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Role of the fluid-structure interaction in mixed convection in a vented cavity
Accepted Manuscript
Role of the Fluid-Structure Interaction in Mixed Convection in a Vented Cavity Muneer A. Ismael , Haider F. Jasim PII: DOI: Reference:
S0020-7403(17)32234-8 10.1016/j.ijmecsci.2017.11.001 MS 4017
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
12 August 2017 12 October 2017 1 November 2017
Please cite this article as: Muneer A. Ismael , Haider F. Jasim , Role of the Fluid-Structure Interaction in Mixed Convection in a Vented Cavity, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.11.001
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Highlights Fluid-solid interaction with mixed convection in a vented cavity is investigated.
Arbitrary Lagrangian- Eularian approach is used.
Elasticity of the flexible fin can highly improve the heat transfer.
Proximity of the flexible fin to the inlet opening enhances the heat transfer.
At high Richardson number, a highly elastic fin is not favorable.
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Role of the Fluid-Structure Interaction in Mixed Convection in a Vented Cavity Muneer A. Ismael *
Mechanical Engineering Department, Engineering College, University of Basrah, Basrah,
Haider F. Jasim
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Iraq, E-mail: [email protected], [email protected]
Mechanical Engineering Department, Engineering College, University of Basrah, Basrah,
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Iraq. Email: [email protected]
Abstract: The current paper investigates the role of fluid-structure interaction (FSI) in the mixed convection inside a square cavity having two inlet and outlet openings. Flexible elastic fin is attached to the bottom wall of the cavity. The cavity is differentially heated by
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maintaining the vertical walls at two different temperatures. Equations govern the unsteady fields of fluid, thermal and stresses are solved numerically using the Galerkin Finite Element
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Method implemented in Arbitrary Lagrangian- Eularian (ALE) approach. The governing parameters of the present geometry are Cauchy number, which reflects the inertia to elastic
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forces, (Ca = 10-12 – 2×10-4), proximity of fin to the inlet opening (Xf = 0.2 – 0.8), Richardson number (Ri = 0.1 – 100) and the Reynolds number (Re = 50 – 250). The results
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show that the flexible fin enhances the Nusselt number better than the rigid fin. The shape of the fin and the Nusselt number reach the steady periodic state at higher values of Cauchy and
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Richardson numbers. It is also found that the average Nusselt number increases with Cauchy number. However, at very high values of Richardson number, the fin material should not be very elastic.
Keywords: FSI; mixed convection; cavity; flexible fin; ALE. Nomenclatures Ca ⁄ Cauchy number, b width of the inlet and outlet openings ds displacement vector 2
E Fv g H Kr k Lf n Nu P Pr
dimensional Young’s modulus of the fin body force vector gravitational acceleration cavity side length thermal conductivity ratio thermal conductivity fin length normal vector Nusselt number Pressure ⁄ Prandtl number,
Re
Reynolds number
⁄ ⁄
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Ri Richardson number t Time T temperature x,y Cartesian coordinates Xf distance of fin from the inlet opening velocity vector u moving coordinate velocity w Greek symbols α thermal diffusivity β thermal expansion coefficient ν kinematic viscosity ρ density ρr density ratio σ stress tensor τ dimensionless time υ Poisson’s ratio dimensionless temperature stream function ψ Subscripts av average over length c cold f fluid h hot in inlet r ratio s solid Superscripts * dimensional parameters
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ACCEPTED MANUSCRIPT 1. Introduction Convection is an efficient mechanism by which heat is carried and transported by fluid motion. During the last few years, insistent demands have been raised to control convective heat transfer in enclosures as this topic has many engineering applications such as cooling of electronics, solar collectors, lubrication, and food processing. Researchers, therefore, are increasingly developing several routes of controlling (or enhancing) convective heat transfer.
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These routes may be passive and active, examples of these are rigid fins or baffles with natural convection inside enclosures [1-3], inserting of thermally conductive (or adiabatic) rigid bodies inside enclosures [4-8] or by lid-driving cavities [9]. There is an alternative growing version of controlling the convective heat transfer utilizing the interaction between the fluid motion and the deformation of solid or what is known as the fluid-structure
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interaction (FSI). During the last decade, the role of FSI has become prominent topic in many important applications such as cooling of electronic components using the fan-piezoelectric effect [10-11], enhancement of heat transfer using flow-induced vibration [12], designing the heat exchanger [13-14], analysis of aortic valve [15] and prosthetic heart design [16]. In such
of flow and heat transfer fields.
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problem, the solid deforms under the action of fluid motion resulting in changing the domain
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The gained capacity of computers has motivated Küttler and Wall [17] to develop simple and robust FSI solver qualified for wide range of applications. Then, enclosures incorporated with
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the FSI have been achieved by assuming one of the enclosure walls to be deformable. AlAmiri and Khanafer [18] utilized the fully coupled FSI analysis to investigate steady mixed convection in a lid-driven cavity with flexible bottom wall. They reported that the modulus of
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elasticity of the flexible bottom wall plays an important role in enhancing the heat transfer. In addition, they pointed out that the mixed convection contributes in significant deformation of
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the flexible wall along with the momentum and energy exchange inside the cavity. Khanafer [19] has adapted the configuration of the heated bottom wall of [18] by considering rigid rectangular and sinusoidal wavy profiles using the FSI analysis. His results revealed more gain in heat transfer incorporated with the flexible wall than flat wall especially at higher Grashof number. Selimefendigil and Oztop [20] have investigated steady MHD mixed convection in lid-driven cavity having an elastic sidewall. Their cavity was filled with CuOwater nanofluid and differentially heated by a flexible wall. They recorded an increase in the absolute heat transfer with low values of Young’s modulus of the elastic hot wall. In another 4
ACCEPTED MANUSCRIPT study, Selimefendigil and Oztop [21] have considered the steady natural convection in a triangular cavity with flexible inclined wall under the action of inclined magnetic field. They observed the same effect of the elasticity modulus of the flexible wall on the flow and heat transfer characteristics. In a novel aspect of FSI configuration, Jamesahar et al. [22] have studied unsteady natural convection in a square cavity divided diagonally by a thermally conductive flexible membrane. They took into account the weight of the membrane. They found that the heat transfer with flexible membrane is greater than with rigid one, also the
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effect of natural convection has experienced highly deformation of the membrane. It is worth noted that the hopeful findings of Jamesahar et al. [22] has raised the tendency of researchers to include further parameters and geometries of this topic. Mehryan et al. [23] have incorporated the effect of externally applied magnetic field imposed into a square cavity divided by a vertical flexible membrane of high thermal conductivity. They showed
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significance effects of the magnitude and orientation of the magnetic field on the heat transfer and the shape of the membrane. In other study, Mehryan et al. [24] have investigated the effects of the sinusoidal heating and inclination of the square cavity partitioned by a vertical highly conductive membrane taking into account the parameter of the body force acting on
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the membrane. Their results showed that at low Rayleigh number the membrane shape is a function of the imposed body force while at high Rayleigh number, the shape of membrane is
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a function of buoyancy force. Ghalambaz et al. [25] have studied the unsteady convection in a square cavity including an elastic fin attached to the vertical hot wall. The fin was subjected to the buoyancy force and an oscillating excitation. They showed that the Nusselt number is
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significantly enhanced with increasing the oscillation amplitude of the fin. The best length of the fin was found to be 20 percent of the cavity side length. They indicated that the fin of less
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modulus of elasticity gives maximum heat transfer. They found also that the required work for the find does not depend on the modulus of elasticity of the fin only, but also on the fluid
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interaction.
The contribution of FSI in aircraft wings, micro-air vehicle with flapping wings, wind turbines, etc. can be found in the literature of simulating a flexible plate attached to a fixed cylinder in air stream. Lee et al. [26] have presented a computational FSI analysis for flapping flexible plate in quiescent fluid. They focused on the effect of flexibility on the enhancement of propulsion of micro-air vehicles, which have flapping wings. They found that the elasticity of the flexible plate reduces the resistance from the flow and hence improve the propulsion efficiency. Kalmbach and Breuer [27] presented an FSI benchmark 5
ACCEPTED MANUSCRIPT experimental study of a structure consisting of a rigid cylinder with a flexible para-rubber tail in a uniform turbulent water flow. In order to guarantee homogeneous displacements in spanwise direction, they attached a rear mass to the tail. They provided reference data sets for next numerical predictions. Their main finding is that the flexibility of the splitter plate serves in further reduction of the drag coefficients than the stiff case. De Nayer and Breuer [28] introduced a comprehensive numerical simulation for the same case of Kalmbach and Breuer [27]. Their numerical simulation showed a similar unsteady behavior of the flow and the
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structure as the experiment of [27]. Moreover, the streamwise and transverse velocity components were in very good agreement between the numerical and the experiment. They reported that the Young’s modulus has the major influence on the FSI phenomenon for the setup used. Soti et al. [29] demonstrated a numerical FSI study of large-scale flow-induced deformation of an elastic thin plate attached to lee side of a rigid cylinder in a heated channel
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with laminar flow to simulate an efficient passive augmentation of heat transfer. They demonstrated that the vortices generated due to the motion of the flexible plate drive higher sources of vorticity generated on the channel walls out into the high velocity regions, which helps in the mixing of the fluid and therby enhancing the heat transfer. They found also that the enhancement of heat transfer is impeded and the pumping power is increased when
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Young’s modulus of the plate becomes larger.
