalla Al-Amiri, Khalil Khanafer, Joseph Bull, Ioan Pop, Effect of sinusoidal wavy bottom surface on mixed convection heat transfer in a lid-driven cavity, Int. J. Heat Mass Transfer 50 (2007) 1771–1780. [7] K. Khanafer, B. Al-Azmi, Alia Marafie, I. Pop, Non-Darcian effects on natural convection heat transfer in a wavy porous enclosure, Int. J. Heat Mass Transfer 52 (2009) 1887–1896. [8] K.M. Khanafer, A.J. Chamkha, Mixed convection flow in a lid-driven enclosure with a fluid-saturated porous medium, Int. J. Heat Mass Transfer 42 (1999) 2465–2481. [9] Abdalla Al-Amiri, K. Khanafer, I. Pop, Numerical simulation of unsteady mixed convection in a driven cavity using an externally excited sliding lid, Eur. J. Mech. B/Fluids 26 (2007) 669–687. [10] R. Iwatsu, J.M. Hyun, Three-dimensional driven-cavity flows with a vertical temperature gradient, Int. J. Heat Mass Transfer 38 (1995) 3319–3328. [11] R. Iwatsu, J.M. Hyun, K. Kuwahara, Convection in a differentially-heated square cavity with a torsionally-oscillating lid, Int. J. Heat Mass Transfer 35 (1992) 1069–1076. [12] R. Iwatsu, J.M. Hyun, K. Kuwahara, Numerical simulation of flows driven by a torsionally oscillating lid in a square cavity, J. Fluids Eng. 114 (1992) 143–149. [13] V.C. Mariani, A.T. Prata, A Eulerian–Lagrangian method applied to fluid flow in lid-driven cavities with irregular bottom walls, Numer. Heat Transfer, Part B: Fundam. 53 (2008) 206–233. [14] S.B. Paramane, A. Sharma, Consistent implementation and comparison of FOU, CD, SOU, and QUICK convection schemes on square, skew, trapezoidal, and triangular lid-driven cavity flow, Numer. Heat Transfer, Part B: Fundam. 54 (2008) 84–102. [15] K. Khanafer, K. Vafai, Microcantilevers in biomedical and thermo/fluid applications, Front. Heat Mass Transfer (in press). [16] K. Khanafer, A. AlAmiri, I. Pop, Fluid–structure interaction analysis of flow and heat transfer characteristics around a flexible microcantilever in a fluidic cell, Int. J. Heat Mass Transfer 53 (2010) 1646–1653. [17] K. Khanafer, R. Berguer, Fluid–structure interaction analysis of turbulent pulsatile flow within a layered aortic wall as related to aortic dissection, J. Biomech. 42 (2009) 2642–2648.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Fluid–structure interaction analysis of mixed convection heat transfer in a lid-driven cavity with a flexible bottom wall Abdalla Al-Amiri a, Khalil Khanafer b,⇑ a b
Mechanical Engineering Department, United Arab Emirates University, United Arab Emirates Vascular Mechanics Lab, Biomedical Engineering Department and Section of Vascular Surgery, University of Michigan, Ann Arbor, MI 48109, USA
a r t i c l e
i n f o
Article history: Received 19 January 2011 Received in revised form 21 April 2011 Accepted 21 April 2011 Available online 18 May 2011 Keywords: Elasticity Fluid–structure interaction Lid-driven cavity Mixed convection
a b s t r a c t A numerical investigation of steady laminar mixed convection heat transfer in a lid driven cavity with a flexible bottom surface is analyzed. A stable thermal stratification configuration was considered by imposing a vertical temperature gradient while the vertical walls were considered to be insulated. In addition, the transport equations were solved using a finite element formulation based on the Galerkin method of weighted residuals. In essence, a fully coupled fluid–structure interaction (FSI) analysis was utilized in this investigation. Moreover, the fluid domain is described by an Arbitrary-Lagrangian– Eulerian (ALE) formulation that is fully coupled to the structure domain. Comparisons of streamlines, isotherms, bottom wall displacement and average Nusselt number were made between rigid and flexible bottom walls. The results of this investigation revealed that the elasticity of the bottom wall surface plays a significant role on the heat transfer enhancement. Furthermore, the contribution of the forced convection heat transfer to that offered by natural convection heat transfer has a profound effect on the behavior of the flexible wall as well as the momentum and energy transport processes within the cavity. This investigation paves the road for future research studies to consider flexible walls when augmentation of heat transfer is sought. