Numerical Simulation about Reconstruction of the ... - Semantic Scholar

energies Article

Numerical Simulation about Reconstruction of the Boundary Layer † Yan Li *

ID

, Chuan Li, Yajie Wu, Cong Liu, Han Yuan and Ning Mei

Department of Mechanical Engineering, Ocean University of China, Qingdao 266100, Shandong, China; [email protected] (C.L.); [email protected] (Y.W.); [email protected] (C.L.); [email protected] (H.Y.); [email protected] (N.M.) * Correspondence: [email protected]; Tel.: +86-0532-6678-2073 (ext. 2407) † This paper is an extended version of our paper published in OF096, 4th International Conference on Energy and Environment Research, ICEER 2017, Porto, Portugal, 17–20 July 2017. Received: 31 October 2017; Accepted: 4 December 2017; Published: 6 December 2017

Abstract: In this paper, the reconstruction mechanism of the boundary layer in the channel is studied using the lattice Boltzmann method (LBM). By comparing the distribution of velocity in the channel, the conclusion that LBM has feasibility and superiority is obtained. Based on this, a physical model of square cylinders is set up to simulate the velocity distribution and the effect on the thickness of boundary layer. When the square cylinder moves at a certain speed, the velocity distribution in the flow field changes drastically. As well, it is found that the thickness of the boundary layer decreases with the cylinders’ height increasing in the given range. Furthermore, double cylinders model is also set up, and the results show that the optimal interval distance of the cylinders is between 90 and 140 lattice units. It is found that the moving cylinders have a significant effect on the thickness of the boundary layer, which will change the fluid flow and enhance the heat transfer. Keywords: lattice Boltzmann method (LBM); moving cylinders; reconstruction; the optimal interval distance; thickness of boundary layer

1. Introduction Heat transfer enhancement technology is very important in engineering problems such as compact heat exchangers, the cooling of electronic equipment, nuclear reactor fuel rods, cooling towers, chimney stacks and offshore structures, because it can improve the efficiency of heat exchangers, while reducing overall energy consumption. There are many ways to improve the efficiency of heat transfer, such as using porous media [1,2], adding nanoparticles [3–6], setting various effective structures [7–10], and so on. Thus, enormous work has been done analytically, experimentally and numerically. Many authors devoted their efforts to the effect of the arrangement of the cylinder in the equipment, on the fluid flow and heat transfer. Saha [11] found that the thermal performance factor increases with the increase in pitch of the cylinders. Sharman [12] analyzed flows over two cylinders in tandem arrangement by collocated unstructured computational fluid dynamics code. Rashidi [13] built a model to track the discrete nature of particles in a flow field with two cylinders placed side by side. Lu [14] simulated the fluid flow pass two square cylinder in a tandem arrangement and investigated heat transfer around the two cylinders. Kotcioglu [15] indicated that hexagonal pin-fins can lead to an advantage in terms of heat transfer enhancement via experiments. Shi [16] presented an analysis of convective heat transfer in fluid flow. It can be observed that the nature of the flow and heat transfer depend strongly on the arrangement of cylinders. In the meantime, some works focused on the effect of agitation in the flow field. Chang [17] performed the lattice Boltzmann simulation for the convective heat transfer problems of three types of motion of an impeller stirred tank. Agrawal [18] made experiments to study heat transfer property Energies 2017, 10, 2074; doi:10.3390/en10122074

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In the meantime, some works focused on the effect of agitation in the flow field. Chang [17] Boltzmann simulation for the convective heat transfer problems of three types 2 of 11 of motion of an impeller stirred tank. Agrawal [18] made experiments to study heat transfer property of a channel flow by active stirring and stationary spoilers. Yoon [19] presented a of a channel flow numerical by active stirring and stationary spoilers.fluid Yoonflows [19] presented a two-dimensional two-dimensional investigation that the laminar through two rotating circular numerical investigation that the laminar fluid flows through two rotating circular cylinders placed sidea cylinders placed side by side. Yan [20] studied and simulated that a viscous fluid flows past by side. Yan [20] studied andby simulated that a viscousmethod. fluid flows past a rotating isothermal cylinder rotating isothermal cylinder the lattice Boltzmann by the lattice Boltzmann method. The lattice Boltzmann method (LBM) is a mesoscopic numerical method to simulate fluid flow. The lattice method (LBM) a mesoscopic numerical method simulate fluid flow. This method hasBoltzmann second-order accuracy foristhe Navier-Stokes equation. In the to LBM, the well-known This has second-order accuracyGross, for the and Navier-Stokes equation. the LBM, BGKmethod model (presented by Bhatnagar, Krook [21]) is often In adopted by the Hewell-known [22], and it BGK model (presented by Bhatnagar, Gross, and Krook [21]) is often adopted by He [22], and it makes makes the lattice Boltzmann equation (LBE) into a linearized form. In our work, considering the the lattice Boltzmann equation (LBE) into a linearized form. In our work, considering the effect of the effect of the arrangement of cylinders and agitation on the fluid flow and heat transfer, we present a arrangement of cylindersofand the fluid flow and heat transfer, numerical numerical investigation theagitation effect ofon square cylinders moving from thewe leftpresent to the aright in the investigation of the effect of square cylinders moving from the left to the right in the channel on the channel on the boundary layer, and then analyze the contribution to the heat transfer. boundary layer, and then analyze the contribution to the heat transfer. performed the2074 lattice Energies 2017, 10,