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From the aforementioned literature review, it can be concluded that despite the efficient improvement of using flexible materials in heat transfer enhancement, little attempt has been
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paid to use this technique in enclosures where many studies focused on reducing the drag effect on a plate-like-flapping wings, and fewer studies focused on enhancing heat transfer
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topic. Different from the existing works, the present paper investigates the FSI represented by a flexible elastic fin incorporated inside a ventilated cavity for two main goals. The first is to
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compare the present problem with a ventilated cavity including a rigid fin and with a ventilated cavity without any fin. The second goal is to conduct a comprehensive study on the dimensionless parameters that govern the interaction between the deformable elastic fin and the mixed convection in a ventilated cavity. The importance of this problem arises when assuming one of the cavity walls at constant high temperature. In this case, we can simulate and improve the process of cooling of electronic components, which is a crucial factor in developing of computers processors.
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ACCEPTED MANUSCRIPT 2. Mathematical Modeling The geometry of the current problem is shown in Fig. 1, it is a two-dimensional square cavity of H side length with two openings (of size b) for inlet and outlet located at the vertical left and right walls, respectively. A flexible thin fin with length lf is attached to the bottom wall at a distance xf from the inlet opening. The left vertical wall is maintained at a high temperature Th while the right vertical wall is maintained at a low temperature Tc. The horizontal walls are
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kept adiabatic. The thermal conductivity of the fin is considered much larger than that of the fluid.
The dimensional unsteady governing equations of mass, momentum, and energy of the fluid and elasto-dynamic structural written in dimensional vector form in the Arbitrary
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Lagrangian-Eulerian method (ALE) as:
Fig. 1 Schematic diagram and coordinates arrangements
For fluid domain: (1) 7
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(2)
(3)
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For elastic structure domain: The equations of the nonlinear elastic displacement and the energy of the fin can be written
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as:
(4)
(5)
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Where u* is the velocity vector, w* is velocity of the moving coordinate, p* is the fluid pressure, T* is the fluid/solid temperature,
*
is ignored in this study as the elastic fin is fixed at one end and free from the other, ds is the
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fin.
is the gravitational force per unit volume acting on
fin displacement vector and σ* is the stress tensor, ρs and ρf represent the densities of solid
respectively.
is the kinematic viscosity of the fluid and β is volumetric thermal expansion
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coefficient.
f
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and fluid, respectively. αs and αf represent thermal diffusivities of the solid and the fluid
The elastic fin exposed to a stress tensor due to the fluid flow pressure taking the nonlinear
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geometry variation, the following form of equations can describe this stress:
Where
(6)
,
, S is the Piola-Kirchhoff stress tensor which is related
with strain ε by the following relation;
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,
(7)
Where C = C(E,ν). C is the elasticity tensor, E is the modulus of the fin elasticity, is the Poisson ratio and the olon “ ” is the double-dot tensor product. The boundary conditions for the fluid-solid
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interaction at the fin surfaces are continuity of kinematic forces and dynamic movements. Considering the regular no-slip boundary condition for fluid at the solid interface results in:
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and
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For the energy equation, the energy balance at interface of solid-fluid:
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(9)
Where ks and kf are thermal conductivities of solid and fluid, respectively.
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The elastic fin in the present work located on the bottom horizontal wall with different positions and free displacement fixed with
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is applied at the upper left point of the cavity i.e.
. The condition of the pressure constraint .
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The dimensional boundary conditions can be written as follow:
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at
at at
For basis condition of the fin
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,
,
,
,
,
Using the previous non-dimensional parameters, the dimensional governing equations (1)-(5)
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can be re-written in the non-dimensional form as:
(11)
(12)
(13)
(14)
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For the elastic fin;
is the Prandtl number (the ratio of viscous to thermal diffusion rates),
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where
(10)
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For fluid;
is the Richardson number (the ratio of the natural to forced convections),
is the Reynolds number (the ratio of inertia to viscous forces), Cauchy number which is defined as the ratio of inertia to elastic forces, ratio, and
is the is density
is thermal difusivity ratio.
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at at at
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is the thermal conductivity ratio.
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For basis condition of the fin
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The boundary conditions of the fluid-structure interaction also written in the following form:
The initial temperatures of the fin and the fluid inside the cavity in the dimensionless form
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are zero. In addition, the initial dimensionless inlet velocity is uin = 0. The dimensionless pressure is equal to zero (p = 0). The convective heat transfer is characterized by the average Nusselt number, which is calculated instantaneously by integrating the instantaneous local temperature gradient at the hot left wall of the cavity.
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∫
(15)
To obtain the time averaged Nusselt number, the expression above is averaged over a cyclic time period (τ) as follow: ∫
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(16)
3. Numerical solution and validations
The non-dimensional equations (10)-(14) along with respective boundary conditions govern
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the unsteady fluid-structure interaction problem with deformable solid and domains are to be solved numerically. This requires crucial numerical technique able to deal with moving boundaries. There are several methods can be used in FSI problems [30] such as immersed boundary method (which consider a finite difference grid) and fictitious domain method (which evolved from the field of finite elements). In the former method, the solid boundary
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interacts with the fluid by means of local body forces applied to the fluid at the position of the points, while the later one is closely related to the immersed boundary method. However,
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amongst several related methods, there is a robust and accurate numerical technique, known as Arbitrary Lagrangian-Eulerian (ALE). It is an efficient approach that mitigates the
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weakness associated with the Lagrangian and Eulerian methods individually, where the mesh nodes of the computational domain may be moved (according to the Lagrangian procedure),
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held fixed (according to the Eulerian procedure), or moved in arbitrary procedure. Detailed demonstrations of this method are discussed in [31-35]. We used this method in the current
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study by transforming the governing equations into the weak form and discretizing them using Galerkin finite element method (see [31]). The mesh of the current computational domains is achieved by applying non-uniform triangular grid. In any time step, the computations are stopped when the error is less than 10-7. To ensure the grid independency solution, we have tested five mesh sizes with tracking the values of the average Nusselt number and the computation time taken by the computer. The parameters of this test are set at Reynolds number =150, Cauchy number = 10-4, fin location Xf = 0.2 and Richardson number is varied from 0.1 to 100. To obtain accurate results with 12
ACCEPTED MANUSCRIPT reasonable time taken by the processor of the computer, we adjusted the convergence criterion to the error of 10-7. The results are shown in Fig. 2, which led us to adopt the mesh size of 6816 elements thorough the calculations of this paper. Furthermore, the numerical solution with this grid size was validated by a comparison with the published work of Ghalambaz et al. [25]. We adapted our numerical code to re-calculate results presented in [25], which is a natural convection inside a square enclosure with a flexible fin attached at the mid span of the left hot wall. The end of this fin is subjecting to a sinusoidal force. We
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achieved the comparison for various Rayleigh numbers and examined the variations of the stream and isothermal lines as shown in Fig. 3. We obtained very good agreement between the current and those presented in [25]. As such, we ascertain that the present numerical
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solution can give results of high confidence level.
Fig. 2 Grid independency test with Richardson number
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Isotherms
Present (a) Ra = 106
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Ghalambaz et al. [25]
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Streamlines
Ghalambaz et al. [25]
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Present (b) Ra = 107
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Fig. 3 Comparison with natural convection problem in a square enclosure with a flexible fin subjected to a periodic force at its free end (Ghalambaz et al. [25]) for (a) Ra = 106 and (b) Ra = 107
4. Results and discussion
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In this study, we focused on the fluid and thermal fields; therefore, the stress analysis in the deformable fin is omitted and deferred to a future study. However, the governing parameters of this study are; the Richardson number (Ri = 0.l, 1, 10, 100), Cauchy number (Ca = 10 -12,
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10-5, 10-4, 2×10-4), Reynolds number (Re = 50, 100, 150, 200, 250), fin location (Xf = 0.2, 0.4, 0.6, 0.8). The fluid entering and leaving the inlet and exit openings is taken to be water
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with Pr = 5.8. The fin is considered of high thermal conductivity, equal to ten times that of water (Kr =10). Fin length, Lf is fixed at 20 percent of the cavity length. In order to
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comprehend the results easily, the numerical results are gathered in the following subsections.
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4.1 Evolution of the fluid and thermal fields Since the present problem is unsteady, it is useful to estimate the evolution of the transient behavior of the fluid and thermal fields to quantify weather the fin reaches a stable profile or oscillates cyclically with time evolution. Several computations have led us to ascertain that a time step of 0.01 can give the minimum deviation of the average Nusselt number, so this time step was adopted through the entire computations. Figure 4 (a) shows the evolution of the instantaneous Nusselt number (Equ. (15)) for Re = 250, Ca = 10-4 and different values of Richardson number. Under these parameters range, it is expected that the flexible fin 15
ACCEPTED MANUSCRIPT undergoes a high deformation rate. Unless at Ri = 100, the Nusselt number reaches the constant steady state values beyond t = 10. While For Ri = 100, a cyclic behavior of the Nusselt number is observed to start a few seconds after the initial time. This cyclic behavior is also reaches its steady periodic state within t ≥ 10 as portrayed in Fig. 4 (a). Through many other implementations of governing parameters, we found no periodic state of the average Nusselt number. For example, there is no fin deformation with lower values of Cauchy number as shown in Fig. 4 (b). This due to that the fluid circulation is insufficient to deform
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such stiff fin. On the other hand, the evolution of the fluid field was pursued by examining the streamlines and the fin shape for selected periods. Based on Fig. 4 (a), the time of one cycle is 1.24, so the selected periods were chosen to cover this cycle. Figure 5 shows the evolution of streamlines within the initial state and within three cycles. Due to the convective heat transfer resulting from the wall temperature difference and the fluid entering the cavity,
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a main clock-wise vortex forms within the cavity. The size of the outlet opening is insufficient to pass out all the circulated fluid, so some fluid fall down forming a circulating vortex. The circulating fluid hit the right side of the flexible fin, thus it bends to the left towards the inlet opening. Now, the fluid entering the cavity will acts as impinging jet and
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forcing the fin towards its original position, but during this action, the falling fluid hit the fin again towards the inlet opening resulting in cycles of fin oscillation. Three oscillation cycles
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are presented in Fig. 5; the first, fifth and the eighth. It is observed that after five oscillation cycles, the streamlines become identical and hence the fin shape reaches the steady periodic state. Therefore, the following instantaneous contour maps are presented at the eighth cycle
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equation (16).