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Augmentation of flow and heat transfer in geometries with irregular surfaces is a topic of fundamental importance. This interest stems from its significance in many engineering and industrial applications such as flat-plate solar collectors, flat plate condensers in refrigerators [1], micro-electronic devices, cooling system of micro-electronic devices, and cooling of electrical components [2]. Further, all the studies on mixed convection flow and heat transfer in geometries assuming regular and irregular surfaces have been restricted to rigid walls [1–14]. For example, Al-Amiri et al. [2] conducted a study to analyze mixed convection heat transfer in a lid-driven cavity with a sinusoidal wavy bottom surface. Their results had illustrated that the average Nusselt number would increase with an increase in both the amplitude of the wavy surface and Reynolds number. Moreover, optimum heat transfer was achieved when the wavy surface was designated with two undulations while subjected to low Richardson numbers. Khanafer et al. [7] carried out a numerical study on natural convection heat transfer inside a porous cavity with a sinusoidal vertical wavy wall.
⇑ Corresponding author. Tel.: +1 (734) 763 5240; fax: +1 (734) 647 4834. E-mail address: [email protected] (K. Khanafer). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.04.047
Their results showed that the amplitude of the wavy surface and the number of undulations affect heat transfer characteristics inside the cavity. Owing to the difficulties associated with machining geometries with wavy surfaces; the authors propose in this investigation the use of flexible surfaces with known elasticity to enhance heat transfer characteristics as the fluid motion will cause the deformation of the flexible solid structure. The authors have proposed a liddriven cavity to demonstrate their case owing to its fundamental nature. The scenario of a flexible wall disturbing a fluid motion, which involves the coupling of fluid mechanics and structural mechanics, is referred to in the literature as the fluid–structure interaction (FSI). FSI approach has received a great attention in recent years and its importance is of a growing interest in mechanical, aerospace and biomedical engineering applications [15–18]. In an FSI situation, the stresses and deformations of a given structure are computed simultaneously with the fluid flow and heat transfer variables that surrounds the structure. The lid-driven cavity with a flexible bottom surface was used by several authors as a benchmark for validating their respective numerical codes for fluid– structure interaction problems [19–23]. It is worth noting that the energy transport was ignored in these cited studies as the prime motivation has been centered around proper modeling of the fluid motion in the presence of a flexible structure.
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Nomenclature cp €s d
specific heat local acceleration of the solid region
df
displacement vector of the fluid domain
ds
displacement vector of the solid domain
E
Young’s Modulus
B ff B fs
fluid body force per unit volume
g Gr h H Nu Pr Re Si T u
gravitational acceleration Grashof number, gbðT H T C ÞH3 =m2 thickness of the flexible bottom wall side length of the cavity Nusselt number Prandtl number, m=a Reynolds number, U 0 H=m interface of the fluid and solid domains temperature fluid velocity vector
ug U0 x, y Xb, Yb
moving coordinate velocity sliding lid velocity Cartesian coordinates dimensionless displacements
Greek symbols a thermal diffusivity, k=ðqf cp Þ b volumetric expansion coefficient m kinematic viscosity # Poisson ratio h dimensionless temperature, ðT T C Þ=ðT H T C Þ q density r stress tensor
solid externally applied body force vector
Subscript f fluid domain s solid domain
Table 1 Comparison of the average Nusselt number at the top wall between various studies at Gr = 102. Re
Present FEM
100 2.02 400 4.05 1000 6.45
Ref. [32] FDM
Ref. [33] FVM
Ref. [34]
Ref. [35] FVM
Ref. [36] FDM
Ref. [37] FVM
1.94 3.84 6.33
2.01 3.91 6.33
1.99 3.88 6.35
NA 4.05 6.55
2.03 4.02 6.48
2.01 3.91 6.33
Table 2 Comparison of the average Nusselt number at the top wall between various studies at Gr = 104 and 106. Re
Gr = 104
Gr = 106
Present
Ref. [32]
Ref. [35]
Present
Ref. [32]
Ref. [35]
100 400 1000
1.38 3.76 6.56
1.34 3.62 6.29
– 3.82 6.50
1.02 1.17 1.72
1.02 1.22 1.77
– 1.17 1.81
⁄ FEM: finite element method; FDM: finite difference method; FVM: finite volume method.