2. Physical Model and Numerical Method 2. Physical Model and Numerical Method 2.1. Physical Physical Model Model 2.1. In the the design design and and manufacture manufacture of of heat heat exchange exchange equipment, equipment, plug-in plug-in technology technology is is aa simple simple and and In effective way to strengthen heat exchange, and its essence is to enhance fluid turbulence. However, effective way to strengthen heat exchange, and its essence is to enhance fluid turbulence. However, it is is possible possible to to form form aa reflow reflow area area in in the the rear rear of of the the static static spoiler, spoiler, and and that that can can hinder hinder convection convection it heat transfer. Meanwhile, the thickness of boundary layer is an important factor that affects the heat heat heat transfer. Meanwhile, the thickness of boundary layer is an important factor that affects the transfer efficiency. efficiency.Convection Convection heat transfer is weak the boundary layer because of the low transfer heat transfer is weak in theinboundary layer because of the low velocity. velocity. Therefore, this paper hopes to drive the spoiler movement mechanically to destroy the Therefore, this paper hopes to drive the spoiler movement mechanically to destroy the reflow area reflow area and thin boundary layer. As shown in Figure 1, a square cylinder moves from left to and thin boundary layer. As shown in Figure 1, a square cylinder moves from left to right along the right along lower wall. the lower wall.

Figure 1. Physical model of moving square cylinder. cylinder.

2.2. Lattice Boltzmann Method

The lattice latticeBoltzmann Boltzmannmethod method is based on kinetic theory. And the fundamental principle The is based on kinetic theory. And the fundamental principle of LBM of is f ( x , t ) using distribution functions f ( x, t ) to simulate the fluid flow at the mesoscopic scale. The distribution LBM is using distribution functions to simulate the fluid flow at the mesoscopic scale. function f (x, t) represents the of the particles at position x and at time t moving with a f ( x,probability t) The distribution function represents the probability of the particles at position x and at time velocity ei during the interval time ∆t along each lattice line (i.e., direction i). The lattice Boltzmann t moving with a velocity ei during the interval time  t along each lattice line (i.e., direction i ). equation (LBE) incorporating the single relaxation BGK approximation is given by Qian [23]: The lattice Boltzmann equation (LBE) incorporating the single relaxation BGK approximation is eq given by Qian [23]: f i (x + ei ∆t, t + ∆t) − f i (x, t) = ω [ f (x, t) − f i (x, t)] (1) i

f i ( x  ei  t , t   t )  f i ( x , t )   [√f i eq ( x , t )  f i ( x , t )]

(1) where ω = 1/τ denotes the relaxation factor, cs = c/ 3 is the speed of sound, and c = ∆x/∆t. eq The kinematic viscosity υ is given by the relaxation f (x, t) c factor  c / υ3 = (2/ω − 1)∆x · c/6. Aditionally, where   1/  denotes the relaxation factor, s is the speed of sound, and c = i x / t . is the equilibrium distribution, and is given by: The kinematic viscosity  is given by the relaxation factor   (2 /   1)x  c / 6 . Aditionally, and is given f i eq ( x , t ) is the equilibrium distribution, e · u by: ( e · u)2 u2 eq f i (x, t) = wi ρ[1 + i 2 + i 4 − 2 ] cs 2cs 2cs

(2)

For the D2Q9 model, the weightings in√ Equation (2) are assigned as follows: wi = 4/9 for ei = 0, wi = 1/9 for ei = 1, and wi = 1/36 for ei = 2.

fi eq ( x, t )  wi [1 

ei  u (ei  u)2 u2   2] cs2 2cs4 2cs

(2).

For the D2Q9 model, the weightings in Equation (2) are assigned as follows: wi  4 / 9 for 3 of 11 w  1/ 9 ei  0 , i for ei  1 , and wi  1 / 36 for ei  2 . The density  and momentum  u of fluid are obtained: The density ρ and momentum ρu of fluid are obtained:    f i ,  u   ei f i (3). i ρ = ∑i f i , ρu = ∑ ei f i (3)

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In the problem of simulating the curvedi boundary,i the solid boundary surface is either located xf Insolid the problem the curved boundary, s and is either on the node ( xofs )simulating or on the middle between solidthe andsolid fluidboundary nodes ( xsurface ). Aslocated shown on in q the solid node (x ) or on the middle between solid and fluid nodes (x and x ). As shown in 2, s s f can be used to describe the fraction of the link located in the fluidFigure region, Figure 2, a parameter a parameter q can be and can be defined as:used to describe the fraction of the link located in the fluid region, and can be defined as: | x|xf −xxw | |//∆x x (4) q q=||xxf f − xxww ||//||xxf f−xxss| |= (4) w f

0  q 1 when 0 ≤ q ≤ 1..

Figure 2. 1D projection projection of ofcurved curvedboundary boundarybased basedononregular regular grid (arrows indicate direction of grid (arrows indicate thethe direction of the the fluid particle movement). fluid particle movement).