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and exactly at t = 10, whereas the time averaged Nusselt number is calculated according to
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(a)
(b) Fig. 4 Evolution of the average Nusselt number with time for (a) different Richardson number and (b) different Cauchy numbers
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t = 0.01 (a) Initial time
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0.4,
0.6
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t = 0.2,
1.0, (b) First cycle
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t = 5,
5.2,
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6.0
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5.8, (c) Fifth cycle
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t = 5.6,
9.2,
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t = 9.0,
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t = 9.6,
9.8, (d) Eighth cycle
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Fig. 5 Evolution of the streamlines and the fin shape with time for Ri = 100, Re = 250, Ca = 10-4, and Xf = 0.2
4.2 The essence of using flexible fin The objective of using flexible elastic fin is to enhance the mixed convection in a ventilated cavity. Thus, before starting to explain the results, it is necessary to distinguish the role of flexible fin from other methods. For this purpose, we conducted a comparison with two other ventilated cavities, namely a cavity with a rigid fin and another cavity without fin as illustrated in Figs. 6 and 7. 19
ACCEPTED MANUSCRIPT Figure 6 presents the streamlines and isotherms of the three cases for Ri =100, Re = 150. It demonstrates that the flexible fin cavity experiences thinner thermal boundary layer. Moreover, smaller stagnant and isothermal zones are observed behind the flexible elastic fin. Figure 7 shows the variations of the average Nusselt number with Richardson number for three cavities, cavity with a flexible fin, with a rigid fin and a cavity free of any fin. Overall, Fig. 7 demonstrates a clear positive role for the use of flexible fin in raising the average Nusselt number. However, compared with rigid fin cavity, the role of using flexible fin takes
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place within the predominance natural convection (Ri > 1). When Re = 50 (Fig. 7 (a)), the enhancements compared to the no fin case are 48.6% and 8% at Ri = 0.1 and Ri = 100, respectively. While compared to the rigid fin, the enhancement is 10.2% at Ri = 100. On the other hand, when Re increased to 250, the enhancements compared to the no fin case are 230% and 12.5% at Ri = 0.1and Ri = 100, respectively. While compared to the rigid fin
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cavity, the enhancement was 8.5% at Ri = 100 as depicted in Fig. 7 (b). The physical
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Isotherms
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Streamlines
attributions of the role of flexible fin will be explained in the following sections.
(a) Without fin
(b) Rigid fin, Xf = 0.2
(c) Flexible fin, Xf = 0.2,Ca = 10-4
Fig. 6 Streamlines and isotherms at Ri = 100 and Re = 150 for three different cavities; (a) cavity without fin, (b) cavity with rigid fin and (c) cavity with flexible fin 20
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Fig. 7 Variations of the average Nusselt number for three different cavities at Xf = 0.2 and (a) Re = 50 and (b) Re = 250
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4.3 Effect of Cauchy number
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To study the effect of the elasticity modulus (E) of the flexible fin, the Cauchy number is varied in the range shown in Fig. 8 for Ri = 100, Re = 150 and Xf = 0.2. It is worth noting that higher Cauchy number reflects more elastic fin and vice versa. Hence, decrease of Ca
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number means increasing the elasticity modulus of the fin, which means stiffer fin, so it is clear how the fin shape is significantly influenced by the fluid circulation for higher Ca
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values (Fig. 8 (a)). On the other hand, the fin looks rigid for very low value of Ca number as shown in Fig. 8 (d). It is observed from Fig. 8 that a stagnant fluid region is formed when Ca
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≥ 10-4 in the space trapped between the fin and the inlet opening. Nevertheless, the main circulation of fluid does not influenced by Ca number except with the number of core eyes, where double-eye circulation is seen when Ca = 2 ×10-4 and triple-eye fashions are seen with other Ca values. Because of the dominant buoyancy forces (Ri = 100), the isotherms looks mostly horizontal with steeper temperature gradients close to the left hot wall at higher Ca numbers. The variations of the time averaged Nusselt number with Ca number are depicted in Fig. 9. It can be seen that the elasticity modulus (Ca number) becomes effective in the ranges of 21
ACCEPTED MANUSCRIPT predominant natural convection (Ri > 1) as shown in Fig. 9 (a). For higher values of Cauchy numbers, the fin becomes flexible and as such, it bends towards the inlet opening at higher Ri values. This lead to redirect the inlet fluid towards the left vertical wall and as a result the fluid conveys more heat, which in turn increases the value of the Nusselt number. It is worth noting that the impact of elasticity modulus agrees with the results of [20, 21, 25, and 29]. However, for Ri = 100 and Ca = 2×10-4, the fin extremely bends in such a way that the inlet fluid is obstructed, so Nusselt number decreases. Figure 9 (b) demonstrates that the variation
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of Ca is effective within most ranges of the Reynolds number (Re > 50) and the stiffer fin has the lower Nusselt number. For example, at Re = 250, the enhancements of the Nusselt number at Ca = 2×10-4 are 7.8% and 10.8% compared to the cases of Ca = 10-4 and 10-12, respectively. The increase of either Ri or Re lead to increase of natural convection or inertia
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force, respectively. In both situations, the time averaged Nusselt number increases violently.
Isotherms
(a) Ca = 2×10-4
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Streamlines
(b) Ca = 10-4
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(c) Ca = 10-5
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(d) Ca = 10-12
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Fig. 8 Streamlines and isotherms for different values of Ca at Ri = 100, Re = 150 and Xf = 0.2
(a)
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Fig. 9 Variation of the instantaneous Nusselt number with (a) Ri and (b) Re for different values of Cauchy number. 23
ACCEPTED MANUSCRIPT 4.4 Effect of fin location The location of the flexible fin affects the streamlines and the isotherms as shown in Fig. 10 for Ri =10, Re = 150 and Ca = 10-4. The fin experiences more bending to the left when its fixed end is positioned closer to the right wall (higher Xf values). This trend can be attributed to that when the fin becomes far away from the effect of the impinging jet of the inlet opening and closer to the situation of the falling fluid, so it will experience appropriate
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conditions responsible for more bending to the left. As a result, when Xf exceeds the mid length of the horizontal wall, the core of the main clockwise circulation shifts up slightly and the stagnant zone close the fin expands. Moreover, for lower values of Xf, the impinging jet is partially blocked; hence, fluid rises up and serves in augmenting the convective currents. Therefore, the horizontal isotherms looks more uniform when Xf = 0.2 (Fig. 10 (a)) than the
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isotherms associated with other fin locations. Figure 11 demonstrates how the time averaged Nusselt number decays with Xf up to 0.6. It is worth noting that when Ri ≥ 10, the natural convection is crucial, therefore, when the fin becomes far away from the inlet opening, the convective heat transfer is slightly influenced by the blockage effect, so the Nusselt number looks constant beyond Xf ≥ 0.4 as shown in Fig. 11 (a). On the other hand, Fig. 11 (b) shows
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that at high Reynolds numbers, the proximity of the fin to the inlet opening significantly
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increases the average Nusselt number.
Isotherms
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(a) Xf = 0.2
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(b) Xf = 0.4
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(c) Xf = 0.6
(d) Xf = 0.8
Fig. 10 Streamlines and isotherms for different values of fin location Xf at Ri = 10, Re = 150 and Ca = 10-4.
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Fig. 11 Variations of the average Nusselt number with fin position Xf for (a) different Richardson number and (b) different Reynolds number 26
ACCEPTED MANUSCRIPT 4.5 Effect of Richardson and Reynolds numbers From the aforementioned discussion, we have pointed out to the physical meaning of Reynolds number. However, Richardson number can imply to the dominant natural convection (Ri < 1), dominant forced convection (Ri > 1), or equivalent convection modes (Ri = 1). In the presence of the partially blockage effect accompanying with the flexible fin, when Re is increased for constant Ri or when Ri is increased for constant Re, the convective
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heat transfer increases due to the rise of buoyancy and inertia forces. The streamlines presented in Fig. 12 (a) shows that for dominant forced convection (Ri = 0.1), the fin undergoes to the impinging jet only, thus it bends to the right. In addition, most of input fluid exits from the outlet opening and little fluid circulate below the diagonal that connects the inlet and outlet openings. For equivalent convection mode (Ri = 1), the circulating fluid
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expand and reinforce the fin to reserve its initial vertical shape (Fig. 12 (b)). For dominant natural convection (Ri ≥ 10), the fluid circulation becomes stronger, so the fin bends to the left and the amount of outlet fluid is restricted as shown in Fig. 12 (c) and (d). The respective isotherms of Fig. 12 demonstrate the alteration from mostly vertical lines at low Ri values to
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mostly horizontal lines at high Ri values.
Figure 13 shows the effect of Reynolds number on the streamlines and isotherms when Ri is
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fixed at 10 and Ca at 10-4. In general, the streamlines show that when Re increases, the amount of fluid leaving the outlet port increases also. Thus, the remaining circulated fluid is
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insufficient for further fin bending. Therefore, the fin shape demonstrates indistinct deformation. The isotherms develop to be mostly horizontal with Re as an indication to the
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convective heat transfer.
Although the impact of Richardson and Reynolds numbers on the Nusselt number can be
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recognized from some earlier graphs, nevertheless, we set apart an individual one to demonstrate this effect as portrayed in Fig. 14, where there is a monotonic increase of Nusselt number with Richardson number. Prominent increase in Nusselt number with Re is associated at Ri = 100, that is, the augmentation in Nusselt number when Reynolds number is raised from 50 to 100 are 43% at Ri = 0.1 and 145% at Ri = 100.