Fig. 1. Schematic diagram of the physical model.
There are many studies reported in the literature on investigating the effect of moving boundaries on the flow and heat transfer characteristics [24–28]. These studies involved boundaries that
move with a prescribed motion (e.g., displacement, pressure, or velocity). For example, Khaled and Vafai [24] studied the effects of both external squeezing and internal pressure pulsations on
0.12 Present [20] Bathe and Zhang [36]
Y-Displacement (m)
0.06
0.00
-0.06
-0.12
-0.18 0
10
20
30
40
50
60
70
Time (s) Fig. 2. Comparison of the y-displacement between the present results and that of Bathe and Zhang [20] in the absence of heat transfer.
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Fig. 3. Comparison of the streamlines between rigid and flexible bottom wall at various Re (Gr = 102, E = 25,000 N/m2).
flow and heat transfer inside non-isothermal and incompressible thin film that has a small thickness compared to its length. The lower plate of the thin film was fixed while the vertical motion of the upper plate was considered to assume a sinusoidal behavior when the thin film gap is not charged with the working fluid. Nakamura et al. [26] investigated numerically the influence of the wall oscillation on the heat transfer characteristics of a two dimensional channel. In their study, the moving boundary problem was transformed into a fixed boundary problem using the coordinate transformation method. Based on the literature survey conducted by the authors and up to their knowledge, there has not been any attention given to investigate the momentum and energy transport processes in liddriven cavities while incorporating flexible walls. Therefore, this fundamental investigation focuses on studying the effect of flexible bottom wall on the heat transfer characteristics for various pertinent parameters such as Reynolds number, elasticity of the bottom wall and Grashof number.
2. Mathematical formulation Consider a two-dimensional square cavity with the physical dimensions shown in Fig. 1. The cavity side length is considered to be H while the thickness of the flexible bottom wall was set to be 0.002 of the side length. The working fluid is assumed to be an incompressible Newtonian fluid with a Prandtl number of 0.71 that is operating under steady-state laminar heat transfer regime. Furthermore, the thermophysical properties of the working fluid are taken to be constant except for the density variation, which is modeled according to the Boussinesq approximation. The mechanically induced lid motion (top wall) is assumed to move from left to right at a constant speed Uo and is maintained at a higher temperature TH than the flexible bottom wall TC. Such an imposed temperature condition will render a thermal stratification in the cavity. Meanwhile, the two vertical walls are assumed rigid and are subjected to an insulated boundary condition. Finally, the conduction through the flexible wall is assumed to be negligible.
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Fig. 4. Comparison of the isotherms between rigid and flexible bottom wall at various Re (Gr = 102, E = 25,000 N/m2).