For with no-slip no-slip condition, condition, the For aa rigid rigid wall wall with the bounce-back bounce-back (BB) (BB) scheme scheme is is the the most most easily easily implemented boundary condition. Therefore, it is widely used in LBM simulations. The implemented boundary condition. Therefore, it is widely used in LBM simulations. The particle particle c andreverses with velocity ci , andi ,then from the surface flow with c−i =with − ci moves from fromflow flowtotosurface surface with velocity then reverses from the to surface to flow c-i the  solid ci atsurface, the solidthat surface, is, the BB scheme. at is, thethat BB scheme. For treating [24] proposed a scheme by combining the BB with For treating curved curvedboundaries, boundaries,Bouzidi Bouzidi [24] proposed a scheme by combining thescheme BB scheme an interpolation approach. By applying quadratic interpolation, the model takes the following form: with an interpolation approach. By applying quadratic interpolation, the model takes the following form: f i (x f , t ) =

(

q(1 + 2q) f i (x f + ei ∆t, t) + (1 − 4q2 ) f i (x f , t) − q(1 − 2q) f i (x f − ei ∆t, t), q < 1/2 2 −1 1  q (1  2 q ) f i ( x f  ei t , t2q f i ( ex f∆t, , t ) t)q− (1 2q2− q )1f i ( x f  ei t , t ), q  1 / 2 f (x + ei ∆t, t) + ) q(1 f4i (qx f) − i 2q + 1 f i (x f − 2ei ∆t, t ), q ≥ 1/2 q(2q −  1) i f

(5)

fi ( x f , t )   (5). 1 2q  1 2q  1  q (2 q  1) f i ( x f  ei t , t )  q f i ( x f -ei t , t )  2 q  1 f i ( x f -2ei  t , t ), q  1 / 2 For a moving wall treatment, a certain amount of momentum transfer between fluid and solid 

surfaces must be considered. Lallemand [25] applied Bouzidi’s model to treat the problem. So to reflect For a moving wall treatment, a certain amount of momentum transfer between fluid and solid the fluid–solid interaction (i.e., the momentum transfer) at the boundary surface (xs ), one additional surfaces must be considered. Lallemand [25] applied Bouzidi’s model to treat the problem. So to term need be added to the distribution function. δ f i , the additional term, is given by: reflect the fluid–solid interaction ( (i.e., the momentum transfer) at the boundary surface ( x s ), one 3ωi ρ f ei · uω , 0 ≤ q < 1/2 additional term need be added function.  fi , the additional term, is given by: (6) δ f i to = the distribution 3 ω ρ e · uω , 1/2 ≤ q ≤ 1 q(2q + 1) i f i

0  q  1/ 2 3i  f ei  u ,    f 3method [26], the drag and lift forces that are imposed  (6). i By applying the momentum exchange by i  f ei  u , 1 / 2  q  1 

 q (2q  1)by the formula: the fluid on the solid body can be determined

By applying the momentum exchange 8 method [26], the drag and lift forces that are imposed by the fluid on the solid body can be determined F= −∑ ∑ ei [by f i (xthe + f i (xb , t + δt )] (7) b , t )formula: xb i =1

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F eei [[ ffi (( xxb ,, tt))  ffi (( xxb ,, tt   t )]  F    i b b t )] i x  i 1 i b

xb

i 1

(7). (7).

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3. 3. Results Results and and Discussion Discussion 3. Results and Discussion 3.1. Model 3.1. Model Validation Validation In order order to validate the feasibility of Bouzidi’s interpolation-based dealing with with aa In model in in dealing order to to validate validate the the feasibility feasibility of of Bouzidi’s Bouzidi’s interpolation-based interpolation-based model moving boundary, the verification simulations considering two cases of channel flow (i.e., moving verification simulations considering two cases channel flow (i.e.,flow a stationary moving boundary, boundary,thethe verification simulations considering two ofcases of channel (i.e., aa stationary square cylinder positioned in the channel and a square cylinder moving in the channel) square cylinder positioned the channel a square and cylinder moving in the moving channel)in arethe performed. stationary square cylinderin positioned in and the channel a square cylinder channel) are performed. As shown in Figure 3, a 1000 × 160 lattice structure is chosen for computation, and As in Figure 3, a 1000 160 lattice structure chosenstructure for computation, square cylinder areshown performed. As shown in×Figure 3, a 1000 × 160islattice is chosenand forthe computation, and the square cylinder on the axis of the channel is assumed to have a side length of 20 lattice units on axis of the channel is assumed to have a side of 20 article alwaysunits use thethe square cylinder on the axis of the channel is length assumed tolattice have units a side(this length of will 20 lattice (this article will always use lattice units). In this model, each lattice spacing corresponds to 1 mm, lattice units).will In this model, lattice spacing corresponds to 1lattice mm, and each corresponds time step corresponds (this article always useeach lattice units). In this model, each spacing to 1 mm, and each step corresponds to 0.165 At upper lower aa no-slip boundary to 0.165 At the upper and lower condition is imposed. In Figure 3a, and eachs. time time step corresponds towalls, 0.165 as. s.no-slip At the theboundary upper and and lower walls, walls, no-slip boundary U condition is In square cylinder is and velocity U  0.1 0.1,velocity square cylinder is stationary and3a, inlet velocity U = 0.1, and in Figure 3b, square cylinder , and and in in condition is imposed. imposed. In Figure Figure 3a, square cylinder is stationary stationary and inlet inlet velocity