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ACCEPTED MANUSCRIPT Isotherms
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(a) Ri = 0.1
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Streamlines
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(b) Ri = 1
(c) Ri = 10
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(d) Ri = 100
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Fig. 12 Streamlines and isotherms for different values of Richardson number at Re = 150, Ca
(a) Re = 50
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= 10-4, and Xf = 0.2
(b) Re = 150
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(c) Re = 250
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Fig. 13 Streamlines and isotherms for different values of Reynolds number at Ri = 10, Ca =
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10-4, and Xf = 0.2
Fig. 14 Variations of the average Nusselt number with Richardson number and different Reynolds number at Ca = 10-4 and Xf =0.2
5. Conclusions The effect of fluid-structure interaction represented by a flexible fin attached to the bottom wall of a square cavity with two openings is investigated numerically. The present work is 30
ACCEPTED MANUSCRIPT focused on the interaction between the fin deformation shape and fluid and thermal fields, which govern the mixed convection inside the cavity. ALE approach is used in the numerical analysis to deal with moving domains. The following conclusions are accomplished from the numerical results. 1- The use of flexible elastic fin inside a vented cavity promotes the benefit of the fluid circulation to enhance the heat transfer greater than a cavity without fin and even than
compared with a cavity without fin.
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a cavity with a rigid fin. For specified case, the enhancement reaches to 230%
2- The average Nusselt number increases with increasing Cauchy number when Richardson number is greater than 0.1 where at Ri =10 and Re =250, the
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enhancements of the average Nusselt number at Ca = 2×10-4 are 7.8% and 10.8% compared to the cases of Ca = 10-4 and 10-12, respectively. However, at very high values of Richardson number, the fin material should not be very elastic (high Cauchy
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number) in order to avoid the excessive blockage effect.
3- The flexible fin undergoes large deformation when it fixed far away from the inlet
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opening.
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4- The fin location has a significant effect on the average Nusselt number at higher Reynolds number, and a negligible effect at higher Richardson number. Globally, the
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proximity of the fin to the inlet opening can increase the average Nusselt number.
5- The Richardson number affects the fin deformation shape greater than Reynolds
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number does.
6- For equivalent convection modes, forces on the fin sides are approximately equivalent, thus the fin looks vertical, not distorted.
References [1] A. Ben-Nakhi, A. J. Chamkha, Conjugate natural convection in a square enclosure with inclined thin fin of arbitrary length, Int. J. Therm. Sci. 46 (2007) 467-478. 31
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[2] A. Ben-Nakhi, A. J. Chamkha, Effect of length and inclination of a thin fin on natural convection in a square enclosure, Numer. Heat Transfer A 50 (2006) 381-399. [3] N. Ben-Cheikh, A. J. Chamkha, B. Ben Beya, Effect of inclination on heat transfer and fluid flow in a finned enclosure filled with a dielectric liquid, Numer. Heat
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Transfer A 56 (2009) 286-300. [4] A. J. Chamkha, F. Selimefendigil, M. A. Ismael, Mixed convection in a partially layered porous cavity with an inner rotating cylinder”, Numer. Heat Transfer A 69 (2016) 659-675.
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[5] F. Selimefendigil, M.A. Ismael, A. J. Chamkha, Mixed convection in superposed nanofluid and porous layers in square enclosure with inner rotating cylinder, Int. J. Mech. Sci. 124–125 (2017) 95-108.
[6] Y. Shih, J. Khodadadi, K.Weng, A. Ahmed, Periodic fluid flow and heat transfer in a
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square cavity due to an insulated or isothermal rotating cylinder, J. Heat Transfer 131 (2009) 1-11.
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[7] J.M. Lee, M.Y. Ha, H.S. Yoon, Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations, Int. J. Heat Mass
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Transfer 53 (2010) 5905-5919.
[8] H.F. Oztop, Z. Zhao, B. Yu, Fluid flow due to combined convection in lid-driven
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enclosure having circular body, Int. J. Heat Fluid Flow 30 (2009) 886-901.
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[9] M. A. Ismael, Numerical solution of mixed convection in a lid-driven cavity with arcshaped moving wall, Eng. Computation 34 (2017) 869-891.
[10] M. Kimber, S. V. Garimella, Measurement and prediction of the cooling characteristics of a generalized vibrating piezoelectric fan, Int. J. Heat Mass Transfer 52 (2009) 4470-4478. [11] W. J. Sheu, G. J. Chen, C. C. Wang, Performance of piezoelectric fins for heat dissipation, Int. J. Heat Mass Transfer 86 (2015) 72-77. 32
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[12] W.S. Fu, W.J. Shieh, A study of thermal convection in an enclosure induced simultaneously by gravity and vibration. Int. J. Heat Mass Transfer 35 (1992) 16951710 [13] L. Cheng, T. Luan, W. Du, M. Xu M, Heat transfer enhancement by flow-induced
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vibration in heat exchangers, Int. J. Heat Mass Transfer (2009) 1053-1057. [14] J. Felicjancik, P. Ziółkowski, J. Badur, An advanced thermal-FSI approach of an evaporation of air heat pump, Transactions of the Institute of Fluid-Flow Machinery, No. 129, 2015, 111–141.
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[15] J. De Hart, G.W.M. Peters, P.J.G. Schreurs, F.P.T. Baaijens, A three-dimensional computational analysis of fluid–structure interaction in the aortic valve, J. Biomech. 36 (2003) 103–112.
[16] K. Dumont, Experimental and numerical modeling of heart valve dynamics, Ph.D.
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Dissertation, de Toegepaste Wetenschappen, (2004).
[17] U. Küttler, W. A. Wall, Fixed-point fluid–structure interaction solvers with dynamic
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relaxation, Comput. Mech. 43 (2008) 61–72. [18] A. Al-Amiri, K. Khanafer, Fluid–structure interaction analysis of mixed convection
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heat transfer in a lid-driven cavity with a flexible bottom wall, Int. J. Heat Mass
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Transfer 54 (2011) 3826–3836. [19] K. Khanafer, Comparison of flow and heat transfer characteristics in a lid-driven cavity between flexible and modified geometry of a heated bottom wall, Int. J. Heat
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Mass Transfer 78 (2014) 1032–1041.
[20] F. Selimefendigil, H. F. Öztop, Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation, Int. J. Mech. Sci. 118 (2016) 113–124.
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ACCEPTED MANUSCRIPT [21] F. Selimefendigil, H. F. Öztop, Natural convection in a flexible sided triangular cavity with internal heat generation under the effect of inclined magnetic field, J. MMM 417 (2016) 327–337. [22] E. Jamesahar, M. Ghalambaz, A. J. Chamkha, Fluid-solid interaction in natural convection heat transfer in a square cavity with a perfectly thermal conductive
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flexible diagonal partition, Int. J. Heat Mass Transfer 100 (2016) 303-319. [23] S. A. M. Mehryan, M. Ghalambaz, M. A. Ismael, A. J. Chamkha, Analysis of fluidsolid interaction in MHD natural convection in a square cavity equally partitioned by a vertical flexible membrane, J. MMM 424 (2017) 161–173.
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[24] S. A. M. Mehryan, A. J. Chamkha, M. A. Ismael, M. Ghalambaz, Fluid–structure interaction analysis of free convection in an inclined square cavity partitioned by a flexible impermeable membrane with sinusoidal temperature heating, Meccanica 52 (2017) 2685-2073.
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[25] M. Ghalambaz, E. Jamesahar, M. A. Ismael, A. J. Chamkha, Fluid-structure interaction study of natural convection heat transfer over a flexible oscillating fin in a
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square cavity, Int. J. Therm. Sci. 111 (2017) 256-273.
[26] J.S. Lee, J.-Ho Shin, S.-H. Lee, Fluid–structure interaction of a flapping flexible
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plate in quiescent fluid, Comput. Fluids 57 (2012) 124–137.
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[27] A. Kalmbach, M. Breuer, Experimental PIV/V3V measurements of vortex-induced fluid–structure interaction in turbulent flow - A new benchmark FSI-PfS-2a, J. Fluid
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Struct. 42 (2013) 369–387.
[28] G. De Nayer, M. Breuer, Numerical FSI investigation based on LES: Flow past a cylinder with a flexible splitter plate involving large deformations (FSI-PfS-2a), Int. J. Heat Fluid Flow 50 (2014) 300–315. [29] A. K. Soti, R. Bhardwaj, J. Sheridan, Flow-induced deformation of a flexible thin structure as manifestation of heat transfer enhancement, Int. J. Heat Mass Transfer 84 (2015) 1070–1081. 34
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[30] R. van Loon, P.D. Anderson, F.N. van de Vosse, S.J. Sherwin, Comparison of various fluid–structure interaction methods for deformable bodies, Comput. Struct. 85 (2007) 833–843. [31] J. Donea, A. Huerta, Finite element methods for flow problems, John Wiley Sons,
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2003. [32] C. W. Hirt, A.A. Amsden, J. L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14 (1974) 14 227-253.
[33] T. J. R. Hughes, W. K. Liu, T. K. Zimmermann, Lagrangian–Eulerian finite element
(1981) 329–349.
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formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng. 29
[34] J. Donea, S. Giuliani, Jp. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comput. Methods Appl.
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Mech. Engrg. 33 (1982) 689-723.
[35] E. Kuhl, S. Hulsho, R. de Borst, An arbitrary Lagrangian-Eulerian finite-element
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CE
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(2003) 117–142.