2.1. FSI analysis
7
ru¼0
ð1Þ
qf ðu ug Þ ru ¼ r rf þ qf f Bf
ð2Þ
u rT ¼ ar2 T
ð3Þ
6
Flexible Wall Average Nu
An Arbitrary Lagrangian–Eulerian formulation was employed to describe the fluid motion in the FSI model. The governing equations for the fluid domain are the continuity, Navier–Stokes equations, and energy equation which are described as:
5
Rigid Wall 4
3
B
where a = k/(qfcp) is the thermal diffusivity, f f the body force per unit volume, qf the fluid density, rf the fluid stress tensor, T the temperature, u the fluid velocity vector, ug the moving coordinate velocity, and (u ug) the relative velocity of the fluid with respect to the moving coordinate velocity. The governing equation for the
2 0
250
500
750
1000
Re Fig. 5. Comparison of the average Nusselt number between rigid and flexible bottom wall for various Reynolds number (Gr = 102, E = 25,000 N/m2).
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Fig. 6. Effect of varying the elasticity of the bottom wall surface on the streamlines and isotherms (Re = 100, Gr = 102).
solid domain of the FSI model can be described by the following elastodynamics equation:
2.4
qs d€ s ¼ r rs þ f Bs Average Nu
ð4Þ
€ s represents the local acceleration of the solid region where d B € _ ðds ¼ ug Þ; f s the externally applied body force vector at time t, qs the density of the bottom wall, and rs the solid stress tensor, and. In the current investigation, the physical properties of the flexible bottom wall were assumed to be constant and homogeneous. The default values chosen for the flexible wall were as follows: a density qs = 500 kg/m3, a Young’s Modulus E = 25,000 N/m2 and a Poisson’s ratio # = 0.45.
2.3
Flexible Wall 2.2
2.1
Rigid Wall
2.2. Boundary conditions 2 0
20000
40000
60000
80000
100000
E (N/m2) Fig. 7. Effect of varying the elasticity of the bottom wall on the average Nusselt number (Re = 100, Gr = 102).
The following boundary conditions are specified for the fluid domain:
u ¼ v ¼ 0;
T ¼ TC
y ¼ 0;
06x6H
ð5aÞ
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Fig. 8. Effect of varying the elasticity of the bottom wall on the displacement (Re = 100, Gr = 102).
u ¼ Uo ; u¼v ¼
v ¼ 0;
T ¼ TH
@T ¼ 0 x ¼ 0; H; @x
y ¼ H;
06x6H
06y6H
ð5bÞ
ð5cÞ
The final set of boundary conditions is applied to the FSI interfaces such that the conditions of displacement compatibility and traction equilibrium along the structure–fluid interfaces must be satisfied. These conditions can be expressed mathematically as follows [29]: Displacement compatibility:
d f ¼ ds
ðx; tÞ 2 Si
ð6Þ
3. Numerical scheme A finite element formulation based on the Galerkin method was utilized to solve the governing equations of fluid–structure interaction model in a lid-driven cavity with a flexible bottom surface using ADINA software subject to the boundary conditions described above. A variable grid-size system was employed to capture the rapid changes in the dependent variables especially in the vicinity of the walls where the major gradients occur inside the boundary layer. As such, a non-uniform mesh of 120 120 nodes was used for the solution of the Navier–Stokes equations [30]. Further, the bottom surface was modeled using iso-parametric beam elements [31].
Traction equilibrium: 4. Code validation
rf ¼ rs ðx; tÞ 2 Si
ð7Þ
where df and ds are the displacements, rf and rs the tractions of the fluid and solid, respectively, and Si the interface of the fluid and solid domains.