0 Uww  0.1 0.1 and inlet velocity U Figure 3b, square U inletcylinder velocityvelocity U = 0. U w = 0.1 and inlet velocity U  0 .. Figure 3b,and square cylinder velocity numerical results moving boundary treatment the two cases are The ofof thethe moving boundary treatment in thein two compared in Figurein 4, The numerical numericalresults results of the moving boundary treatment in thecases twoare cases are compared compared in Figure 4, which show the distribution of the X-direction velocity component (U) on longitudinal which show the distribution of the X-direction velocity component (U) on longitudinal section and Figure 4, which show the distribution of the X-direction velocity component (U) on longitudinal section and Y-direction velocity component (V) cross center of the Y-direction component on cross section the containing center of thethe square cylinder. section and the thevelocity Y-direction velocity(V) component (V) on oncontaining cross section section containing the center of the the square cylinder. To make the comparison results more clearly and comparable, instead of the actual To make the comparison results more clearly and comparable, instead of the actual velocity, the relative square cylinder. To make the comparison results more clearly and comparable, instead of the actual velocity, the relative velocity of the fluid, is to the flow of velocity the fluid, relative is used to denote the flow of the fluid. In Figure is velocity,of the relative velocityto ofthe thesquare, fluid, relative relative to the the square, square, is used used to denote denote the flow 3, of itthe the fluid. In Figure 3, it is obvious that the numerical results of both U and V are in excellent agreement obvious that the numerical results of both U and V are in excellent agreement except for the results of fluid. In Figure 3, it is obvious that the numerical results of both U and V are in excellent agreement except for the longitudinal section. In order of of V on the longitudinal section. In addition, the order of magnitude V on longitudinal section is on so except for the results results of of V V on on the the longitudinal section. In addition, addition,ofthe the order of magnitude magnitude of V V on − 13 −13 longitudinal so small that the volatility no It small (i.e., 10 section ) thatis volatility no difference. canmake be concluded that, although longitudinal section isthe so slight small (i.e., (i.e., 10 10−13))will thatmake the slight slight volatility Itwill will make no difference. difference. It can can be be concluded that, although physical models of the two cases are different, the numerical results physical models of the two cases are different, the numerical results of the two direction velocities on concluded that, although physical models of the two cases are different, the numerical results of of the the two velocities on sections both sections are consistent, respectively. two direction direction velocities on both both sections are are consistent, consistent, respectively. respectively.

U  0.1 Figure 3. model of square cylinder around the flow. (a) Inlet velocity square 0.1,, square 3. Physical of square square cylinder cylinder around aroundthe theflow. flow.(a) (a)Inlet Inletvelocity velocityU U= 0.1, square Figure 3. Physical model of Uw  0 U  0.1 U  0 w  0.1 . Uww =0 0 and U (b) Inlet velocity , square cylinder’s velocity velocity cylinder’s U and (b) Inlet velocity U = 0, square cylinder’s velocity U = 0.1. and (b) Inlet velocity U  0 , square cylinder’s velocity ww . cylinder’s velocity

(a) (a) Figure 4. Cont.

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(b) (b) Figure 4.4.Comparison Figure4. Comparisonof ofnumerical numericalresults resultsof ofU andV oncross crosssection sectionand andlongitudinal longitudinalsection. section. Figure Comparison of numerical results of UUand and VVon on cross section and longitudinal section. (a) Comparison of U and V on longitudinal section and (b) Comparison of U and VVon (a)Comparison ComparisonofofUUand andVVon onlongitudinal longitudinalsection sectionand and(b) (b)Comparison ComparisonofofUUand andV oncross crosssection. section. (a) on cross section.

C C InIn this simulation, the lift coefficient the drag coefficient this simulation, simulation, the the lift lift coefficient coefficient CCl l l and and the the drag drag coefficient coefficient CCddd are aretwo twoimportant important dimensionless are given by [22]: parameters for results validation, which are given by [22]: dimensionless parameters for results validation, which are given by [22]: F l l FF 2 CCl Cl=l11/ /22lDU DU0202  DU 1/2ρ∞ 0

(8) (8)

FF CCd d Fd d 2 2 Cd = 11/ /22dDU DU020

(9). (9). (9)

1/2ρ∞ DU0

CCl l is constant equal to 0, for the square cylinder at the middle of the The lift coefficient C is constant equalequal to 0, for thefor square cylindercylinder at the middle the channel Thelift liftcoefficient coefficient is constant to 0, the square at the of middle of the l

CCd d 5. throughout the entire simulation. The comparison of the drag coefficient Cdrag in Figure d is shown isis channel the The comparison result ofofthe coefficient channelthroughout throughout theentire entiresimulation. simulation. Theresult comparison result thedrag coefficient Inshown general, the results of the two cases show a good agreement. shown ininFigure 5. In general, the results of the two cases show a good agreement. Figure 5. In general, the results of the two cases show a good agreement.

CC Figure Figure Comparison results drag coefficient Figure5.5. 5.Comparison Comparisonresults resultsofof ofdrag dragcoefficient coefficient Cdd.d. .