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approach for fluid–structure interaction phenomena, Int. J. Numer. Meth. Engng. 57
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Graphical Abstract
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Role of the Fluid-Structure Interaction in Mixed Convection in a Vented Cavity Muneer A. Ismael , Haider F. Jasim PII: DOI: Reference:
S0020-7403(17)32234-8 10.1016/j.ijmecsci.2017.11.001 MS 4017
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
12 August 2017 12 October 2017 1 November 2017
Please cite this article as: Muneer A. Ismael , Haider F. Jasim , Role of the Fluid-Structure Interaction in Mixed Convection in a Vented Cavity, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.11.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights Fluid-solid interaction with mixed convection in a vented cavity is investigated.
Arbitrary Lagrangian- Eularian approach is used.
Elasticity of the flexible fin can highly improve the heat transfer.
Proximity of the flexible fin to the inlet opening enhances the heat transfer.
At high Richardson number, a highly elastic fin is not favorable.
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Role of the Fluid-Structure Interaction in Mixed Convection in a Vented Cavity Muneer A. Ismael *
Mechanical Engineering Department, Engineering College, University of Basrah, Basrah,
Haider F. Jasim
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Iraq, E-mail: [email protected], [email protected]
Mechanical Engineering Department, Engineering College, University of Basrah, Basrah,
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Iraq. Email: [email protected]
Abstract: The current paper investigates the role of fluid-structure interaction (FSI) in the mixed convection inside a square cavity having two inlet and outlet openings. Flexible elastic fin is attached to the bottom wall of the cavity. The cavity is differentially heated by
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maintaining the vertical walls at two different temperatures. Equations govern the unsteady fields of fluid, thermal and stresses are solved numerically using the Galerkin Finite Element
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Method implemented in Arbitrary Lagrangian- Eularian (ALE) approach. The governing parameters of the present geometry are Cauchy number, which reflects the inertia to elastic
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forces, (Ca = 10-12 – 2×10-4), proximity of fin to the inlet opening (Xf = 0.2 – 0.8), Richardson number (Ri = 0.1 – 100) and the Reynolds number (Re = 50 – 250). The results
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show that the flexible fin enhances the Nusselt number better than the rigid fin. The shape of the fin and the Nusselt number reach the steady periodic state at higher values of Cauchy and
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Richardson numbers. It is also found that the average Nusselt number increases with Cauchy number. However, at very high values of Richardson number, the fin material should not be very elastic.
Keywords: FSI; mixed convection; cavity; flexible fin; ALE. Nomenclatures Ca ⁄ Cauchy number, b width of the inlet and outlet openings ds displacement vector 2
E Fv g H Kr k Lf n Nu P Pr
dimensional Young’s modulus of the fin body force vector gravitational acceleration cavity side length thermal conductivity ratio thermal conductivity fin length normal vector Nusselt number Pressure ⁄ Prandtl number,
Re
Reynolds number
⁄ ⁄
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Ri Richardson number t Time T temperature x,y Cartesian coordinates Xf distance of fin from the inlet opening velocity vector u moving coordinate velocity w Greek symbols α thermal diffusivity β thermal expansion coefficient ν kinematic viscosity ρ density ρr density ratio σ stress tensor τ dimensionless time υ Poisson’s ratio dimensionless temperature stream function ψ Subscripts av average over length c cold f fluid h hot in inlet r ratio s solid Superscripts * dimensional parameters
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ACCEPTED MANUSCRIPT 1. Introduction Convection is an efficient mechanism by which heat is carried and transported by fluid motion. During the last few years, insistent demands have been raised to control convective heat transfer in enclosures as this topic has many engineering applications such as cooling of electronics, solar collectors, lubrication, and food processing. Researchers, therefore, are increasingly developing several routes of controlling (or enhancing) convective heat transfer.
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These routes may be passive and active, examples of these are rigid fins or baffles with natural convection inside enclosures [1-3], inserting of thermally conductive (or adiabatic) rigid bodies inside enclosures [4-8] or by lid-driving cavities [9]. There is an alternative growing version of controlling the convective heat transfer utilizing the interaction between the fluid motion and the deformation of solid or what is known as the fluid-structure
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interaction (FSI). During the last decade, the role of FSI has become prominent topic in many important applications such as cooling of electronic components using the fan-piezoelectric effect [10-11], enhancement of heat transfer using flow-induced vibration [12], designing the heat exchanger [13-14], analysis of aortic valve [15] and prosthetic heart design [16]. In such
of flow and heat transfer fields.
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problem, the solid deforms under the action of fluid motion resulting in changing the domain
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The gained capacity of computers has motivated Küttler and Wall [17] to develop simple and robust FSI solver qualified for wide range of applications. Then, enclosures incorporated with
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the FSI have been achieved by assuming one of the enclosure walls to be deformable. AlAmiri and Khanafer [18] utilized the fully coupled FSI analysis to investigate steady mixed convection in a lid-driven cavity with flexible bottom wall. They reported that the modulus of
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elasticity of the flexible bottom wall plays an important role in enhancing the heat transfer. In addition, they pointed out that the mixed convection contributes in significant deformation of
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the flexible wall along with the momentum and energy exchange inside the cavity. Khanafer [19] has adapted the configuration of the heated bottom wall of [18] by considering rigid rectangular and sinusoidal wavy profiles using the FSI analysis. His results revealed more gain in heat transfer incorporated with the flexible wall than flat wall especially at higher Grashof number. Selimefendigil and Oztop [20] have investigated steady MHD mixed convection in lid-driven cavity having an elastic sidewall. Their cavity was filled with CuOwater nanofluid and differentially heated by a flexible wall. They recorded an increase in the absolute heat transfer with low values of Young’s modulus of the elastic hot wall. In another 4
ACCEPTED MANUSCRIPT study, Selimefendigil and Oztop [21] have considered the steady natural convection in a triangular cavity with flexible inclined wall under the action of inclined magnetic field. They observed the same effect of the elasticity modulus of the flexible wall on the flow and heat transfer characteristics. In a novel aspect of FSI configuration, Jamesahar et al. [22] have studied unsteady natural convection in a square cavity divided diagonally by a thermally conductive flexible membrane. They took into account the weight of the membrane. They found that the heat transfer with flexible membrane is greater than with rigid one, also the
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effect of natural convection has experienced highly deformation of the membrane. It is worth noted that the hopeful findings of Jamesahar et al. [22] has raised the tendency of researchers to include further parameters and geometries of this topic. Mehryan et al. [23] have incorporated the effect of externally applied magnetic field imposed into a square cavity divided by a vertical flexible membrane of high thermal conductivity. They showed
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significance effects of the magnitude and orientation of the magnetic field on the heat transfer and the shape of the membrane. In other study, Mehryan et al. [24] have investigated the effects of the sinusoidal heating and inclination of the square cavity partitioned by a vertical highly conductive membrane taking into account the parameter of the body force acting on
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the membrane. Their results showed that at low Rayleigh number the membrane shape is a function of the imposed body force while at high Rayleigh number, the shape of membrane is
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a function of buoyancy force. Ghalambaz et al. [25] have studied the unsteady convection in a square cavity including an elastic fin attached to the vertical hot wall. The fin was subjected to the buoyancy force and an oscillating excitation. They showed that the Nusselt number is
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significantly enhanced with increasing the oscillation amplitude of the fin. The best length of the fin was found to be 20 percent of the cavity side length. They indicated that the fin of less
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modulus of elasticity gives maximum heat transfer. They found also that the required work for the find does not depend on the modulus of elasticity of the fin only, but also on the fluid
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interaction.
The contribution of FSI in aircraft wings, micro-air vehicle with flapping wings, wind turbines, etc. can be found in the literature of simulating a flexible plate attached to a fixed cylinder in air stream. Lee et al. [26] have presented a computational FSI analysis for flapping flexible plate in quiescent fluid. They focused on the effect of flexibility on the enhancement of propulsion of micro-air vehicles, which have flapping wings. They found that the elasticity of the flexible plate reduces the resistance from the flow and hence improve the propulsion efficiency. Kalmbach and Breuer [27] presented an FSI benchmark 5
ACCEPTED MANUSCRIPT experimental study of a structure consisting of a rigid cylinder with a flexible para-rubber tail in a uniform turbulent water flow. In order to guarantee homogeneous displacements in spanwise direction, they attached a rear mass to the tail. They provided reference data sets for next numerical predictions. Their main finding is that the flexibility of the splitter plate serves in further reduction of the drag coefficients than the stiff case. De Nayer and Breuer [28] introduced a comprehensive numerical simulation for the same case of Kalmbach and Breuer [27]. Their numerical simulation showed a similar unsteady behavior of the flow and the
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structure as the experiment of [27]. Moreover, the streamwise and transverse velocity components were in very good agreement between the numerical and the experiment. They reported that the Young’s modulus has the major influence on the FSI phenomenon for the setup used. Soti et al. [29] demonstrated a numerical FSI study of large-scale flow-induced deformation of an elastic thin plate attached to lee side of a rigid cylinder in a heated channel
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with laminar flow to simulate an efficient passive augmentation of heat transfer. They demonstrated that the vortices generated due to the motion of the flexible plate drive higher sources of vorticity generated on the channel walls out into the high velocity regions, which helps in the mixing of the fluid and therby enhancing the heat transfer. They found also that the enhancement of heat transfer is impeded and the pumping power is increased when
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Young’s modulus of the plate becomes larger.
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From the aforementioned literature review, it can be concluded that despite the efficient improvement of using flexible materials in heat transfer enhancement, little attempt has been
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paid to use this technique in enclosures where many studies focused on reducing the drag effect on a plate-like-flapping wings, and fewer studies focused on enhancing heat transfer
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topic. Different from the existing works, the present paper investigates the FSI represented by a flexible elastic fin incorporated inside a ventilated cavity for two main goals. The first is to
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compare the present problem with a ventilated cavity including a rigid fin and with a ventilated cavity without any fin. The second goal is to conduct a comprehensive study on the dimensionless parameters that govern the interaction between the deformable elastic fin and the mixed convection in a ventilated cavity. The importance of this problem arises when assuming one of the cavity walls at constant high temperature. In this case, we can simulate and improve the process of cooling of electronic components, which is a crucial factor in developing of computers processors.