The validation of the current algorithm is carried out in 2-fold. First, the present numerical solution using ADINA software is verified against the average Nusselt number calculated at the top wall of the cavity for rigid bottom surface and the results were found to
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Fig. 9. Effect of varying Reynolds number on the displacement (Gr = 103, E = 25 103 N/m2).
be in excellent agreement between the present results and others reported in the literature as shown in Tables 1 and 2, respectively. The second fold of validating the accuracy of the present numerical scheme examines the predictions of the vertical displacement at mid-section of the flexible bottom wall in a driven cavity problem. In this regard, the comparison between the present results against those reported by Bathe et al. [29] in the absence of heat transfer as shown in Fig. 2. The comparison manifests a striking agreement between both results as the maximum relative difference between the two models does not exceed 7%.
average Nusselt number and fluid displacement in the subsequent sections. In addition, the implications of varying the Reynolds number Re = U0H/m and Grashof number Gr ¼ gbðT H T C ÞH3 =m2 on the transport phenomena will be investigated. In essence, the Reynolds number is considered to vary between 100 and 1000, Grashof number between 100 and 1000, and the elasticity of the bottom wall is varied in the range of 103 and 105 N/m2.
5. Results and discussion
Figs. 3 and 4 depict the streamlines and isotherms for both the FSI model and the rigid model for various Reynolds number values. The results show an augmentation in heat transfer when the flexible bottom wall is invoked. As can be seen from the presented results, the shape of the flexible bottom wall is found to depend significantly on the employed Reynolds number value. At high Reynolds number (Re = 1000), the shape of the bottom wall of the cavity approaches a sinusoidal wave pattern. Moreover, Fig. 5 demonstrates that the flexible bottom wall has a profound effect
As mentioned earlier, the main objective of the present investigation is to explore the sensitivity of the elasticity behavior of the flexible bottom wall on the momentum and energy transport in a square lid-driven cavity, which is operating under a steady-state laminar mixed convection heat transfer regime. The transport phenomena are presented by demonstrating and analyzing the streamlines, isotherms [dimensionless temperature; h = (T TC)/(TH TC)],
5.1. Comparison of isotherms, streamlines, and Nusselt number between FSI and rigid models
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Fig. 10. Effect of varying Reynolds number on the displacement of the bottom wall (Gr = 104 and E = 25 103 N/m2).
on the average Nusselt number. As such, flexible bottom wall exhibits higher Nusselt numbers than rigid wall model for various Reynolds numbers. It is worth noting that the maximum relative difference in Nusselt number prediction between both models is 9.4% and occurs at Re = 100. As the Reynolds number increases, it is interesting to note that the Nusselt number difference between both considered models decreases. This is likely attributed to the behavior of the flexible bottom wall with increasing Re. The ‘parabolic’ nature of the flexible wall at Re 6 400 augments heat transfer in as it facilitate the motion of the cold fluid layers from the bottom towards the center of the cavity. Such an augmentation becomes less pronounced when Re = 1000 (i.e., presents a dominant forced convection mechanism as Gr/Re2 1) where the flexible wall assumes a sinusoidal pattern. The sinusoidal pattern brings about an adverse effect which depreciates the contribution of the flexible wall in augmenting the energy transport. Hence, the magnitude of Nusselt number enhancement at higher employed Reynolds number values tends to decrease when compared to the rigid model. This suggests that heat transfer enhancement can be achieved using flexible walls rather than incorporating, for example, irregular geometries.
5.2. Effect of varying the Young’s Modulus of the bottom wall on streamlines, isotherms and Nusselt numbers Fig. 6 shows the effect of varying the elasticity of the bottom wall on the streamlines and isotherms when using Re = 100 and Gr = 102. The variation of the elasticity in the considered range in this investigation appears to have less dramatic impact on the shape of the flexible wall as compared to the considered range of the Reynolds number. In addition, it can be seen from this figure that the bottom surface bulges significantly inward at a small value of elasticity, i.e., E = 1000 N/m2. As the elasticity of the bottom wall increases, its stiffness will consequently increase. As a result, less momentum and thermal penetrations into the cavity are manifested. Furthermore, the impact of elasticity on the average Nusselt number is appraised as depicted in Fig. 7. As can be shown from the figure, the average Nusselt number decreases and asymptotically approaches the value of the rigid model case as the elasticity of the bottom wall increases. It is worth noting that the relative difference in the Nusselt number predictions between both considered models was found to be 15.3% and 4.5% at E = 103 and 105, respectively.