3.2. Height on Boundary Layer 3.2.Effect Effectof SquareCylinder Cylinderwith withDifferent DifferentHeight Heighton onBoundary BoundaryLayer Layer 3.2. Effect ofofSquare Square Cylinder with Different When field, field changes Whenthe thesquare squarecylinder cylinderis placedin theflow flowfield, field,the thevelocity velocityof theflow flowfield fieldchanges changes When the square cylinder isisplaced placed ininthe the flow the velocity ofofthe the flow significantly, layer tototo change. As shown ininFigure 6,6,the significantly,which whichcauses causesthe thethickness the boundary layer change. AsAs shown Figure the significantly, which causes the thicknessofofthe theboundary boundary layer change. shown in Figure 6, center line of the square cylinder with a height of 0.75D is set at 100 lattices in the calculation region center lineline of the square cylinder withwith a height of 0.75D is setisatset 100atlattices in thein calculation region the center of the square cylinder a height of 0.75D 100 lattices the calculation of proven tests 380 400 ××600, of400 400×of ×160, 160, which proven tobe besuitable suitable aftergrid-independence grid-independence testswith with 380×with ×152, 152,380 400× 600, region 400which × 160,isiswhich isto proven to be after suitable after grid-independence tests 152, 420 × 168, and the inlet velocity of the flow field is set to 0.05 lattice units. At the upper and lower 420 × 168, and the inlet velocity of the flow field is set to 0.05 lattice units. At the upper and lower 400 × 600, 420 × 168, and the inlet velocity of the flow field is set to 0.05 lattice units. At the upper walls, walls,no-slip no-slipboundary boundarycondition conditionisisimposed. imposed.InInthis thismodel, model,each eachlattice latticecorresponds correspondstoto2.5 2.5mm, mm, and andeach eachtime timestep stepcorresponds correspondstoto4.95 4.95s.s.

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and lower walls, no-slip boundary condition is imposed. In this model, each lattice corresponds to It can seen thatstep thecorresponds velocity is to faster in the area above the square cylinder, and is the 2.5 mm, andbeeach time 4.95 s. smallest in be theseen front regions of the square cylinder, in theand rearis of square It can thatand therear velocity is faster in the area above the especially square cylinder, thethe smallest in column, which will cause the heat transfer capacity to deteriorate. Moreover, as shown in Figure 6b, the front and rear regions of the square cylinder, especially in the rear of the square column, which under the same conditions, a squaretocylinder moves from leftastoshown right along the 6b, lower wall. will cause the heat transfer capacity deteriorate. Moreover, in Figure under theWhen same the square cylinder starts to move, the velocity distribution changes drastically in the flow field. conditions, a square cylinder moves from left to right along the lower wall. When the square cylinder Figure 6bmove, showsthe that the square cylinder moves drastically to 300 lattices. hugefield. vortex is formed in the starts to velocity distribution changes in theAflow Figure 6b shows thatarea the where the square cylinder passes, resulting in dramatic change in the flow field and an effective square cylinder moves to 300 lattices. A huge vortex is formed in the area where the square cylinder reduction of the boundary layer. passes, resulting in dramatic change in the flow field and an effective reduction of the boundary layer. Figure 7a is about Re varies with Y-axis on different sections when thecylinder square is cylinder is Figure 7a is about Re varies with Y-axis on different sections when the square stationary. stationary. 0.25,flows the fluid flowsthe only fromside the of upper side ofcylinder, the square and When x/LxWhen = 0.25,x/Lx the =fluid only from upper the square andcylinder, the Reynolds the Reynolds number has the maximum value. When x/Lx = 0.50, Re has two peaks: the first peak is number has the maximum value. When x/Lx = 0.50, Re has two peaks: the first peak is near the near the ordinate of 20. This is related to the surrounding vortex, and it will not improve the heat ordinate of 20. This is related to the surrounding vortex, and it will not improve the heat transfer transfer efficiency; Thepeak second peak near the ordinate of 120. Because the flow cross-sectional efficiency; The second is near theisordinate of 120. Because the flow cross-sectional area decreases, area decreases, the flow rate increases, resulting in the upper wall boundary layer When the flow rate increases, resulting in the upper wall boundary layer thinning. When x/Lxthinning. = 0.75, the flow x/Lx = 0.75, the flow field tends to be stable. Figure 7b is about Re varies with Y-axis on different field tends to be stable. Figure 7b is about Re varies with Y-axis on different sections when the square sections the cylinder moves towith 300 the lattices. with thepin flow of stationary cylinderwhen moves tosquare 300 lattices. Compared flow Compared field of stationary fin,field the magnitude of pin fin, the magnitude of Re is basically the same. However, As the square cylinder moves, Re Re is basically the same. However, As the square cylinder moves, Re changes faster in the area near changes the solid walls, fluid velocity increases theverifies main velocity the solidfaster walls,insothe thearea fluidnear velocity increases toso thethe main velocity rapidly. Thistoalso that the rapidly. This also verifies that the boundary layer is thinner in Figure 6b. Therefore, by the boundary layer is thinner in Figure 6b. Therefore, by reducing the boundary layer andreducing eliminating boundary layer and eliminating the vortex behind the cylinder to improve the heat transfer the vortex behind the cylinder to improve the heat transfer efficiency is the basic starting point in efficiency is the basic starting point in this section. this section.