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ACCEPTED MANUSCRIPT 2. Mathematical Modeling The geometry of the current problem is shown in Fig. 1, it is a two-dimensional square cavity of H side length with two openings (of size b) for inlet and outlet located at the vertical left and right walls, respectively. A flexible thin fin with length lf is attached to the bottom wall at a distance xf from the inlet opening. The left vertical wall is maintained at a high temperature Th while the right vertical wall is maintained at a low temperature Tc. The horizontal walls are
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kept adiabatic. The thermal conductivity of the fin is considered much larger than that of the fluid.
The dimensional unsteady governing equations of mass, momentum, and energy of the fluid and elasto-dynamic structural written in dimensional vector form in the Arbitrary
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Lagrangian-Eulerian method (ALE) as:
Fig. 1 Schematic diagram and coordinates arrangements
For fluid domain: (1) 7
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(2)
(3)
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For elastic structure domain: The equations of the nonlinear elastic displacement and the energy of the fin can be written
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as:
(4)
(5)
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Where u* is the velocity vector, w* is velocity of the moving coordinate, p* is the fluid pressure, T* is the fluid/solid temperature,
*
is ignored in this study as the elastic fin is fixed at one end and free from the other, ds is the
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fin.
is the gravitational force per unit volume acting on
fin displacement vector and σ* is the stress tensor, ρs and ρf represent the densities of solid
respectively.
is the kinematic viscosity of the fluid and β is volumetric thermal expansion
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coefficient.
f
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and fluid, respectively. αs and αf represent thermal diffusivities of the solid and the fluid
The elastic fin exposed to a stress tensor due to the fluid flow pressure taking the nonlinear
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geometry variation, the following form of equations can describe this stress:
Where
(6)
,
, S is the Piola-Kirchhoff stress tensor which is related
with strain ε by the following relation;
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,
(7)
Where C = C(E,ν). C is the elasticity tensor, E is the modulus of the fin elasticity, is the Poisson ratio and the olon “ ” is the double-dot tensor product. The boundary conditions for the fluid-solid
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interaction at the fin surfaces are continuity of kinematic forces and dynamic movements. Considering the regular no-slip boundary condition for fluid at the solid interface results in:
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and
(8)
For the energy equation, the energy balance at interface of solid-fluid:
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(9)
Where ks and kf are thermal conductivities of solid and fluid, respectively.
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The elastic fin in the present work located on the bottom horizontal wall with different positions and free displacement fixed with
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is applied at the upper left point of the cavity i.e.
. The condition of the pressure constraint .
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The dimensional boundary conditions can be written as follow:
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at
at at
For basis condition of the fin
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ACCEPTED MANUSCRIPT In order to generalize the present study, the governing equations along with mixed convection problem are re-casted using the following non-dimensional parameters and definitions: ,
,
,
,
,
,
Using the previous non-dimensional parameters, the dimensional governing equations (1)-(5)
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can be re-written in the non-dimensional form as:
(11)
(12)
(13)
(14)
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For the elastic fin;
is the Prandtl number (the ratio of viscous to thermal diffusion rates),
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where
(10)
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For fluid;
is the Richardson number (the ratio of the natural to forced convections),
is the Reynolds number (the ratio of inertia to viscous forces), Cauchy number which is defined as the ratio of inertia to elastic forces, ratio, and
is the is density
is thermal difusivity ratio.
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at at at
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is the thermal conductivity ratio.
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For basis condition of the fin
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The boundary conditions of the fluid-structure interaction also written in the following form:
The initial temperatures of the fin and the fluid inside the cavity in the dimensionless form
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are zero. In addition, the initial dimensionless inlet velocity is uin = 0. The dimensionless pressure is equal to zero (p = 0). The convective heat transfer is characterized by the average Nusselt number, which is calculated instantaneously by integrating the instantaneous local temperature gradient at the hot left wall of the cavity.
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∫
(15)
To obtain the time averaged Nusselt number, the expression above is averaged over a cyclic time period (τ) as follow: ∫
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(16)
3. Numerical solution and validations
The non-dimensional equations (10)-(14) along with respective boundary conditions govern
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the unsteady fluid-structure interaction problem with deformable solid and domains are to be solved numerically. This requires crucial numerical technique able to deal with moving boundaries. There are several methods can be used in FSI problems [30] such as immersed boundary method (which consider a finite difference grid) and fictitious domain method (which evolved from the field of finite elements). In the former method, the solid boundary
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interacts with the fluid by means of local body forces applied to the fluid at the position of the points, while the later one is closely related to the immersed boundary method. However,
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amongst several related methods, there is a robust and accurate numerical technique, known as Arbitrary Lagrangian-Eulerian (ALE). It is an efficient approach that mitigates the
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weakness associated with the Lagrangian and Eulerian methods individually, where the mesh nodes of the computational domain may be moved (according to the Lagrangian procedure),
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held fixed (according to the Eulerian procedure), or moved in arbitrary procedure. Detailed demonstrations of this method are discussed in [31-35]. We used this method in the current
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study by transforming the governing equations into the weak form and discretizing them using Galerkin finite element method (see [31]). The mesh of the current computational domains is achieved by applying non-uniform triangular grid. In any time step, the computations are stopped when the error is less than 10-7. To ensure the grid independency solution, we have tested five mesh sizes with tracking the values of the average Nusselt number and the computation time taken by the computer. The parameters of this test are set at Reynolds number =150, Cauchy number = 10-4, fin location Xf = 0.2 and Richardson number is varied from 0.1 to 100. To obtain accurate results with 12
ACCEPTED MANUSCRIPT reasonable time taken by the processor of the computer, we adjusted the convergence criterion to the error of 10-7. The results are shown in Fig. 2, which led us to adopt the mesh size of 6816 elements thorough the calculations of this paper. Furthermore, the numerical solution with this grid size was validated by a comparison with the published work of Ghalambaz et al. [25]. We adapted our numerical code to re-calculate results presented in [25], which is a natural convection inside a square enclosure with a flexible fin attached at the mid span of the left hot wall. The end of this fin is subjecting to a sinusoidal force. We
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achieved the comparison for various Rayleigh numbers and examined the variations of the stream and isothermal lines as shown in Fig. 3. We obtained very good agreement between the current and those presented in [25]. As such, we ascertain that the present numerical
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solution can give results of high confidence level.
Fig. 2 Grid independency test with Richardson number
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Isotherms
Present (a) Ra = 106
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Ghalambaz et al. [25]
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Streamlines
Ghalambaz et al. [25]
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Present (b) Ra = 107
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Fig. 3 Comparison with natural convection problem in a square enclosure with a flexible fin subjected to a periodic force at its free end (Ghalambaz et al. [25]) for (a) Ra = 106 and (b) Ra = 107
4. Results and discussion
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In this study, we focused on the fluid and thermal fields; therefore, the stress analysis in the deformable fin is omitted and deferred to a future study. However, the governing parameters of this study are; the Richardson number (Ri = 0.l, 1, 10, 100), Cauchy number (Ca = 10 -12,
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10-5, 10-4, 2×10-4), Reynolds number (Re = 50, 100, 150, 200, 250), fin location (Xf = 0.2, 0.4, 0.6, 0.8). The fluid entering and leaving the inlet and exit openings is taken to be water
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with Pr = 5.8. The fin is considered of high thermal conductivity, equal to ten times that of water (Kr =10). Fin length, Lf is fixed at 20 percent of the cavity length. In order to
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comprehend the results easily, the numerical results are gathered in the following subsections.
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4.1 Evolution of the fluid and thermal fields Since the present problem is unsteady, it is useful to estimate the evolution of the transient behavior of the fluid and thermal fields to quantify weather the fin reaches a stable profile or oscillates cyclically with time evolution. Several computations have led us to ascertain that a time step of 0.01 can give the minimum deviation of the average Nusselt number, so this time step was adopted through the entire computations. Figure 4 (a) shows the evolution of the instantaneous Nusselt number (Equ. (15)) for Re = 250, Ca = 10-4 and different values of Richardson number. Under these parameters range, it is expected that the flexible fin 15
ACCEPTED MANUSCRIPT undergoes a high deformation rate. Unless at Ri = 100, the Nusselt number reaches the constant steady state values beyond t = 10. While For Ri = 100, a cyclic behavior of the Nusselt number is observed to start a few seconds after the initial time. This cyclic behavior is also reaches its steady periodic state within t ≥ 10 as portrayed in Fig. 4 (a). Through many other implementations of governing parameters, we found no periodic state of the average Nusselt number. For example, there is no fin deformation with lower values of Cauchy number as shown in Fig. 4 (b). This due to that the fluid circulation is insufficient to deform
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such stiff fin. On the other hand, the evolution of the fluid field was pursued by examining the streamlines and the fin shape for selected periods. Based on Fig. 4 (a), the time of one cycle is 1.24, so the selected periods were chosen to cover this cycle. Figure 5 shows the evolution of streamlines within the initial state and within three cycles. Due to the convective heat transfer resulting from the wall temperature difference and the fluid entering the cavity,
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a main clock-wise vortex forms within the cavity. The size of the outlet opening is insufficient to pass out all the circulated fluid, so some fluid fall down forming a circulating vortex. The circulating fluid hit the right side of the flexible fin, thus it bends to the left towards the inlet opening. Now, the fluid entering the cavity will acts as impinging jet and
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forcing the fin towards its original position, but during this action, the falling fluid hit the fin again towards the inlet opening resulting in cycles of fin oscillation. Three oscillation cycles
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are presented in Fig. 5; the first, fifth and the eighth. It is observed that after five oscillation cycles, the streamlines become identical and hence the fin shape reaches the steady periodic state. Therefore, the following instantaneous contour maps are presented at the eighth cycle
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equation (16).