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Fig. 11. Effect of varying Reynolds number on the streamlines and isotherms (Gr = 103, E = 25 103 N/m2).
5.3. Impact of the Young’s Modulus on the displacement of the bottom wall (Xb, Yb) The effect of manipulating the elasticity magnitude of the bottom wall is documented in terms of the incurring Xb wall and Yb displacement in the x- and y-directions, respectively when using Re = 100 and Gr = 102. Fig. 8 shows relatively large positive displacement of the bottom wall in the vertical direction when E = 25 103 N/m2. As the elasticity of the bottom wall increases, the vertical displacement Yb depreciates in value and consequently impedes its influence on the fluid layers in the vicinity of the bottom wall and, hence, retards heat transfer augmentation. It is worth noting that an increase of the elasticity of the flexible wall by 100-fold brings about a reduction in the vertical displacement by around 3-fold. 5.4. Effect of varying Reynolds and Grashof numbers on the streamlines, isotherms, displacement, and Nusselt number Figs. 9–13 display the effect of varying Reynolds number on the bottom wall displacement, streamlines, isotherms, and average Nusselt number predictions at relatively higher Grashof number
values, i.e., Gr = 103 and 104. The Young’s Modulus of elasticity was set equal to 25 103 N/m2 in all these figures. Giving that the Grashof number reflects the strength of the thermal buoyancy force; its value has a significant effect on the shape of the bottom wall and consequently on the heat transfer characteristics within the lid-driven cavity. This can be depicted from the results outlined in Figs. 9 and 10, respectively. This effect is more pronounced at low Reynolds number (Re = 100) when both the buoyancy contribution and the mechanical force of the sliding lid are equally important. At higher Reynolds numbers, the forced convection effect offered by the sliding mechanical lid overwhelms the natural convection sustained by the imposed Grashof number. Hence, the sliding lid tends to dampen the displacement of the flexible bottom wall. As can be seen from the results, the increase of Grashof number from 103 to 104 renders an appreciated increase in the vertical displacement by 1.71, 1.91 and 2.05 for assigned Reynolds number values of 100, 400 and 1000, respectively. Next, the streamlines and isotherms are presented for Grashof numbers of 103 and 104, respectively. The increase of Grashof number pushes the center of the main vortex closer to the sliding lid. Also, it confines the energy carried away from the sliding lid to
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Fig. 12. Effect of varying Reynolds number on the streamlines and isotherms (Gr = 104, E = 25 103 N/m2).
of the thermal currents to the vicinity of the bottom wall. Finally, Fig. 13 documents the appreciated increase in the average Nusselt number using the flexible wall as compared to the rigid model. Such an enhancement gap closes, however, as the Reynolds number value picks up.
8
6
Average Nu
Flexible Wall
6. Conclusions
4
Rigid Wall
2
0 0
200
400
600
800
1000
Re Fig. 13. Comparison of the average Nusselt number between flexible and rigid bottom wall for various Reynolds numbers (Gr = 104, E = 25 103 N/m2).
the upper half of the cavity. Such an effect becomes less pronounced when the Reynolds number increases. This is vivid from the enlargement of the main central vortex and the penetration
Flow and heat transfer in a lid-driven cavity was investigated numerically for various pertinent parameters using a fluid–structure interaction model. The bottom wall of the cavity was considered to be flexible whereas the vertical walls were taken to be rigid. In addition, the elasticity of the bottom wall was found to have a profound impact on the shape of the bottom wall and consequently on the heat transfer characteristics within the cavity. Moreover, the effect of increasing Grashof number was also found to significantly impact the flexible wall shape and displacement, especially at low Reynolds number. It was also noted that the increase of the Reynolds number brings about a acounter effect to that of the Grashof number. Hence, the considered values of the elasticity, Reynolds number and Grashof number will primarily depend on the applciation at hand. The current investigation invites
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