(a)

(b) Figure6.6.The Thedistribution distributionof ofvelocity velocitywith withaapin pinfin. fin.(a) (a)Static Staticpin pinfin, fin,(b) (b)Moving Movingpin pinfin. fin. Figure

In this this section, section, the the effect effect of of square square cylinder cylinder moving moving from from left left to to right right along along the the lower lower wall wall isis In studied.As Asshown shownin inFigure Figure88below, below, the the computational computational domain domain is is mapped mapped using using aa 400 400 ××160 160mesh, mesh, studied. the height of the square cylinder is 0.75D (D is the channel diameter). The inlet velocity of the the height of the square cylinder is 0.75D (D is the channel diameter). The inlet velocity of the fluid Uw Uw are fixed at 0.05 and 0.1. Figure 8 shows the fluid and square cylinder’s movement velocity U andUsquare cylinder’s movement velocity are fixed at 0.05 and 0.1. Figure 8 shows the reconstruction of the boundary layer of the fluid in the process of aof square cylinder moving from left to reconstruction of the boundary layer of the fluid in the process a square cylinder moving from right channel. It is clear as the comes close andclose goesand away, theaway, boundary left toalong right the along the channel. It isthat clear thatsquare as the cylinder square cylinder comes goes the layer is destroyed, and then returns to its previous state, respectively. The boundary layer of the front boundary layer is destroyed, and then returns to its previous state, respectively. The boundary and rear of front the cylinder become obviously thinnerobviously through the movement. layer of the and rear of the cylinder become thinner through the movement.

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(a)(a)

(b) (b)

Figure Re varies with Y-axis different sections. fin 0.25); (b)Moving Moving pin Figure 7.7.Re with Y-axis onon different sections. (a) (a) Static pinpin fin (x/Lx = 0.25); (b) (b) Moving pinpin fin Figure 7.varies Re varies with Y-axis on different sections. (a)Static Static pin fin (x/Lx (x/Lx == 0.25); fin (x/Lx = 0.75). (x/Lx 0.75). fin=(x/Lx = 0.75).

(a) (a)

(b) (b)

(c)

(c) flow field at different n (n is the simulation steps). Figure 8. Contour of the boundary layer Figure 8. Contour lineline of the boundary layer in in thethe flow field at different n (n is the simulation steps). (a) n = 900; (b) n = 1400; (c) n = 2400. (a) n = 900; (b) n = 1400; (c) n = 2400. Figure 8. Contour line of the boundary layer in the flow field at different n (n is the simulation steps). (a) n = 900; (b) n = 1400; (c) n = 2400.

In order to explore the effect of different the moving cylinder thefield, flow the field, the In order to explore the effect of different heightheight of theofmoving cylinder on theon flow height height of the square cylinder is varied from 0.25D to 0.75D in increments of 0.25D. In Figure 9, of theIn square is varied fromof0.25D to 0.75D in of increments of 0.25D. In Figure X-direction ordercylinder to explore the effect different height the moving cylinder on the 9, flow field, the X-direction indicates the ratio between the distance (x) from the inlet of the channel and the length indicates between theisdistance (x) from thetoinlet of in theincrements channel and the length of that, height ofthe theratio square cylinder varied from 0.25D 0.75D of 0.25D. In Figure 9, of that, and Y-direction indicates thickness of the boundary layer. Figure 9a–c shows, as the square and Y-direction indicates of thethe boundary Figure as the square cylinder X-direction indicates the thickness ratio between distancelayer. (x) from the 9a–c inlet shows, of the channel and the length cylinder with different heights H (0.25D, 0.5D, 0.75D) moves to different locations (x/Lx = 3/8, 4/8, with different heights H indicates (0.25D, 0.5D, 0.75D) to different (x/Lx = 3/8, 5/8), of that, and Y-direction thickness of moves the boundary layer.locations Figure 9a–c shows, as 4/8, the square 5/8), the thickness of the boundary layer along the channel varies with that. It is obvious that the the thickness of the boundary layer along the channel varies with that. It is obvious that the boundary cylinder with layer different heights H where (0.25D,the 0.5D, 0.75D) movesmoves to different locations 3/8,the 4/8, boundary in the location square cylinder to is the thinnest,(x/Lx and = with layer in location of where the squarelayer cylinder moves to is the varies thinnest, andthat. withItthe square cylinder 5/8),square thethe thickness the boundary along the channel with is obvious that the cylinder going away, the boundary layer becomes gradually thicker until it returns to the boundary layer in the location where the square cylinder moves to is the thinnest, and with the square cylinder going away, the boundary layer becomes gradually thicker until it returns to the

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of 11 layer becomes gradually thicker until it returns to the original state it8 was. In Figure 9d, the results of that three square cylinders with different height move to the same location original state it was. In Figure 9d, the results of that three square cylinders with different height (x/Lx = 3/8) are compared. The conclusion can be drawn that the thickness of the boundary layer move to the same location (x/Lx = 3/8) are compared. The conclusion can be drawn that the decreases with the square cylinder’s height increasing. thickness of the boundary layer decreases with the square cylinder’s height increasing.