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and exactly at t = 10, whereas the time averaged Nusselt number is calculated according to
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(a)
(b) Fig. 4 Evolution of the average Nusselt number with time for (a) different Richardson number and (b) different Cauchy numbers
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t = 0.01 (a) Initial time
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0.4,
0.6
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t = 0.2,
1.0, (b) First cycle
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t = 5,
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6.0
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5.8, (c) Fifth cycle
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t = 5.6,
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t = 9.0,
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t = 9.6,
9.8, (d) Eighth cycle
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Fig. 5 Evolution of the streamlines and the fin shape with time for Ri = 100, Re = 250, Ca = 10-4, and Xf = 0.2
4.2 The essence of using flexible fin The objective of using flexible elastic fin is to enhance the mixed convection in a ventilated cavity. Thus, before starting to explain the results, it is necessary to distinguish the role of flexible fin from other methods. For this purpose, we conducted a comparison with two other ventilated cavities, namely a cavity with a rigid fin and another cavity without fin as illustrated in Figs. 6 and 7. 19
ACCEPTED MANUSCRIPT Figure 6 presents the streamlines and isotherms of the three cases for Ri =100, Re = 150. It demonstrates that the flexible fin cavity experiences thinner thermal boundary layer. Moreover, smaller stagnant and isothermal zones are observed behind the flexible elastic fin. Figure 7 shows the variations of the average Nusselt number with Richardson number for three cavities, cavity with a flexible fin, with a rigid fin and a cavity free of any fin. Overall, Fig. 7 demonstrates a clear positive role for the use of flexible fin in raising the average Nusselt number. However, compared with rigid fin cavity, the role of using flexible fin takes
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place within the predominance natural convection (Ri > 1). When Re = 50 (Fig. 7 (a)), the enhancements compared to the no fin case are 48.6% and 8% at Ri = 0.1 and Ri = 100, respectively. While compared to the rigid fin, the enhancement is 10.2% at Ri = 100. On the other hand, when Re increased to 250, the enhancements compared to the no fin case are 230% and 12.5% at Ri = 0.1and Ri = 100, respectively. While compared to the rigid fin
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cavity, the enhancement was 8.5% at Ri = 100 as depicted in Fig. 7 (b). The physical
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Isotherms
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Streamlines
attributions of the role of flexible fin will be explained in the following sections.
(a) Without fin
(b) Rigid fin, Xf = 0.2
(c) Flexible fin, Xf = 0.2,Ca = 10-4
Fig. 6 Streamlines and isotherms at Ri = 100 and Re = 150 for three different cavities; (a) cavity without fin, (b) cavity with rigid fin and (c) cavity with flexible fin 20
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Fig. 7 Variations of the average Nusselt number for three different cavities at Xf = 0.2 and (a) Re = 50 and (b) Re = 250
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4.3 Effect of Cauchy number
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To study the effect of the elasticity modulus (E) of the flexible fin, the Cauchy number is varied in the range shown in Fig. 8 for Ri = 100, Re = 150 and Xf = 0.2. It is worth noting that higher Cauchy number reflects more elastic fin and vice versa. Hence, decrease of Ca
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number means increasing the elasticity modulus of the fin, which means stiffer fin, so it is clear how the fin shape is significantly influenced by the fluid circulation for higher Ca
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values (Fig. 8 (a)). On the other hand, the fin looks rigid for very low value of Ca number as shown in Fig. 8 (d). It is observed from Fig. 8 that a stagnant fluid region is formed when Ca
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≥ 10-4 in the space trapped between the fin and the inlet opening. Nevertheless, the main circulation of fluid does not influenced by Ca number except with the number of core eyes, where double-eye circulation is seen when Ca = 2 ×10-4 and triple-eye fashions are seen with other Ca values. Because of the dominant buoyancy forces (Ri = 100), the isotherms looks mostly horizontal with steeper temperature gradients close to the left hot wall at higher Ca numbers. The variations of the time averaged Nusselt number with Ca number are depicted in Fig. 9. It can be seen that the elasticity modulus (Ca number) becomes effective in the ranges of 21
ACCEPTED MANUSCRIPT predominant natural convection (Ri > 1) as shown in Fig. 9 (a). For higher values of Cauchy numbers, the fin becomes flexible and as such, it bends towards the inlet opening at higher Ri values. This lead to redirect the inlet fluid towards the left vertical wall and as a result the fluid conveys more heat, which in turn increases the value of the Nusselt number. It is worth noting that the impact of elasticity modulus agrees with the results of [20, 21, 25, and 29]. However, for Ri = 100 and Ca = 2×10-4, the fin extremely bends in such a way that the inlet fluid is obstructed, so Nusselt number decreases. Figure 9 (b) demonstrates that the variation
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of Ca is effective within most ranges of the Reynolds number (Re > 50) and the stiffer fin has the lower Nusselt number. For example, at Re = 250, the enhancements of the Nusselt number at Ca = 2×10-4 are 7.8% and 10.8% compared to the cases of Ca = 10-4 and 10-12, respectively. The increase of either Ri or Re lead to increase of natural convection or inertia
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force, respectively. In both situations, the time averaged Nusselt number increases violently.
Isotherms
(a) Ca = 2×10-4
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Streamlines
(b) Ca = 10-4
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(c) Ca = 10-5
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(d) Ca = 10-12
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Fig. 8 Streamlines and isotherms for different values of Ca at Ri = 100, Re = 150 and Xf = 0.2
(a)
(b)
Fig. 9 Variation of the instantaneous Nusselt number with (a) Ri and (b) Re for different values of Cauchy number. 23
ACCEPTED MANUSCRIPT 4.4 Effect of fin location The location of the flexible fin affects the streamlines and the isotherms as shown in Fig. 10 for Ri =10, Re = 150 and Ca = 10-4. The fin experiences more bending to the left when its fixed end is positioned closer to the right wall (higher Xf values). This trend can be attributed to that when the fin becomes far away from the effect of the impinging jet of the inlet opening and closer to the situation of the falling fluid, so it will experience appropriate
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conditions responsible for more bending to the left. As a result, when Xf exceeds the mid length of the horizontal wall, the core of the main clockwise circulation shifts up slightly and the stagnant zone close the fin expands. Moreover, for lower values of Xf, the impinging jet is partially blocked; hence, fluid rises up and serves in augmenting the convective currents. Therefore, the horizontal isotherms looks more uniform when Xf = 0.2 (Fig. 10 (a)) than the
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isotherms associated with other fin locations. Figure 11 demonstrates how the time averaged Nusselt number decays with Xf up to 0.6. It is worth noting that when Ri ≥ 10, the natural convection is crucial, therefore, when the fin becomes far away from the inlet opening, the convective heat transfer is slightly influenced by the blockage effect, so the Nusselt number looks constant beyond Xf ≥ 0.4 as shown in Fig. 11 (a). On the other hand, Fig. 11 (b) shows
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that at high Reynolds numbers, the proximity of the fin to the inlet opening significantly
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increases the average Nusselt number.
Isotherms
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(a) Xf = 0.2
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(b) Xf = 0.4
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(c) Xf = 0.6
(d) Xf = 0.8
Fig. 10 Streamlines and isotherms for different values of fin location Xf at Ri = 10, Re = 150 and Ca = 10-4.
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(a)
(b)
Fig. 11 Variations of the average Nusselt number with fin position Xf for (a) different Richardson number and (b) different Reynolds number 26
ACCEPTED MANUSCRIPT 4.5 Effect of Richardson and Reynolds numbers From the aforementioned discussion, we have pointed out to the physical meaning of Reynolds number. However, Richardson number can imply to the dominant natural convection (Ri < 1), dominant forced convection (Ri > 1), or equivalent convection modes (Ri = 1). In the presence of the partially blockage effect accompanying with the flexible fin, when Re is increased for constant Ri or when Ri is increased for constant Re, the convective
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heat transfer increases due to the rise of buoyancy and inertia forces. The streamlines presented in Fig. 12 (a) shows that for dominant forced convection (Ri = 0.1), the fin undergoes to the impinging jet only, thus it bends to the right. In addition, most of input fluid exits from the outlet opening and little fluid circulate below the diagonal that connects the inlet and outlet openings. For equivalent convection mode (Ri = 1), the circulating fluid
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expand and reinforce the fin to reserve its initial vertical shape (Fig. 12 (b)). For dominant natural convection (Ri ≥ 10), the fluid circulation becomes stronger, so the fin bends to the left and the amount of outlet fluid is restricted as shown in Fig. 12 (c) and (d). The respective isotherms of Fig. 12 demonstrate the alteration from mostly vertical lines at low Ri values to
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mostly horizontal lines at high Ri values.
Figure 13 shows the effect of Reynolds number on the streamlines and isotherms when Ri is
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fixed at 10 and Ca at 10-4. In general, the streamlines show that when Re increases, the amount of fluid leaving the outlet port increases also. Thus, the remaining circulated fluid is
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insufficient for further fin bending. Therefore, the fin shape demonstrates indistinct deformation. The isotherms develop to be mostly horizontal with Re as an indication to the
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convective heat transfer.
Although the impact of Richardson and Reynolds numbers on the Nusselt number can be
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recognized from some earlier graphs, nevertheless, we set apart an individual one to demonstrate this effect as portrayed in Fig. 14, where there is a monotonic increase of Nusselt number with Richardson number. Prominent increase in Nusselt number with Re is associated at Ri = 100, that is, the augmentation in Nusselt number when Reynolds number is raised from 50 to 100 are 43% at Ri = 0.1 and 145% at Ri = 100.