(a)

(b)

(c)

(d)

Figure 9. Changes of the thickness 5/8). Figure thickness of of the theboundary boundarylayer layerinindifferent differentlocations locations(x/Lx (x/Lx==3/8, 3/8, 4/8, 4/8, 5/8). (a) H == 0.25D; 0.25D; (b) (b) H H ==0.50D; 0.50D;(c) (c)HH==0.75D; 0.75D;(d) (d)x/Lx x/Lx= = 3/8. 3/8.

3.3. Double Double Cylinders Cylinders Model Model From From the above simulation, simulation, the the boundary boundary layer layer will will be reconstructed reconstructed after the cylinder passes by. So destroy thethe boundary layer again, which can can bothboth lengthen the Soanother anothercylinder cylinderisisadded addedtoto destroy boundary layer again, which lengthen domain withwith thinner boundary layer andand strengthen thethe mixture the domain thinner boundary layer strengthen mixtureand anddisturbance disturbanceofofthe theflow flow to enhance the the heat heattransfer transferfurther. further.AsAs shown in Figure 10, computational the computational domain is mapped shown in Figure 10, the domain is mapped using using a 1000 × 160 mesh, the side length of the cylinders is 20, the height of both cylinders is 0.75D. a 1000 × 160 mesh, the side length of the cylinders is 20, the height of both cylinders is 0.75D. As well, U  cylinders 0.05 . Twowith thewell, velocity the liquid in the channel U = 0.05. a certain distance move at the As the of velocity of the liquid in theischannel is Two cylinders with a certain distance same velocity of Uw = 2U to destroy Uw  2Uthe boundary layer. move at the same velocity of to destroy the boundary layer. Figure 11 shows the thickness of boundary layer changing with the non-dimensional time. With the movement of the cylinders from left to right, the thickness of the boundary layer changes with the increase of value n along X axis, which are compared with the thickness of boundary layer of Poiseuille flow. It is obvious that, in most part of the channel, the boundary layer of double cylinders model is much thinner than that of Poiseuille flow. However, at the domain behind the cylinders, the thickness of boundary layer become thicker, which increases the thermal resistance to some extent, regardless of increasing the disturbance. Taken together, the effect of the double cylinders on thinning boundary layer and enhancing heat transfer is highlighted. Figure 10. Physical model of double cylinders.

Figure 11 shows the thickness of boundary layer changing with the non-dimensional time. With the movement of the cylinders from left to right, the thickness of the boundary layer changes

by. So another cylinder is added to destroy the boundary layer again, which can both lengthen the cylinders is much thinner than Poiseuille However, at the domain domainmodel with thinner boundary layer that and of strengthen theflow. mixture and disturbance of thebehind flow tothe cylinders, of boundary layer become thicker, increases the thermal resistance enhancethe thethickness heat transfer further. As shown in Figure 10,which the computational domain is mapped to some extent, increasing the Taken of the double using a 1000regardless × 160 mesh,ofthe side length of disturbance. the cylinders is 20, thetogether, height of the botheffect cylinders is 0.75D. U  0.05 cylinders thinning layer andchannel enhancing highlighted. As well,onthe velocityboundary of the liquid in the is heat transfer . Two is cylinders with a certain distance Energies 2017, 10, 2074 of the doubleUcylinders 9 of 11 The distance has an obvious effect on the thickness of boundary layer.  2U move at the same velocity of w to destroy the boundary layer. In some cases, it did not reduce the thickness of boundary layer effectively. Therefore, the optimized distance of the double cylinders is needed for design by simulation. The thickness curves of boundary with the different interval distance of the double cylinders at the same time and Energies 2017,layer 10, 2074 9 of 11 the same Re number are shown in Figure 12. At the domain from 400 to 500 in the X coordinate, the with theofincrease of value n along axis, which are compared with the value thickness of boundary layer thickness the boundary layer mayXhas crests or troughs with different of interval distance of Poiseuille flow. It There is obvious that, in most part of the in channel, the boundary of double of the double cylinders. is an optimal interval distance the domain of 60~180.layer The effect of cylinders model is much thinner than that of Poiseuille flow. However, at the domain behind the interval distance is obvious from 400 to 500 in the X coordinate. The Figure 13 shows thethe cylinders, the boundary thickness of boundary layer become thicker, which increases thermalat resistance thickness of the layer with different interval distance of the doublethe cylinders x = 450. to some ofFigure increasing the disturbance. Taken together, the effect of the double Thus, theextent, optimalregardless interval distance of the double cylinders is between 90 and 140. 10. Physical model of double cylinders. Figure 10. Physical model of double cylinders. cylinders on thinning boundary layer and enhancing heat transfer is highlighted. The distance of the cylinders has anlayer obvious effect with on the of boundary Figure 11 shows thedouble thickness of boundary changing thethickness non-dimensional time.layer. In some cases, it did reducefrom the left thickness layer effectively. Therefore, With the movement of thenot cylinders to right,of theboundary thickness of the boundary layer changes the optimized distance of the double cylinders is needed for design by simulation. The thickness curves of boundary layer with the different interval distance of the double cylinders at the same time and the same Re number are shown in Figure 12. At the domain from 400 to 500 in the X coordinate, the thickness of the boundary layer may has crests or troughs with different value of interval distance of the double cylinders. There is an optimal interval distance in the domain of 60~180. The effect of the interval distance is obvious from 400 to 500 in the X coordinate. The Figure 13 shows the thickness of the boundary layer with different interval distance of the double cylinders at x = 450. Thus, the optimal interval distance of the double cylinders is between 90 and 140.

Figure 11. 11. Thickness Thicknessof ofthe theboundary boundarylayer layer of of double double cylinders cylinders at at different different n. n. Figure

The distance of the double cylinders has an obvious effect on the thickness of boundary layer. In some cases, it did not reduce the thickness of boundary layer effectively. Therefore, the optimized distance of the double cylinders is needed for design by simulation. The thickness curves of boundary layer with the different interval distance of the double cylinders at the same time and the same Re number are shown in Figure 12. At the domain from 400 to 500 in the X coordinate, the thickness of the boundary layer may has crests or troughs with different value of interval distance of the double cylinders. There is an optimal interval distance in the domain of 60~180. The effect of the interval distance is obvious from 400 to 500 in the X coordinate. The Figure 13 shows the thickness of the boundary layer with different interval distance of the double cylinders at x = 450. Thus, the optimal interval distance of the double cylinders is boundary between 90 and Figure 11. Thickness of the layer of 140. double cylinders at different n.

Figure 12. The thickness of the boundary layer with different interval distance of the double cylinders (interval distance of the double cylinders is 60, 100, 140, 180).

Figure The thickness boundary layer with different interval distance of the double cylinders Figure 12.12. The thickness of of thethe boundary layer with different interval distance of the double cylinders (interval distance of the double cylinders is 60, 100, 140, 180). (interval distance of the double cylinders is 60, 100, 140, 180).

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(a)

Figure 13. The thickness of the boundary layer with different interval distance of the double cylinders Figure 13. The thickness of the boundary layer with different interval distance of the double at x = 450 in Figure 10. cylinders at x = 450 in Figure 10.

4. Conclusions 4. Conclusions In thiswe paper, wecharacterized have characterized movingboundary boundary with lattice Boltzmann method. In this paper, have thethe moving withthe the lattice Boltzmann method. In order to provide the basis for the realization of the heat exchanger’s performance optimization In order to provide the basis for the realization of the heat exchanger’s performance optimization and and solid attachments flowing out of the heat exchanger with the water, the boundary layer solid attachments flowing out of the heat exchanger with the water, the boundary layer reconstruction reconstruction model is established. According to the simulation results, conclusions can be drawn model is established. as followed. According to the simulation results, conclusions can be drawn as followed.

(1)

(2)

(3)

(1) By comparing the changes of the X-direction velocity component (U) and the Y-direction By comparing the changes of the X-direction velocity component (U) and the Y-direction velocity velocity component (V) and lift and drag in the center of the square cylinder, the conclusion component (V) and lift and drag in the center of the square cylinder, the conclusion that lattice that lattice Boltzmann method has the feasibility and superiority in dealing with moving Boltzmann method has the feasibility and superiority in dealing with moving boundary is obtained. boundary is obtained. The (2) boundary layer reconstruction model has been established. It isIt found that when The boundary layer reconstruction model has been established. is found that whenthe thesquare cylindersquare is present, theisfluid boundary willlayer be destroyed to enhance the total heatheat transfer cylinder present, the fluid layer boundary will be destroyed to enhance the total transfer coefficient, makes the optimization of the heat exchanger possible. thesame same time, coefficient, which makeswhich the optimization of the heat exchanger possible. AtAtthe time, the of the boundary layer increases the height of the squarecylinder cylinder decreases. the thickness of thickness the boundary layer increases as theasheight of the square decreases. (3) As the square cylinder passes by, the boundary layer away from the square cylinder will As the square cylinder passes by, the boundary layer away from the square cylinder will gradually thicken and be reconstructed. Double cylinders are effective to make the boundary gradually thicken and bethin, reconstructed. are effective to make the boundary layer continuously and there is anDouble optimumcylinders interval distance of the cylinders. layer continuously thin, and there is an optimum interval distance of the cylinders.

Acknowledgements: This work was supported by the National Natural Science Foundation of china under Grant No. 51376164 and Natural Science Foundation of Shandong Province under Grant No. ZR2013EEQ034. Acknowledgments: This work was of supported by the National Natural ScienceEnergy Foundation In addition, this is an extension the work presented at ICEER2017 and published Procedia.of china under

Grant No. 51376164 and Natural Science Foundation of Shandong Province under Grant No. ZR2013EEQ034. Author Contributions: Yan Li, Chuan Li and Cong Liu designed the numerical simulation and analyzed the In addition, this is an extension of the work presented at ICEER2017 and published Energy Procedia. data; Yajie Wu, Han Yuan and Ning Mei revised the manuscript.

Author Contributions: Yan Li, Chuan Li and Cong Liu designed the numerical simulation and analyzed the data; Conflicts of Interest: The authors declare no conflict of interest. Yajie Wu, Han Yuan and Ning Mei revised the manuscript. Conflicts of Interest: The authors declare no conflict of interest. References 1.

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