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(a) Ri = 0.1
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Streamlines
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(b) Ri = 1
(c) Ri = 10
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(d) Ri = 100
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Fig. 12 Streamlines and isotherms for different values of Richardson number at Re = 150, Ca
(a) Re = 50
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= 10-4, and Xf = 0.2
(b) Re = 150
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(c) Re = 250
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Fig. 13 Streamlines and isotherms for different values of Reynolds number at Ri = 10, Ca =
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10-4, and Xf = 0.2
Fig. 14 Variations of the average Nusselt number with Richardson number and different Reynolds number at Ca = 10-4 and Xf =0.2
5. Conclusions The effect of fluid-structure interaction represented by a flexible fin attached to the bottom wall of a square cavity with two openings is investigated numerically. The present work is 30
ACCEPTED MANUSCRIPT focused on the interaction between the fin deformation shape and fluid and thermal fields, which govern the mixed convection inside the cavity. ALE approach is used in the numerical analysis to deal with moving domains. The following conclusions are accomplished from the numerical results. 1- The use of flexible elastic fin inside a vented cavity promotes the benefit of the fluid circulation to enhance the heat transfer greater than a cavity without fin and even than
compared with a cavity without fin.
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a cavity with a rigid fin. For specified case, the enhancement reaches to 230%
2- The average Nusselt number increases with increasing Cauchy number when Richardson number is greater than 0.1 where at Ri =10 and Re =250, the
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enhancements of the average Nusselt number at Ca = 2×10-4 are 7.8% and 10.8% compared to the cases of Ca = 10-4 and 10-12, respectively. However, at very high values of Richardson number, the fin material should not be very elastic (high Cauchy
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number) in order to avoid the excessive blockage effect.
3- The flexible fin undergoes large deformation when it fixed far away from the inlet
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opening.
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4- The fin location has a significant effect on the average Nusselt number at higher Reynolds number, and a negligible effect at higher Richardson number. Globally, the
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proximity of the fin to the inlet opening can increase the average Nusselt number.
5- The Richardson number affects the fin deformation shape greater than Reynolds
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number does.
6- For equivalent convection modes, forces on the fin sides are approximately equivalent, thus the fin looks vertical, not distorted.
References [1] A. Ben-Nakhi, A. J. Chamkha, Conjugate natural convection in a square enclosure with inclined thin fin of arbitrary length, Int. J. Therm. Sci. 46 (2007) 467-478. 31
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[2] A. Ben-Nakhi, A. J. Chamkha, Effect of length and inclination of a thin fin on natural convection in a square enclosure, Numer. Heat Transfer A 50 (2006) 381-399. [3] N. Ben-Cheikh, A. J. Chamkha, B. Ben Beya, Effect of inclination on heat transfer and fluid flow in a finned enclosure filled with a dielectric liquid, Numer. Heat
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Transfer A 56 (2009) 286-300. [4] A. J. Chamkha, F. Selimefendigil, M. A. Ismael, Mixed convection in a partially layered porous cavity with an inner rotating cylinder”, Numer. Heat Transfer A 69 (2016) 659-675.
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[5] F. Selimefendigil, M.A. Ismael, A. J. Chamkha, Mixed convection in superposed nanofluid and porous layers in square enclosure with inner rotating cylinder, Int. J. Mech. Sci. 124–125 (2017) 95-108.
[6] Y. Shih, J. Khodadadi, K.Weng, A. Ahmed, Periodic fluid flow and heat transfer in a
M
square cavity due to an insulated or isothermal rotating cylinder, J. Heat Transfer 131 (2009) 1-11.
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[7] J.M. Lee, M.Y. Ha, H.S. Yoon, Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations, Int. J. Heat Mass
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Transfer 53 (2010) 5905-5919.
[8] H.F. Oztop, Z. Zhao, B. Yu, Fluid flow due to combined convection in lid-driven
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enclosure having circular body, Int. J. Heat Fluid Flow 30 (2009) 886-901.
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[9] M. A. Ismael, Numerical solution of mixed convection in a lid-driven cavity with arcshaped moving wall, Eng. Computation 34 (2017) 869-891.
[10] M. Kimber, S. V. Garimella, Measurement and prediction of the cooling characteristics of a generalized vibrating piezoelectric fan, Int. J. Heat Mass Transfer 52 (2009) 4470-4478. [11] W. J. Sheu, G. J. Chen, C. C. Wang, Performance of piezoelectric fins for heat dissipation, Int. J. Heat Mass Transfer 86 (2015) 72-77. 32
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[12] W.S. Fu, W.J. Shieh, A study of thermal convection in an enclosure induced simultaneously by gravity and vibration. Int. J. Heat Mass Transfer 35 (1992) 16951710 [13] L. Cheng, T. Luan, W. Du, M. Xu M, Heat transfer enhancement by flow-induced
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vibration in heat exchangers, Int. J. Heat Mass Transfer (2009) 1053-1057. [14] J. Felicjancik, P. Ziółkowski, J. Badur, An advanced thermal-FSI approach of an evaporation of air heat pump, Transactions of the Institute of Fluid-Flow Machinery, No. 129, 2015, 111–141.
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[15] J. De Hart, G.W.M. Peters, P.J.G. Schreurs, F.P.T. Baaijens, A three-dimensional computational analysis of fluid–structure interaction in the aortic valve, J. Biomech. 36 (2003) 103–112.
[16] K. Dumont, Experimental and numerical modeling of heart valve dynamics, Ph.D.
M
Dissertation, de Toegepaste Wetenschappen, (2004).
[17] U. Küttler, W. A. Wall, Fixed-point fluid–structure interaction solvers with dynamic
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relaxation, Comput. Mech. 43 (2008) 61–72. [18] A. Al-Amiri, K. Khanafer, Fluid–structure interaction analysis of mixed convection
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heat transfer in a lid-driven cavity with a flexible bottom wall, Int. J. Heat Mass
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Transfer 54 (2011) 3826–3836. [19] K. Khanafer, Comparison of flow and heat transfer characteristics in a lid-driven cavity between flexible and modified geometry of a heated bottom wall, Int. J. Heat
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Mass Transfer 78 (2014) 1032–1041.
[20] F. Selimefendigil, H. F. Öztop, Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation, Int. J. Mech. Sci. 118 (2016) 113–124.
33
ACCEPTED MANUSCRIPT [21] F. Selimefendigil, H. F. Öztop, Natural convection in a flexible sided triangular cavity with internal heat generation under the effect of inclined magnetic field, J. MMM 417 (2016) 327–337. [22] E. Jamesahar, M. Ghalambaz, A. J. Chamkha, Fluid-solid interaction in natural convection heat transfer in a square cavity with a perfectly thermal conductive
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flexible diagonal partition, Int. J. Heat Mass Transfer 100 (2016) 303-319. [23] S. A. M. Mehryan, M. Ghalambaz, M. A. Ismael, A. J. Chamkha, Analysis of fluidsolid interaction in MHD natural convection in a square cavity equally partitioned by a vertical flexible membrane, J. MMM 424 (2017) 161–173.
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[24] S. A. M. Mehryan, A. J. Chamkha, M. A. Ismael, M. Ghalambaz, Fluid–structure interaction analysis of free convection in an inclined square cavity partitioned by a flexible impermeable membrane with sinusoidal temperature heating, Meccanica 52 (2017) 2685-2073.
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[25] M. Ghalambaz, E. Jamesahar, M. A. Ismael, A. J. Chamkha, Fluid-structure interaction study of natural convection heat transfer over a flexible oscillating fin in a
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square cavity, Int. J. Therm. Sci. 111 (2017) 256-273.
[26] J.S. Lee, J.-Ho Shin, S.-H. Lee, Fluid–structure interaction of a flapping flexible
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plate in quiescent fluid, Comput. Fluids 57 (2012) 124–137.
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[27] A. Kalmbach, M. Breuer, Experimental PIV/V3V measurements of vortex-induced fluid–structure interaction in turbulent flow - A new benchmark FSI-PfS-2a, J. Fluid
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Struct. 42 (2013) 369–387.
[28] G. De Nayer, M. Breuer, Numerical FSI investigation based on LES: Flow past a cylinder with a flexible splitter plate involving large deformations (FSI-PfS-2a), Int. J. Heat Fluid Flow 50 (2014) 300–315. [29] A. K. Soti, R. Bhardwaj, J. Sheridan, Flow-induced deformation of a flexible thin structure as manifestation of heat transfer enhancement, Int. J. Heat Mass Transfer 84 (2015) 1070–1081. 34
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[30] R. van Loon, P.D. Anderson, F.N. van de Vosse, S.J. Sherwin, Comparison of various fluid–structure interaction methods for deformable bodies, Comput. Struct. 85 (2007) 833–843. [31] J. Donea, A. Huerta, Finite element methods for flow problems, John Wiley Sons,
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2003. [32] C. W. Hirt, A.A. Amsden, J. L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14 (1974) 14 227-253.
[33] T. J. R. Hughes, W. K. Liu, T. K. Zimmermann, Lagrangian–Eulerian finite element
(1981) 329–349.
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formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng. 29
[34] J. Donea, S. Giuliani, Jp. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comput. Methods Appl.
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Mech. Engrg. 33 (1982) 689-723.
[35] E. Kuhl, S. Hulsho, R. de Borst, An arbitrary Lagrangian-Eulerian finite-element
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CE
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(2003) 117–142.
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approach for fluid–structure interaction phenomena, Int. J. Numer. Meth. Engng. 57
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Graphical Abstract
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